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arXiv:cond-mat/0308393 · Physical review. B, Condensed matter

Domain wall dynamics of the Ising chain in a transverse field

V. Subrahmanyam

Published Dec 19, 2003• 4 citations
arXivHTMLPDF

Abstract

We show that the dynamics of an Ising spin chain in a transverse field conserves the number of domains (strings of down-spins in an up-spin background) at discrete times. This enables the determination of the eigenfunctions of the time-evolution operator, and the dynamics of initial states with domains. The transverse magnetization is shown to be identically zero in all sectors with a fixed number of domains. For an initial state with a single string of down-spins, the local magnetization, the equal time and double-time spin-spin correlation functions, are calculated analytically as functions of time and the initial string size. The domain size distribution function can be expressed as a simple integral involving Bessel functions.

DOI
10.1103/physrevb.68.212407

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We show that the dynamics of an Ising spin chain in a transverse field conserves the number of domains (strings of down-spins in an up-spin background) at discrete times. This enables the determination of the eigenfunctions of the time-evolution operator, and the dynamics of initial states with domains. The transverse magnetization is shown to be identically zero in all sectors with a fixed number of domains. For an initial state with a single string of down-spins, the local magnetization, the equal time and double-time spin-spin correlation functions, are calculated analytically as functions of time and the initial string size. The domain size distribution function can be expressed as a simple integral involving Bessel functions.
We investigate analytically and numerically the dynamics of domain walls in a spin chain with ferromagnetic Ising interaction and subject to an external magnetic field perpendicular to the easy magnetization axis (transverse field Ising model). The analytical results obtained within the continuum approximation and numerical simulations performed for the discrete classical model are used to analyze the quantum properties of domain walls using the semiclassical approximation. We show that the domain wall spectrum shows a band structure consisting of 2S2S2S nonintersecting zones.
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.
We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions with which we can obtain numerically exact results for system sizes L as large as 256. The probability distribution of the local imaginary time correlation function $S(\ensuremath{\tau})$ is investigated and found to be simply a function of $\ensuremath{\alpha}\ensuremath{\equiv}\ensuremath{-}\mathrm{ln}S(\ensuremath{\tau})/\mathrm{ln}\ensuremath{\tau}.$ This scaling behavior implies that the typical correlation function decays algebraically, ${S}_{\mathrm{typ}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{typ}}},$ where the exponent ${\ensuremath{\alpha}}_{\mathrm{typ}}$ is determined from $P(\ensuremath{\alpha}),$ the distribution of $\ensuremath{\alpha}.$ The precise value for ${\ensuremath{\alpha}}_{\mathrm{typ}}$ depends on exactly how the ``typical'' correlation function is defined. The form of $P(\ensuremath{\alpha})$ for small $\ensuremath{\alpha}$ gives a contribution to the average correlation function ${S}_{\mathrm{av}}(\ensuremath{\tau})$ namely, ${S}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}(\mathrm{ln}\ensuremath{\tau}{)}^{\ensuremath{-}{2x}_{m}},$ where xm{x}_{m}xm​ is the bulk magnetization exponent, which was reported recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.
We study the time dependent behavior of the four-spin interactions Ising chain in the presence of a disordered transverse magnetic field. In order to analyze the effects of field randomness in the dynamical behavior of the system we calculate the time-dependent correlation function and the longitudinal relaxation function of the infinite chain. We consider a bi-modal distribution for the field energy strength, where one of its values equals the exchange coupling energy while the other value can be lower or higher than the coupling energy. We find that an increasing concentration of weaker fields produces an enhancement of the central mode behavior. However, when the concentration of stronger fields increases, the system undergoes a crossover from central mode to collective-type dynamics.
We study the dynamic response of magnetic domain walls in low-lying excited states of an Ising chain to an oscillating transverse field. Based on the exact instantaneous eigenstates, we find that when the frequency of the external field is in off-resonant regions, the domain wall exhibits Bloch oscillation, maintaining the magnetic order. However, the magnetic order is destroyed when the field is at resonant frequency. Numerical simulations of the dynamics of magnetization and entanglement entropy for initial states with single and double domain walls accord with the predictions. These findings reveal the nontrivial effect of a monochromatic electromagnetic field on quantum spin dynamics.
Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as log(t)\mathrm{log}(t)log(t) in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.