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quantum many - body system out of equilibrium has been subjected to upsurging interest in recent times . extensive studies have been done over decades giving rise to many new findings in this field ( see @xcite for a review ) . lots of questions still remain unresolved too . from the results obtained so far , it is impossible to draw a generic picture of the dynamics out of equilibrium . emergence of current of a physical quantity generally renders a system to a nonequilibrium one . transport of that quantity due to presence of current thereby gets attention as an important aspect of nonequilibrium dynamics . transport properties , especially magnetization transport in low dimensional spin systems have received attention of the experimentalists too gaining importance in the area of spintronics , nano - device applications @xcite . earlier theoretical studies @xcite have confirmed the fact that the properties of transportation in closed quantum system differ significantly from its classical counterpart . a number of works , both analytical and numerical , have been done on introducing current of a quantity in the integrable hamiltonian to observe its relaxation dynamics @xcite . in these papers , current - carrying states have been generated in the system in two ways , either by adding current to the hamiltonian or by preparing an inhomogeneous initial state . introduction of magnetization current results in oscillatory nature and power - law decay of correlations in space . a typical initial state with a steplike inhomogeneity in magnetization even yields a scaling form of the same in the large - time limit . analytical as well as simulative studies have also been made on nonequilibrium transport of magnetization in open quantum spin chain @xcite . these studies have been able to shed some light upon the behavior of physical quantities in an integrable system while trying to equilibrate . + integrability has become a very crucial perspective in these explorations of nonequilibrium phenomena . integrable systems with many local conserved quantities have restricted dynamics and unlike non - integrable systems , their relaxation process is dependent on the initial state . another behavior believed to be the line of division between integrable and non - integrable system is thermalization . it was claimed to be absent in integrable systems @xcite . numerical studies on non - integrable systems has been done recently in this regard @xcite . evidences are however present where non - integrable systems are shown to have weak thermalization or even no thermalization @xcite . owing to some more elaborate studies the role of integrability on the behavior of response functions have been put under question . it has been shown @xcite that in a completely integrable system , apart from the usual non - thermal behavior of some quantities , there do exist some other quantities that exhibit thermal behavior which is counterintuitive to the preceding ideas on thermalization . another important question is whether their lies any generic relation between integrability and transport properties . efforts have been given in that part too by some preceding works @xcite . + a scattered context like this has been the motivation of more investigation in relaxation dynamics of integrable quantum system generally followed by a quench @xcite ( see @xcite for review ) . in this article , we study the dynamics due to transport of transverse magnetization in a quantum ising chain . transportation of magnetization is induced by preparing an initial state which produces inhomogeneity in transverse magnetization and thus puts the system into nonequilibrium . we let it evolve in presence of a homogeneous and constant ( in time ) transverse field with a view to studying whether the transport properties finally make the magnetization homogeneous or not and how the magnetization relaxes with time . somewhat similar study has been made on xxz spin chain showing oscillatory behavior and power - law decay @xcite . power - law relaxation of local magnetization has been observed in heisenberg spin chain too @xcite . study of temporal behavior of transverse magnetization from a thermally inhomogeneous state has also been done in quantum xx chain @xcite . in our case , to prepare such an initial configuration , a new protocol is introduced : we include two odd - occupation basis states in fermionic momentum