Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions

We study quenches in integrable spin-$1/2$ chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench-action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench-action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit.

Figure 1

(a),(b) Density functions ${\mathit{\rho }}_1$ and ${\mathit{\rho }}_2$ for the quench to different values of $\mathrm{\Delta }>1$ of both the quench-action saddle-point state (solid lines) and the GGE equilibrium state (dashed lines). (c) Difference between the GGE prediction for ${\mathit{\rho }}_1$ and the quench-action saddle-point result. All distributions are symmetric functions of $\lambda \in [-\pi /2,\pi /2)$.

Figure 2

Correlator $⟨{\sigma }_1^z{\sigma }_2^z⟩$ evaluated on the quench-action steady state (solid lines) and on the GGE (dashed lines). The energy sum rule $2⟨{\sigma }_1^x{\sigma }_2^x⟩+\mathrm{\Delta }⟨{\sigma }_1^z{\sigma }_2^z⟩=-\mathrm{\Delta }$ explains the exact value of $-1/3$ at the isotropic point $\mathrm{\Delta }=1$. Numerical errors are ${10}^{-5}$ or smaller. Both sets of data are in agreement with the finite-size computations of Ref. [19], within the numerical precision of the latter. Inset: Relative difference between the GGE prediction and the quench-action saddle-point result, $\delta ⟨{\sigma }_1^z{\sigma }_2^z⟩=(⟨{\sigma }_1^z{\sigma }_2^z{⟩}_{\mathrm{GGE}}-⟨{\sigma }_1^z{\sigma }_2^z{⟩}_{\mathrm{sp}})/|⟨{\sigma }_1^z{\sigma }_2^z{⟩}_{\mathrm{sp}}|$.

Figure 3

(a) The same as Fig. 2 for $⟨{\sigma }_1^z{\sigma }_3^z⟩$. (b) Comparison between the quench action, the GGE prediction, and the NLCE result close to the isotropic point. Error bars in the NLCE data display an interval of confidence that includes all resummation results (except for $\mathrm{\Delta }=1.015$) (Supplemental Material [58]).