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[ "The goal of this thesis is the development and implementation of a\nnon-perturbative solution method for Wegner's flow equations. We show that a\nparameterization of the flowing Hamiltonian in terms of a scalar function\nallows the flow equation to be rewritten as a nonlinear partial differential\nequation. The implementation is non-perturbative in that the derivation of the\nPDE is based on an expansion controlled by the size of the system rather than\nthe coupling constant. We apply this method to the Lipkin model and obtain very\naccurate results for the spectrum, expectation values and eigenstates for all\nvalues of the coupling and in the thermodynamic limit. New aspects of the phase\nstructure, made apparent by this non-perturbative treatment, are also\ninvestigated. The Dicke model is treated using a two-step diagonalization\nprocedure which illustrates how an effective Hamiltonian may be constructed and\nsubsequently solved within this framework.", "The goal of this thesis is the development and\nimplementation of a non-perturbative solution method for Wegner’s flow\nequations. We show that a parameterization of the flowing Hamiltonian in\nterms of a scalar function allows the flow equation to be rewritten as a\nnonlinear partial differential equation. The implementation is\nnon-perturbative in that the derivation of the PDE is based on an\nexpansion controlled by the size of the system rather than the coupling\nconstant. We apply this method to the Lipkin model and obtain very\naccurate results for the spectrum, expectation values and eigenstates\nfor all values of the coupling and in the thermodynamic limit. New\naspects of the phase structure, made apparent by this non-perturbative\ntreatment, are also investigated. The Dicke model is treated using a\ntwo-step diagonalization procedure which illustrates how an effective\nHamiltonian may be constructed and subsequently solved within this\nframework.\n" ]	##### Contents - 1 Flow equations - 1.1 Overview - 1.2 The role of @xmath - 1.2.1 Wegner’s choice - 1.2.2 @xmath - 1.2.3 The structure preserving generator - 1.3 Flow equations in infinite dimensions - 2 Solving the flow equation - 2.1 Expansion in fluctuations - 2.2 Moyal bracket approach - 3 The Lipkin Model - 3.1 Introduction - 3.2 The model - 3.3 The flow equation approach - 3.4 Expansion method - 3.4.1 The flow equation in the @xmath limit. - 3.4.2 Finding the spectrum - 3.4.3 Calculating expectation values - 3.5 Moyal bracket method - 3.5.1 Probing the structure of eigenstates - 3.6 Numerical results - 3.6.1 Spectrum and expectation values - 3.6.2 Structure of the eigenstates - 3.7 Solution in the local approximation - 4 The Dicke Model - 4.1 Introduction - 4.2 The model - 4.3 The flow equation approach - 4.4 Flow equations in the @xmath limit - 4.4.1 Variables and representations - 4.4.2 Solution in the local approximation - 4.4.3 Flow equation for the Hamiltonian - 4.4.4 Flow equation for an observable - 4.5 Perturbative solutions - 4.6 Diagonalizing @xmath - 4.7 Numeric results - A Representations by irreducible sets - B Calculating expectation values with respect to coherent states - C Scaling behaviour of fluctuations - D Decomposing operators in the Dicke model - E Dicke model flow coefficients ###### List of Figures - 1.1 The structure of band block diagonal matrices. - 3.1 The domain of @xmath . - 3.2 The solutions of the Lipkin model flow equation ( 3.42 ). - 3.3 Eigenvalues and gaps between successive states as functions of @xmath . - 3.4 Expectation values as functions of @xmath . - 3.5 Aspects of the phase structure of the Lipkin model. - 3.6 Results obtained for the ground state wavefunction using the flow equation. - 3.7 Results of the local approximation - 4.1 A schematic representation of the basis states spanning the Dicke model Hilbert space. - 4.2 Flow of the coupling constants @xmath and @xmath in the local approximation. - 4.3 The eigenvalue @xmath and excitation energy @xmath as functions of @xmath . - 4.4 Various excitation energies as functions of @xmath . - 4.5 Eigenstates of the Dicke Hamiltonian obtained using different methods. Expectation values of @xmath as functions of @xmath . - 4.6 Spectrum and expectation values of the RWA Hamiltonian \specialhead Introduction The renormalization group and its associated flow equations [ 1 ] have become an indispensable tool in the study of modern physics. Its applications range from the construction of effective theories to the study of phase transitions and critical phenomenon. It also constitutes one of the few potentially non-perturbative techniques available for the treatment of interacting quantum systems. Our interest lies with a recent addition to this framework proposed by Wegner [ 2 ] and separately by Glazek and Wilson [ 3 ] , namely that of flow equations obtained from continuous unitary transformations. The flow equation in question describes the evolution of an operator, typically a Hamiltonian, under the application of a sequence of successive infinitesimal unitary transformations. These transformations are constructed so as to steer the evolution, or flow, of the operator towards a simpler, possibly diagonal, form. The major advantage of this approach is that no prior knowledge of these transformations is needed as they are generated dynamically at each point during the flow. These attractive properties have led to applications to several diverse quantum mechanical problems, including that of electron-phonon coupling [ 2 ] , boson and spin-boson models [ 4 , 42 , 43 ] , the Hubbard model [ 5 ] , the Sine-Gordon model [ 6 ] and the Foldy-Wouthuysen transformation [ 7 ] . The Lipkin model has also been particularly prominent among applications [ 8 , 9 , 10 , 11 , 12 ] . More recently the flow equations have been used to construct effective Hamiltonians which conserve the number of quasi-particles or elementary excitations in a system [ 13 ] . Examples of quasi-particles treated in this manner include triplet bonds on a dimerized spin chain [ 14 , 15 , 16 , 17 ] and particle-hole excitations [ 18 , 19 ] in Fermi systems. Studies of these equations in a purely mathematical context, where they are known as double bracket flows, have also been conducted [ 20 , 21 ] . The versatility of this approach should be clear from the references above. Unfortunately the practical implementation is hampered by the fact that the Hamiltonian typically does not preserve its form under the flow, and that additional operators, not present in the original Hamiltonian are generated [ 1 , 2 ] . In general we are confronted with an infinite set of coupled nonlinear differential equations, the truncation of which is a highly non-trivial task. Perturbative approximations do allow one to make progress, although the validity of the results is usually limited to a single phase. It is the aim of this thesis to develop methods for the non-perturbative treatment of the flow equations. We will do so via two routes, both of which involve the representation of the flowing Hamiltonian as a scalar function and a systematic expansion in @xmath , where @xmath represents the system size. This allows the original operator equation to be rewritten as a regular partial differential equation amenable to a numeric or, in some instances, analytic treatment. The bulk of this work comprises of the detailed application of these methods to two simple but non-trivial models. These calculations are found to reproduce known exact results to a very high accuracy. The material is organized as follows. Chapter 1 provides a brief overview of the flow equation formalism with particular emphasis on the role of the generator. In Chapter 2 we present the two solution methods. The first is based on an expansion in fluctuations controlled by the system size, while the second makes use of non-commutative coordinates to rewrite the flow equation in terms of the Moyal bracket [ 22 ] . The application of these methods to the Lipkin model constitutes the third chapter. We are able to calculate both eigenvalues and expectation values non-perturbatively and in the thermodynamic limit. New aspects of the phase structure, made apparent by the non-perturbative treatment, are also investigated. These results, published in [ 23 ] , have subsequently led to further studies of the phase structure in [ 24 ] . In Chapter 4 the Dicke model is treated using a novel two-step diagonalization procedure. Although we derive the flow equation non-perturbatively its complexity necessitates a partially perturbative solution. Despite this our approach serves as a valuable example of how an effective Hamiltonian may be constructed and subsequently solved within this framework. ## Chapter 1 Flow equations ### 1.1 Overview The central notion in Wegner’s flow equations [ 2 ] is the transformation of a Hamiltonian @xmath through the application of a sequence of consecutive infinitesimal unitary transformations. It is the continuous evolution of @xmath under these transformations that we refer to as the flow of the Hamiltonian. These transformations are constructed to bring about decoupling in @xmath , leading to a final Hamiltonian with a diagonal, or block diagonal, form. The major advantage of this approach is that the relevant transformations are determined dynamically during the flow, and no a priori knowledge about them is required. We are led to consider a family of unitary transformations @xmath which is continuously parametrized by the flow parameter @xmath . @xmath constitutes the net effect of all the infinitesimal transformations applied up to the point in the flow labelled by @xmath . At the beginning of the flow @xmath equals the identity operator. The evolution of @xmath is governed by -- -------- -- ------- @xmath (1.1) -- -------- -- ------- where @xmath is the anti-hermitian generator of the transformation. Applying @xmath to @xmath produces the transformed Hamiltonian @xmath for which the flow equation reads -- -------- -- ------- @xmath (1.2) -- -------- -- ------- To be consistent in the calculation of expectation values the same transformation needs to be applied to the relevant observables. An eigenstate @xmath of @xmath transforms according to -- -------- -- ------- @xmath (1.3) -- -------- -- ------- while the flow of a general observable @xmath is governed by -- -------- -- ------- @xmath (1.4) -- -------- -- ------- Expectation values in the original and transformed basis are related in the usual way: -- -------- -- ------- @xmath (1.5) -- -------- -- ------- ### 1.2 The role of @xmath Much of the versatility of the flow equation method stems from the freedom that exists in choosing the generator @xmath . Several different forms have been employed in the literature, and we will explore the consequences of some of these next. For the moment we restrict ourselves to the finite dimensional case. #### 1.2.1 Wegner’s choice In Wegner’s original formulation [ 2 ] @xmath was chosen as the commutator of the diagonal part of @xmath , in some basis, with @xmath itself, i.e. -- -------- -- ------- @xmath (1.6) -- -------- -- ------- It was shown that in the @xmath limit @xmath converges to a final Hamiltonian @xmath for which -- -------- -- ------- @xmath (1.7) -- -------- -- ------- We conclude that the effect of the flow is to decouple those states which correspond to differing diagonal matrix elements. In general this leads to a block-diagonal structure for @xmath . We will not use this formulation as other choices exist which offer greater control over both the type of decoupling present in @xmath (i.e. the fixed point of the flow) and the form of @xmath during flow (i.e. the path followed to the fixed point). #### 1.2.2 @xmath An alternative to Wegner’s formulation is -- -------- -- ------- @xmath (1.8) -- -------- -- ------- where @xmath is a fixed ( @xmath -independent) hermitian operator of our choice. It is straightforward to show that @xmath converges to a final Hamiltonian which commutes with @xmath . The proof rests on the observation that -- -------- -- ------- @xmath (1.9) -- -------- -- ------- where the positivity of the trace norm has been used. It follows that @xmath is a monotonically decreasing function of @xmath that is bounded from below by @xmath , and so its derivative must vanish in the @xmath limit. The right-hand side of ( 1.9 ) is simply the trace norm of @xmath , and so we conclude that @xmath . Choosing a diagonal @xmath clearly leads to a block-diagonal structure for @xmath where only states corresponding to equal diagonal matrix elements of @xmath are connected. Put differently, @xmath assigns weights to different subspaces through its diagonal matrix elements. The flow generated by @xmath then decouples subspaces with differing weights. In particular, a non-degenerate choice of @xmath will lead to a complete diagonalisation of @xmath . Furthermore, it can be shown [ 20 ] that the eigenvalues of @xmath , as they appear on the diagonal of @xmath , will have the same ordering as the eigenvalues (diagonal matrix elements) of @xmath . We can summarize this by saying that the flow equation generates a transformation that maps the eigenstates of @xmath onto the eigenstates of @xmath in an order preserving fashion. It is worth noting that this ordering can only take place within subspaces that are irreducible under @xmath and @xmath . The reason for this is that the flow equation clearly cannot mix subspaces that are not connected by either @xmath or @xmath . We will see several examples of this later on. #### 1.2.3 The structure preserving generator Whereas the formulation above provides a good deal of control over the structure of @xmath , the form of @xmath at finite @xmath is generally unknown. This is due to non-zero off-diagonal matrix elements appearing at finite @xmath that are not present in either the initial or final Hamiltonian. For example, a band-diagonal Hamiltonian may become dense ¹ ¹ 1 A dense matrix is one of which the majority of elements is non-zero. during flow and still converge to a diagonal form. From a computational point of view it is clearly desirable that @xmath assumes as simple a form as possible. Of particular interest in this regard are generators which preserve a band diagonal, or more generally band block diagonal structure present in the original Hamiltonian. These types of generators have been applied to a wide range of models [ 14 , 15 , 16 , 17 , 18 , 19 ] , and are particularly attractive in that they allow for a clear physical interpretation of the transformed Hamiltonian @xmath . First we introduce an operator @xmath with integer eigenvalues which will serve as a labelling device for different subspaces in the Hilbert space. Associated with each distinct eigenvalue @xmath of @xmath is the corresponding subspace of eigenstates @xmath where @xmath . We also assume, without loss of generality, that @xmath for all @xmath . The Hamiltonian @xmath is said to possess a band block diagonal structure with respect to @xmath if there exists an integer @xmath such that @xmath for all @xmath and @xmath whenever @xmath . This selection rule clearly places a bound on the amount by which @xmath can change @xmath . The matrix representation of such an operator in the @xmath basis typically has a form similar to that shown in Figure 1.1 (a). For the cases we will consider, and for those treated in the literature, it is possible to group terms in the Hamiltonian together based on whether they increase, decrease or leave unchanged the value of @xmath . This leads to the form -- -------- -- -------- @xmath (1.10) -- -------- -- -------- where @xmath changes @xmath by @xmath , i.e. @xmath . Clearly @xmath is hermitian, while @xmath and @xmath are conjugates. @xmath is responsible for scattering within each @xmath -sector. Flow equations are used to bring this Hamiltonian into a form which conserves @xmath , i.e. for which @xmath . This effective Hamiltonian will be block diagonal, similar to Figure 1.1 (b), with each block containing new interactions generated during the flow. We require that the band block diagonal structure of @xmath is retained at finite @xmath , and so the flowing Hamiltonian should be of the form -- -------- -- -------- @xmath (1.11) -- -------- -- -------- The generator which achieves this is -- -------- -- -------- @xmath (1.12) -- -------- -- -------- The corresponding flow equation reads -- -------- -- -------- @xmath (1.13) -- -------- -- -------- The combinations of @xmath and @xmath appearing in ( 1.13 ) clearly cannot generate scattering between @xmath -sectors differing by more than @xmath , and so the band block diagonal structure of @xmath is preserved. It can be shown [ 14 ] that this generator guarantees convergence to the desired form of @xmath which conserves @xmath . Before proceeding, let us point out why the choice of the previous section @xmath does not lead to a flow of the form ( 1.11 ), although it does produce the correct fixed point. Using the form of @xmath we see that -- -------- -- -------- @xmath (1.14) -- -------- -- -------- where @xmath , and so -- -------- -- -------- @xmath (1.15) -- -------- -- -------- Whereas commutators between terms which increase (decrease) @xmath dropped out in ( 1.13 ) this is not the case in ( 1.15 ). In general @xmath will contain terms which change @xmath by up to @xmath , thus destroying the band block diagonal structure. This process will continue until, at finite @xmath , @xmath will connect all possible @xmath -sectors. Although our treatment has dealt with @xmath largely in abstract terms, this is not generally the case in the literature. Previous applications of this method present a physical picture of @xmath as a counting operator for the fundamental excitations describing the low energy physics of a system. Examples of these include triplet bonds on a dimerized spin chain [ 14 , 15 , 16 , 17 ] and quasi-particles (particle-hole excitations) [ 18 , 19 ] in Fermi systems. The flow equation is used to obtain an effective Hamiltonian that conserves the number of these excitations. These “effective particle-conserving models” [ 13 ] have been studied in considerable detail within a perturbative framework. Finally we point out that the approaches of Sections 1.2.2 and 1.2.3 coincide when @xmath only connects sectors for which @xmath differs by some fixed amount @xmath , i.e. @xmath when @xmath . ### 1.3 Flow equations in infinite dimensions Proofs concerning the convergence properties of the flow equations discussed thus far rely on the invariance of the trace under unitary transformations. In infinite dimensions the trace is not generally well defined, and the question of convergence depends on special properties of the Hamiltonian. The most important property in this regard is the boundedness of the spectrum. It has been shown that all the methods described above will converge provided that the Hamiltonian possesses a spectrum bounded from below [ 2 , 14 ] . Aspects of the flow equations in infinite dimensions were investigated in [ 25 ] and [ 26 ] using the bosonic Hamiltonian -- -------- -- -------- @xmath (1.16) -- -------- -- -------- and we digress for a moment to discuss this case in more depth. It is well known that for @xmath this model possesses a harmonic spectrum which can be found by applying the Bogoliubov transformation. When @xmath the spectrum forms a continuum which is unbounded from both above and below. We will contrast the behaviour of the flow equations resulting from Wegner’s choice of @xmath and the choice @xmath . In both these cases @xmath has the simple form -- -------- -- -------- @xmath (1.17) -- -------- -- -------- and the flow equation closes on a coupled set of three differential equations for @xmath and @xmath . When @xmath in the starting Hamiltonian we find that both choices of @xmath lead to the fixed point -- -------- -- -------- @xmath (1.18) -- -------- -- -------- where @xmath ; a result matching that of the Bogoliubov transformation. Next, consider the @xmath case. When we solve the flow equation for @xmath we no longer find convergence. Surprisingly, Wegner’s choice for the generator still produces a convergent flow, although not one leading to a diagonal form: -- -------- -- -------- @xmath (1.19) -- -------- -- -------- Note that this does not contradict the assertion made earlier that @xmath . Admittedly this example does represent an extreme case. For physically sensible Hamiltonians the spectrum is always bounded from below, as one would expect in order to have a well-defined ground state. For the models that we will consider the flow equations may be safely applied without further modification. ## Chapter 2 Solving the flow equation We have seen the theoretical capabilities of flow equations with regard to the diagonalization of Hamiltonians and the construction of effective operators. The practical implementation of this method is, however, hampered by the difficulty of solving the resulting operator differential equation. On the operator level this is due to the generation of additional operators during the flow that were not present in the original Hamiltonian. This leads to an extremely large set of coupled differential equations for the coupling constants of these terms. Generally some kind of approximation is required in order to continue. The usual approach consists of replacing @xmath by a simpler parametrized form for which the flow equation closes on a set of equations of a tractable size. A particular parametrization is usually selected on the basis of a perturbative approximation, or by using some knowledge of the relevant degrees of freedom in the problem. In general this approach is only valid for a limited range of the coupling constant, and tends to break down when the system exhibits non-perturbative features, i.e. non-analytic behaviour in the coupling constant. We will introduce two new approaches to this problem which allow us to treat the flow equation in a non-perturbative way. The first involves a systematic expansion in fluctuations, which is controlled by the size of the system rather than the coupling constant. The second makes use of non-commutative variables to recast the flow equation as a regular partial differential equation. This approach also involves an expansion controlled by the system size. Although there are similarities between these two methods we will treat them separately and then show how they produce the same results in specific cases. ### 2.1 Expansion in fluctuations We consider the flow of a time reversal invariant hermitian operator @xmath which acts on a Hilbert space @xmath . We assume @xmath to be finite dimensional since the expansion ( 2.1 ) below is easily proved in this case (see Appendix A ). This restriction is, however, by no means essential. If the Hilbert space is infinite dimensional the expansion ( 2.1 ) still applies to bounded operators [ 27 ] and subsequently one may approach the infinite dimensional case by studying the flow of a bounded function of the Hamiltonian, instead of the Hamiltonian itself. Alternatively one can introduce a cut-off in some basis, e.g., a momentum cut-off and study the behaviour of the system as a function of the cut-off. The aim of this section is to develop a parameterization of the flowing Hamiltonian which allows for a systematic expansion controlled by the fluctuations, rather than the coupling constant. This is in the same spirit as the semi-classical expansion of quantum mechanics, based on an expansion in orders of @xmath , and corresponds to resumming certain classes of diagrams in the perturbation series of the coupling constant. Let @xmath and @xmath be hermitian operators acting on @xmath which together form an irreducible set. What follows holds for any irreducible set, however, the case of two operators appears naturally in flow equations as the Hamiltonian can usually be written in the form @xmath where the spectrum and eigenstates of @xmath are known. Note that if @xmath and @xmath are reducible on @xmath , but irreducible on a proper subspace of @xmath , the problem can be restricted to this smaller subspace making the irreducibility of @xmath and @xmath a very natural requirement. In Appendix A it is shown that any operator acting on @xmath can be written as a polynomial in @xmath and @xmath . In particular this holds for @xmath , and so we may write -- -------- -- ------- @xmath (2.1) -- -------- -- ------- where each @xmath coefficient is a function of @xmath and repeated indices indicate sums over @xmath and @xmath . Since @xmath is both hermitian and real it follows that the @xmath ’s are invariant under the reversal of indices: -- -------- -- ------- @xmath (2.2) -- -------- -- ------- Now define @xmath as the fluctuation of @xmath around the expectation value @xmath , where @xmath is some arbitrary state. By setting @xmath in equation ( 2.1 ) we obtain @xmath as an expansion in these fluctuations -- -------- -- ------- @xmath (2.3) -- -------- -- ------- where -- -------- -------- -------- -- ------- @xmath @xmath @xmath (2.4) @xmath @xmath @xmath (2.5) @xmath @xmath @xmath (2.6) -- -------- -------- -------- -- ------- and so forth. Note that @xmath , when viewed as a function of @xmath , @xmath and @xmath , encodes information about all the expansion coefficients @xmath appearing in equation ( 2.1 ). A natural strategy that presents itself is to set up an equation for @xmath as a function of @xmath , @xmath and @xmath in some domain @xmath of the @xmath – @xmath plane, determined by the properties of the operators @xmath and @xmath . In particular we require that this domain includes values of @xmath ranging from the largest to the smallest eigenvalues of @xmath . For this purpose we require a family of states @xmath , parameterized by a continuous set of variables @xmath , such that as @xmath is varied over the domain @xmath , @xmath and @xmath range continuously over the domain @xmath . A very general set of states that meets these requirements are coherent states [ 28 ] . In this representation we may consider @xmath and @xmath as continuous variables with each @xmath a function of @xmath , @xmath and the following relations hold generally: -- -------- -- ------- @xmath (2.7) -- -------- -- ------- Coefficients with three or more indices cannot be written in this way, since, for example, @xmath need not be equal to @xmath . The general relationship between the true coefficients and derivatives of @xmath are -- -------- -- ------- @xmath (2.8) -- -------- -- ------- where the sum over @xmath is over all the distinct ways of ordering @xmath zeros and @xmath ones. For example -- -------- -- ------- @xmath (2.9) -- -------- -- ------- Replacing @xmath by the left-hand side of the equation above is equivalent to approximating @xmath by the average of @xmath , @xmath and @xmath . In general this amounts to approximating @xmath by the average of the coefficients corresponding to all the distinct reorderings of @xmath . To set up an equation for @xmath we insert the expansion ( 2.3 ) into the flow equation with @xmath and take the expectation value with respect to the state @xmath . The left-and right-hand sides become a systematic expansion in orders of the fluctuations @xmath : -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- Note that writing @xmath or @xmath in the first position of the double commutator on the right-hand side of equation ( 2.1 ) is a matter of taste. The expectation values appearing above are naturally functions of @xmath and may be written as functions of @xmath and @xmath by inverting the equations @xmath to obtain @xmath . With an appropriately chosen state, such as a coherent state, the higher orders in the fluctuation can often be neglected, as the expansion is controlled by the inverse of the number of degrees of freedom (see Appendix C for an explicit example). A useful analogy is the minimal uncertainty states in quantum mechanics which minimizes the fluctuations in position and momentum. The choice of state above aims at the same goal for @xmath and @xmath . Clearly it is difficult to give a general algorithm for the construction of these states and this property has to be checked on a case by case basis. If this is found to be the case we note that replacing the @xmath ’s with more than two indices by a derivative will introduce corrections in ( 2.1 ) of an order higher than the terms already listed, or, on the level of the operator expansion, corrections higher than second order in the fluctuations. Working to second order in the fluctuations we can therefore safely replace @xmath by the derivatives of a function @xmath and write for @xmath : -- -------- -------- -------- -------- -------- @xmath @xmath @xmath (2.11) @xmath @xmath -- -------- -------- -------- -------- -------- where @xmath denotes the anti-commutator and -- -------- -- -------- @xmath (2.12) -- -------- -- -------- This turns the flow equation ( 2.1 ) into a nonlinear partial differential equation for @xmath , correct up to the order shown in ( 2.1 ). The choice of coherent state, the corresponding calculation of the fluctuations appearing in equation ( 2.1 ) and the identification of the parameter controlling the expansion are problem specific. There are, however, a number of general statements that can be made about the flow equation and the behaviour of @xmath . The first property to be noted is that since @xmath is diagonal in the eigenbasis of @xmath it should become a function of only @xmath , provided that the spectrum of @xmath is non-degenerate. This is reflected in the behaviour of @xmath by the fact that @xmath should be a function of @xmath only. This is indeed borne out to high accuracy in our later numerical investigations. This function in turn provides us with the functional dependence of @xmath on @xmath . Keeping in mind the unitary connection between @xmath and @xmath this enables us to compute the eigenvalues of @xmath straightforwardly by inserting the supposedly known eigenvalues of @xmath into the function @xmath . A second point to note is that the considerations above apply to the flow of an arbitrary hermitian operator with time reversal symmetry. It is easily verified that the transformed operator @xmath satisfies the flow equation: -- -------- -- -------- @xmath (2.13) -- -------- -- -------- where the same choice of @xmath as for the Hamiltonian has to be made, i.e., @xmath . The expansion ( 2.11 ) can be made for both operators @xmath and @xmath . Denoting the corresponding function for @xmath by @xmath the flow equation ( 2.13 ) turns into a linear partial differential equation for @xmath containing the function @xmath , which is determined by ( 2.1 ). The expectation value of @xmath with respect to an eigenstate @xmath of @xmath can be expressed as -- -- -- -------- (2.14) -- -- -- -------- where @xmath . In the limit @xmath the states @xmath are simply the eigenstates of @xmath , which are supposedly known. In this way the computation of the expectation value @xmath can be translated into the calculation of expectation values of the operator @xmath , obtained by solving the flow equation ( 2.13 ), in the known eigenstates of @xmath . ### 2.2 Moyal bracket approach Next we present an approach based on the Moyal bracket formalism [ 22 ] . Let @xmath denote the @xmath -dimensional Hilbert space of the Hamiltonian under consideration. We define two unitary operators @xmath and @xmath that act irreducibly on @xmath and satisfy the exchange relation -- -------- -- -------- @xmath (2.15) -- -------- -- -------- Since @xmath is unitary its eigenvalues are simply phases. Let @xmath be one such eigenvalue, and consider the action of @xmath on the corresponding eigenstate @xmath : -- -------- -- -------- @xmath (2.16) -- -------- -- -------- We see that @xmath is again an eigenstate of @xmath with eigenvalue @xmath . Since @xmath and @xmath act irreducibly on @xmath all the eigenstates of @xmath can be obtained by the repeated application of @xmath to @xmath . Furthermore, we may scale @xmath so that it has an eigenvalue equal to one. It follows that the eigenvalues and eigenstates of @xmath take the form -- -------- -- -------- @xmath (2.17) -- -------- -- -------- while @xmath acts as a ladder operator between these states: -- -------- -- -------- @xmath (2.18) -- -------- -- -------- The allowed values of @xmath are found by taking the trace on both sides of @xmath , which leads to the requirement -- -------- -- -------- @xmath (2.19) -- -------- -- -------- This fixes @xmath at an integer multiple of @xmath . We choose @xmath as this ensures that @xmath is non-degenerate, which is crucial for the