Taking the temperature of quantum many-body scars

TL;DR

The paper addresses how thermalization emerges in closed quantum many-body systems and why quantum many-body scars (QMBS) evade the eigenstate thermalization hypothesis (ETH). It introduces the eigenstate subsystem temperature $\beta_S(|E\rangle)$, defined by minimizing the trace distance between the eigenstate's reduced density matrix and a corresponding reduced canonical state, and compares it to the canonical temperature $\beta_C(E)$. Using a projector-embedding construction (Shiraishi-Mori) with random two-spin terms from the Gaussian unitary ensemble, the authors generate QMBS and analyze both QMBS and nearby thermal eigenstates across many instances, quantifying correlations via $e_C$, $e_S$, the Pearson coefficient, and the distance $d_1$. They find that QMBS tend to have $\beta_S(|E\rangle)$ correlated with $\beta_C(E)$ despite a large $d_1$ distance to $\hat{\sigma}_S(\beta_C)$, whereas thermal states exhibit a tighter, size-dependent correlation and smaller $d_1$. This reveals that QMBS retain approximate information about their spectral position in the state structure, offering a nuanced view of thermalization and suggesting directions for exploring nonthermal states in broader contexts.

Abstract

A quantum many-body scar is an eigenstate of a chaotic many-body Hamiltonian that exhibits two seemingly incongruous properties: its energy eigenvalue corresponds to a high temperature, yet its entanglement structure resembles that of low-temperature eigenstates, such as ground states. Traditionally, a temperature is assigned to an energy \emph{eigenvalue} through the textbook canonical temperature-energy relationship. However, in this work, we use the \emph{eigenstate subsystem temperature} -- a recently developed quantity that assigns a temperature to an energy eigenstate, based on the structure of its reduced density matrix. For a thermal state, the eigenstate subsystem temperature is approximately equal to its canonical temperature. Given that quantum many-body scars have a ground-state-like entanglement structure, it is not immediately clear that their eigenstate subsystem temperature would be close to their canonical temperature. Surprisingly, we find that this is the case: the quantum many-body scars have approximate ``knowledge'' of their position in the spectrum encoded within their state structure.

Figures (6)

• Figure 1: (a) An example of a canonical temperature ($\beta_C)$ versus energy ($E$) curve (Eq. \ref{['eq_canonical_beta']}) for the 'projected' uniform-field XXZ spin-1/2 chain with $N=18$, within the $k=0$ momentum sector. (b) Illustration of a spin chain, divided into the subsystem $S$ and its complement $\bar{S}$.
• Figure 2: We generate the Hamiltonian $\hat{H}$ (Eq. \ref{['eq:H']}) of a chain of $N=16$ spin-1/2 particles, choosing the local two-spin Hermitian matrix $\hat{h}$ at random from the Gaussian unitary ensemble (GUE). (a) For a typical instance of the Hamiltonian $\hat{H}$, the entanglement entropy of each eigenstate is plotted against its corresponding eigenvalue. (b) Each data point $(\beta_S (|E_{\rm QMBS}\rangle)$, $\beta_C (E_{\rm QMBS}))$ is calculated for the QMBS $|E_{\rm QMBS}\rangle$ of a randomly generated Hamiltonian $\hat{H}$. (c) Each data point $(\beta_S (|E_{\rm thermal}\rangle), \beta_C (E_{\rm thermal}))$ is calculated for the representative thermal eigenstate $|E_{\rm thermal}\rangle$ of a randomly generated Hamiltonian $\hat{H}$. The upper insets show the histograms of $\delta\beta = \beta_S - \beta_C$. The dashed vertical lines show that the average $\langle \delta\beta \rangle \approx 0$ is close to zero in both cases. For thermal states, the distribution of $\delta\beta$ decays exponentially around $\delta\beta \approx 0$, while for QMBS it decays as a Gaussian (solid black lines). The lower-right insets show the statistical variance $\langle (\delta \beta)^2\rangle$ versus system size $N$.
• Figure 3: (a) $\delta\beta$ versus $N$, (b) $\min(d_1)$ versus $N$ -- Grey lines indicate individual random instances of the random local model, and the solid black line denotes the average over the $\sim300$ realizations. The dashed blue and dotted red lines correspond to a QMBS state and an average over highly excited thermal states of the uniform field XXZ chain respectively.
• Figure S1: The same data as in Fig. \ref{['fig:delta_beta_for_random_models']}(b,c), translated into the "fraction of the spectrum": $e_S$ replacing $\beta_S$ and $e_C$ replacing $\beta_C$. The marker color indicates the density of surrounding data points: yellow for more density of data points and blue for lower density.
• Figure S2: The Pearson correlation coefficient for the canonical temperature $\beta_C$ and the subsystem temperature $\beta_S$, as a function of system size $N$. The $N=16$ points here correspond to the data in Figs. \ref{['fig:delta_beta_for_random_models']}(b,c).