Complete Generalized Gibbs Ensembles in an Interacting Theory

In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady-state predictions. Here we solve this long-standing problem for the case of the spin-$1/2$ Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behavior after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasilocal charges. As a test, we reproduce the exact postquench steady state of the Néel quench problem obtained previously by means of the Quench Action method.

Figure 1

Comparison of methods: QA method versus improved GGE predictions. Colored lines pertain to the refined GGE calculation with the systematic addition of higher-spin families of quasilocal charges $\{H_s^{(n)}\}$ for the local correlation function $⟨{\sigma }_1^z{\sigma }_3^z⟩$ in the regime $\mathrm{\Delta }\gtrsim 1$ (left panel). Labels in ${\mathrm{GGE}}_{\overline{s}}$ indicate the maximal auxiliary spin $\overline{s}$ for the charges $\{H_s^{(n)}\}$ being included in the GGE computation. The right panel displays the relative differences ${\delta }_{⟨{\sigma }_1^z{\sigma }_3^z⟩}=(⟨{\sigma }_1^z{\sigma }_3^z{⟩}_{\mathrm{QA}}-⟨{\sigma }_1^z{\sigma }_3^z{⟩}_{{\mathrm{GGE}}_{\overline{s}}})/⟨{\sigma }_1^z{\sigma }_3^z{⟩}_{\mathrm{QA}}$ in logarithmic scale. We used the mapping between correlation functions and the set of densities $\mathit{\rho }$ given in [51].