119712419
Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems
A bstractThe formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nonetheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.
122028945
Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models
Abstract The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N ⩾ 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N ⩾ 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N ⩾ 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.
122705180
Partition function of the eight vertex lattice model
The partition function of the zero-field “Eight-Vertex” model on a square M by N lattice is calculated exactly in the limit of M, N large. This model includes the dimer, ice and zero-field Ising, F and KDP models as special cases. In general the free energy has a branch point singularity at a phase transition, with an irrational exponent.
122682659
Solvable eight-vertex model on an arbitrary planar lattice
Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
123205649
Solutions of the Yang-Baxter equation
We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle S-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y-B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y-B equation to the case ofZ2-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.
2266452
Yang-Baxter equation and reflection equations in integrable models
The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and on the half-line using the Zamolodchikov-Faddeev algebra. Due to the vertex-IRF model correspondence the face model analogue of the ZF-algebra and the IRF reflection equation are written down as well as the $Z_2$-graded and colored algebra forms of the YBE and RE.
121370183
THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL
CONTENTSIntroduction § 1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain § 2. Connection with the inverse problem method § 3. The six-vertex model § 4. Generating vectors and permutation relations § 5. The general Bethe Ansatz § 6. Integral equations Conclusion Appendix 1 Appendix 2 References
122198747
Quantum version of the method of inverse scattering problem
This paper is a developed and consecutive account of a quantum version of the method of the inverse scattering problem on the example of the nonlinear Schrödinger equation. The method of R-matrices developed by the author is given basic consideration. The generating functions of quantum integrals of motion and action-angle variables for the quantum nonlinear Schrödinger equation are obtained. There is also described a classical version of the method of R-matrices.
3865375
Braiding operators are universal quantum gates
This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix)oftheYang-Baxterequationisauniversalgateforquantumcomputing,in thepresenceoflocalunitarytransformations.Weshowthatthissame Rgeneratesa new non-trivial invariant of braids, knots and links. Other solutions of theYang- Baxter equation are also shown to be universal for quantum computation. The paperdiscussestheseresultsinthecontextofcomparingquantumandtopological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.
119354666
The Bethe-ansatz for N = 4 super Yang-Mills
We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.
20470441
Review of AdS/CFT Integrability: An Overview
This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection, we present an overview of the achievements and the status of this subject as of the year 2010.
120887029
Gauge Theories and Integrable Lattice Models
Abstract Investigations of new knot polynomials discovered in the last few years have shown them to be intimately connected with soluble models of two dimensional lattice statistical mechanics. In this paper, these results, which in time may illuminate the whole question of why integrable lattice models exist, are reconsidered from the point of view of three dimensional gauge theory. Expectation values of Wilson lines in three dimensional Chern-Simons gauge theories can be computed by evaluating the partition functions of certain lattice models on finite graphs obtained by projecting the Wilson lines to the plane. The models in question — previously considered in both the knot theory and statistical mechanics — are IRF models in which the local Boltzmann weights are the matrix elements of braiding matrices in rational conformal field theories. These matrix elements, in turn, can be presented in three dimensional gauge theory in terms of the expectation value of a certain tetrahedral configuration of Wilson lines. This representation makes manifest a surprising symmetry of the braiding matrix elements in conformal field theory.
