Entropy of Isolated Quantum Systems after a Quench

A diagonal entropy, which depends only on the diagonal elements of the system’s density matrix in the energy representation, has been recently introduced as the proper definition of thermodynamic entropy in out-of-equilibrium quantum systems. We study this quantity after an interaction quench in lattice hard-core bosons and spinless fermions, and after a local chemical potential quench in a system of hard-core bosons in a superlattice potential. The former systems have a chaotic regime, where the diagonal entropy becomes equivalent to the equilibrium microcanonical entropy, coinciding with the onset of thermalization. The latter system is integrable. We show that its diagonal entropy is additive and different from the entropy of a generalized Gibbs ensemble, which has been introduced to account for the effects of conserved quantities at integrability.

Figure 1

Entropies vs $t^{\prime }=V^{\prime }$. Left: bosons; right: fermions; top: quench from $t_{\mathrm{ini}}=0.5$, $V_{\mathrm{ini}}=2.0$; bottom: quench from $t_{\mathrm{ini}}=2.0$, $V_{\mathrm{ini}}=0.5$. Filled symbols: $d$ entropy (1); empty symbols: $S_s$ (2); ○ $T=1.5$; □ $T=2.0$; △ $T=3.0$. All panels: $1/3$ filling and $L=24$; insets of panels (a) and (c) show $S_d/S_s$ for $L=24$, thick (red) line, and $L=21$, thin (black) line for $T=3.0$. Solid lines in the insets of panels (b) and (d), from bottom to top: microcanonical entropy; canonical entropy $S_c$ for eigenstates with $k=0$ and the same parity as the initial state; $S_c$ for eigenstates with $k=0$ and both parities; and $S_c$ for all eigenstates with $N=8$.

Figure 2

Normalized distribution function of energy. Bosonic system, $L=24$, $T=3.0$ and the values of $t^{\prime }=V^{\prime }$ are indicated. Top panels: quench from $t_{\mathrm{ini}}=0.5$, $V_{\mathrm{ini}}=2.0$; bottom panels: quench from $t_{\mathrm{ini}}=2.0$, $V_{\mathrm{ini}}=0.5$. Solid smooth line: best Gaussian fit $(\sqrt{2\pi }a{)}^{-1}{e}^{-(E-b{)}^2/(2a^2)}$ for parameters $a$ and $b$; dashed line: $(\sqrt{2\pi }\delta E{)}^{-1}{e}^{-(E-E_{\mathrm{ini}}{)}^2/(2\delta E^2)}$.

Figure 3

Entropy vs system size. Panel (a): from top to bottom, quench to $A_{\mathrm{fin}}=4$, 8, 12, 16; panel (b): quench from $A_{\mathrm{ini}}=4$, 8, 12, 16, curves closely superpose. Lower panels: the quench type is indicated as (initial $A$)-(final $A$). Symbols: $S_d$; dashed lines: GCE entropy (the closest to the $d$ entropy in all cases studied); dotted lines: GGE entropy; dashed double-dotted line: canonical entropy; and dash-dotted line: microcanonical entropy.