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arXiv:1106.5248 · Physical Review B

Semiclassical theory for quantum quenches in finite transverse Ising chains

Heiko Rieger, Ferenc Iglói

Published Oct 19, 2011• 133 citations
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Abstract

We present a quantitative semiclassical theory for the nonequilibrium dynamics of transverse Ising chains after quantum quenches, in particular, sudden changes of the transverse field strength. We obtain accurate predictions for the quench-dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time-dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantitative predictions of our semiclassical theory (local magnetization, equal-time bulk-bulk and surface-to-bulk correlations, and bulk autocorrelations) with the results from exact free-fermion calculations, and discuss the range of applicability of the semiclassical theory and possible generalizations and extensions.

DOI
10.1103/physrevb.84.165117

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We present a quantitative semiclassical theory for the nonequilibrium dynamics of transverse Ising chains after quantum quenches, in particular, sudden changes of the transverse field strength. We obtain accurate predictions for the quench-dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time-dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantitative predictions of our semiclassical theory (local magnetization, equal-time bulk-bulk and surface-to-bulk correlations, and bulk autocorrelations) with the results from exact free-fermion calculations, and discuss the range of applicability of the semiclassical theory and possible generalizations and extensions.
We consider global quenches in the quantum XY chain in a transverse field and study the nonequilibrium relaxation of the magnetization and the correlation function as well as the entanglement entropy in finite systems. For quenches in the ordered phase, the exact results are well described by a semiclassical theory (SCT) in terms of ballistically moving quasiparticles. In the thermodynamic limit the SCT is exact for the entanglement entropy and its modified version following the method of Calabrese, Essler and Fagotti: arXiv:1204.3911 is exact for the magnetization and the correlation function, too. The stationary correlation function is shown to be described by a generalized Gibbs ensemble.
We use the semi-classical approach to study the non-equilibrium dynamics of the O(3) non-linear sigma model. For a class of quenches defined in the text, we obtain the order-parameter dynamical correlator in the thermodynamic limit. In particular we predict quench-dependent relaxation times and correlation lengths. The approach developed for the O(3) non-linear sigma model can also be applied to the transverse field Ising chain, where the semi-classical results can be directly compared to both the exact and the numerical ones, revealing the limits of the method.
We investigate the time evolution of the density of kinks in the spin-1/2 quantum Ising spin chain after a sudden quench in the transverse field strength, and find that it relaxes to a value which depends on the initial and the final values of the transverse field, with an oscillating power-law decay. We provide analytical estimates of the long-time behavior and of the asymptotic value reached after complete relaxation, and discuss the role of quantum criticality in the quench dynamics. We show that, for a dynamics at the critical point, the residual density of kinks after the quench can be described by equilibrium statistical mechanics at a finite temperature dictated by the energy of the state after the quench. On the other hand, outside of criticality it does not exhibit thermalization.
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 study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.
We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalized Gibbs ensemble. Then we study the full time evolution of the transverse magnetization by means of stationary phase methods. The equal-time two-point longitudinal correlation function is analytically derived for a particular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches.