Basis dependence of eigenstate thermalization

Abstract

Eigenstate thermalization refers to the property that an energy eigenstate of a many-body system is indistinguishable from a thermal equilibrium ensemble at the same energy as far as expectation values of local observables are concerned. In systems with degeneracies, the choice of an energy eigenbasis is not unique and the fraction of basis states exhibiting eigenstate thermalization can vary. We present a simple example where this fraction vanishes in the thermodynamic limit for one basis choice, but remains nonzero for another choice. In other words, the weak eigenstate thermalization hypothesis is satisfied in the first, but violated in the second basis. We furthermore prove that degeneracies must abound whenever a system is simultaneously symmetric under spatial translations and reflection. Finally, we derive general bounds on how strongly eigenstate thermalization may depend on the choice of the basis, and we reveal some interesting implications regarding the temporal relaxation properties of such systems.

Figures (4)

• Figure 1: ETH quantifier $\Delta_{A,\mathrm{eq}}^2$ from \ref{['18']} vs. $L^{-1}$ for the canonical ensemble \ref{['eq:rhoCan']}, Hamiltonian $H$ from \ref{['55']}, and observable $A = s_1^z$, for two basis choices: one where $A$ is diagonal in all energy eigenspaces (maximal ETH violation, orange) and a translationally invariant one (provably satisfying weak ETH, blue). Panels correspond to different values of the inverse temperature $\beta$. Symbols are numerical results obtained via dynamical typicality with error bars indicating the corresponding standard error (cf. Appendix \ref{['app:DynTyp']}; most error bars are actually not discernible on the scale of these plots); lines are linear fits and orange numbers the resulting extrapolation of $\Delta_{A,\mathrm{eq}}^2$ to $L \to \infty$ ($y$-axis intercept, including, in parentheses, the uncertainty in the last digit).
• Figure 2: (a): Numerically obtained time-dependent expectation values $\langle A\rangle_t - A_{\mathrm{eq}}$ for the observable $A = s_1^z$, the Hamiltonian \ref{['55']}, the initial condition (\ref{['57']}), and the canonical ensemble \ref{['42']} with parameter values $L=15$, $g=1$, and $\beta=0.2$, see also Eqs. (\ref{['17']}), (\ref{['19']}), and (\ref{['56']}). Main plot: Initial relaxation behavior for $t\in[0,5]$. Inset: Long-time behavior for $t\in[0,100]$. (b): Long-time average of the same expectation values as in (a) versus $\beta$. Orange crosses, blue crosses, and red squares correspond to $L=11$, $13$, and $15$, respectively. Black circles with error bars: Extrapolations for $L\to\infty$. For further details regarding the numerical methods and the extrapolations we refer to Fig. \ref{['fig:ETHVio']} and its discussion in the main text.
• Figure 3: (a) Same as the middle panel ($\beta = 0.2$) of Fig. \ref{['fig:ETHVio']}, but now for the perturbed, nondegenerate Hamiltonian $H'$ from Eq. \ref{["eq:H'"]}. (b) Same as Fig. \ref{['fig2']}(a), but now again for the perturbed system $H'$.
• Figure 4: Same as in Fig. \ref{['fig:ETHVio']} but now employing the maximally ETH-violating basis within various magnetization ($m$) and dipole-moment ($p$) subsectors (see also Appendix \ref{['app:DynTyp']}).