space which has zero eigenvalue . a surprising observation is that the later dynamics driven by the external transverse field can not make the magnetization homogeneous even after infinite time which is counterintuitive to our ideas in similar context of classical counterpart . this immediately renders the strength of the external field less important than the presence of odd - occupation states . although the system relaxes with time , the local magnetization at different sites do not attain the same value . the underlying dynamics at each site are shown to possess two different timescales of oscillation . the smaller timescale contributes insignificant undulations whereas the oscillations of larger timescale exhibit generic power - law decay . the exponent of the decay is proved to be independent of the strength of the external field . this type of decay also indicates absence of thermalization , in agreement with previous works @xcite , whereas in contrast with that work , no scaling form is observed in the large - time limit for this type of initial configuration . a very recent work on thermalization also supports the fact that such type of decay indicates lack of thermalization @xcite . + another important feature of our work is that the phenomena of quantum phase transition is manifested in the dynamics in two ways as can be established analytically , ( i ) the frequency of the characteristic oscillation is a monotonic function of the external field as long as the system is in the ordered phase and becomes independent of the field when it crosses the critical value to the disordered phase , ( ii ) the transverse magnetization after infinite time at each site shows different functional behavior in the ordered and disordered phase and thus produces nonanalyticity at the critical point . + lastly , our initial configuration contains several parameters and all the features of the dynamics we observe , is true for arbitrary values of those parameters . thus , inspite of the fact that the hamiltonian is integrable , some characteristics of the dynamics is valid for a class of initial configurations . + in the next section we give detailed description of the model and of the initial state . section iii contains the exact analytical treatment of transverse magnetization and its dynamics . in the last section , conclusions are drawn from the results obtained along with general discussions . one dimensional spin-@xmath1 quantum ising chain of @xmath2 sites is described by the hamiltonian = -_i=1^n s^x_i s^x_i+1 - _ i=1^n s^z_i where @xmath0 , scaled by the coupling constant , is the external transverse field and @xmath3 are two components of pauli spin matrices . + the well - known jordan - wigner transformation enables us to transform the hamiltonian into a direct sum of hamiltonians ( @xmath4 ) of nonlocal free fermions of momenta @xmath5 @xcite _ k = ( -2 i k ) - 2(+ k ) [ hk_def ] where @xmath6 and @xmath7 are fermionic creation and annihilation operator in momentum space and @xmath8 , with @xmath9 . the operator @xmath4 has four eigenstates . the even - occupation eigenstates of @xmath4 are spanned by two basis states namely , @xmath10 and @xmath11 where the numbers signify the occupation status of the fermions having momenta @xmath12 and @xmath13 respectively . we denote the eigenstates of @xmath4 within these subspaces as @xmath14 with eigenvalues @xmath15 , where @xmath16 $ ] . the other basis states @xmath17 and @xmath18 have zero eigenvalues . for any wave function @xmath19 , the transverse magnetization at the @xmath20-th site is given by @xmath21 this magnetization is homogeneous if @xmath19 is the ground state or any other state comprising of @xmath22 and @xmath23 . + we construct an initial configuration incorporating the odd - occupation basis ( @xmath24 and @xmath25 ) in momentum space alongwith the states @xmath22 and @xmath23 . we choose |(0)= |_k |_k= _ k |11_k + _ k |00_k + _ k |10_k + _ k |01_k [ def_psi ] the normalization condition requires    note that this state is not an eigenstate of the fermionic hamiltonian @xmath4 . we shall now show that one can calculate analytically the magnetization at any given site for this initial state under certain conditions . the expression for magnetization eq.