Generalized Thermalization in an Integrable Lattice System

After a quench, observables in an integrable system may not relax to the standard thermal values, but can relax to the ones predicted by the generalized Gibbs ensemble (GGE) [M. Rigol et al., Phys. Rev. Lett. 98, 050405 (2007)]. The GGE has been shown to accurately describe observables in various one-dimensional integrable systems, but the origin of its success is not fully understood. Here we introduce a microcanonical version of the GGE and provide a justification of the GGE based on a generalized interpretation of the eigenstate thermalization hypothesis, which was previously introduced to explain thermalization of nonintegrable systems. We study relaxation after a quench of one-dimensional hard-core bosons in an optical lattice. Exact numerical calculations for up to 10 particles on 50 lattice sites ($\approx {10}^{10}$ eigenstates) validate our approach.

Figure 1

(a),(b) Momentum distribution of the initial state (init), diagonal (DE), generalized microcanonical (GME), generalized Gibbs (GGE), and the microcanonical (ME) ensembles. (c),(d) Relative difference of the GME, GGE and ME from the DE. (e),(f) Conserved quantities, $⟨{\widehat{I}}_n⟩$, in the quenched state (identical to the DE), GME and ME. $⟨I_n⟩$ are ordered in descending occupations in the quenched state. $L=50$, $N=10$, ${\delta }_{\mathrm{ME}}=0.05J$, ${\delta }_{\mathrm{GME}}=0.8$. (a),(c),(e) $\epsilon =0.72J$, $V_0=0.029J$. (b),(d),(f) $\epsilon =1.52J$, $V_0=0.125J$.

Figure 2

(a) $(\Delta n_k{)}_{\mathrm{ME}}$ versus energy per particle of the quenched state. ${\delta }_{\mathrm{ME}}=0.05J$. (b) $(\Delta n_k{)}_{\mathrm{GME}}$ vs $\epsilon$, ${\delta }_{\mathrm{GME}}=0.8$. (c) Integrated difference between the conserved quantities in the quenched state and the ME, $(\Delta I_n{)}_{\mathrm{ME}}$. (d) $(\Delta n_k{)}_{\mathrm{GGE}}$ vs $\epsilon$. Inset: $(\Delta n_k{)}_{\mathrm{GGE}}$ vs $L^{-0.73}$ for $\epsilon =1.07J$, where a fit to $(\Delta n_k{)}_{\mathrm{GGE}}=z{L}^{-\gamma }$ gives $\gamma =0.73\pm 0.02$.

Figure 3

Density plot of the coarse-grained weights with which eigenstates contribute to (a) the DE (sum over diagonal weights, $|c_{\alpha }{|}^2$) and (b) the GME (fractional number of states = number of states/total number of states) as a function of eigenstate energy and $⟨{\widehat{n}}_{k=0}{⟩}_{\alpha }$. The sums are performed over window of width $\delta n_k=0.0067$, and $\delta \epsilon =0.035J$. The horizontal and vertical dotted lines are the expectation values of ${\widehat{n}}_{k=0}$ and $\epsilon$ in each ensemble. $L=45$, $N=9$, $V_0=0.036J$, $\epsilon =0.72J$, ${\delta }_{\mathrm{GME}}=0.85$. Inset in (a) Histogram of DE weights (green), fractional number of GME states (blue) and fractional number of ME states (black) summed over all energies. Bin width, $\delta n_k=0.0067$. Vertical lines give the mean, $⟨{\widehat{n}}_{k=0}⟩$ within each ensemble. Inset in (b) Fluctuations of $⟨{\widehat{n}}_{ka=0}⟩$ (●) and $⟨{\widehat{n}}_{ka=2\pi /5}⟩$ (▲) within the DE (green) and GME (blue) as a function of inverse system size. $\epsilon =0.72J$.