Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT

TL;DR

This work tests the subsystem weak ETH in a 2D CFT with large central charge by comparing a highly excited heavy primary state to a canonical-ensemble thermal state, focusing on single-interval entanglement and Rényi entropies in the short-interval limit. It employs two replica-based methods to compute the excited-state Rényi entropy and uncovers a nontrivial $1/c$ correction at order $(\ell/L)^8$ driven by the level-4 vacuum quasiprimary $\mathcal{A}$, signaling subleading structure in the large-$c$ expansion. By relating entropy differences to the relative entropy and applying Fannes–Audenaert and Pinsker inequalities, the authors derive bounds on the trace distance and show that subsystem weak ETH is compatible with power-law suppression only if the effective dimension scales appropriately (potentially as $d\sim e^{\mathcal{O}(c)}$), though the exact exponent remains undetermined. An additional byproduct is the explicit computation of the relative entropy between reduced density matrices of two different heavy primary states, illustrating rich $c$-dependence beyond leading order.

Abstract

We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval Rényi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of Rényi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central charge expansion. We then evaluate the relative entropy of the reduced density matrices to measure the difference between the heavy primary state and thermal state of canonical ensemble, and find that the aforementioned nontrivial eighth order results make the relative entropy unsuppressed in the large central charge limit. By using Pinsker's and Fannes-Audenaert inequalities, we can exploit the results of relative entropy to yield the lower and upper bounds on trace distance of the excited-state and thermal-state reduced density matrices. Our results are consistent with subsystem weak ETH, which requires the above trace distance is of power-law suppression by the large central charge. However, we are unable to pin down the exponent of power-law suppression. As a byproduct we also calculate the relative entropy to measure the difference between the reduced density matrices of two different heavy primary states.

Figures (4)

• Figure 1: Relative entropy (\ref{['j45']}) as a function of $\ell/L$ in comparing the reduced density matrices for chiral primary state and thermal state of 2D massless scalar. Note that the higher the temperature is, the larger the relative entropy becomes.
• Figure 2: This figure illustrates how OPE of twist operators on a cylinder and that on a complex plane are related. The one-fold CFT on a one-fold CFT on an $n$-fold cylinder is equivalent to an $n$-fold CFT on a one-fold cylinder. The boundary conditions of the $n$-fold CFT can be replaced by the insertion of a pair of twist operators Calabrese:2004eu. The cylinder with twist operators can be mapped to a complex plane with twist operators.
• Figure 3: This figure illustrates the replica method of multi-point function on complex plane Alcaraz:2011tnBerganza:2011mhLashkari:2014yvaLashkari:2015diaSarosi:2016oksSarosi:2016atxRuggiero:2016khg. Firstly, one has the one-fold CFT on an $n$-fold cylinder in excited state $|\Phi\rangle$. Then, by state/operator correspondence one gets a two-point function on an $n$-fold complex plane $\mathcal{C}^n$. Lastly, by a conformal transformation one gets a $2n$-point function on a one-fold complex plane. In the last part of the figure we use $n=5$ as an example.
• Figure 4: The calculation of $\textrm{tr}_A (\rho_{A,\phi}\rho_{A,\psi}^{n-1})$. Here $\Psi_\phi\equiv\phi_0\prod_{j=1}^{n-1}\psi_j$, with $\phi$ existing in one copy and $\psi$ existing in the other $n-1$ copies.