([mnz ] ) involves jump of fermions from any site @xmath26 to any site @xmath27 . let us first assume that @xmath27 and @xmath26 are both positive and @xmath28 . for such a jump , the sign of the resulting term is then determined by whether the fermion crosses an even or odd number of fermions during its flight . @xmath29 \nonumber \\    & &   \left(\alpha_{k_2 } |01\rangle_{k_2 } + \gamma_{k_2 } |00\rangle_{k_2 } \right )      \end{aligned}\ ] ] the sign will be plus ( minus ) if the number of fermions in the range @xmath30 for a given term in the expanded form of the portion within square brackets is even ( odd ) . however , one observes that , the correct sign is obtained from the equality ,    @xmath31 \nonumber \\    & &   \left(\alpha_{k_2 } |01\rangle_{k_2 } + \gamma_{k_2 } |00\rangle_{k_2 } \right )   \label{ak }     \end{aligned}\ ] ]    in order to calculate the magnetization @xmath32 at site @xmath20 , we need to operate @xmath33 on eq.([ak ] ) . however , a simplification occurs if we note that ( _ k 11| + _ k 00|_k + _ k 10|_k + _ k 01|_k ) ( _ k |11_k + _ k |00_k - _ k |10_k - _ k |01_k ) = |_k|^2 + |_k|^2 - 2|_k|^2 . hence , if we ensure that |_k|^2 + |_k|^2 = 2|_k|^2 [ zero ] for all @xmath34 , the quantity @xmath35 will be non - zero _ only when _ either @xmath36 or @xmath27 and @xmath26 are two successive points so that @xmath37 ( say ) . note that the constraints ( [ normalization ] ) and ( [ zero ] ) need @xmath38 and @xmath39 for all @xmath34 . we assume additionally , that @xmath40 for all @xmath34 . one can now obtain the expression for magnetization @xmath41 as , @xmath42 here the sum runs over @xmath43 values between @xmath44 and @xmath45 and @xmath46 stands for complex conjugate . in order to study the dynamics we first note that , from eq.([def_psi ] ) the wave function at time @xmath47 may be obtained as @xmath48 + \beta_k[-sin\theta_k e^{i\lambda_k t}|(\gamma , k)_{-}\rangle + \cos\theta_k e^{-i\lambda_k t}|(\gamma , k)_{+}\rangle ]   \nonumber \\ % & & + \ ; \gamma_k|10\rangle + \ ; \gamma_k|01\rangle \\ & = & \alpha_k^{\prime } |11\rangle + \beta_k^{\prime } |00\rangle + \gamma |10\rangle + \gamma |01\rangle \end{aligned}\ ] ] where @xmath49 @xmath50 with @xmath47 is scaled by @xmath51 . a very crucial point is that the coefficients @xmath52 , @xmath53 and @xmath54 also satisfy the condition eq.([zero ] ) . hence , the magnetization at time @xmath47 can also be calculated following the previous procedure : m_z(n , t ) = -1 + _ k=0^ m_k [ m1 ] where @xmath55 is given by the expression in eq.([mn0 ] ) with @xmath56 , @xmath57 replaced by @xmath58 , @xmath59 . after this replacement one can express @xmath55 as m_k = a_k + b_k ( 2_k t + b_k ) + c_k(n ) 2(_k+u+_k)t + c_k(n ) + d_k(n)2(_k+u-_k)t + d_k(n ) [ mk ] the precise expressions for the amplitudes @xmath60 and the phases @xmath61 , @xmath62 , @xmath63 in terms of @xmath34 , @xmath20 , @xmath56 , @xmath57 and @xmath64 can be obtained but is not required for our subsequent calculations . + from now on we shall be restricted to the case where @xmath56 and @xmath57 are real and equal for all @xmath34 , i.e. @xmath65 and @xmath66 . then , in the thermodynamic limit , @xmath41 can be expressed as @xmath67 \ ; dk % m_k & = & 2\gamma^2(\sin^4\theta_k + \cos^4\theta_k)\left[2(\beta^2-\alpha^2)\cos un + 2(\beta^2+\alpha^2)\cos 2kn \right]\cos u\lambda'_k t \nonumber \\ % & & - 8\gamma^2\alpha\beta\cos 2\theta_k \sin 2kn \sin u\lambda'_k t   + 2\gamma^2 \sin^2 2\theta_k \left[(\beta^2-\alpha^2)\cos un \right](\cos^2\lambda_k t-3\sin^2\lambda_k t ) \nonumber \\ % & & - 8\gamma^2\alpha\beta\sin 2\theta_k \cos un \sin 2\lambda_k t + 4\gamma^2(\beta^2+\alpha^2)\sin 4\theta_k\sin un \sin^2\lambda_k t \nonumber \\ % & & + 2\gamma^2\left[(\beta^2+\alpha^2)\sin^2 2\theta_k + 1\right]\cos 2kn \nonumber \\ % & & + \alpha^2 \sin^2 2\theta_k\cos 2\lambda_k t + 2\alpha\beta \sin 2\theta_k \sin 2\lambda_k t + 2\beta^2 \sin^2 2\theta_k \sin^2\lambda_k t + 2\alpha^2 ( \cos^4\theta_k + \sin^4\theta_k ) \label{m_int}\ ] ] where @xmath68\cos 2\lambda_k t + [ 2\alpha\beta\sin 2\theta_k(1-\cos un ) ] \sin 2\lambda_k t \\ m_k(t / n)&= & \frac{1}{2}(\cos^4\theta_k + \sin^4\theta_k)[2(\beta^2-\alpha^2)\cos un + \cos 2kn ] \cos 2\pi\lambda'_k\frac{t}{n } \;-\ ; 2\alpha\beta\cos 2\theta_k \sin 2kn \sin 2\pi\lambda'_k\frac{t}{n}\end{aligned}\ ] ] we have written @xmath69 as in the thermodynamic limit the difference between consecutive @xmath34-points become infinitesimally small . + the dynamics is characterized by the behavior of transverse magnetization @xmath41 at a time @xmath47 at site @xmath20 , as given by eqs ( [ m_int ] ) . at any given time ( including @xmath70 ) magnetization is a continuously varying function of @xmath20 .    , starting from initial state ( [ def_psi ] ) with @xmath71 for all @xmath34 and @xmath72.,title="fig:",width=377 ]    initial magnetization as derived from eq.([m_int ] ) is given by m_z(n , t=0)=-+2 ^ 2+(^2-^2)un [ mz - init ] eqs ( [ m_int ] ) and ( [ mz - init ] ) give clear indication that the magnetization is inhomogeneous at all time and is shown in fig [ mzvsn ] . + the temporal behavior of transverse magnetization at a given site is shown in fig.[mzvst ] for different external fields . it exhibits oscillatory behavior with the envelope decaying algebraically in the large time limit . it is evident that eq.([m_int ] ) has two time - dependent parts , @xmath73 and @xmath74 of two separate timescales . the former part gives tiny undulations ( fig . [ mzvst](a ) ) and the other is responsible to produce larger oscillation with algebraic decay of envelope and dominates in the large time limit . hence , for @xmath75 and @xmath76 , we have @xmath77 \cos 2\pi\lambda'_k\frac{t}{n } \nonumber \\   & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\ ; 2\alpha\beta\cos 2\theta_k \sin 2kn \sin 2\pi\lambda'_k\frac{t}{n } \;]\;dk \nonumber \\ & \approx & m_{\infty } + \frac{1}{\pi}\int_0^\pi \mathit{g}_k(n)\cos(\omega_k\tau + \phi_k(n))\;dk   \label{mz - tau }   \end{aligned}\ ] ] where @xmath78 and @xmath79 . the quantities @xmath80 , @xmath81 can be written in terms of @xmath82 and @xmath34 from the last equation , but the explicit form is not necessary for our calculations below . we call the @xmath83-independent term @xmath84 because it will give us the magnetization at @xmath85 which we shall discuss later . and @xmath86     at @xmath87 against @xmath88 , starting from initial state ( [ def_psi ] ) with @xmath89 for all @xmath34 . ( a ) @xmath32 vs @xmath83 alongwith small undulations due to time scale @xmath47 . variation of @xmath32 for very small change of @xmath83 given as inset shows oscillations in small timescale . ( b ) oscillations in large timescale are shown in @xmath32 vs @xmath83 plot for @xmath72 and @xmath90 . the envelopes of oscillations decay as @xmath91 both for @xmath92 and @xmath93 . ( c ) & ( d ) frequency of large timescale oscillation is shown to increase with @xmath0 for @xmath94 in @xmath32 vs @xmath83 plot whereas for @xmath95 , it becomes constant.,title="fig:",width=604 ]    we first observe that for sufficiently large @xmath83 , the quantity @xmath96 will be large ( so that its cosine will fluctuate very rapidly and vanish on integration ) unless @xmath97 is very small . hence the region where @xmath97 is minimum with respect to @xmath34 will only contribute to the integral . this minima is found to occur at k = k_0 = \ {    rl ^-1(- ) & + ^-1 ( - ) &    . [ k0 ]    it is hence sufficient to integrate the oscillatory term over a small region @xmath98 ( @xmath99 becoming smaller and smaller with increasing @xmath83 ) where @xmath100 and @xmath81 do not vary appreciably . expanding @xmath97 about @xmath101 , _ k = _ k_0 + f(k - k_0)^2 [ omega - taylor ] ( where @xmath102 ) we get the expression for magnetization at large @xmath83 as , @xmath103 where @xmath104 . thus it is evident that for the chosen initial configuration , the transverse magnetization always exhibits @xmath91 decay in the envelope of oscillation irrespective of the magnitude of the transverse field . + as shown in fig.[mzvst](c ) and ( d ) , the frequency of oscillation varies monotonically with @xmath0 in the ferromagnetic phase and becomes independent of @xmath0 in paramagnetic region . it can be explained analytically . we note from eq.([k0 ] ) that @xmath105 changes from @xmath106 to @xmath107 as @xmath0 crosses the critical point . this means that the frequency in the large @xmath83 limit behaves as |_k| = 2|| = \ {    rl 4 & + 4 &    . [ omegak0 ] the significance of this nonanalytic behavior of frequency is that it coincides with the occurrence of order - disorder quantum phase transition of the system . + behavior at @xmath85 ( and hence @xmath108 as well ) can also be obtained form eq.