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arXiv:cond-mat/0301473 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Magnetization distribution in the transverse Ising chain with energy flux

Viktor Eisler, Zoltàn Ràcz, Frédéric van Wijland

Published May 27, 2003• 41 citations
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Abstract

The zero-temperature transverse Ising chain carrying an energy flux j(E) is studied with the aim of determining the nonequilibrium distribution functions, P(M(z)) and P(Mx) of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M(z)) is a Gaussian both at j(E)=0 and at j(E) not equal to 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(Mx), is evaluated numerically for spin chains of up to 20 spins. For the equilibrium case (j(E)=0), we find the expected Gaussian fluctuations away from the critical point, while the critical order-parameter fluctuations are shown to be non-Gaussian with a scaling function Phi(x)=Phi(M(x)/<Mx>)=<Mx>P(Mx) strongly dependent on the boundary conditions. When j(E) not equal to 0, the system displays long-range, oscillating correlations but P(Mx) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j(E). In particular, we find that, at critical transverse field, the width has a j(-3/8)(E) asymptotic in the j(E)-->0 limit.

DOI
10.1103/physreve.67.056129

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The zero-temperature transverse Ising chain carrying an energy flux j(E) is studied with the aim of determining the nonequilibrium distribution functions, P(M(z)) and P(Mx) of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M(z)) is a Gaussian both at j(E)=0 and at j(E) not equal to 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(Mx), is evaluated numerically for spin chains of up to 20 spins. For the equilibrium case (j(E)=0), we find the expected Gaussian fluctuations away from the critical point, while the critical order-parameter fluctuations are shown to be non-Gaussian with a scaling function Phi(x)=Phi(M(x)/<Mx>)=<Mx>P(Mx) strongly dependent on the boundary conditions. When j(E) not equal to 0, the system displays long-range, oscillating correlations but P(Mx) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j(E). In particular, we find that, at critical transverse field, the width has a j(-3/8)(E) asymptotic in the j(E)-->0 limit.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
We consider the critical and off-critical properties at the boundary of the random transverse-field Ising spin chain when the distribution of the couplings and/or transverse fields, at a distance l from the surface, deviates from its uniform bulk value by terms of order ${l}^{\ensuremath{-}\ensuremath{\kappa}}$ with an amplitude A. Exact results are obtained using a correspondence between the surface magnetization of the model and the surviving probability of a random walk with time-dependent absorbing boundary conditions. For slow enough decay, $\ensuremath{\kappa}<1/2,$ the inhomogeneity is relevant: Either the surface stays ordered at the bulk critical point or the average surface magnetization displays an essential singularity, depending on the sign of A. In the marginal situation $\ensuremath{\kappa}=1/2,$ the average surface magnetization decays as a power law with a continuously varying, A-dependent, critical exponent which is obtained analytically. The behavior of the critical and off-critical autocorrelation functions as well as the scaling form of the probability distributions for the surface magnetization and the first gaps are determined through a phenomenological scaling theory. In the Griffiths phase, the properties of the Griffiths-McCoy singularities are not affected by the inhomogeneity. The various results are checked using numerical methods based on a mapping to free fermions.
We study the nonequilibrium steady states of an Ising chain in a transverse field hhh by investigating the effect of a field $\ensuremath{\lambda}$, which drives the current of energy. The zero-temperature, hhh- $\ensuremath{\lambda}$ phase diagram is determined exactly, and it is found that the energy current appears continuously above a threshold $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}(h)$. The long-range magnetic order (present for $h<{h}_{c},\ensuremath{\lambda}<{\ensuremath{\lambda}}_{c}$) is destroyed in the current-carrying state where the correlations are characterized by power-law, oscillatory decay. The mechanism which generates the power-law correlations in the current-carrying state is discussed.
Within the universality class of ferromagnetic vector models with O(n) symmetry and purely dissipative dynamics, we study the non-equilibrium critical relaxation from a magnetized initial state. Transverse correlation and response functions are exactly computed for Gaussian fluctuations and in the limit of infinite number n of components of the order parameter. We find that the fluctuation–dissipation ratios (FDRs) for longitudinal and transverse modes differ already at the Gaussian level. In these two exactly solvable cases we completely describe the crossover from the short-time to the long-time behaviour, corresponding to a disordered and a magnetized initial condition, respectively. The effects of non-Gaussian fluctuations on longitudinal and transverse quantities are calculated in the first order in the -expansion ( = 4−d) and reliable three-dimensional estimates of the two FDRs are obtained.
Within the universality class of ferromagnetic vector models with O( n ) symmetry and purely dissipative dynamics, we study the non-equilibrium critical relaxation from a magnetized initial state. Transverse correlation and response functions are exactly computed for Gaussian fluctuations and in the limit of infinite number n of components of the order parameter. We find that the fluctuation–dissipation ratios (FDRs) for longitudinal and transverse modes differ already at the Gaussian level. In these two exactly solvable cases we completely describe the crossover from the short-time to the long-time behaviour, corresponding to a disordered and a magnetized initial condition, respectively. The effects of non-Gaussian fluctuations on longitudinal and transverse quantities are calculated in the first order in the -expansion ( = 4− d ) and reliable three-dimensional estimates of the two FDRs are obtained.