Abstract We investigate the eigenstate thermalization hypothesis (ETH) in d + 1 dimensional conformal field theories by studying the reduced density matrices in energy eigenstates. We show that if the local probes of the finitely excited primary eigenstates satisfy ETH, then any finite energy observable with support on a subsystem of finite size satisfies ETH. In two dimensions, we discover that if ETH holds locally, the finite size reduced density matrix of states created by heavy primary operators is well-approximated by a projection to the Virasoro identity block.
A bstract Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates. This can be done efficiently by representing zero modes of these operators in terms of the Virasoro algebra generators. In this paper we present a pedagogical introduction explaining how this calculation can be performed analytically or using computer algebra. We illustrate the computation of zero modes by a number of examples and list explicit expressions for all local operators from the vacuum family with the dimension of less or equal than eight. Finally, we derive an explicit expression for the quantum KdV generator Q 7 in terms of the Virasoro algebra generators. The obtained results can be used for quantitative studies of ETH at finite value of central charge.
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.
Abstract We study probe corrections to the Eigenstate Thermalization Hypothesis (ETH) in the context of 2D CFTs with large central charge and a sparse spectrum of low dimension operators. In particular, we focus on observables in the form of non-local composite operators
Oobs(x)=OL(x)OL(0) O o b s x = O L x O L 0 with h L ≪ c. As a light probe,
Oobs(x) O o b s x is constrained by ETH and satisfies
⟨Oobs(x)⟩hH≈⟨Oobs(x)⟩micro O o b s x h H ≈ O o b s x micro for a high energy energy eigenstate |h H 〉. In the CFTs of interests,
⟨Oobs(x)⟩hH O o b s x h H is related to a Heavy-Heavy-Light-Light (HL) correlator, and can be approximated by the vacuum Virasoro block, which we focus on computing. A sharp consequence of ETH for
Oobs(x) O o b s x is the so called “forbidden singularities”, arising from the emergent thermal periodicity in imaginary time. Using the monodromy method, we show that finite probe corrections of the form
O(hL/c) O h L / c drastically alter both sides of the ETH equality, replacing each thermal singularity with a pair of branch-cuts. Via the branch-cuts, the vacuum blocks are connected to infinitely many additional “saddles”. We discuss and verify how such violent modification in analytic structure leads to a natural guess for the blocks at finite c: a series of zeros that condense into branch cuts as c → ∞. We also discuss some interesting evidences connecting these to the Stoke’s phenomena, which are non-perturbative e −c effects. As a related aspect of these probe modifications, we also compute the Renyi-entropy S n in high energy eigenstates on a circle. For subsystems much larger than the thermal length, we obtain a WKB solution to the monodromy problem, and deduce from this the entanglement spectrum.
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.
Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.