Integrability-Protected Adiabatic Reversibility in Quantum Spin Chains

We consider the out-of-equilibrium dynamics of the Heisenberg anisotropic quantum spin-$1/2$ chain threaded by a time-dependent magnetic flux. In the spirit of the recently developed generalized hydrodynamics (GHD), we exploit the integrability of the model for any flux values to derive an exact description of the dynamics in the limit of slowly varying flux: the state of the system is described at any time by a time-dependent generalized Gibbs ensemble. Two dynamical regimes emerge according to the value of the anisotropy $\mathrm{\Delta }$. For $|\mathrm{\Delta }|>1$, reversibility is preserved: the initial state is always recovered whenever the flux is brought back to zero. On the contrary, for $|\mathrm{\Delta }|<1$, instabilities of quasiparticles produce irreversible dynamics as confirmed by the dramatic growth of entanglement entropy. In this regime, the standard GHD description becomes incomplete and we complement it via a maximum entropy production principle. We test our predictions against numerical simulations finding excellent agreement.

Figure 1

Top: The Heisenberg spin-$1/2$ chain is threaded by a time dependent magnetic field inducing a magnetic flux density $\mathrm{\Phi }(t)$. The instantaneous Hamiltonian Eq. (1) describing the dynamics is always integrable and in the adiabatic limit, the state of the system is always locally described by a GGE ensemble. Bottom: Phase diagram of the $XXZ$ spin chain as a function of the anisotropy $\mathrm{\Delta }$ and magnetic field $B$. In the region $|\mathrm{\Delta }|\ge 1$, the model supports an infinite number of stable bound states, which preserve the reversibility of the dynamics. For $|\mathrm{\Delta }|<1$, the number of stable bound states strongly depends on $\mathrm{\Delta }$ and their momentum support does not cover the full Brillouin zone: this instability leads to irreversibility.

Figure 2

Expectation value of the instantaneous energy and spin current vs flux density. The GHD prediction is compared against exact diagonalization (ED) and time-dependent variational principle (TDVP) [108], the flux density being changed as $\mathrm{\Phi }(t)=2\pi t/T$. Choices of larger $T$ result in a more faithful hydrodynamic description. Panels (a) and (b), we choose $\mathrm{\Delta }=\cosh (1.5)>1⟨{\widehat{s}}_j^z⟩=0.1$ (ED with 25 sites, blue triangle in Fig. 1). In this case the evolution is perfectly $2\pi$ periodic [94]. In (c) and (d) we rather consider $\mathrm{\Delta }=0.5$ and $⟨{\widehat{s}}_j^z⟩=0.1$ (ED with 25 sites, blue square in Fig. 1), while in (e) and (f) $⟨{\widehat{s}}_j^z⟩=0.4$ (ED with 50 sites, blue circle in Fig. 1). While for $\mathrm{\Delta }>1$ we testified an excellent convergence even for relatively fast flux changes and small system sizes, significantly longer time scales and larger systems are needed for $\mathrm{\Delta }<1$. This can be understood looking at the spin current: the GHD for $\mathrm{\Delta }<1$ displays nonanalyticity points, which are located at those values of the flux density such that the Fermi sea hits the boundaries $\lambda =\pm \infty$. In order for the smooth time evolution to well approximate such a behavior, very long times are needed.

Figure 3

Entanglement entropy relative to the initial state vs flux density for $\mathrm{\Delta }=0.5$ and $⟨{\widehat{s}}_j^z⟩=0.1$. The infinitely large system is divided in two halves and different velocities of the flux density are considered $\mathrm{\Phi }(t)=2\pi t/T$. We interpret the plot as follows: the initial filling is a Fermi sea in the first string and a finite change in flux density is needed for translating it up to the boundaries $\lambda \pm \infty$. As long as the boundaries are not involved, no entropy is produced. As soon $\lambda =\pm \infty$ is reached, the thermodynamic entropy of the GGE starts to increase, accordingly with Eq. (7). GHD predicts an infinite entanglement entropy of the infinite half as soon as $\mathrm{\Phi }$ overcomes the critical value: consistently, the slopes in the plot increase with $T$.

Figure 4

The irreversibility of the GHD equations is displayed for the density of energy for $\mathrm{\Delta }=0.5$ and $⟨{\widehat{s}}_j^z⟩=0.4$. The flux density is changed from 0 to $2\pi$, then back.