Integrability-breaking-induced Mpemba effect in spin chains

Abstract

We show that there are two distinct mechanisms that can cause the symmetry-restoration Mpemba effect in spin chains with \textit{weakly broken} integrability, such that the asymptotic equilibration is diffusive, but the lifetime of anomalously fast spin hydrodynamics at low temperature is parametrically large. In particular, we consider isotropic spin chains quenched out of equilibrium by suppressing the $z$-components, without inducing any net magnetisation. Initially, the restoration of isotropy is faster in hotter systems -- because they have more phase space available to scramble their initial conditions -- which may cause the equilibration curves to cross at early times in both integrable and non-integrable systems. At later times, however, the equilibration is effectively hydrodynamic, and the \textit{colder} systems start to equilibrate faster as the lifetime over which they evince superdiffusive spin hydrodynamics is parametrically larger -- but only in \textit{non}-integrable models. Depending on the details of the temperatures and the extent of the initial symmetry-breaking, two isotropy-restoration curves may have a crossing at early time, late time, neither, or both.

Figures (5)

• Figure 1: An example of each of the four possible scenarios for the equilibration curves, all for the classical Heisenberg chain \ref{['eq:H_Heisenberg']}. Dashed lines are fits to the hydrodynamic form \ref{['eq:hydro_eq']} for $10^3 < t < 10^4$. From left to right: (I) No crossings -- the hotter chain starts closer to isotropy so there is no early-time crossing, and the colder chain crosses over to diffusive equilibration before the curves cross. (II) Early-time crossing only -- the hotter system starts further from isotropy but initially equilibrates faster; but there is no late time crossing because both chains cross over to diffusive equilibration relatively quickly. (III) Early- and late-time crossings -- the hotter chain starts further from isotropy and quickly crosses the colder equilibration curve; but the colder system equilibrates superdiffusively over a long enough timescale that there is a second crossing at late time. (IV) Late-time crossing only -- this is a clean demonstration of the integrability-breaking-induced Mpemba effect. The colder system starts further from isotropy and the early-time symmetry-restoration Mpemba effect does not occur; but the difference in the timescale associated to the superdiffusive equilibration is large enough that the curves cross at late time. In (I), (II), and (III) we use a system size of $L = 8192$ and ensembles of $10^5$ states; in (IV), we use $L = 32\,768$, and ensembles of $30\,000$ states.
• Figure 2: Early-time equilibration coefficients and crossings of the equilibration curves. (a) The initial equilibration rate $\Gamma(T, \eta)$ in the classical Heisenberg chain (cf. Eq. \ref{['eq:init_eq_rate']}), showing that $\Gamma$ is not purely a function of $\eta$, and thus a system with an initially higher degree of symmetry-breaking may equilibrate faster than a system with lower initial symmetry-breaking. (b), (c), and (d) show example crossings of the equilibration curves for the Heisenberg chain, $\gamma = 0.5$ chain (cf. Eq. \ref{['eq:H_delta']}), and Ishimori chain, respectively. In some cases, the oscillations can even cause multiple crossings, as seen in (d).
• Figure 3: Dependence of the crossing times on the degree of integrability-breaking $\delta$, in the case where there are both early- and late-time crossings. The early-time crossing depends only very weakly on $\delta$, though there are pronounced oscillations of $\delta K_z(t)$ as the integrable point $\delta = 0$ is approached. The late-time crossing depends very strongly on $\delta$, occurring (for these temperatures and quench parameters) around (a) $t^* \approx 10^2$ for $\delta = 1$ (the Heisenberg chain); (b) $t^*\approx 10^3$ for $\delta = 2/3$; and (c) $t^* \approx 10^4$ for $\delta = 1/3$. There is no late-time crossing for (d) $\delta = 0$ (the Ishimori chain), as integrability means the hydrodynamic equilibration is perfectly superdiffusive at all temperatures even as $t \to \infty$. In all cases here we use a system size of $L = 8192$ and average over $10^5$ states.
• Figure 4: The integrability-breaking-induced Mpemba effect. (a) For these temperatures and quench parameters the crossing has disappeared at $\delta = 1$ (the Heisenberg chain), but is present for (b) $\delta = 2/3$ and (c) $\delta = 1/3$, and the Mpemba effect is observed in these cases. (d) The crossing is forbidden in the integrable case $\delta = 0$ (the Ishimori chain). The curves are extrapolated by fitting the hydrodynamic form Eq. \ref{['eq:hydro_eq']} between $t = 3000$ and $t = 10\,000$; the validity of this extrapolation can be inferred from the agreement between the curves for $100 < t < 3000$, where they are not fitted, implying we have indeed entered the hydrodynamic equilibration regime after $t \approx 100$. In all cases here we use a system size of $L = 8192$ and average over $10^5$ states.
• Figure S1: Evolution of the full one-spin probability distribution function $\rho_t(S^z)$ of the $z$-components, evolving from the initial out-of-equilibrium uniform distribution over the interval $[-\eta, \eta]$ towards, asymptotically, the equilibrium uniform distribution over the interval $[-1, 1]$. At all times, we observe, $\rho_t(z)$ is a non-increasing function of $|z|$; thus, at all times, the variance $\left\langle z^2 \right\rangle < 1/3$ is less than the equilibrium value. Indeed, all the even moments $\left\langle z^{2n} \right\rangle < 1/(2n+1)$ are below their equilibrium values. It follows that $\left\langle z^2 \right\rangle$ only attains its equilibrium value of $1/3$ if $\rho_t(z)$ attains the equilibrium distribution, which justifies the use of $\delta K_z(t)$ as a measure of anisotropy. In this figure, $T = 1.0$, $\eta = 0.4$, the system size is $L = 8192$, and we use an ensemble of $10^4$ states. The dotted line shows the equilibrium distribution.