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arXiv:2110.05972 · Physical Review A

Dynamics of spin helices in the one-dimensional XX model

Darren Pereira, Erich J. Mueller

Published Oct 11, 2022• 11 citations
arXivHTMLPDF

Abstract

Motivated by cold-atom experiments and a desire to understand far-from-equilibrium quantum transport, we analytically study the dynamics of spin helices in the one-dimensional $\mathit{XX}$ model. We use a Jordan-Wigner transformation to map the spin chain onto a noninteracting Fermi gas with simple equations of motion. The resulting dynamics are nontrivial, however, as the spin-helix initial condition corresponds to a highly nonequilibrium distribution of the fermions. We find a separation of timescales between the in-plane and out-of-plane spin dynamics. We gain insights from analyzing the case of a uniform spin chain and from a semiclassical model. One of our key findings is that the spin correlation functions decay as ${t}^{\ensuremath{-}1/2}$ at long time, in contrast to the experimentally observed exponential decay.

DOI
10.1103/physreva.106.043306

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Motivated by cold-atom experiments and a desire to understand far-from-equilibrium quantum transport, we analytically study the dynamics of spin helices in the one-dimensional XX\mathit{XX}XX model. We use a Jordan-Wigner transformation to map the spin chain onto a noninteracting Fermi gas with simple equations of motion. The resulting dynamics are nontrivial, however, as the spin-helix initial condition corresponds to a highly nonequilibrium distribution of the fermions. We find a separation of timescales between the in-plane and out-of-plane spin dynamics. We gain insights from analyzing the case of a uniform spin chain and from a semiclassical model. One of our key findings is that the spin correlation functions decay as t\ensuremath−1/2{t}^{\ensuremath{-}1/2}t\ensuremath−1/2 at long time, in contrast to the experimentally observed exponential decay.
Motivated by cold atom experiments, we analytically study the dynamics of spin helices in the 1D XX model. Using the Jordan-Wigner transformation to map spins into fermions, we calculate the evolution of the spin helix over time. The fermions are non-interacting, with simple equations of motion. Nonetheless, the calculation is non-trivial, as the spin-helix initial condition corresponds to a highly non-equilibrium distribution of the fermions. We compare the resulting dynamics to a semiclassical model, and find qualitatively similar out-of-plane dynamics. The in-plane dynamics, however, are dramatically different. The spins decohere over time, resulting in a t−1/2t^{-1/2}t−1/2 decay of the spin-spin correlation function, which can be contrasted to exponential decay seen in recent experiments.
In the study of quantum transport, much has been known about dynamics near thermal equilibrium. However, quantum transport far away from equilibrium is much less well understood---the linear response approximation does not hold for physics far out of equilibrium in general. In this work, motivated by recent cold atom experiments on probing quantum many-body dynamics of a one-dimensional XXZ spin chain, where a transition from ballistic to diffusive dynamics has been established by increasing the interaction strengths, we study the strong interaction limit of the one-dimensional spinless fermion model, which is dual to the XXZ spin chain. We develop a highly efficient computation algorithm for simulating the nonequilibrium dynamics of this system exactly, and examine the nonequilibrium dynamics starting from a density modulation quantum state. We find ballistic transport in this strongly correlated setting and show that a plane-wave description emerges at long-time evolution. We also observe a sharp distinction between transport velocities in short and long times as induced by interaction effects and provide a quantitative interpretation for the long-time transport velocity. We expect our results to shed light on the understanding of the dynamics of the XXZ spin chain in the strong interaction regime.
Motivated by recent cold-atom experiments, we study the relaxation of spin helices in quantum XXZ spin chains. The experimentally observed relaxation of spin helices follows scaling laws that are qualitatively different from linear-response transport. We construct a theory of the relaxation of such spin helices, combining generalized hydrodynamics with diffusive corrections and a generalized form of the local density approximation. Although helices are far from local equilibrium, our hydrodynamic approach reproduces the experimentally observed relaxational dynamics and also predicts the late-time relaxation, which is outside the experimentally accessible timescales. In particular, our theory explains the existence of temporal regimes with apparent anomalous diffusion, as well as the asymmetry between positive and negative anisotropy regimes at short and intermediate times.
Long-lived spin-helix states facilitate the study of non-equilibrium dynamics in quantum magnets. We consider the decay of transverse spin-helices in antiferromagnetic spin-SSS XXZ chains with single-ion anisostropy. The spin-helix decay is observable in the time evolution of the local magnetization that we calculate numerically for the system in the thermodynamic limit using infinite time-evolving block decimation simulations. Although the single-ion anisotropy prevents helix states from being eigenstates of the Hamiltonian, they still can be long-lived for appropriately chosen wave numbers. In case of an easy-axis exchange anisotropy the single-ion anisotropy may even stabilize the helices. Within a spin-wave approximation, we obtain a condition giving an estimate for the most stable wave number QQQ that agrees qualitatively with our numerical results.
In this Letter, we investigate spin dynamics of a two-component Bose gas with spin-orbit coupling realized in cold atom experiments. We derive coupled hydrodynamic equations for number and spin densities as well as their associated currents. Specializing to the quasi-one-dimensional situation, we obtain analytic solutions of the spin helix structure and its dynamics in both adiabatic and diabatic regimes. In the adiabatic regime, the transverse spin decays parabolically in the short-time limit and exponentially in the long-time limit, depending on initial polarization. In contrast, in the diabatic regime, transverse spin density and current oscillate in a way similar to the charge-current oscillation in an undamped LC circuit. The effects of Rabi coupling on the short-time spin dynamics is also discussed. Finally, using realistic experimental parameters for ^{87}Rb, we show that the timescales for spin dynamics is of the order of milliseconds to a few seconds and can be observed experimentally.
We describe the dynamics of a transverse spin-helix state---a product state with spatially rotating magnetization---under XXZ spin chain evolution. Due to experimental relevance we especially focus on magnetization dynamics. At long times the U(1)U(1)U(1) symmetry of the Hamiltonian is restored, leading to the decay of transverse magnetization, which can be described as an exponential decay of a spatially harmonic profile. We show that the dependence of the short- and intermediate-time decay timescale, which in principle depends on all different parameters, like the wave vector of the initial helix, the anisotropy, etc., can be described well by a single scaling function. We also briefly discuss the evolution of magnetization current.