([m_int ] ) by performing the integrations involved : m _ ( n ) = - + 2 ^ 2 - + ( 2 - un ) ^p - ( - ) ^2 ^p(2n-1 ) [ mz - inf ] where the index @xmath109 is @xmath110 for @xmath111 and @xmath112 for @xmath113 . the persistence of inhomogeneity of magnetization at @xmath85 is evident . another important aspect of the long - time behaviour is that the quantity @xmath114 shows different behavior as a function of @xmath0 in ferromagnetic and paramagnetic region and is non - analytic at the critical point . such behaviour originates from two integrals involved in the calculation of @xmath115 . they can be written as a contour integral over the unit circle and the integrand has poles at @xmath116 and @xmath117 . for @xmath111 , the former pole is inside the unit circle and for @xmath113 , the latter is inside . this change of pole structure leads to a change in functional behavior and to non - analyticity , as has been observed elsewhere @xcite . vs @xmath0 plot & ( b ) @xmath118 vs @xmath0 plot , for @xmath119 with the initial state ( [ def_psi ] ) taking @xmath71 for all @xmath34.,title="fig:",width=529 ] we have shown analytically that in a transverse ising chain at zero temperature , a state can be manipulated in the fermioninc momentum space so that it produces spatial variation in transverse magnetization . such inhomogeneity gives rise to transport of transverse magnetization from one site to another as a result of which both spatial and temporal variations take place in presence of homogeneous and constant ( in time ) external transverse field . at each site the magnetization evolves in an oscillatory manner with the envelope decaying algebraically . the oscillation and decay however do not lead to homogeneous magnetization because the odd - occupation states have zero energy and their coefficients , which are the prime factor for inhomogeneity , remain nonzero forever . the exponent of decay is independent of the field i.e. , it remains unaltered in ferromagnetic and paramagnetic phases . the characteristic change is shown by the frequency of oscillation at long time and the final magnetization at each site . they show different behavior in ferromagnetic and paramagnetic phases . thus we find no signature of criticality in the exponent of decay whereas two new quantities are found to bear it . moreover , starting with such configuration , the inhomogeneity in transverse magnetization can not be removed by the external field . both the phenomenon are counterintuitive to the prevailing ideas . the scenario may be thought of as a quench from an initial state and the preceding works on quench dynamics report the change of the exponent of decay in different phases @xcite although in our case , the presence of odd - occupation states becomes an important factor in the dynamics . although the initial configuration we have worked with contains real and @xmath34-independent values of the coefficients , it is important to note that , all our observations are valid for arbitrary phases of these quantities and arbitrary modulus of @xmath56 or @xmath57 . thus , although the hamiltonian is integrable , we observe a generic behaviour for a wide range of initial states . + a question arises regarding other observables . integrable system has been reported to have some non - local observables which show typically different behaviors including even thermalization in course of quench dynamics @xcite . now in our system , with this initial configuration , how do non - local observables behave ? can thermalization or field - dependent decay exponent be found out in any of them ? how do the non - integrable systems behave starting from such inhomogeneity ? search for answers of many such open questions may thus be quite interesting to investigate in future .
we study the dynamics caused by transport of transverse magnetization in one dimensional transverse ising chain at zero temperature . 
 we observe that a class of initial states in fermionic momentum - space produce spatial variation in transverse magnetization . starting from such a state 
 , we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field @xmath0 . in contradiction with general expectation , 
 whatever be the strength of the field , the magnetization of the system does not become homogeneous even after infinite time . in each site , the dynamics is associated with oscillations having two different timescales . 
 the envelope of the larger timescale oscillation decays algebraically having an exponent which is universal for the entire class of such initial states . 
 signature of quantum criticality is also embedded in the dynamics in two aspects : behavioral change shown by the frequency of large timescale oscillation at any site and the corresponding magnetization at infinity when they cross the quantum critical point .