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.
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.
We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length ℓ 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 ℓ9 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.
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 R\\'enyi 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 ell9 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.
Abstract We investigate the Rényi entropy and entanglement entropy of an interval with an arbitrary length in the canonical ensemble, microcanonical ensemble and primary excited states at large energy density in the thermodynamic limit of a two-dimensional large central charge c conformal field theory. As a generalization of the recent work [17], the main purpose of the paper is to see whether one can distinguish these various large energy density states by the Rényi entropies of an interval at different size scales, namely, short, medium and long. Collecting earlier results and performing new calculations in order to compare with and fill gaps in the literature, we give a more complete and detailed analysis of the problem. Especially, we find some corrections to the recent results for the holographic Rényi entropy of a medium size interval, which enlarge the validity region of the results. Based on the Rényi entropies of the three interval scales, we find that Rényi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can do the job for both medium and long intervals. At the leading order of large c the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for all interval lengths, but the difference of entanglement entropy for a long interval between the two states would appear with 1/c corrections. We also discuss Rényi entropy and entanglement entropy differences between the thermal states and primary excited state. Overall, our work provide an up-to-date picture of distinguishing different thermal or primary states at various length scales of the subsystem.
Abstract We investigate the eigenstate thermalization hypothesis (ETH) of highly excited descendant states in two-dimensional large central charge c conformal field theory. We use operator product expansion of twist operators to calculate the short interval expansions of entanglement entropy and relative entropy for an interval of length ℓ up to order ℓ 12. Using these results to ensure ETH of a heavy state when compared with the canonical ensemble state up to various orders of c, we get the constraints on the expectation values of the first few quasiprimary operators in the vacuum conformal family at the corresponding order of c. Similarly, we also obtain the constraints from the expectation values of the first few Korteweg-de Vries charges. We check these constraints for some types of special descendant excited states. Among the descendant states we consider, we find that at most only the leading order ones of the ETH constraints can be satisfied for the descendant states that are slightly excited on top of a heavy primary state. Otherwise, the ETH constraints are violated for the descendant states that are heavily excited on top of a primary state.
We study subsystem entropies in 2d CFTs for subsystems constituting a finite fraction of the full system. We focus on the extensive contribution, which scales linearly with the subsystem size in the thermodynamic limit. We employ the so-called diagonal approximation to evaluate subsystem entropy for chaotic CFTs in the thermal state (canonical ensemble), the microcanonical ensemble, and in a primary state, matching previously known results. We then proceed to find analytic expressions for the subsystem entropy at leading order in c , when the global CFT state is the KdV-generalized Gibbs ensemble or the KdV-microcanonical ensemble. Previous studies of primary eigenstates have shown that, akin to the fixed-area states in AdS/CFT, the corresponding subsystem entanglement spectrum is flat. This behavior is seemingly in sharp contradiction with that of the thermal (microcanonical) state, and thus in apparent contradiction with the subsystem eigenstate thermalization hypothesis (ETH). In this paper, we resolve this issue by comparing the primary state with the KdV-(micro)canonical ensemble. We show that the results are consistent with the KdV-generalized version of the subsystem ETH, in which local properties of quantum eigenstates are governed by their values of conserved KdV charges. Our paper solidifies evidence for the KdV-generalized ETH in 2d CFTs and emphasizes Rényi entropy as a sensitive probe of the reduced-density matrix.
The eigenstate thermalization hypothesis (ETH) provides a way to understand how an isolated quantum mechanical system can be approximated by a thermal density matrix. We find a class of operators in (1+1)-dimensional conformal field theories, consisting of quasiprimaries of the identity module, which satisfy the hypothesis only at the leading order in large central charge. In the context of subsystem ETH, this plays a role in the deviation of the reduced density matrices, corresponding to a finite energy density eigenstate from its hypothesized thermal approximation. The universal deviation in terms of the square of the trace-square distance goes as the eighth power of the subsystem fraction and is suppressed by powers of inverse central charge (c). Furthermore, the nonuniversal deviations from subsystem ETH are found to be proportional to the heavy-light-heavy structure constants which are typically exponentially suppressed in sqrt[h/c], where h is the conformal scaling dimension of the finite energy density state. We also examine the effects of the leading finite-size corrections.