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Physical review. B./Physical review. B

Magnetization dynamics in the XX spin chain model via multiparticle state

Saptorshi Ghosh, Manvendra Singh, Suhas Gangadharaiah

Published Oct 29, 2021• 1 citation
Publisher

Abstract

We consider a multiparticle fermionic state to study the magnetization dynamics in an XX spin chain model. The advantage this state provides is the freedom in choosing the initial magnetization profile at each spin site. We characterize the $t=0$ properties of this state, in particular, the probability for the state to have $n$ fermions, fidelity, the two-point and the four-point correlators, and spin-spin correlations. The dynamics of the domain-wall state formed by considering the ground states with magnetizations ${m}_{0}$ and $\ensuremath{-}{m}_{0}$ in the left and right half of the chain, respectively, results in the growth of a flat plateau $(m=0)$ away from the origin. In contrast, the magnetization dynamics of the domain wall state initialized via the multiparticle state reveals the absence of such a region. Our studies of the particle-number fluctuation for the multiparticle state with all sites having identical magnetization reveals linear dependence with respect to the system size, as opposed to the $lnl$ dependence which is obtained for the ground state with the same magnetization.

DOI
10.1103/physrevb.104.144306

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We consider a multiparticle fermionic state to study the magnetization dynamics in an XX spin chain model. The advantage this state provides is the freedom in choosing the initial magnetization profile at each spin site. We characterize the t=0t=0t=0 properties of this state, in particular, the probability for the state to have nnn fermions, fidelity, the two-point and the four-point correlators, and spin-spin correlations. The dynamics of the domain-wall state formed by considering the ground states with magnetizations m0{m}_{0}m0​ and \ensuremath−m0\ensuremath{-}{m}_{0}\ensuremath−m0​ in the left and right half of the chain, respectively, results in the growth of a flat plateau (m=0)(m=0)(m=0) away from the origin. In contrast, the magnetization dynamics of the domain wall state initialized via the multiparticle state reveals the absence of such a region. Our studies of the particle-number fluctuation for the multiparticle state with all sites having identical magnetization reveals linear dependence with respect to the system size, as opposed to the lnllnllnl dependence which is obtained for the ground state with the same magnetization.
We study the connection between magnetization transport and magnetization profiles in zero-temperature XX chains. The time evolution of the transverse magnetization m(x,t) is calculated using an inhomogeneous initial state that is the ground state at fixed magnetization but with m reversed from -m(0) for x<0 to m(0) for x>0. In the long-time limit, the magnetization evolves into a scaling form m(x,t)=Phi(x/t) and the profile develops a flat part (m=Phi=0) in the (x/t)<or=c(m(0)) region. The flat region shrinks to zero if m(0)-->1/2 while it expands with the maximum velocity c(0)=1 for m(0)-->0. The states emerging in the scaling limit are compared to those of a homogeneous system where the same magnetization current is driven by a bulk field, and we find that the expectation values of various quantities (energy, occupation number in the fermionic representation) agree in the two systems.
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
In spin chains with local unitary evolution preserving the magnetization Sz{S}^{\mathrm{z}}Sz, the domain-wall state $\left|\ensuremath{\cdots}\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\downarrow}\ensuremath{\downarrow}\ensuremath{\downarrow}\ensuremath{\downarrow}\ensuremath{\cdots}\right\ensuremath{\rangle}$ typically ``melts.'' At large times, a nontrivial magnetization profile develops in an expanding region around the initial position of the domain wall. For nonintegrable dynamics, the melting is diffusive, with entropy production within a melted region of size t\sqrt{t}t​. In contrast, when the evolution is integrable, ballistic transport dominates and results in a melted region growing linearly in time, with no extensive entropy production: The spin chain remains locally in states of zero entropy at any time. Here we show that, for the integrable spin-1/21/21/2 XXZ chain, low-energy quantum fluctuations in the melted region give rise to an emergent Luttinger liquid which, remarkably, differs from the equilibrium one. The striking feature of this emergent Luttinger liquid is its quasiparticle charge (or Luttinger parameter KKK), which acquires a fractal dependence on the XXZ chain anisotropy parameter $\mathrm{\ensuremath{\Delta}}$.
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 present the results obtained on the magnetization relaxation properties of an XX quantum chain in a transverse magnetic field. We first consider an initial thermal kink-like state where half of the chain is initially thermalized at a very high temperature T b while the remaining half, called the system, is put at a lower temperature T s . From this initial state, we derive analytically the Green function associated with the dynamical behaviour of the transverse magnetization. Depending on the strength of the magnetic field and on the temperature of the system, different regimes are obtained for the magnetic relaxation. In particular, with an initial droplet-like state, that is a cold subsystem of the finite size in contact at both ends with an infinite temperature environment, we derive analytically the behaviour of the time-dependent system magnetization.
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.