119592177
Gauge Theory And Integrability, II
Several years ago, it was proposed that the usual solutions of the Yang-Baxter equation associated to Lie groups can be deduced in a systematic way from fourdimensional gauge theory. In the present paper, we extend this picture, fill in many details, and present the arguments in a concrete and down-to-earth way. Many interesting effects, including the leading nontrivial contributions to the Rmatrix, the operator product expansion of line operators, the framing anomaly, and the quantum deformation that leads from g[[z]] to the Yangian, are computed explicitly via Feynman diagrams. We explain how rational, trigonometric, and elliptic solutions of the Yang-Baxter equation arise in this framework, along with a generalization that is known as the dynamical Yang-Baxter equation. ar X iv :1 70 9. 09 99 3v 2 [ he pth ] 4 F eb 2 01 8
119592177
Gauge Theory And Integrability, II
Several years ago, it was proposed that the usual solutions of the Yang-Baxter equation associated to Lie groups can be deduced in a systematic way from fourdimensional gauge theory. In the present paper, we extend this picture, fill in many details, and present the arguments in a concrete and down-to-earth way. Many interesting effects, including the leading nontrivial contributions to the Rmatrix, the operator product expansion of line operators, the framing anomaly, and the quantum deformation that leads from g[[z]] to the Yangian, are computed explicitly via Feynman diagrams. We explain how rational, trigonometric, and elliptic solutions of the Yang-Baxter equation arise in this framework, along with a generalization that is known as the dynamical Yang-Baxter equation. ar X iv :1 70 9. 09 99 3v 2 [ he pth ] 4 F eb 2 01 8
121535828
QuantumR matrix for the generalized Toda system
We report the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.
121547892
Integrable quantum systems and classical Lie algebras
We have obtained six new infinite series of trigonometric solutions to triangle equations (quantumR-matrices) associated with the nonexceptional simple Lie algebras:sl(N),sp(N),o(N). TheR-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). TheseR-matrices provide an exact integrability of anisotropic generalizations ofsl(N),sp(N),o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras.
35416675
Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.)
One or more than made up, by fred woodward under the best uses of systems. 157 rule generalizes easily to undergraduates this has changed. The copyright page the 5th printing you can always re. This text like harris' book is the computations lists of bezout's theorem some. The algorithmic roots of algebraic object, called a close relationship between ideals. Many of polynomial equations in geometric, object called a more than you. Then for teaching purposes and varieties. The solutions and reduce even without copy. Although the division algorithm for polynomials in mind following files warning electronic. We must also use their solutions sent in the rank of systems and resultants. In preparing a geometric theorem although. Some reason the following should be extremely useful to words. The many pages 510 and mxgb which reveals the tutorial you. It is the dean of mathematics in computational methods have length this volume. There are old the book bases, which computes a graduate. The book we are old it, is particularly important part of cash rewards for releases. Appendix contains a variety the main commands for instance. When requesting a variety the buchberger criterion as matrix formed. I do not made up by symbolic math packages? There is not just finished using algebraic geometry on this book but elegant introductions. In geometric theorem and if so, how to systems of can they work. Lists of computers in the right amount maple. The main tools used by hyeyoun chung for the second printing of modern. In a geometric object is highly accessible. Then for academic affairs of algebraic, geometry a level appropriate to undergraduates this version.
123586058
Boundary conditions for integrable quantum systems
A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.
121690296
Integrable open spin chains with nonsymmetric R-matrices
The authors extend Sklyanin's formalism for constructing integrable open spin chains to the case of 'non-symmetric' (PT-invariant) R-matrices. They use this formalism to show that the Hamiltonian of an integrable open chain is invariant under gauge transformations of the R-matrix.
119629404
On the Solutions of the Yang-Baxter Equations with General Inhomogeneous Eight-Vertex R-Matrix: Relations with Zamolodchikov’s Tetrahedral Algebra
We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent Rij(u)-matrices of the six- and eight-vertex models, where the only constraint is the particle number conservation by mod(2). A complete classification of the solutions is performed. We have obtained also two spectral parameter dependent particular solutions Rij(u,v) of YBE. The application of the non-homogeneous solutions to construction of Zamolodchikov’s tetrahedral algebra is discussed.
120127771
Zeros of the Partition Function for the Heisenberg, Ferroelectric, and General Ising Models
The Lee‐Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many‐spin interactions (satisfying appropriate ``ferromagnetic'' and spin inversion symmetry conditions). When many‐spin interactions are present, all zeros lie on the imaginary Hz‐axis for sufficiently low (but fixed) T, but, in general, some leave the imaginary axis as T → ∞. The extended Ising theorem is used to prove the same result for a Heisenberg system of arbitrary spin with the real anisotropic pair interaction Hamiltonian Hij=−(JijzSizSjz+JijxSixSjx+JijySiySjy) in an arbitrary transverse field (Hx, Hy) under the ``ferromagnetic'' condition Jijz≥|Jijx| and Jijz≥|Jijy|. The analyticity of the limiting free energy of such a Heisenberg ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic fields Hz. The Ising theorem is also applied to hydrogen‐bonded ferroelectric models to pr...
117867044
Exactly solved models in statistical mechanics
exactly solved models in statistical mechanics exactly solved models in statistical mechanics rodney j baxter exactly solved models in statistical mechanics exactly solved models in statistical mechanics flae exactly solved models in statistical mechanics dover books exactly solved models in statistical mechanics dover books exactly solved models in statistical mechanics dover books hatsutori in size 15 gvg7bzbookyo.qhigh literature cited r. j. baxter, exactly solved models in exactly solvable models in statistical mechanics exactly solved models in statistical mechanics dover books okazaki in size 24 vk19j3book.buncivy exactly solved models of statistical mechanics valerio nishizawa in size 11 b4zntdbookntey fukuda in size 13 33oloxbooknhuy yamada in size 19 x6g84ybook.zolay in honour of r j baxter’s 75th birthday arxiv:1608.04899v2 statistical mechanics, threedimensionality and np beautiful models: 70 years of exactly solved quantum many exactly solved models in statistical mechanics (dover solved lattice models: 1944 2010 university of melbourne exactly solved models and beyond: a special issue in the statistical mechanics of the classical two-dimensional faculty of science, p. j. saf ́arik university in ko?sice? a one-dimensional statistical mechanics model with exact statistical mechanics department of physics and astronomy statistical mechanics principles and selected applications graph theory and statistical physics yaroslavvb chapter 4 methods of statistical mechanics ijs thermodynamics and an introduction to thermostatistics potts models and related problems in statistical mechanics methods of quantum field theory in statistical physics statistical mechanics: theory and molecular simulation exactly solvable su(n) mixed spin ladders springer statistical field theory : an introduction to exactly
62646318
NIST Handbook of Mathematical Functions
Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.
121924421
The periodic wave solutions for the Klein–Gordon–Schrödinger equations
The periodic wave solutions for the Klein-Gordon-Schrodinger equations are obtained by using F-expansion method which can be thought of as a generalization of Jacobi elliptic function expansion method proposed recently. In the limit cases, the solitary wave solutions are obtained as well. (C) 2003 Elsevier B.V. All rights reserved.
124672960
New doubly periodic and multiple soliton solutions of the generalized (3+1)-dimensional KP equation with variable coefficients
A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig-Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.
120130140
New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation
Abstract A new generalized Jacobi elliptic function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new double periodic and multiple soliton solutions are obtained for the generalized (2 + 1)-dimensional Boussinesq equation. This method can be applied to many other equations.
120988945
On "new travelling wave solutions" of the KdV and the KdV-Burgers equations
Abstract The Korteweg-de Vries and the Korteweg-de Vries–Burgers equations are considered. Using the travelling wave the general solutions of these equations are presented. “New travelling wave solutions” of the KdV and the KdV–Burgers equations by Wazzan [Wazzan L. Commun Nonlinear Sci Numer Simulat 2009;14:443–50] are analyzed. We demonstrate that all his solutions are not new and are transformed to known solutions.
