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arXiv:1905.08756 · SciPost Physics

Domain wall problem in the quantum XXZ chain and semiclassical behavior close to the isotropic point

Grégoire Misguich, Nicolas Pavloff, Vincent Pasquier

Published Aug 29, 2019• 48 citations
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Abstract

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.

DOI
10.21468/scipostphys.7.2.025

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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 study the nonequilibrium spin dynamics in the nonintegrable spin-1 XXZ chains, emerging from inhomogeneous quenches, with two halves initialized in distinct maximally polarized states. Employing the time-dependent variational principle (TDVP) we are able to reach times that are inaccessible by standard time-dependent density matrix renormalization group methods. We identify three distinct phases depending on the anisotropy of the chain, corresponding to diffusive, logarithmic superdiffusive, and insulating. In the easy-axis regime we observe a transient ballistic behavior with a crossover time that diverges as the isotropic point is approached. This leads us to the conclusion that on the intermediate-large timescales the spin dynamics is well described by the integrable Landau-Lifshitz classical differential equation.
We study the dynamics of an isotropic spin-12\frac{1}{2}21​ Heisenberg chain starting in a domain-wall initial condition where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization-group simulations (up to t=350t=350t=350) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.
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.
The transport of magnetization is analyzed for the classical Heisenberg chain at and especially above the isotropic point. To this end, the Hamiltonian equations of motion are solved numerically for initial states realizing harmonic-like magnetization profiles of small amplitude and with random phases. Above the isotropic point, the resulting dynamics is observed to be diffusive in a hydrodynamic regime starting at comparatively small times and wave lengths. In particular, hydrodynamic regime and diffusion constant are both found to be in quantitative agreement with close-to-equilibrium results from quantum S =1/2 autocorrelations at high temperatures. At the isotropic point, the resulting dynamics turns out to be non-diffusive at the considered times and wave lengths.
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 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.