construction that follows. In similar fashion to the previous section, we wish to represent flowing operators in terms of @xmath and @xmath . The main result in this regard is that the set -- -------- -- -------- @xmath (2.20) -- -------- -- -------- forms an orthogonal basis for the space of linear operators acting on @xmath . The orthogonality of @xmath follows from applying the trace inner product in the @xmath -basis to two members of @xmath : -- -------- -- -------- @xmath (2.21) -- -------- -- -------- where @xmath equals one if @xmath and is zero otherwise. This, together with the observation that the dimension of the linear operator space equals @xmath , proves the claim. Consider two arbitrary operators @xmath and @xmath expressed in the @xmath basis as -- -------- -- -------- @xmath (2.22) -- -------- -- -------- where @xmath and @xmath are scalar coefficients. We use the convention of always writing the @xmath ’s to the right of the @xmath ’s. The product of @xmath and @xmath then gives -- -------- -- -------- @xmath (2.23) -- -------- -- -------- Note the similarity in form between this product and the product of functions of regular commuting variables. Only the phase factor, the result of imposing our ordering convention on the product, distinguishes the two. In fact, we may treat @xmath and @xmath as regular scalar variables provided that we modify the product rule to incorporate this phase. Convenient variables for this procedure are @xmath and @xmath , which are related to @xmath and @xmath (now treated as scalars) through @xmath and @xmath . Having replaced operators by functions of @xmath and @xmath the modified product rule reads -- -------- -- -------- @xmath (2.24) -- -------- -- -------- where the @xmath and @xmath derivatives act to the right and left respectively. This is seen to be of the required form by using the fact that both @xmath and @xmath are eigenfunctions of @xmath and @xmath : -- -------- -------- -------- -- -------- @xmath @xmath @xmath (2.25) -- -------- -------- -------- -- -------- which agrees with ( 2.23 ). The @xmath -operation is known as the Moyal product [ 22 ] , while the corresponding commutator @xmath is the Moyal bracket. When @xmath and @xmath are represented in this manner the flow equation becomes a partial differential equation in @xmath , @xmath and @xmath : -- -------- -- -------- @xmath (2.26) -- -------- -- -------- In its exact form this formulation is not of much practical value, since the operator exponent involved in the Moyal product is very difficult to implement numerically. A significant simplification is achieved by expanding the operator exponent to first order in @xmath , which is known to scale like one over the dimension @xmath of the Hilbert space. We expect this to be a very good approximation provided that the derivatives do not bring about factors of the order of @xmath . This translates into a smoothness condition: we require that the derivatives of the relevant functions remain bounded in the thermodynamic limit as @xmath goes to infinity. Using this approximation the Moyal product becomes, to leading order, -- -------- -- -------- @xmath (2.27) -- -------- -- -------- while the Moyal bracket reads -- -------- -- -------- @xmath (2.28) -- -------- -- -------- Partial derivatives are indicated by the subscript shorthand. The form of the flow equation is now largely fixed, up to the specific choice of the generator. As an example, consider the generator @xmath which we will use later on. In this case the flow equation becomes -- -------- -- -------- @xmath (2.29) -- -------- -- -------- The remaining problem is that of constructing the initial conditions, i.e. @xmath , in terms of @xmath and @xmath (or equivalently @xmath and @xmath ) in such a way that the smoothness conditions are satisfied. The reader may have noticed that we have not specified how the realization of @xmath and @xmath should be constructed on @xmath . Put differently, there is no obvious rule which associates a specific basis of @xmath with the eigenstates of @xmath . It seems reasonable that this freedom may allow us to construct smooth initial conditions through an appropriate choice of basis, whereas a malicious choice could produce very poorly behaved functions. We know of no way to proceed on such general terms, and we will instead tackle this problem on a case-by-case basis. In all of these we will use the algebraic properties of operators appearing in the Hamiltonian to reduce this problem to one of representation theory. Although two operators are clearly the minimum required to construct a complete operator basis, it is also possible to introduce multiple such pairs. This would be a natural choice when @xmath is a tensor product of Hilbert spaces @xmath @xmath , each of which is of a high dimension. We can introduce @xmath pairs of operators @xmath which satisfy @xmath and @xmath for all @xmath . In the same way as before this leads to @xmath pairs of scalar variables @xmath for which the product rule, to first order in the @xmath ’s, is -- -------- -- -------- @xmath (2.30) -- -------- -- -------- Note that @xmath are analogous to conjugate position and momenta coordinates representing the independent degrees of freedom of the system. The Moyal bracket acts like the Poisson bracket for these coordinates: -- -------- -- -------- @xmath (2.31) -- -------- -- -------- This formulation strongly suggests an analogy with semi-classical approximation schemes. Our approach to solving the flow equation is indeed very closely related to the Wigner-Weyl-Moyal [ 22 , 29 ] formalism, which describes the construction of a mapping between quantum operators and functions of classical phase space coordinates. This allows for the description of a quantum system in a form formally analogous to classical dynamics. When applied to the flow equations this formalism produces results similar to those obtained before. The central approximation again involves a non-perturbative expansion, but which is now controlled by @xmath , and so is semi-classical in nature. Let us formalize some of these notions in the context of a single particle in @xmath dimensions. The relevant Hilbert space is @xmath and the position and momentum operators satisfy the standard commutation relations -- -------- -- -------- @xmath (2.32) -- -------- -- -------- We first introduce the characteristic operator [ 30 ] -- -------- -- -------- @xmath (2.33) -- -------- -- -------- where @xmath , @xmath and similar for @xmath and @xmath . Varying the arguments of @xmath over their domains produces a set of operators analogous to @xmath (equation ( 2.20 )), where the discrete powers @xmath and @xmath correspond to the continuous labels @xmath and @xmath . We again find both completeness and orthogonality with respect to the trace norm: -- -------- -------- -------- -------- -------- @xmath @xmath @xmath (2.34) @xmath @xmath @xmath @xmath -- -------- -------- -------- -------- -------- Using this, an operator @xmath can be represented as -- -------- -- -------- @xmath (2.35) -- -------- -- -------- where -- -------- -- -------- @xmath (2.36) -- -------- -- -------- is a scalar function. Now consider the product of two operators represented in this manner: -- -------- -- -------- @xmath (2.37) -- -------- -- -------- The non-commutativity of @xmath and @xmath gives rise to the scalar factor @xmath , which is the only element distinguishing this product from one of regular scalar functions. We conclude, as before, that the position and momentum operators may be treated as scalar variables provided that we modify the product rule to incorporate this phase. This leads to the Moyal product -- -------- -- -------- @xmath (2.38) -- -------- -- -------- where @xmath and @xmath . Note that an expansion of the exponential is now controlled by @xmath instead of @xmath . To leading order the Moyal bracket is given by -- -------- -- -------- @xmath (2.39) -- -------- -- -------- where the subscripts denote partial derivatives. We conclude that in a semi-classical approximation the Hamiltonian and generator may be replaced by scalar functions @xmath and @xmath , and that the flow equation is given in terms of the Moyal bracket by -- -------- -- -------- @xmath (2.40) -- -------- -- -------- When solved to leading order in @xmath this equation describes the renormalization of the Hamiltonian within a semi-classical approximation. Further quantum corrections can be included by simply expanding the Moyal bracket to higher orders in @xmath . ## Chapter 3 The Lipkin Model ### 3.1 Introduction Since its introduction in 1965 as a toy model for two shell nuclear interactions the Lipkin-Meshov-Glick model [ 31 ] has served as a testing ground for new techniques in many-body physics. Here we will use it to illustrate both the working of the flow equations and the solution methods presented in the previous chapter. While the simple structure of the model will allow many calculations to be performed exactly, its non-trivial phase structure will provide a true test for our non-perturbative approach. We begin with an overview of the model, its features and the quantities we are interested in calculating. After pointing out some specific aspects of the flow equations for the Lipkin model we proceed to treat the equations using the methods developed earlier. Finally we present the results obtained from the numerical solutions of the resulting PDE’s and compare them with some known results. New aspects of the model, brought to the fore by this treatment, will also be discussed. ### 3.2 The model The Lipkin model describes @xmath fermions distributed over two @xmath -fold degenerate levels separated by an energy of @xmath . For simplicity we shall take @xmath to be even. Fermi statistics require that @xmath , and accordingly the thermodynamic limit should be understood as @xmath followed by @xmath . The interaction @xmath introduces scattering of pairs between levels. Labelling the two levels by @xmath , the Hamiltonian reads -- -------- -- ------- @xmath (3.1) -- -------- -- ------- where the indices @xmath and @xmath run over the level degeneracy @xmath . A spin representation for @xmath can be found by introducing the @xmath generators -- -------- -- ------- @xmath (3.2) -- -------- -- ------- Together with the second order Casimir operator @xmath , these satisfy the regular @xmath commutation relations: -- -------- -- ------- @xmath (3.3) -- -------- -- ------- We divide @xmath by @xmath and define the dimensionless coupling constant @xmath to obtain -- -------- -- ------- @xmath (3.4) -- -------- -- ------- where all energies are now expressed in units of @xmath . The factor of @xmath in the definition of @xmath brings about the @xmath in front of the second term, which ensures that the Hamiltonian as a whole is extensive and scales like @xmath . Since @xmath the Hamiltonian acts within irreducible representations of @xmath where states are labelled by the eigenvalues of @xmath and @xmath , i.e., @xmath and @xmath for @xmath . The Hamiltonian thus assumes a block diagonal structure of sizes @xmath . The low-lying states occur in the multiplet @xmath , and we fix @xmath at this value throughout, using the shorthand @xmath for the basis states. The Hamiltonian can be reduced further by noting that it leaves the subspaces of states with either odd or even spin projection invariant. States belonging to one of these subspaces are referred to as having either odd or even parity. We denote the eigenstate of @xmath with energy @xmath by @xmath , where @xmath . When @xmath the ground state is simply @xmath which is written as @xmath in the spin basis. Non-zero values of @xmath cause particle-hole excitations across the gap, and at @xmath the model exhibits a phase transition from an undeformed first phase to a deformed second phase. To distinguish the two phases we use the order parameter @xmath where @xmath is the expectation value of @xmath in the ground state. As we will show, @xmath is non-zero only within the second phase. The phases can be characterized further by the energy gap @xmath which is positive in the first phase and vanishes like @xmath as @xmath approaches @xmath . In the second phase the ground states of the odd and even parity subspaces become degenerate, causing the parity symmetry to be broken and the corresponding energy gap to vanish. Further discussion of this model and its features can be found in [ 31 ] . ### 3.3 The flow equation approach We will follow the formulations of Sections 1.2.2 and 1.2.3 and choose the generator as -- -------- -- ------- @xmath (3.5) -- -------- -- ------- Let us consider the consequences of this choice. Firstly, since @xmath is non-degenerate, we expect the final Hamiltonian @xmath to be diagonal in the known spin basis. Furthermore the eigenvalues of @xmath appear on the diagonal of @xmath in the same order as in @xmath , i.e. increasing from top to bottom. Secondly, note that @xmath possesses a band diagonal structure with respect to @xmath , which plays the role of @xmath in Section 1.2.3 . Based on our earlier discussion we expect this structure to be conserved during flow, and so @xmath will only connect states of which the spin projection differs by two. Let @xmath denote the transformed eigenstates of @xmath . From the ordering property of eigenvalues in @xmath we conclude that @xmath , and so a general expectation value may be calculated through -- -------- -- ------- @xmath (3.6) -- -------- -- ------- Before continuing we mention some of the previous applications of flow equations to the Lipkin model. The first of these was by Pirner and Friman [ 8 ] , who dealt with newly generated terms by linearizing them around their ground state expectation values. This yielded good results in the first phase but lead to divergences in the second; a common ailment of these types of approximations. Subsequent work by Mielke [ 10 ] relied on an ansatz for the form of @xmath ’s matrix elements, while Stein [ 9 ] followed a bosonization approach. Dusuel and Vidal [ 11 , 12 ] used flow equations together with the Holstein-Primakoff boson representation to compute finite-size scaling exponents for a number of physical quantities. Their approach was also based on a @xmath expansion. A method which produced reliable results in both phases was that of [ 32 ] . This method also relied on the linearization of newly generated operators but, in contrast with [ 8 ] , did so around a dynamically changing (“running”) expectation value taken with respect to the flowing ground state. This “running” expectation value could be solved for in a self consistent manner, and then used in the flow equation for @xmath . This lead to non-perturbative results for the ground state, excitation gap and order parameter in both phases. Although it described the low energy physics well, this method failed to produce the correct spectrum for the higher energy states. ### 3.4 Expansion method In order to apply the method of Section 2.1 we require an irreducible set of operators @xmath in terms of which the flowing Hamiltonian will be constructed. For the purposes of the Lipkin model we will choose @xmath and @xmath , i.e. the diagonal and off-diagonal parts of the Hamiltonian respectively. The reader may well remark that this does not constitute an irreducible set on the entire Hilbert space, since the subspaces of odd and even spin projection are left invariant, as pointed out in Section 3.2 . Indeed, it is only within a subspace of definite parity (i.e. odd or even spin projection) that this set is irreducible. However, we note that the flow equation will never mix these odd and even subspaces, and so the flow will proceed independently within each subspace. This ensures that our representation of @xmath will never require an operator which connects odd and even states, and it turns out that this choice of @xmath is indeed sufficient. The same conclusion extends to @xmath by equation ( 1.1 ), and then to any flowing operator @xmath , provided that @xmath can be represented in this way. We will calculate the averages of @xmath and @xmath with respect to the coherent state [ 28 , 33 ] -- -------- -- ------- @xmath (3.7) -- -------- -- ------- where @xmath . See Appendix B for details on the properties of these states and the method by which the averages are calculated. We find