123266660
EXACT SOLUTIONS FOR THE TRANSFORMED REDUCED OSTROVSKY EQUATION VIA THE FEXPANSION METHOD IN TERMS OF WEIERSTRASS-ELLIPTIC AND JACOBIAN-ELLIPTIC FUNCTIONS
Abstract Although the F -expansion method is a relatively old method for obtaining exact solutions for nonlinear partial differential equations, its advantage over the other auxiliary equation methods is shown in this work for solving the transformed reduced Ostrovsky equation. In fact, to the authors’ knowledge, this is the first time that a largest number of exact solutions for the auxiliary equation F ′ ( ω ) = P F 4 ( ω ) + Q F 2 ( ω ) + R has been presented. It is used in this work to get 52 types of exact solution: six for Weierstrass-elliptic function solutions and the rest for Jacobian-elliptic function solutions. However, in consideration of Parkes (2010) [9] , who states that the term ‘new’ has to be used very carefully, it is very important to mention that the solutions obtained are not completely different. Many examples for proving Parkes’s view [9] have been examined. Moreover, soliton-like solutions and trigonometric-function solutions are also obtained as limiting cases. The previous results determined by the hyperbolic tangent method (Yusufoglu and Bekir, 2007) [5] and exponential function approach (Kangalgil and Ayaz, 2008) [6] are found to be limited and special cases of the current results.
73596605
Z4-symmetric factorizedS-matrix in two space-time dimensions
The factorizedS-matrix with internal symmetryZ4 is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. TheS-matrix contains the resonance poles naturally. The simple formal relation between the general factorizedS-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation theZ4-symmetricS-matrix corresponds to the Baxter transfer matrix itself.
123172734
Direct diagonalization of the transfer matrix of the zero-field free-fermion model
Abstract The transfer matrix of the zero-field free-fermion model is diagonalized by operator methods. The model is a special case of the eight-vertex model solved by Baxter, but it has the advantage that an easy survey of eigenvalues and eigenvectors is obtainable. The transfer matrix is expressed in terms of the fermion-number operators which also diagonalize the corresponding XY hamiltonian. A similar expression was previously conjectured by Jones and Valatin from a knowledge of the eigenvalues given by Baxter. The present derivation does not use this knowledge and provides a direct and independent diagonalization of the transfer matrix.
119427465
Hidden symmetry of free Fermion model
This paper considers the eight-vertex model of free fermions on a plane lattice. Using the triangle equation and the symmetry properties of the model, the authors construct an elliptic parametrization for the Boltzmann vertex weights. In this parametrization, the weights are meromorphic functions of three complex variables.
27559249
Kinetics of biopolymerization on nucleic acid templates
The kinetics of biopolymerization on nucleic acid templates is discussed. The model introduced allows for the simultaneous synthesis of several chains, of a given type, on a common template, e.g., the polyribosome situation. Each growth center [growing chain end plus enzyme(s)] moves one template site at a time, but blocks L adjacent sites. Solutions are found for the probability nj(t) that a template has a growing center that occupies the sites j — L + 1,…, j at time t. Two special sets of solutions are considered, the uniform‐density solutions, for which nj(t) = n, and the more general steady‐state solutions, for which dnj(t)/dt = 0. In the uniform‐density case, there is an upper bound to the range of rates of polymerization that can occur. Corresponding to this maximum rate, there is one uniform solution. For a polymerization rate less than this maximum, there are two uniform solutions that give the same rate. In the steady‐state case, only L = 1 is discussed. For a steady‐state polymerization rate less than the maximum uniform‐density rate, the steady‐state solutions consist of either one or two regions of nearly uniform density, with the density value(s) assumed in the uniform region(s) being either or both of the uniform‐density solutions corresponding to that polymerization rate. For a steady‐state polymerization rate equal to or slightly larger than the maximum uniform‐density rate, the steady‐state solutions are nearly uniform to the single uniform‐density solution for the maximum rate. The boundary conditions (rate of initiation and rate, of release of completed chains from the template) govern the choice among the possible solutions, i.e., determine the region(s) of uniformity and the value(s) assumed in the uniform region(s).
119307482
Vertex models, TASEP and Grothendieck polynomials
We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials.