Temperature and conditions for thermalization after canonical quenches

TL;DR

This work develops a systematic perturbative approach to quenches from canonical quantum equilibria, connecting post-quench temperature and observables to pre-quench equilibrium data via expansions in the quench strength $g$. It derives explicit formulas for the post-quench inverse temperature $\beta$ and for equilibrium expectation values, enabling predictions of late-time observables if thermalization occurs. A hierarchical set of necessary thermalization conditions is formulated in the presence of additional conserved quantities, with a geometric interpretation based on the canonical covariance that determines when quench operators couple to conserved charges. Numerical demonstrations in both integrable (Ising) and nonintegrable (XXZ) models illustrate how these conditions diagnose when a simple canonical ensemble suffices or when a generalized Gibbs ensemble is required, particularly highlighting higher-order effects that can reveal nonthermal behavior even when first-order criteria seem satisfied.

Abstract

We consider quenches of a quantum system that is prepared in a canonical equilibrium state of one Hamiltonian and then evolves unitarily in time under a different Hamiltonian. Technically, our main result is a systematic expansion of the pre- and post-quench canonical ensembles in the quench strength. We first demonstrate how this can be used to predict the system's temperature after the quench from equilibrium properties at the pre-quench temperature. For a thermalizing post-quench system, it furthermore allows us to calculate equilibrium observable expectation values. Finally, in the presence of additional conserved quantities besides the Hamiltonian, we obtain a hierarchy of necessary conditions for thermalization towards the (post-quench) canonical ensemble. At first order, these thermalization conditions have a nice geometric interpretation in operator space with the canonical covariance as a semi-inner product: The quench operator (difference between post- and pre-quench Hamiltonians) and the conserved quantity must be orthogonal in the orthogonal complement of the post-quench Hamiltonian. We illustrate the results numerically for a variety of setups involving integrable and nonintegrable models.

Figures (5)

• Figure 1: Geometric interpretation of the first-order thermalization condition \ref{['eq:CondTherm:1:Geometric']} [see also Eq. \ref{['eq:CondTherm:1']}]. (a) No special relationship between $H$, $Q$, and $V$; the condition is violated. (b) $V$ is orthogonal to $Q$ (with respect to $\operatorname{cov}(\cdot, \cdot)$), but the respective projections $V^\perp$ and $Q^\perp$ onto the orthogonal complement of $H$ are not orthogonal to each other; the condition is violated. (c) The projections $V^\perp$ and $Q^\perp$ are orthogonal (and $V$ and $Q$ may or may not be orthogonal to each other); the condition is satisfied.
• Figure 2: (a) Post-quench inverse temperature $\beta$ and (b) LIOM expectation values $\langle Q^+_n\rangle_{\!\rho}$ as a function of the pre-quench inverse temperature $\tilde{\beta}$ for canonical quenches in the transverse-field Ising model \ref{['eq:H:TFIM']}. Post-quench: $h = \frac{1}{2}$; pre-quench: $h = \frac{1}{2} + g$, $g = 0.25$ (left), $0.5$ (middle), $1$ (right). Solid lines are exact values; dotted, dashed, and dot-dashed lines correspond, respectively, to the zeroth-, first-, and second-order approximations in the quench strength $g$. In (b), gray-shaded lines (with different dashings to indicate different approximation orders, see above) correspond to the initial value $\langle Q^+_n \rangle_{\!\tilde{\rho}_{\tilde{\beta}}}$, while blue-shaded lines (with different dashings) correspond to the (thermal) expectation value $\langle Q^+_n \rangle_{\!\rho_\beta}$ of the post-quench ensemble, where $\beta = \beta(\tilde{\beta})$ is the respective post-quench temperature [as shown in panel (a)]. Note that the different line dashings may be hard to distinguish in the regimes where they overlap, indicating good agreement of approximations and exact values.
• Figure 3: (a) Post-quench inverse temperature $\beta$ and (b) LIOM expectation values $\langle Q^\pm_n\rangle_{\!\rho}$ as a function of the pre-quench inverse temperature $\tilde{\beta}$ for canonical quenches to the transverse-field Ising model $H = H_1$ from \ref{['eq:H:TFIM']} with $h = \frac{1}{2}$. Pre-quench: $\tilde{H} = H_1 - g J^z$ with $g = 0.25$ (left), $0.5$ (middle), $1$ (right). Line styles and colors as indicated in the legends and the same as in Fig. \ref{['fig:TFIM1']}.
• Figure 4: (a) Post-quench inverse temperature $\beta$ and (b) $z$-magnetization expectation values $\langle M^z\rangle_{\!\rho}$ as a function of the pre-quench inverse temperature $\tilde{\beta}$ for canonical quenches in the nonintegrable XXZ model. Post-quench Hamiltonian $H = H_2$ from \ref{['eq:H:XXZ+']}, quench operator $V = W$ from \ref{['eq:V:XXZ+:9S']}, and strengths $g = 0.125$ (left), $0.25$ (middle), $0.5$ (right). Line styles and colors as indicated in the legends and the same as in Fig. \ref{['fig:TFIM1']}.
• Figure 5: (a) Post-quench inverse temperature $\beta$ as a function of the pre-quench inverse temperature $\tilde{\beta}$ for canonical quenches in the nonintegrable XXZ model with post-quench Hamiltonian $H = H_2$ from \ref{['eq:H:XXZ+']}, quench operator $V = M^x$ from \ref{['eq:V:XXZ+:5']}, and quench strengths $g = 0.25$ (left), $0.5$ (middle), $1$ (right). (b) Same relationship between pre- and post-quench temperatures, but for the post-quench Hamiltonian $H = H_3$ from \ref{['eq:H:XXZ+:5Sa']}. (c) $z$-magnetization expectation values $\langle M^z \rangle_{\!\rho}$ for the initial ($\tilde{\rho}_{\!\tilde{\beta}}$) and thermal ($\rho_\beta$) states, and (d) their difference, for the scenario with post-quench Hamiltonian $H = H_3$. Line styles and colors as indicated in the legends and the same as in Fig. \ref{['fig:TFIM1']}.