Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

TL;DR

This work develops a precise, twist-operator–based framework to quantify how excited states in a 2D CFT with large central charge differ from thermal (canonical, GGE, and microcanonical) states over a small region. By computing short-interval expansions up to $O(\ell^9)$ for vacuum-conformal-family contributions and expressing Renyi/entanglement entropies, relative entropy, Jensen-Shannon divergence, and Schatten norms, the authors illuminate leading ETH behavior and subleading violations, and they explore conditions under which GGE may restore equivalence between reduced density matrices. The results establish explicit, high-order relationships among geometric backgrounds, operator content, and dissimilarity measures, highlighting the role of vacuum charges and the potential for a precise, geometry-dependent formulation of subsystem ETH in 2D CFTs. The work provides a rigorous analytical toolkit for ETH analysis in conformal systems and suggests directions for extending to nonvacuum sectors and more general ensembles.

Abstract

We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length $\ell$ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the Rényi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of $\ell^9$ for the contributions from the vacuum conformal family. The formal forms of these dissimilarity measures and the derived Fisher information metric from contributions of general operators are also given. As an application of the results, we use these dissimilarity measures to compare the excited and thermal states, and examine the eigenstate thermalization hypothesis (ETH) by showing how they behave in high temperature limit. This would help to understand how ETH in 2D CFT can be defined more precisely. We discuss the possibility that all the dissimilarity measures considered here vanish when comparing the reduced density matrices of an excited state and a generalized Gibbs ensemble thermal state. We also discuss ETH for a microcanonical ensemble thermal state in a 2D large central charge CFT, and find that it is approximately satisfied for a small subsystem and violated for a large subsystem.

Figures (5)

• Figure 1: The Riemann surfaces as environments for the interval $A=[0,\ell]$ we consider in this paper. (a) A complex plane $\mathcal{R}(\emptyset)$. (b) A vertical cylinder $\mathcal{R}(L)$. (c) A vertical cylinder capped with operators $\mathcal{R}(L,\phi)$. (d) A horizontal cylinder $\mathcal{R}(\beta)$. (e) A horizontal cylinder capped with operators $\mathcal{R}(\beta,\phi)$. (f) A fat torus $\mathcal{R}(L,q=\mathrm{e}^{-2\pi\beta/L})$. (g) A thin torus $\mathcal{R}(\beta,p=\mathrm{e}^{-2\pi L/\beta})$.
• Figure 2: The seven Rényi entropies we can calculate using OPE of the twist operators. In practice, we only need to calculate $S_n(L,\phi)$ and $S_n(L,\phi)$, as marked in blue, and the other cases can be obtained easily from them.
• Figure 3: The 48 relative entropies we can calculate using OPE of the twist operators. By $\rho_A \rightarrow \sigma_A$ we mean the relative entropy $S(\rho_A\|\sigma_A)$, and by $\rho_A \leftrightarrow \sigma_A$ we mean the relative entropies $S(\rho_A\|\sigma_A)$ and $S(\sigma_A\|\rho_A)$. Note that $q=\mathrm{e}^{-2\pi\beta/L}$ and $p=\mathrm{e}^{-2\pi L/\beta}$ depend on both $L$ and $\beta$. In the figure $\cdots L \cdots \rightarrow \cdots L \cdots$ actually means $\cdots L_1 \cdots \rightarrow \cdots L_2 \cdots$ with generally $L_1 \neq L_2$, $\cdots \phi \cdots \rightarrow \cdots \phi \cdots$ means $\cdots \phi_1 \cdots \rightarrow \cdots \phi_2 \cdots$, and $\cdots \beta \cdots \rightarrow \cdots \beta \cdots$ means $\cdots \beta_1 \cdots \rightarrow \cdots \beta_2 \cdots$. In practice, we only need to calculate the four relative entropies as marked in blue.
• Figure 4: The 27 symmetrized relative entropies we can calculate using OPE of the twist operators. We only need to calculate the three ones marked in blue. This figure also applies to the 2nd symmetrized relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm in the following subsections.
• Figure 5: The 20 relative entropies we can calculate using modular Hamiltonian and entanglement entropy. We only need to calculate the two relative entropies $S(\rho_A(L_1,\phi)\|\rho_A(L_2))$ and $S(\rho_A(L_1,q)\|\rho_A(L_2))$ as marked in blue.