that: -- -- -- ------- (3.8) -- -- -- ------- This constitutes a mapping of the complex plane onto the domain shown in Figure 3.1 . With these definitions in place we conclude that the flowing Lipkin Hamiltonian may be written as -- -------- -- ------- @xmath (3.9) -- -------- -- ------- with @xmath and @xmath respectively the diagonal and off-diagonal parts of the original Hamiltonian. By definition @xmath and so @xmath is defined on the domain pictured in Figure 3.1 . The initial condition becomes -- -------- -- -------- @xmath (3.10) -- -------- -- -------- Due to the coherent nature of the state ( 3.7 ) one might expect that terms corresponding to high order fluctuations in ( 3.9 ) will contribute less significantly to @xmath than the scalar term @xmath . This statement can be made precise as follows: since the flowing Hamiltonian is extensive @xmath should be proportional to @xmath , as is the case with @xmath and @xmath . To keep track of the orders of @xmath we introduce the scaleless variables @xmath , @xmath and @xmath . When taking the inner-product with respect to @xmath on both sides of ( 3.9 ) the linear terms fall away and we obtain -- -------- -- -------- @xmath (3.11) -- -------- -- -------- Each term is, up to a constant factor, of the form -- -------- -- -------- @xmath (3.12) -- -------- -- -------- where @xmath denotes some arbitrary product of @xmath fluctuations. Using the results of Appendix C we see that such a term is at most of order @xmath . The leading order term corresponds to @xmath , i.e. the scalar term @xmath . We conclude that @xmath is the leading order contribution to @xmath , expressed not as a function of @xmath and @xmath , but rather of the averages @xmath and @xmath . #### 3.4.1 The flow equation in the @xmath limit. We consider the flow of the Hamiltonian and an arbitrary observable @xmath . Since @xmath determines @xmath one would expect a one-way coupling between the equations. It is assumed that @xmath is a hermitian operator constructed in terms of @xmath and @xmath . Furthermore @xmath must be a rational function of @xmath , which ensures that when taking derivatives of @xmath no additional factors of @xmath are generated. First we summarize the equations concerned -- -------- -------- -------- -- -------- @xmath @xmath @xmath (3.13) @xmath @xmath @xmath (3.14) @xmath @xmath @xmath (3.15) @xmath @xmath @xmath (3.16) -- -------- -------- -------- -- -------- where @xmath and @xmath is just @xmath up to leading order in @xmath . Next we substitute the expansion of @xmath into the flow equation ( 3.15 ) and take the expectation value on both sides with respect to the coherent state. Arguing as before we identify the leading order term on the left as being @xmath . A general term on the right is of the form -- -------- -- -------- @xmath (3.17) -- -------- -- -------- By transforming to scaleless variables and using the result of Appendix C it is seen to be at most of order @xmath , where @xmath . Since the @xmath and @xmath terms are zero (they involve commutators of scalars) the leading order contributions come from the @xmath terms. These are exactly the terms found by considering the expansion of @xmath up to second order in the fluctuations. The expectation values of the double commutators may be calculated and expressed as functions of @xmath and @xmath using the method outlined in Appendix B . It is found that, due to cancellations, only four of the potential fifteen terms make leading order contributions. The corresponding expectation values are, to leading order -- -------- -------- -------- -- -------- @xmath @xmath @xmath (3.18) @xmath @xmath @xmath (3.19) @xmath @xmath @xmath (3.20) @xmath @xmath @xmath (3.21) -- -------- -------- -------- -- -------- where @xmath , @xmath and @xmath . When keeping only the leading order terms on both sides the flow equation becomes -- -------- -- -------- @xmath (3.22) -- -------- -- -------- where superscripts denote derivatives to the rescaled averages @xmath and @xmath . For further discussion it is convenient to use the variables @xmath , where @xmath and @xmath are related by @xmath . The square domain of these variables also simplifies the numerical solution significantly. We note that the arguments above can be applied, completely unchanged, to the flow equation of @xmath as well. We obtain, now for both @xmath and @xmath , the coupled set -- -------- -- -------- @xmath (3.23) @xmath (3.24) -- -------- -- -------- Note that, in contrast to the equation for @xmath , the equation for @xmath is a linear equation that can be solved once @xmath has been obtained from ( 3.23 ). In Section 3.3 it was mentioned that @xmath retains its band diagonal structure during flow, which means that @xmath only appears linearly in the representation of @xmath . This implies that @xmath should be linear in @xmath , or, in the new variables, linear in @xmath . When the form @xmath is substituted into ( 3.23 ) this is indeed seen to be the case, and we obtain a remarkably simple set of coupled PDE’s for @xmath and @xmath : -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath (3.25) -- -------- -------- -------- -- -------- The initial conditions are -- -------- -- -------- @xmath (3.26) -- -------- -- -------- #### 3.4.2 Finding the spectrum From @xmath it is possible to obtain the entire spectrum of @xmath . For this discussion it is convenient to use the original @xmath variables. Recall that in the @xmath limit @xmath flows toward a diagonal form and that the eigenvalues appear on the diagonal in the same order as in @xmath , i.e. increasing from top to bottom. This implies that the @xmath eigenvalue of @xmath is given by -- -------- -- -------- @xmath (3.27) -- -------- -- -------- where @xmath corresponds to the ground state energy. As @xmath flows towards a diagonal form the terms of expansion ( 3.13 ) containing @xmath will disappear and eventually @xmath and @xmath will become functions only of @xmath and @xmath respectively. The eigenvalues of @xmath are given by -- -------- -- -------- @xmath (3.28) -- -------- -- -------- where @xmath has been set to zero. This can be understood in two ways. Looking at equations ( 2.4 ) and ( 2.1 ) we see that the functional dependence of @xmath on @xmath is the same as that of @xmath (now called @xmath ) on @xmath . Since taking the expectation value of @xmath with respect to @xmath is equivalent to substituting @xmath for @xmath , the result follows. Alternatively, consider equation ( 2.11 ) and note that setting @xmath makes the ( @xmath diagonal element of @xmath zero. Since @xmath does not appear we see that when taking the inner product with @xmath only the scalar term, @xmath , will survive. #### 3.4.3 Calculating expectation values Next we return to the arbitrary operator @xmath introduced earlier. The aim is to calculate @xmath where @xmath is the (unknown) eigenstate of @xmath corresponding to the energy @xmath . We can formulate this calculation in terms of a flow equation by noting that -- -------- -- -------- @xmath (3.29) -- -------- -- -------- where @xmath , @xmath and @xmath is the unitary operator associated with the flow of @xmath . This equation holds for all @xmath , and particularly in the @xmath limit. Since @xmath it follows that -- -------- -- -------- @xmath (3.30) -- -------- -- -------- and we conclude that the expectation value of @xmath in the @xmath eigenstate of @xmath is simply the @xmath diagonal element of @xmath in the @xmath basis. Furthermore @xmath has the character of a generating function in the sense that -- -------- -- -------- @xmath (3.31) -- -------- -- -------- thus knowing @xmath , which is just @xmath up to leading order in @xmath , is sufficient to obtain all the matrix elements of @xmath with high accuracy. However, unless an analytic solution for @xmath is known, we are limited to numerical calculations for the low lying states. In particular, the ground state expectation value is found by setting @xmath . ### 3.5 Moyal bracket method In Section 2.2 the Moyal Bracket method was used to obtain a simple, but general, realization of the flow equation as a partial differential equation in the variables @xmath , @xmath and @xmath . For the present case this equation becomes -- -------- -- -------- @xmath (3.32) -- -------- -- -------- where @xmath is the Moyal bracket to leading order in @xmath . Here @xmath and so an expansion in @xmath is indeed controlled by the size of the system. As was mentioned earlier the model specific information enters through the initial condition, which we need to construct in terms of @xmath and @xmath . We will not perform this construction for the Hamiltonian directly, but rather follow the more general route of constructing an irreducible representation of @xmath , in terms of which both the Hamiltonian and observables can be readily obtained. We wish to find three functions @xmath , @xmath and @xmath which satisfy the @xmath commutation relations with respect to the Moyal bracket @xmath . Note that the Moyal formalism has allowed an essentially algebraic problem, that of constructing a specific representation, to be reduced to that of solving a set of differential equations. We also remind ourselves that, since we use the Moyal bracket in a first order approximation, our representation can only be expected to be correct up to this same order. We begin the construction by making the following ansatz -- -------- -- -------- @xmath (3.33) -- -------- -- -------- which clearly requires the representation to be unitary. This ansatz is based on the interpretation of @xmath as a ladder operator which connects states labelled by the eigenvalues of @xmath . Substituting these forms into the required commutation relations @xmath and @xmath produces the set -- -------- -- -------- @xmath (3.34) -- -------- -- -------- which is easily solved to obtain -- -------- -- -------- @xmath (3.35) -- -------- -- -------- Here @xmath and @xmath are integration constants that we fix by requiring that the second order Casimir operator assumes a constant value corresponding to the @xmath -irrep: -- -------- -- -------- @xmath (3.36) -- -------- -- -------- We can satisfy this constraint to leading order in @xmath by setting @xmath and @xmath . Finally we arrive at -- -------- -- -------- @xmath (3.37) -- -------- -- -------- Before continuing we define some more suitable variables. Consider the operator @xmath which, according to equation ( 2.17 ), has eigenvalues @xmath and is represented by @xmath . This suggests that the natural domain of @xmath is @xmath . We will use the scaleless variable @xmath in what follows. The representation now becomes -- -------- -- -------- @xmath (3.38) -- -------- -- -------- We remark that this construction is by no means unique, although it is expected that other examples are related to this one by a similarity transformation. For example, complex non-unitary representation such as -- -------- -- -------- @xmath (3.39) -- -------- -- -------- may be constructed. In fact, this is an exact representation to all orders in @xmath , since @xmath (and thus @xmath ) only occurs linearly and the Moyal bracket truncates after the first derivative. This illustrates the important role of the association between the eigenstates of @xmath and the particular basis of the Hilbert space. For our purposes the unitary representation ( 3.38 ) will be sufficient, although it is straightforward to check that the same results can be obtained using ( 3.39 ). By substituting the representation ( 3.38 ) into the Lipkin model Hamiltonian, and being careful to use the Moyal product in calculating the squares of @xmath , we obtain the initial condition, to leading order, as: -- -------- -- -------- @xmath (3.40) -- -------- -- -------- The flow equation reads -- -------- -- -------- @xmath (3.41) -- -------- -- -------- Again it is expected that the band diagonality of @xmath will be manifested as a constraint on the form of the solutions of the flow equation. We note that scattering between states of which the spin projection differ by two is associated with the @xmath term also appearing in the initial condition. Motivated by this we try the form @xmath . Note that a factor of @xmath , responsible for the extensivity of the Hamiltonian, has been factored out. Upon substituting this form into the flow equation we obtain -- -------- -- -------- @xmath (3.42) -- -------- -- -------- which agrees with the equations of ( 3.25 ) in the previous section. For this form the flow of an observable is given by -- -------- -- -------- @xmath (3.43) -- -------- -- -------- We have seen that the governing equations obtained using the Moyal bracket method agrees with those of the previous section, although the interpretation of the constituents do differ. Next we turn to the matter of extracting the spectrum and expectation values from the solutions of these equations. As this procedure is largely similar to that of the previous section we will remain brief. First, note that through the representation ( 3.39 ) we may consider @xmath , and thus @xmath itself, to be a function only of @xmath . We arrive at the familiar result that @xmath for @xmath . Expectation values are given by the diagonal elements of the flowed hermitian observable @xmath . The forms of @xmath suggest that in general @xmath may be written as -- -------- -- -------- @xmath (3.44) -- -------- -- -------- where @xmath corresponds to an off-diagonal term proportional to @xmath . (We only expect even powers since there is no mixing between the odd and even subspaces.) Importantly the @xmath term contains the desired information about the diagonal entries. We can isolate @xmath by integrating over @xmath , which projects out the off-diagonal terms. Thus, in summary, -- -------- -------- -------- -------- -------- @xmath @xmath @xmath (3.45) @xmath @xmath @xmath @xmath -- -------- -------- -------- -------- -------- #### 3.5.1 Probing the structure of eigenstates The ability to calculate expectation values for a large class of operators enables us to probe the structure of the eigenstates in a variety of ways. For this purpose we consider diagonal operators of the form @xmath where @xmath is a smooth, @xmath -independent function defined on @xmath . Since @xmath , the initial condition for the flow equation is simply @xmath . Suppose @xmath is the eigenstate under consideration, in which case -- -- -- -------- (3.46) -- -- -- -------- This leads to the interpretation of the expectation value as the average of the expansion coefficients squared @xmath , weighted by the function @xmath . Ideally we would like to choose @xmath as the projection operator onto some basis state @xmath , as this would provide the most direct way of calculating the contributions of individual states. However, the projection operator does not fall within the class of operators corresponding to smooth initial conditions since there is no continuous function @xmath for which @xmath in the large @xmath limit. Instead we choose @xmath where @xmath and @xmath . This weight function focuses on the contribution of those basis states @xmath for which @xmath lies in a narrow region centered around @xmath . Considered as a function of @xmath , it is expected that @xmath would approximate @xmath up to a constant factor, provided that the latter varies slowly on the scale of @xmath in the region of @xmath . Clearly @xmath controls the accuracy of this method, and also determines the scale on which the structure of the eigenstates can be resolved. ### 3.6 Numerical results In this section we analyze the results obtained by solving the equations derived in the previous two sections. We will mainly use the terminology of the Moyal bracket approach in what follows. See [ 23 ] for a treatment in terms of the expansion method. For compactness the @xmath argument of functions will occasionally be suppressed, in which case @xmath should be assumed. #### 3.6.1 Spectrum and expectation values First, we consider some structural properties of @xmath and @xmath . It is known that the matrix elements of the Lipkin Hamiltonian possess the symmetry @xmath , which implies that the spectrum is anti-symmetric around @xmath . This symmetry is respected by the flow equation and manifests itself through an invariance of equations ( 3.42 ) under the substitutions @xmath , @xmath