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arXiv:1002.4446 · Physical Review E

Quantum quenches in an XXZ spin chain from a spatially inhomogeneous initial state

Jarrett L. Lancaster, Aditi Mitra

Published Jun 30, 2010• 115 citations
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Abstract

Results are presented for the nonequilibrium dynamics of a quantum XXZ -spin chain whose spins are initially arranged in a domain wall profile via the application of a magnetic field in the z direction, which is spatially varying along the chain. The system is driven out of equilibrium in two ways: a). by rapidly turning off the magnetic field, b). by rapidly quenching the interactions at the same time as the magnetic field is turned off. The time evolution of the domain wall profile as well as various two-point spin correlation functions are studied by the exact solution of the fermionic problem for the XX chain and via a bosonization approach and a mean-field approach for the XXZ chain. At long times the magnetization is found to equilibrate (reach the ground state value), while the two-point correlation functions in general do not. In particular, for quenches within the gapless XX phase, the spin correlation function transverse to the z direction acquires a spatially inhomogeneous structure at long times whose details depend on the initial domain wall profile. The spatial inhomogeneity is also recovered for the case of classical spins initially arranged in a domain wall profile and shows that the inhomogeneities arise due to the dephasing of transverse spin components as the domain wall broadens. A generalized Gibbs ensemble approach is found to be inadequate in capturing this spatially inhomogeneous state.

DOI
10.1103/physreve.81.061134

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Results are presented for the nonequilibrium dynamics of a quantum XXZ -spin chain whose spins are initially arranged in a domain wall profile via the application of a magnetic field in the z direction, which is spatially varying along the chain. The system is driven out of equilibrium in two ways: a). by rapidly turning off the magnetic field, b). by rapidly quenching the interactions at the same time as the magnetic field is turned off. The time evolution of the domain wall profile as well as various two-point spin correlation functions are studied by the exact solution of the fermionic problem for the XX chain and via a bosonization approach and a mean-field approach for the XXZ chain. At long times the magnetization is found to equilibrate (reach the ground state value), while the two-point correlation functions in general do not. In particular, for quenches within the gapless XX phase, the spin correlation function transverse to the z direction acquires a spatially inhomogeneous structure at long times whose details depend on the initial domain wall profile. The spatial inhomogeneity is also recovered for the case of classical spins initially arranged in a domain wall profile and shows that the inhomogeneities arise due to the dephasing of transverse spin components as the domain wall broadens. A generalized Gibbs ensemble approach is found to be inadequate in capturing this spatially inhomogeneous state.
Out-of-equilibrium behavior is explored in the one-dimensional anisotropic XY model. Initially preparing the system in the isotropic XX model with a linearly varying magnetic field to create a domain-wall magnetization profile, dynamics is generated by rapidly changing the exchange interaction anisotropy and external magnetic field. Relaxation to a nonequilibrium steady state is studied analytically at the critical transverse Ising point, where correlation functions may be computed in closed form. For arbitrary values of anisotropy and external field, an effective generalized Gibbs' ensemble is shown to accurately describe observables in the long-time limit. Additionally, we find spatial oscillations in the exponentially decaying, transverse spin-spin correlation functions with wavelength set by the magnetization jump across the initial domain wall. This wavelength depends only weakly on anisotropy and magnetic field in contrast to the current, which is highly dependent on these parameters.
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.
We investigate the real-time dynamics of a critical spin-1/2 chain (XXZ model) prepared in an inhomogeneous initial state with different magnetizations on the left and right halves. Using the time-evolving block decimation method, we follow the front propagation by measuring the magnetization and entanglement entropy profiles. At long times, as in the free fermion case [Antal et al., Phys. Rev. E 59, 4912 (1999)], a large central region develops where correlations become time independent and translation invariant. The shape and speed of the fronts is studied numerically and we evaluate the stationary current as a function of initial magnetic field and as a function of the anisotropy $\ensuremath{\Delta}$. We compare the results with the conductance of a Tomonaga-Luttinger liquid, and with the exact free-fermion solution at $\ensuremath{\Delta}=0$. We also investigate the two-point correlations in the stationary region and find a good agreement with the ``twisted'' form obtained by Lancaster and Mitra [Phys. Rev. E 81, 061134 (2010)] using bosonization. Some deviations are nevertheless observed for strong currents.
We show that long-range ferromagnetic interactions in quantum spin chains can induce spatial quasilocalization of topological magnetic defects, i.e., domain walls, even in the absence of quenched disorder. Utilizing matrix-product-states numerical techniques, we study the nonequilibrium evolution of initial states with one or more domain walls under the effect of a transverse field in variable-range quantum Ising chains. Upon increasing the range of these interactions, we demonstrate the occurrence of a sharp transition characterized by the suppression of spatial diffusion of the excitations during the accessible time scale: the excess energy density remains localized around the initial position of the domain walls. This quasilocalization is accurately reproduced by an effective semiclassical model, which elucidates the crucial role that long-range interactions play in this phenomenon. The predictions of this Rapid Communication can be tested in current experiments with trapped ions.
We study the dynamics of a spin- \frac{1}{2} 1 2 XXZ chain which is initially prepared in a domain-wall state. We compare the results of time-dependent Density Matrix Renormalization Group simulations with those of an effective description in terms of a classical anisotropic Landau-Lifshitz (LL) equation. Numerous quantities are analyzed: magnetization ( x x , y y and z z components), energy density, energy current, but also some spin-spin correlation functions or entanglement entropy in the quantum chain. Without any adjustable parameter a quantitative agreement is observed between the quantum and the LL problems in the long time limit, when the models are close to the isotropic point. This is explained as a consequence of energy conservation. At the isotropic point the mapping between the LL equation and the nonlinear Schrödinger equation is used to construct a variational solution capturing several aspects of the problem.
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 generalised Gibbs ensemble (GGE). Then we study the full time evo-lution of the transverse magnetisation by means of stationary phase methods. The equal time two-point longitudinal correlation function is analytically derived for a par-ticular 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. ar