and @xmath . This implies that @xmath and @xmath , and so we may restrict ourselves to the interval @xmath in the @xmath dimension. This will be seen to correspond to the negative half of the spectrum, from which the entire spectrum can easily be obtained. We also note that @xmath , and that this remains the case at finite @xmath since @xmath is proportional to @xmath . Applying the same argument to @xmath seems to suggest that no flow occurs for @xmath at @xmath and that @xmath for all @xmath . This conclusion is, however, incorrect since it is found that in the second phase @xmath develops a square root singularity at @xmath , which allows @xmath to flow away from @xmath . Numerically this can be handled easily by solving for @xmath , instead of @xmath . We made use of the well established fourth order Runge-Kutta method [ 34 ] to integrate the PDE’s to sufficiently large @xmath -values. It was shown that the eigenvalues of @xmath are given by @xmath . Furthermore, for sufficiently large @xmath it holds that -- -------- -------- -------- -------- -------- @xmath @xmath @xmath (3.47) @xmath @xmath -- -------- -------- -------- -------- -------- Figures 3.2 (a) and (b) show @xmath and @xmath at @xmath for three values of @xmath . At this point in the flow @xmath , which represents the off-diagonal part of @xmath , is already of the order of @xmath , and may be neglected completely. As a comparison with exact results we calculated the spectrum for @xmath using direct diagonalization and plotted the pairs @xmath for @xmath as dots. We observe an excellent correspondence for all states and in both phases, with an average error of about @xmath . This small discrepancy can be attributed to numerical errors and finite size effects, since we are comparing exact results obtained at finite @xmath with those of the flow equation which was derived in the @xmath limit. This is again illustrated in Figure 3.2 (d) which shows the relative error in the first five eigenvalues as a functions of @xmath for @xmath . The lines fall almost exactly on one another, so no legend is given. The log-log inset shows a set of straight lines with gradients equal to one, clearly illustrating the @xmath behaviour of the errors. Next we consider the dependence of the eigenvalues on the coupling @xmath . Figure 3.3 (a) shows the ground state energy as a function of @xmath together with the exact result for @xmath . We see that @xmath is fixed at @xmath in the first phase and begins to decrease linearly with @xmath at large coupling. The phase transition in the Lipkin model is known to be of second order, and is characterized by a large number of avoided level crossings [ 35 ] . This brings about complex non-analytic behaviour in the gaps between energy levels. The non-perturbative treatment of the flow equation enables us to reproduce much of this behaviour correctly. As examples we display the gaps @xmath , @xmath and @xmath in Figure 3.3 . One quantity that is not reproduced correctly are the gaps between successive states belonging to subspaces of differing parity. (Recall that subspaces of odd and even spin projection (parity) are not mixed by the Hamiltonian, as was shown in Section 3.2 .) As a definite case consider the gap @xmath which is known to vanish like @xmath in the second phase, while the flow equation produces a gap which grows linearly with @xmath . This can be attributed to the fact that the gaps are not extensive quantities and so they depend on higher order corrections in @xmath , which were neglected in our derivation. The function @xmath found by solving equations ( 3.42 ) can now be substituted into equation ( 3.43 ) to obtain the flow equation for a general observable @xmath . On a technical note, we found it advantageous to use expansion ( 3.44 ) to write the flow equation for @xmath as a coupled set of @xmath dimensional equations for the @xmath ’s, rather than treating it as a general @xmath dimensional PDE. First we consider the flow of @xmath , which corresponds to the initial condition @xmath . Figure 3.4 (a) shows the results obtained for the order parameter together with exact results for @xmath . We again find excellent agreement in both phases. Other observables can be considered by a simple modification of the initial conditions. For example @xmath produces the second moment of @xmath in the ground state, as shown in Figure 3.4 (b). The solution to equation ( 3.43 ) provides us with expectation values corresponding to excited states as well, which we find by evaluating @xmath , as defined in ( 3.44 ), at the values @xmath for @xmath . Figure 3.4 (c) shows @xmath for @xmath at different values of the coupling strength. The non-perturbative application of the flow equation method to the Lipkin model has provided some interesting new insights into aspects of the phase transition. We will investigate some of these next. First we fix some notation. In the large @xmath limit we may label states with the continuous label @xmath through the association @xmath . In this way a state is labelled according to its fractional position in the spectrum, for example @xmath always corresponds to the ground state, while @xmath denotes the state lying one quarter way up the spectrum. Now consider the behaviour of @xmath and @xmath depicted in Figures 3.2 (a) and (c) respectively. We note that for @xmath there always exists a value of @xmath , denoted by @xmath , where the derivative of @xmath very nearly vanishes. Furthermore, @xmath must be a point of inflection of @xmath since the derivative of @xmath to @xmath cannot change sign. If @xmath were to become negative the flow equation would become unstable and cause @xmath to grow exponentially, contradicting the results of Section 1.2.2 concerning the form of @xmath . This explains why we only observe a plateau at @xmath , and no more drastic behaviour. At @xmath this happens precisely at the ground state, i.e. @xmath , while in the first phase no such point exists. In [ 36 ] it was found that the energies of the low-lying states obey the scaling law -- -------- -- -------- @xmath (3.48) -- -------- -- -------- at @xmath . This can be confirmed using the flow equations by considering the behaviour of @xmath close to @xmath . Figure 3.5 (d) shows a log-log plot of @xmath versus @xmath together with a linear fit which reproduces the power of @xmath to within @xmath . Earlier a direct link was established between the gaps separating successive eigenvalues and the derivative of @xmath . This suggests that @xmath corresponds to a point in the spectrum with a very high density of states, brought about by a large number of avoided level crossings. Interestingly this point always occurs at the same absolute energy, namely @xmath , although this corresponds to increasingly highly excited energies relative to the ground state. We believe that in the thermodynamic limit, contrary to Figure 3.2 (c), the derivative of @xmath , and the corresponding gap, should vanish completely at this point. This has been confirmed in [ 36 ] where it was shown that the gap is given by -- -------- -- -------- @xmath (3.49) -- -------- -- -------- where @xmath . That our solution does not reflect this can be attributed to numerics, as this concerns a single point which is effectively “invisible” to the finite discretization used in the numerical method. The off-diagonal part of the Hamiltonian, represented by @xmath , also exhibits striking behaviour at @xmath . In fact, upon returning to the flow equation ( 3.42 ) we make the interesting observation that at the point where the derivative of @xmath vanishes, @xmath is not forced to flow to zero, but may in fact attain a non-trivial fixed point value. This is shown in Figure 3.2 (b) which shows @xmath as a function of @xmath . One clearly sees a sharp peak at the point where the derivative of @xmath nearly vanishes. The peak only occurs for @xmath and moves to the right as @xmath is increased. This is also consistent with the known result [ 2 ] that an off-diagonal element @xmath of @xmath decays roughly as @xmath at large @xmath . This suggests a connection between quantum phase transitions, the corresponding disappearance of an energy scale (gap) [ 35 ] in the thermodynamic limit and the absence of decoupling in the Hamiltonian, also in the thermodynamic limit. The occurrence of this point in the second phase lends itself to the interesting interpretation of a “quantum phase transition” at higher energies. Indeed, apart from possessing some notable properties itself, it separates regions of the spectrum with markedly different characteristics. It is well known that states alternate between odd and even parity as one moves up in the spectrum. In the first phase these odd-even pairs are separated by a finite gap. In the second phase one finds a degeneracy between these successive odd and even states developing below the @xmath point. In particular this implies a degenerate ground state with broken symmetry in the second phase, although, as we have seen, this description may be applied to all states below @xmath . Figure 3.5 (c) shows a subset of eigenvalues as a function of the coupling, clearly illustrating this coalescing of pairs. Turning to Figure 3.4 (d), which shows the expectation value @xmath as a function of the state label @xmath , we see a sharp local minimum occurring at @xmath which separates the two phases. At the point @xmath itself the corresponding state @xmath is characterized by sharp localization around the @xmath basis state, also noted by [ 36 ] . This is clearly illustrated by Figure 3.4 (d) which shows a pronounced decrease in the spread of the state @xmath in the @xmath basis occurring at the point where @xmath . We end this section with two “phase diagrams” which we hope will further clarify the discussion above. Figure 3.5 (a) shows the @xmath dependency of a subset of the negative, even eigenvalues. For @xmath the eigenvalues are confined between @xmath and @xmath . As @xmath increases first the ground state (shown in bold) and then the excited states begin to cross the @xmath phase boundary until eventually, in the large coupling limit, only the @xmath state retains its first phase character. For finite @xmath successive eigenvalues show avoided level crossings on the phase boundary. In the thermodynamic limit one would, however, expect that successive eigenvalues will coalesce as they cross the phase boundary; signaling a vanishing gap. A similar diagram, based in the label @xmath rather than the energy itself, is shown in Figure 3.5 (b). Keeping in mind the symmetries @xmath and @xmath this discussion can easily be adapted to apply to the positive half of the spectrum as well. These findings, published in [ 23 ] , have stimulated further investigation of these phenomenon in the contexts of exceptional points [ 24 ] and semi-classical approximations [ 36 ] . These have shed light on the origins of the high density of states and localization occurring at @xmath , as well as the mechanism responsible for the degeneracy between states of differing parity. #### 3.6.2 Structure of the eigenstates In Section 3.5.1 we outlined a strategy whereby the structure of the eigenstates could be probed using the expectation values of a class of specially constructed operators. In this manner it is possible to approximate the modulus of the expansion coefficients of the state in the @xmath basis. Here we present the numeric results of this procedure using the operators @xmath with @xmath . We will consider the ground state at @xmath and hope to find that @xmath . Figure 3.6 shows that this is indeed the case. As expected larger values of @xmath produce a more accurate reproduction of the form of the absolute wavefunction. ### 3.7 Solution in the local approximation The numeric treatment of the previous section, although very effective as a computational tool, obscures some of the more subtle properties of the flow equation. In this regard an analytic approach provides some valuable insights, and we consider one such treatment next. At @xmath the eigenstates of the Lipkin Hamiltonian are simply the @xmath basis states. Of course, this is no longer the case at non-zero coupling, although we expect that, in the first phase at least, the eigenstates will remain localized in the spin basis. The unitary transformation diagonalising @xmath will reflect this by mainly mixing states of which the spin projections differ only slightly. This is manifested in the flow equations through the locality of the evolution of @xmath . By this we mean that the flow at a point @xmath is very weakly affected by the values of @xmath at points far from @xmath . We first consider the low lying states which can be investigated by solving @xmath within a neighbourhood of @xmath . Since @xmath (from representation ( 3.38 )) this is equivalent to restricting the calculation to states for which @xmath ; an approximation commonly used in this context. We begin by linearizing the initial condition about the point @xmath , which gives -- -------- -- -------- @xmath (3.50) -- -------- -- -------- The form of @xmath is chosen to coincide with the parametrization of @xmath we introduce next. From equation ( 3.42 ) it is clear that the linearity of @xmath is preserved during flow, which allows for the simple parametrization of @xmath as -- -------- -- -------- @xmath (3.51) -- -------- -- -------- where @xmath and @xmath . The flow of the coefficients is given by -- -------- -- -------- @xmath (3.52) -- -------- -- -------- where @xmath has been rescaled by a factor of four. These equations leave @xmath invariant, and since @xmath we conclude that @xmath . This correctly predicts the characteristic square root behaviour of the gap @xmath , as well as the low lying spectrum within a harmonic approximation. The natural question arising here is whether this approach can be extended to allow for the treatment of the highly excited states as well. Indeed, if the flow at a point @xmath exhibits this locality property it seems reasonable that a local solution about @xmath would provide a good result for the corresponding energy eigenvalue @xmath . We will show how this local solution can be found analytically, and that by combining the solutions at various points an approximation for @xmath may be constructed. The first step is again the linearization of @xmath , now about an arbitrary point @xmath . The linearity of @xmath in @xmath is conserved during flow, allowing for the parametrization -- -------- -- -------- @xmath (3.53) -- -------- -- -------- where the flow of the coefficients are given by -- -------- -- -------- @xmath (3.54) -- -------- -- -------- The initial conditions are @xmath , @xmath , @xmath and @xmath . These equations leave @xmath invariant, and since @xmath we conclude that @xmath . The presence of @xmath in this equation is significant, as it indicates that the validity of the local approximation is a function both of the coupling and the specific point under consideration. The stability condition of the local solution is -- -------- -- -------- @xmath (3.55) -- -------- -- -------- which holds for all @xmath in the first phase. In the second phase the stable domain is restricted to the interval @xmath . This is reminiscent of the discussion in the previous section concerning the transition appearing at higher energies in the second phase. It was seen that there exists points @xmath which separate regions of the spectrum for which the states exhibit either first or second phase behaviour. In fact, all the states lying within @xmath have a first phase character, as @xmath is always found to be greater than @xmath . This suggests a connection between the phase structure, the properties of eigenstates, and the locality, or lack thereof, exhibited by the flow equation. Continuing with the derivation, we find that the flow equations can be solved exactly: -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath (3.56) -- -------- -------- -------- -- -------- The @xmath ’s are integration constants that can be fixed as follows: -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- The value of @xmath at @xmath is expected to provide a good approximation for the exact solution @xmath . We find that -- -------- -- -------- @xmath (3.58) -- -------- -- -------- Figure 3.7 shows this analytic result together with the exact eigenvalues. While it reproduces the spectrum of the first phase well, the local solution fares progressively worse at stronger coupling. This does not signal a lack of locality in the flow, but simply reflects the low order to which the flow equation was solved. The inclusion of higher order terms @xmath in the solution leads to much improved results, as is clear from the figure. Note that the point of breakdown for the fourth order solution in (b) has moved closer to the actual phase boundary at @xmath which separates regions of first and second phase character. In conclusion, we have seen that the local approximation provides non-perturbative results for states which possess a first phase character. In particular, the linear case can be solved analytically and reproduces the entire spectrum of the normal phase to good accuracy; the first such result that we are aware of. ## Chapter 4 The Dicke Model ### 4.1 Introduction Since its introduction in 1954 in the study of collective phenomenon in quantum optics, the Dicke model [ 37 ] has received considerable attention in a wide range of fields, including that of quantum chaos and quantum phase transitions. We refer the reader to [ 38 ] for an extensive list of references to these and related studies. Here we present a flow equation treatment of the Dicke model. Other boson and spin-boson problems have been considered in this context [ 39 , 4 ] , and a special case of the Dicke model was treated perturbatively in [ 40 ] . Vidal and Dusuel [ 41 ] have calculated finite-size scaling exponents for various quantities in the context of the Dicke model using flow equations. Compared to the Lipkin model this case presents a greater challenge to the flow equation approach, mainly due to the presence of two independent degrees of freedom. In order to keep the equations manageable we implement a two step procedure, first bringing the Hamiltonian into a block diagonal form before diagonalizing it completely. Unfortunately the complexity of the resulting PDE still prevents us from performing the first step exactly, and so a perturbative approach will be followed. We begin with an overview of the model and its structure relevant to the flow equation treatment. After introducing the necessary variables and representations we digress briefly to first treat a special case analytically before proceeding with the derivation of the full flow equation. Using this equation we will construct an effective form of Hamiltonian in which certain degrees of freedom have been decoupled. Finally the different approaches to diagonalizing this effective form are introduced and compared using numerical results. ### 4.2 The model The Dicke model describes the interaction of @xmath two-level atoms with a number of bosonic fields via a dipole interaction. For simplicity we will take @xmath to be even; the odd case requires only minor modifications. Following [ 38 ] , we consider one such bosonic mode with frequency @xmath and coupling strength @xmath . The corresponding bosonic creation and annihilation operators are @xmath and @xmath . Associated with each atom are the spin-1/2 operators @xmath which obey the standard @xmath commutation relations. All @xmath atoms have equal level splitting @xmath . The Dicke Hamiltonian reads -- -------- -- ------- @xmath (4.1) -- -------- -- ------- The @xmath factor ensures that the Hamiltonian remains extensive when the bosonic mode is macroscopically occupied, i.e. when @xmath . By introducing the collective spin operators @xmath and @xmath we obtain the simplified form -- -------- -- ------- @xmath (4.2) -- -------- -- ------- A natural basis for the Hilbert space @xmath is given by the eigenstates of the total collective spin operator @xmath together with @xmath and the boson number operator @xmath : -- -------- -- ------- @xmath (4.3) -- -------- -- ------- Here @xmath , @xmath and @xmath . Since @xmath the Hamiltonian does not mix different @xmath -sectors of the total spin representation. The relevant sector for the low-lying states is @xmath , and we fix @xmath at this value throughout, dropping the @xmath label in the basis states. We will focus on the resonant case where @xmath . As before, other cases simply correspond to different initial conditions for the equations we will derive. For these parameter values the model undergoes a quantum phase transition at @xmath from the normal to the so-called super-radiant phase. The second phase is characterized by the macroscopic occupation of the bosonic mode in the ground state, i.e. @xmath . Next we consider the structure of @xmath relevant to our application of the flow equations. Central to this discussion is the operator @xmath . With each distinct eigenvalue @xmath of @xmath we associate the corresponding subspace of eigenstates -- -------- -- ------- @xmath (4.4) -- -------- -- ------- which we refer to as a @xmath -sector. The dimension of @xmath increases linearly from a minimum of one at @xmath to a maximum value of @xmath at @xmath , from where it remains constant. A schematic representation of the basis states and @xmath -sectors appears in Figure 4.1 . Grouping the terms in the Hamiltonian according to equation ( 1.10 ) leads to -- -------- -- ------- @xmath (4.5) -- -------- -- ------- which makes it clear that @xmath possesses a band block diagonal structure with respect to @xmath , as defined in Section 1.2.3 . Since @xmath only changes @xmath by two it leaves the subspaces corresponding to odd or even @xmath invariant. This implies a parity symmetry, similar to that of the Lipkin model, which may be compactly expressed in terms of the operator @xmath as @xmath . This symmetry gets broken in the second phase where the ground states of the odd and even sectors become degenerate. Early studies [ 44 , 45 ] of the model’s thermodynamic properties were performed in the rotating wave approximation (RWA) which amounts to dropping the @xmath term. We will consider this case in more detail later on, as it reappears as the first order contribution to the solution of the flow equation. ### 4.3 The flow equation approach We have seen that the Hamiltonian contains interactions responsible for scattering both between different @xmath -sectors and within a fixed sector. A complete diagonalization of @xmath would require a transformation which eliminates both these interactions. Although the flow equation is certainly capable of generating such a transformation in a single application, it is conceptually clearer and technically much simpler to perform this diagonalization in two separate steps. As suggested in the previous section we will first eliminate interactions which connect different @xmath -sectors through the structure preserving flow generated by @xmath . As before this is done in the thermodynamic or large @xmath limit. A physical interpretation of @xmath is that of a counting operator for the elementary excitations or energy quanta present in the system. The flow will produce an effective Hamiltonian @xmath which conserves the number of these excitations. We expect there to be new terms, both diagonal and off-diagonal, appearing in @xmath . Apart from the provision that they conserve @xmath there is no obvious constraint on these newly generated interactions, and so no band (block) diagonal structure need be present within the @xmath -sectors. Having brought @xmath into a block diagonal form there are two ways in which to proceed. We may construct, for large but finite @xmath , the matrices corresponding to each @xmath -sector and then apply direct diagonalization. Although still numerically intensive this is a greatly reduced problem compared to diagonalizing the original fully interacting Hamiltonian, as the submatrices are at most of size @xmath . Alternatively we may remain at infinite @xmath and apply the flow equations to each sector separately using, for example, the generator @xmath . For a fixed value of @xmath the problem now becomes effectively one dimensional as we may eliminate either the bosonic or spin degree of freedom in favor of @xmath , which acts as a scalar parameter. Both these methods will be demonstrated later on. Finally, a remark on the ordering of eigenvalues. When applying the flow equations to each sector in the second step we expect complete diagonalization and that the eigenvalues will appear on the diagonal in increasing order [ 20 ] . For the first step, which ends in a block diagonal form, the situation is no longer so clear, as the ordering proof in [ 20 ] is only valid for the case of complete diagonalization. In short, it is generally unknown in which sector a certain eigenstate of @xmath will be found when we diagonalize @xmath . In the first phase we expect the same locality encountered in the Lipkin model (Section 3.7 ) and so the low-lying states should belong to the sectors with @xmath . In numerical investigations for small @xmath values we have observed this ordering in both phases. In particular, the ground state is mapped to the single basis state of the @xmath subspace. As this ordering is a non-perturbative phenomenon we do not expect to observe it in our perturbative treatment. ### 4.4 Flow equations in the @xmath limit In the subsections that follow we derive the flow equation for the first step of the diagonalization process using the Moyal bracket method. We begin by introducing the relevant variables and representations. #### 4.4.1 Variables and representations To account for the model’s two independent degrees of freedom we introduce two pairs of operators @xmath and @xmath as described in Section 2.2 . These satisfy the exchange relations -- -------- -- ------- @xmath (4.6) -- -------- -- ------- while operators coming from different pairs commute. The pair @xmath is used to represent the spin degree of freedom through the representation constructed in Section 3.5 in terms of @xmath and @xmath : -- -------- -- ------- @xmath (4.7) -- -------- -- ------- In similar fashion we wish to construct a representation for the boson algebra @xmath in terms of @xmath and @xmath . Two new issues, not encountered in the @xmath case, arise here. Firstly, it is well known that only infinite dimensional representations of the boson algebra exist, whereas @xmath and @xmath are finite dimensional. The second point concerns the scale we should associate with the bosonic operators. If @xmath and @xmath were naively assumed to be scaleless with respect to @xmath (or @xmath ), only the @xmath term in the Hamiltonian would survive when working to leading order in @xmath . We will address these issues in the context of our proposed approximate representation -- -------- -- ------- @xmath (4.8) -- -------- -- ------- where @xmath and @xmath is the dimension of the space. Denoting the eigenstates of @xmath by @xmath we see that @xmath and @xmath for @xmath . (See equations ( 2.17 ) and ( 2.18 ).) The creation operator @xmath maps the highest state @xmath to zero, since @xmath . This amounts to a truncation of the boson Fock-space, and the operators in ( 4.8 ) agree with the truncated forms of the exact infinite-dimensional operators. Proceeding as before, we treat @xmath and @xmath as scalars and define @xmath and @xmath through @xmath and @xmath . The representation now becomes -- -------- -- ------- @xmath (4.9) -- -------- -- ------- which satisfies, to leading order in @xmath , the desired commutation relation with respect to the bosonic Moyal bracket @xmath . For the moment we set @xmath , and return to the role of @xmath later. The second issue concerns the scale of @xmath to be associated with @xmath and @xmath . This is really a question of which values of @xmath are relevant to the low energy physics of the system. In the Dicke model it is known [ 38 ] that ground state occupation of the bosonic mode is microscopic, i.e. @xmath , in the first phase and macroscopic, i.e. @xmath , in the second. We can describe both these cases by considering @xmath . A natural choice of variables is @xmath which is scaleless and analogous to @xmath of the @xmath case. The @xmath -scale now becomes explicit in both the representation -- -------- -- -------- @xmath (4.10) -- -------- -- -------- and bosonic Moyal bracket -- -------- -- -------- @xmath (4.11) -- -------- -- -------- We also observe that the dimension @xmath , which controls the Fock-space cutoff, does not appear explicitly. For practical purposes we may safely assume @xmath to be much larger than the interval of @xmath under consideration. Combining the two representations we obtain the initial condition to leading order in @xmath as -- -------- -- -------- @xmath (4.12) -- -------- -- -------- where the terms are ordered as in equation ( 4.5 ). #### 4.4.2 Solution in the local approximation Before proceeding with the derivation of the general flow equation, which will involve significant behind-the-scenes numeric and symbolic computation, let us consider a case amenable to an analytic treatment. Using the same locality argument put forth for the Lipkin model in Section 3.7 we assert that for the low-lying states only the flow within a small region around @xmath is relevant, and that the evolution of @xmath in this region is governed by its local properties. This is equivalent to the assumption that @xmath for the low-lying states. A similar approximation is made in the bosonization treatment of [ 38 ] , the results of which will be reproduced here using the flow equations. In practice we implement this approximation by simply replacing @xmath by @xmath in the initial condition ( 4.12 ); the result of a Taylor expansion to leading order in @xmath . First we consider the flow generated by @xmath . As with the Lipkin model the local approximation leads to a very simple form for @xmath , with only two new terms, proportional to @xmath and @xmath , being generated. As required the band block diagonal structure of @xmath is conserved. The flowing Hamiltonian may be parametrized as -- -------- -------- -------- -- @xmath @xmath @xmath @xmath -- -------- -------- -------- -- where @xmath @xmath are scalar coefficients. In operator language this amount to -- -------- -------- -------- -- @xmath @xmath @xmath @xmath -- -------- -------- -------- -- Observables linear in @xmath and @xmath may be parametrized in a similar fashion. We will consider @xmath for which @xmath is of the form ( 4.4.2 ) with the flowing coefficients denoted by @xmath for @xmath . Substituting these forms into the flow equations ( 4.31 ) and ( 4.35 ) and then matching the coefficients on both sides yield -- -------- -- -------- @xmath (4.15) -- -------- -- -------- and -- -------- -- -------- @xmath (4.16) -- -------- -- -------- The initial conditions for @xmath and @xmath are @xmath , @xmath , @xmath and @xmath , @xmath respectively. First we consider @xmath . In the @xmath limit @xmath and @xmath is expected to vanish, while @xmath and @xmath will assume new, renormalized values. The initial and final values of the @xmath ’s may be related using the flow invariants @xmath and @xmath . By combining these with the initial conditions we find that -- -------- -- -------- @xmath (4.17) -- -------- -- -------- The renormalized values of @xmath and @xmath correspond to a point where the circle and hyperbole described by these equations intersect. For @xmath no such point exists, and the solution breaks down. For @xmath the fixed point is found to be -- -------- -- -------- @xmath (4.18) -- -------- -- -------- This is illustrated in Figure 4.2 which shows the flow of @xmath for @xmath and @xmath . Next we consider the flow of @xmath . To fix the final values of the @xmath coefficients we require four invariants, some of which must involve the coefficients of @xmath . One such set is -- -------- -- -------- @xmath @xmath (4.19) -- -------- -- -------- Combining @xmath and @xmath with the initial conditions lead to -- -------- -- -------- @xmath @xmath (4.20) -- -------- -- -------- and so -- -------- -- -------- @xmath @xmath (4.21) -- -------- -- -------- The values of @xmath and @xmath can now be solved for using @xmath and @xmath . Note that for the purposes of calculating expectation values with respect to the eigenstates of @xmath only @xmath and @xmath are relevant. This follows from the observation that @xmath and @xmath correspond to terms that change @xmath by two, whereas the eigenstates of @xmath are also eigenstates of @xmath . For simplicity we drop these “between-sector” scattering terms in what follows, keeping only those terms relevant to the calculation of expectation values. The successful construction of the effective @xmath -preserving Hamiltonian @xmath and the observable @xmath marks the end of the first step. We now apply the flow equation a second time, using the generator @xmath . When restricted to a single @xmath -sector @xmath is clearly non-degenerate, and so we expect complete diagonalization in the final Hamiltonian. The local approximation again allows for a simple parametrization -- -------- -- -------- @xmath (4.22) -- -------- -- -------- where @xmath and @xmath . The flow equations for the coefficients are -- -------- -- -------- @xmath (4.23) -- -------- -- -------- which leave @xmath and @xmath invariant. Combining these with the initial conditions and using the fact that @xmath leads to -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath (4.24) -- -------- -------- -------- -- -------- Turning to @xmath we find that its parametrized form is again similar to that of @xmath , and we denote the flowing coefficients by @xmath . The flow equations are found to be -- -------- -- -------- @xmath (4.25) -- -------- -- -------- which leave @xmath and @xmath invariant. Proceeding as before we find that @xmath and @xmath . Since the eigenstates of @xmath are also eigenstates of @xmath we drop the @xmath term. In summary, we have seen that within the local approximation the two step diagonalization procedure can be performed exactly. The final Hamiltonian and transformed observable @xmath is -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath (4.26) -- -------- -------- -------- -- -------- which agrees with the result of [ 38 ] obtained using bosonization. This clearly constitutes a harmonic approximation of the spectrum in terms of two oscillators with frequencies @xmath and @xmath . Only the parts of @xmath which commute with both @xmath and @xmath are shown. Note that the gap between the ground state @xmath and first excited state @xmath exhibits the characteristic square root behaviour @xmath . #### 4.4.3 Flow equation for the Hamiltonian We proceed with the derivation of the general flow equation for the Hamiltonian, beginning with a more in depth study of the structure of @xmath . In Section 1.2.3 it was shown that the flow generated by @xmath will preserve the band block diagonal structure present in @xmath , thus restricting the terms in @xmath to those changing @xmath by either zero or two ¹ ¹ 1 In this equation, and in some that follow, we treat operators as commuting scalars. Since reordering can only bring about @xmath corrections this is sufficient for our purposes, and it simplifies the notation considerably. -- -- -- -------- (4.27) -- -- -- -------- A simpler and much more convenient form can be found by introducing the operators @xmath and -- -------- -- @xmath -- -------- -- We interpret @xmath as the fundamental @xmath -preserving interaction while the elements of @xmath constitute the fundamental interactions which change @xmath by two. As shown in Appendix D we can rewrite all the operators appearing in ( 4.27 ) in terms op @xmath , @xmath , @xmath and the elements of @xmath which only appear linearly. For example: -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath (4.28) -- -------- -------- -------- -- -------- This results in the form -- -------- -- -------- @xmath (4.29) -- -------- -- -------- The factor of @xmath responsible for the extensivity of @xmath has been factored out explicitly, and so each @xmath is a scaleless function. The initial condition is given by -- -------- -- -------- @xmath (4.30) -- -------- -- -------- where @xmath . We proceed by inserting this form into the flow equation -- -------- -- -------- @xmath (4.31) -- -------- -- -------- where @xmath is the full Moyal bracket -- -------- -- -------- @xmath (4.32) -- -------- -- -------- The resulting algebra is easily handled using a symbolic processor such as Mathematica . Matching the coefficients of @xmath , @xmath and @xmath on both sides of ( 4.31 ) produces a set of equations of the form -- -------- -- -------- @xmath (4.33) -- -------- -- -------- where we sum over repeated indices. The superscripts of the @xmath ’s denote derivatives to the scaleless variables @xmath , @xmath and @xmath . The @xmath indices run over the set @xmath where @xmath corresponds to no derivative being taken. Governing the flow of @xmath is the @xmath matrix @xmath of which the entries are simple polynomials of the rescaled variables. The non-zero entries of these, typically sparse matrices appear in Appendix E . Due to the obvious complexity of these non-perturbative equations we have been unsuccessful in finding an exact numerical solution. A perturbative solution can be found, and we present this in Section 4.5 . #### 4.4.4 Flow equation for an observable Next we derive the flow equation for an observable @xmath . This case is complicated by the lack of structure present in @xmath at non-zero @xmath . For example, we do not generally expect band block diagonality to be present, and so @xmath may contain interactions connecting distant @xmath -sectors. In general we are confronted by the form -- -------- -- -------- @xmath (4.34) -- -------- -- -------- where @xmath and @xmath . The @xmath label indicates the amount by which the term changes @xmath , and similar for @xmath with respect to @xmath . We define the index set @xmath . Inserting this form into the flow equation -- -------- -- -------- @xmath (4.35) -- -------- -- -------- yields the coupled set -- -------- -- -------- @xmath (4.36) -- -------- -- -------- We sum over the repeated indices @xmath , @xmath and @xmath . @xmath is a @xmath dimensional matrix containing polynomial functions of the scaleless variables. Note that at @xmath only the @xmath function is of interest for the calculation of expectation values. This follows from the observation that eigenvalues of @xmath are also eigenvalues of @xmath , and that @xmath is the only term leaving @xmath invariant. Again it is clear that a direct numerical approach is intractable, and we instead try to find perturbative solutions. ### 4.5 Perturbative solutions We wish to construct solutions to equations ( 4.33 ) and ( 4.36 ) of the forms -- -------- -- -------- @xmath (4.37) -- -------- -- -------- The forms given in ( 4.29 ) and ( 4.34 ) are still valid, as they apply separately to each @xmath and @xmath . When working to finite order in @xmath the functional dependence of @xmath and @xmath on @xmath are constrained to be polynomial and of finite order. This also limits the type of interactions that can be generated in @xmath since terms for which @xmath is at least of order @xmath , and so only @xmath with @xmath are relevant. This is true at @xmath , provided that @xmath and @xmath , and continues to hold at @xmath since @xmath is always at least of order @xmath and contains only @xmath terms. As all the relevant functions are simple polynomials we may proceed by constructing a set of coupled differential equations for the scalar coefficients appearing in these polynomials as functions of @xmath . The extensive algebraic manipulations involved in this step can be automated for arbitrary @xmath using Mathematica . The resulting set of ordinary differential equations is then solved using the standard Runge-Kutta algorithm [ 34 ] . Solutions were obtained for @xmath up to @xmath and for the observables @xmath and @xmath up to @xmath . These transformed operators will be denoted by @xmath , @xmath and @xmath . We only state the results up to fourth order here: -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath (4.38) -- -------- -------- -------- -- -------- where, to aid interpretation, we temporarily abuse notation by writing @xmath for the scaleless variables @xmath . The rationality of the coefficients, apart from the @xmath factors, have been verified to very high numerical accuracy. In similar fashion the observables are given by -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath (4.39) -- -------- -------- -------- -- -------- and -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath (4.40) -- -------- -------- -------- -- -------- where only the @xmath -preserving terms relevant to the calculation of expectation values are shown. Looking at the results for @xmath we see the “in-sector” scattering interaction @xmath appearing in the higher order terms. As the power to which @xmath can occur is limited by the perturbation order a band diagonal structure is present in each @xmath -sector with respect to @xmath (or equivalently @xmath ). We expect @xmath to provide a good description of the Dicke model at weak coupling, and possibly throughout the first phase. At strong coupling there is no guarantee of the accuracy or stability of @xmath , even when working to arbitrarily high orders in the perturbation. In fact, from the terms given in ( 4.38 ) two things become apparent which signal a breakdown of this approximation at @xmath . Firstly, the matrix element of the one-dimensional @xmath subspace is found to be @xmath for all @xmath and orders of the perturbation considered. As is known from non-perturbative numerical studies the flow equation maps the ground state to precisely this @xmath state. However, it is also known that the ground state energy is only equal to @xmath in the first phase, and that it decreases linearly with @xmath in the second [ 38 ] , contradicting the prediction of @xmath . Secondly @xmath is found to become unstable at large coupling for perturbation orders higher than one. For example, consider the expectation value of @xmath to second order with respect to a simple variational state: @xmath . We note that for @xmath this expectation value is unbounded from below. Together with the variational principle this implies that the spectrum of @xmath becomes unbounded from below at some @xmath . For the fourth order case this instability occurs at a @xmath . We have observed that this point of breakdown continues to move closer to the critical point @xmath as higher order corrections are included. This agrees with the general notion that a perturbation series, even when summed up completely, may produce divergent results beyond the series’ radius of convergence. Finally we point out that to first order the effective Hamiltonian is -- -------- -- -------- @xmath (4.41) -- -------- -- -------- which is equivalent to the RWA approximation in which the model was originally studied [ 44 , 45 ] . To this order we also observe that the @xmath -preserving parts of @xmath and @xmath are left unchanged by the transformation generated by the flow equation. ### 4.6 Diagonalizing @xmath The flow equation treatment of the preceding sections has brought the Dicke Hamiltonian into a block diagonal form by eliminating interactions which connect different @xmath -sectors. What remains is to diagonalize @xmath within each @xmath -sector at a time. One approach, valid for large but finite @xmath , is to apply direct diagonalization to each @xmath -block (submatrix) of @xmath . This is done by first constructing the matrix representations of @xmath , @xmath and @xmath for the @xmath -sector under consideration. Finding the desired submatrix of @xmath is simply a matter of replacing @xmath , @xmath and @xmath by the properly scaled matrix representations of @xmath , @xmath and @xmath respectively. One need not be concerned about the precise ordering of operators in products, as different orderings only bring about @xmath corrections. The result of these substitutions is a matrix @xmath , the size of which ranges from @xmath to @xmath depending on the value of @xmath . Generally @xmath will not be Hermitian, and so we consider the symmetrized form @xmath . Results of the subsequent numeric diagonalization appear in the next section. Alternatively we may remain in the large @xmath limit and apply the flow equation again, this time non-perturbatively, to diagonalize each submatrix. Within each @xmath -sector the problem is effectively one-dimensional as we may eliminate either the bosonic or spin degree of freedom in favor of the scalar parameter @xmath . We choose to work with the spin degree of freedom, which has the natural basis -- -------- -- -------- @xmath (4.42) -- -------- -- -------- By writing @xmath and -- -------- -- @xmath -- -------- -- the effective Hamiltonian can be rewritten purely in terms of spin operators. Let us clarify this last step. Since @xmath and @xmath together fix the number of bosons there is no @xmath label in @xmath . In fact, we completely eliminate the bosonic degree of freedom through the correspondence -- -------- -- -------- @xmath (4.43) -- -------- -- -------- where the operators on the right act on @xmath and are equivalent to the operators on the left acting within a fixed @xmath -sector of the original tensor product space @xmath . For concreteness we consider the fourth order case for the rest of the section. The results can be easily extended to higher orders. The form of the spin-only Hamiltonian is @xmath , which clearly possesses a double band diagonal structure with respect to @xmath . To preserve this form during flow we use the generator of Section 1.2.3 : -- -------- -- -------- @xmath (4.44) -- -------- -- -------- For the numerical treatment of the flow equation it is convenient to transform to the variables of the @xmath representation in terms of which @xmath and @xmath become -- -------- -------- -------- -- -------- @xmath @xmath @xmath (4.45) @xmath @xmath @xmath (4.46) -- -------- -------- -------- -- -------- where we write @xmath for @xmath and the initial conditions are -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath (4.47) @xmath @xmath @xmath -- -------- -------- -------- -- -------- Here @xmath , which is now a continuous scalar parameter labelling the relevant sector. The flow of these functions are given by -- -------- -------- -------- -- -------- @xmath @xmath @xmath @xmath @xmath @xmath (4.48) @xmath @xmath @xmath -- -------- -------- -------- -- -------- The corresponding flow equation for an observable @xmath is -- -------- -- -------- @xmath (4.49) -- -------- -- -------- which we treat using a similar approach to that of equations ( 3.44 ) and ( 3.45 ). In the @xmath limit we expect @xmath and @xmath to vanish, as they represent the off-diagonal parts of the flowing Hamiltonian. Using the correspondence between @xmath and @xmath we may consider @xmath , and thus @xmath itself, to be a function of @xmath . It follows that the eigenvalues of the particular @xmath -sector are given by @xmath for @xmath . There is a subtle issue regarding the domain of these functions that still needs to be addressed. Looking back at the definition of the @xmath basis we see that the maximum value of the spin label @xmath is @xmath which is less than @xmath when @xmath . Correspondingly we need to restrict the domain of the @xmath functions to reflect this. Since @xmath the relevant domain is @xmath . Indeed, outside this region the initial conditions are complex, causing the flow to become unstable as @xmath . ### 4.7 Numeric results During the course of this chapter we have encountered a number of different treatments of the Dicke model. In this section we present the results obtained using these methods and compare their ranges of applicability. First we summarize the different approaches: - Direct Diagonalization : By introducing a cut-off in the maximum value of @xmath we can numerically diagonalize the full Dicke Hamiltonian on a truncated Hilbert space. This generally leads to very large matrices, especially in the second phase where an extensive number of @xmath -sectors must be included in order to describe the low-lying states. For sufficiently large values of the cut-off this method does provide very accurate results for the low-lying states, and we will use these as a benchmark for the flow equation results. - Local Approximation : This analytic method was derived in Section 4.4.2 and constitutes a harmonic approximation of the first phase, low-lying spectrum in terms of two oscillators. - Direct Diagonalization of @xmath : In Section 4.5 we derived the effective @xmath -preserving Hamiltonian @xmath in a perturbative approximation. For finite @xmath the submatrices comprising the block-diagonal structure of @xmath may be constructed and diagonalized numerically. Since these submatrices are maximally of size @xmath this is a greatly reduced problem compared to diagonalizing the original Hamiltonian in some large truncated space. - Flow Equation Diagonalization of @xmath : As described in Section 4.6 the individual @xmath -sectors may be diagonalized in the @xmath limit using a flow equation equipped with an appropriate generator. The continuous variable @xmath appearing in the initial condition labels the relevant @xmath -sector. - The Rotating Wave Approximation (RWA) : Within this approximation the Dicke Hamiltonian is equivalent to the first order form of @xmath . In this sense the RWA is already included in the two previous cases. However, due to its prevalence in the literature we will treat it separately as a reference case. Throughout this section only the subspace corresponding to even values of @xmath , as discussed in Section 4.2 , will be considered. The first results we present are those obtained by diagonalizing @xmath at finite @xmath . The flow equation treatment of the second step will be dealt with later on. Figure 4.3 (a) shows the energy of the third excited state @xmath as a function of @xmath at @xmath . Also shown are the exact values obtained using direct diagonalization, and the RWA result. Although the result of @xmath is a marked improvement over that of the RWA, we still see @xmath errors present at larger values of @xmath . This simply the reflects the order to which the Moyal bracket was expanded in our derivation of the flow equation. However, it quickly becomes apparent that this is a systematic error and largely independent of the particular state. We attribute this to the scalar part of the higher order @xmath -corrections that were neglected in our derivation. This only serves to shift the entire spectrum by some fixed amount relative to the exact results. For a more meaningful comparison we eliminate this shift by considering the excited energies relative to the ground state, i.e. we consider the excitation energy (or gap) @xmath rather than @xmath itself. As seen in Figures 4.3 (b) and 4.4 (a), (b) there is indeed a very good correspondence between the exact results for @xmath and those obtained from @xmath . From previous studies [ 38 ] it is known that the critical point @xmath is characterized by the vanishing of the gap between the ground state and first excited state. This agrees with the result of the local approximation that @xmath . Figure 4.4 (c) shows this gap together with those obtained using @xmath . Whereas the local approximation provides reasonable results for the low-lying states, it fails to do so for the highly excited states, as is clear from Figure 4.4 (d). For these states the assumption that @xmath is no longer valid. This means that the local solution within a neighbourhood of @xmath is insufficient since the flow relevant to these states occur at points distant from @xmath . Figure 4.5 (a)-(d) provides a global view the spectra obtained using the different methods. The qualitative properties of the exact result and that of @xmath are clearly very similar. In particular we note the level repulsion among the low-lying states, which leads to the vanishing of the gap @xmath at the critical point. This behaviour is clearly absent in the RWA case, where there appears to be no correlation between levels belonging to different sectors. As expected the local approximation predicts the correct behaviour for the low-lying states but fails at higher energies. In fact, the local approximation predicts a continuous spectrum at @xmath . The @xmath -preserving parts of @xmath and @xmath may be constructed in the same way as we did for the individual submatrices of @xmath . The calculation of the expectation values is now a straight forward procedure, the results of which appear in Figure 4.5 (e)-(h). We again find good agreement with the exact results. To eliminate any constant shift owing to higher order scalar corrections we consider the expectation values relative to the ground state, i.e. @xmath where @xmath . In the final part of this section we present the results obtained by diagonalizing the @xmath -sectors through a second application of the flow equation. All the results obtained thus far can be reproduced using this method, and we will not restate them here. Instead we focus on the RWA case, particularly with respect to the phase structure. The RWA Hamiltonian -- -------- -- @xmath -- -------- -- exhibits a phase transition at @xmath [ 38 ] , in contrast to critical value of @xmath for the full Dicke Hamiltonian. The two phases are distinguished by the order parameter @xmath that becomes non-zero only in the second phase. This signals a macroscopic occupation of the bosonic mode which is responsible for the phenomenon of super-radiance. We proceed by solving the flow equation ( 4.48 ) for a range of @xmath and @xmath values. In the @xmath limit the off-diagonal part @xmath vanishes and we obtain a set of functions @xmath . Figure 4.6 (a) shows a typical example for @xmath at @xmath , together with the exact result for @xmath . Due to the ordering of the eigenvalues at @xmath we expect each @xmath to be an increasing function with @xmath corresponding to the lowest eigenvalue. The overall ground state energy is given by -- -------- -- -------- @xmath (4.50) -- -------- -- -------- The result of this calculation appears in Figure 4.6 (b). Once it is known to which sector the ground state belongs for a certain @xmath we solve the flow equation for the observables @xmath and @xmath within this sector. Figures 4.6 (c) and (d) show the ground state expectation values calculated in this manner. \specialhead Conclusion and outlook We hope that our presentation has convinced the reader that flow equations obtained from continuous unitary transformations present a versatile and potentially very powerful technique for the treatment of interacting quantum systems. Our results for the Lipkin and Dicke models have clearly illustrated the myriad of information yielded by a non-perturbative solution of the flow equations. This approach is not without its difficulties however. As discussed in Section 2.2 , the construction of smooth initial conditions, although relatively simple for the models considered here, are generally non-trivial and in some cases possibly on par in difficulty with diagonalizing the Hamiltonian itself. Although the introduction of additional variables may aid in this construction, the resulting high dimensional PDE presents its own challenges, as was encountered in the Dicke model. Further studies into these issues are needed if the flow equations are to become a truly general and robust framework for non-perturbative calculations. It also seems inevitable that the treatment of realistic models would require further model-specific approximations which were largely absent in our approach. One possibility is the generalization of the local approximation scheme of Sections 3.7 and 4.4.2 . The flow equation would be solved locally around an appropriate point, possibly determined by a variational method, and using a specialized generator. This may allow for a treatment of the system by considering fluctuations of the degrees of freedom around their classical values. The flow equation is one example of a non-linear operator equation of which there are numerous others appearing in virtually all branches of physics. Although our focus has been on a small subclass of these, the solution methods developed here are much more general in nature and may in future find application in other fields as well. ## Appendix A Representations by irreducible sets In Section 2.1 we made use of the fact that any flowing operator may be expressed in terms of operators coming from an irreducible set which contains the identity. Now we show why this is the case, using a result by von Neumann. First we establish some notation. Let @xmath be a finite dimensional Hilbert space and denote by @xmath the set of all matrix operators acting on it. The commutant @xmath of a set @xmath is defined as the set of operators which commute with all the elements of @xmath . The double commutant of @xmath is defined by @xmath . The following theorem [ 46 ] provides the key: The Double Commutant Theorem: If @xmath is a subalgebra of @xmath which contains the identity and is closed under hermitian conjugation then @xmath . Now suppose @xmath is an irreducible set of hermitian operators, one of which is the identity, and @xmath the subalgebra of @xmath spanned by all the products of operators from @xmath . By Schur’s lemma @xmath and thus @xmath . It follows from the theorem above that @xmath , i.e. any operator acting on @xmath may be written in terms of operators coming from @xmath . ## Appendix B Calculating expectation values with respect to coherent states Define @xmath where @xmath is the unnormalized coherent state. These states are known to provide a basis for the Hilbert state through the resolution of unity -- -- -- ------- (B.1) -- -- -- ------- where the integral ranges over the entire complex plane. Furthermore, for any @xmath the state @xmath has non-zero overlap with every state in the Hilbert space. The action of the @xmath generators on these states may be expressed in terms of differential operators [ 33 ] with respect to @xmath and @xmath : -- -------- -- ------- @xmath (B.2) @xmath (B.3) @xmath (B.4) -- -------- -- ------- We will always consider @xmath and @xmath to be distinct variables, and that @xmath is a function of @xmath only. When calculating expectation values of the double commutators in Section 2.1 , one encounters expressions of the form -- -------- -- ------- @xmath (B.5) -- -------- -- ------- where @xmath . These can easily be computed by replacing each operator by its differential representation to obtain -- -------- -- ------- @xmath (B.6) -- -------- -- ------- where the orders of the operators have been reversed. As a result these expectation values can be expressed as some rational function of @xmath and @xmath . Applying this method to @xmath and @xmath , as defined in Section 2.1 , we obtain -- -------- -- ------- @xmath (B.7) -- -------- -- ------- Using these equations to express @xmath and @xmath in terms of @xmath we may consider expectation values of the form of equation ( B.5 ) as functions of these averages. For example -- -------- -- ------- @xmath (B.8) -- -------- -- ------- where @xmath and @xmath . ## Appendix C Scaling behaviour of fluctuations We wish to show that -- -------- -- ------- @xmath (C.1) -- -------- -- ------- From direct calculation this is found to hold for @xmath , and we employ induction to obtain the general result. Assuming that this holds for all products of @xmath fluctuations, the induction step consists of adding either @xmath or @xmath to a general product @xmath and then proving the result for @xmath . First we add an extra @xmath . Using the results from the previous section we may write -- -------- -------- -------- -------- ------- @xmath @xmath @xmath (C.3) @xmath @xmath @xmath -- -------- -------- -------- -------- ------- where ( B.7 ) and @xmath were used. It should be remembered that @xmath contains a @xmath dependency through the average appearing in each @xmath . Dividing by @xmath leads to -- -------- -- ------- @xmath @xmath (C.4) -- -------- -- ------- The last term can be rewritten using -- -------- -- ------- @xmath (C.5) -- -------- -- ------- which is just the product rule, to obtain -- -------- -- ------- @xmath (C.6) -- -------- -- ------- Note the important cancellation of terms proportional to @xmath . From the induction hypothesis, and the fact that @xmath is polynomial in @xmath , the first term in ( C.6 ) will be of order @xmath . The second term becomes -- -------- -- ------- @xmath (C.7) -- -------- -- ------- Taking into account the expressions for @xmath and @xmath , we conclude that each term in equation ( C.7 ) will have order @xmath , which reduces to @xmath in both the cases where @xmath is odd and even. Thus, for any product @xmath of @xmath fluctuations we arrive at -- -------- -- ------- @xmath (C.8) -- -------- -- ------- which concludes the induction step. Exactly the same procedure is followed when adding a @xmath , although more algebra is required as @xmath now contains second order derivatives to @xmath . The final result remains unchanged: -- -------- -- ------- @xmath (C.9) -- -------- -- ------- In this case these results make exact the general notion that relative fluctuations scale like powers of one over the system size. Finally we mention that when calculating the expectation values of a double commutator there is often a cancellation of leading order terms. This can be seen in equation ( B.8 ), where the sum of terms of order @xmath turns out to be of order @xmath . ## Appendix D Decomposing operators in the Dicke model We define two classes of operators: @xmath and @xmath . Note that @xmath changes @xmath by @xmath while @xmath does so by @xmath . We wish to find an algorithm for writing a given @xmath in terms of @xmath and @xmath ’s for which @xmath . Central to this procedure are the relations -- -- -- ------- (D.1) -- -- -- ------- which follows directly from the definitions and basic properties of spin and boson operators. Given a @xmath we apply these relation repeatedly, until finally only @xmath ’s and @xmath appear. Note that the former can only appear linearly. Furthermore, any @xmath appearing in intermittent steps has @xmath equal to @xmath , and any @xmath has @xmath . ## Appendix E Dicke model flow coefficients The non-zero entries of the @xmath matrix. For clarity the subscript indices have been raised. -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- The non-zero entries of the @xmath matrix. -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- The non-zero entries of the @xmath matrix. -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- The non-zero entries of the @xmath matrix. -- -------- -------- -------- -- @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath @xmath -- -------- -------- -------- -- \specialhead BIBLIOGRAPHY
["This thesis firstly investigates whether D=11 supergravity can be lifted to a\nhigher dimensional (...TRUNCATED)	"##### Contents\n\n- 1 Symmetries and physical fields\n - 1.1 Symmetries in physical theori(...TRUNCATED)
["In the context of A. Eskin and A. Okounkov's approach to the calculation of\nthe volumes of the di(...TRUNCATED)	"# Acknowledgements\n\nI am deeply indebted to my parents, for their unconditional support\nthrougho(...TRUNCATED)
["We investigate different aspects of lattice QCD in Landau gauge using Monte\nCarlo simulations. In(...TRUNCATED)	"###### Contents\n\n- Note added to the e-print version\n- Introduction\n- 1 The various co(...TRUNCATED)
["In earlier work, the Abstract State Machine Thesis -- that arbitrary\nalgorithms are behaviorally (...TRUNCATED)	"## 1. Introduction\n\nTraditional models of computation, like the venerable Turing machine,\nare, d(...TRUNCATED)
["This thesis is devoted to asymptotic norm estimates for oscillatory integral\noperators acting on (...TRUNCATED)	"## Chapter 1 Introduction\n\n### 1.1 Formulation of the problem\n\nMy thesis studies asymptotic nor(...TRUNCATED)
["This thesis describes the development of two independent computer programs, Herwig++ and Effective(...TRUNCATED)	"##### Contents\n\n- 1 Introduction\n - 1.1 Field Theory Introduction\n - 1.1.1 L(...TRUNCATED)
["A graph is circle if there is a family of chords in a circle such that two\nvertices are adjacent (...TRUNCATED)	"### Chapter 1 Introduction\n\nCircle graphs [ 15 ] are intersection graphs of chords in a circle. I(...TRUNCATED)
["In this Diploma-thesis models of gauge field theory on noncommutative spaces\nare studied. On the (...TRUNCATED)	"##### Contents\n\n- Abstract\n- 1 Influence of the Ordering Prescription in the Case @xmath\n(...TRUNCATED)
["In this Thesis we examine the interplay between the encoding of information\nin quantum systems an(...TRUNCATED)	"# Contributions\n\nThe author and his collaborators have made the respective contributions\nto the (...TRUNCATED)

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