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Eigenstate Thermalization Hypothesis in Conformal Field Theory Eigenstate thermalization hypothesis in conformal field theory The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Lashkari, Nima et al. “Eigenstate Thermalization Hypothesis in Conformal Field Theory.” Journal of Statistical Mechanics: Theory and Experiment 2018, 3 (March 2018): 033101 © 2018 IOP Publishing Ltd and SISSA Medialab srl As Published http://dx.doi.org/10.1088/1742-5468/aab020 Publisher IOP Publishing Version Original manuscript Citable link http://hdl.handle.net/1721.1/120293 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Eigenstate Thermalization Hypothesis in Conformal Field Theory Nima Lashkari,1 Anatoly Dymarsky,2 and Hong Liu1 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 2Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow 143026 Russia We investigate the eigenstate thermalization hypothesis (ETH) in d+1 dimensional conformal field theories by studying reduced density matrices in energy eigenstates. We show that if local probes of high energy 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. INTRODUCTION proving local ETH should amount to understanding how the corresponding statement regarding OPE coefficients Whether and how an isolated interacting quantum sys- connects with non-integrability. tem initially out of equilibrium equilibrates is a complex We believe that local ETH provides a powerful techni- dynamical question about which we currently have little cal handle for obtaining many dynamical properties of a analytic control. For a non-integrable system, there is CFT. As a simple application we use it to compute the nevertheless a powerful conjecture{the Eigenstate Ther- Renyi entropies for an interval in an energy eigenstate malization Hypothesis (ETH){which states that expecta- in a (1+1)-dimensional CFT. Interestingly we find that tion values of generic few-body observables in a finitely even though the reduced density matrix for the region excited energy eigenstate should coincide with those in approaches that of the canonical ensemble in the ther- the micro-canonical ensemble up to corrections that are modynamic limit, the Renyi entropies differ by a finite exponentially small in entropy [1, 2]. Assuming ETH, amount. one can then readily deduce that in the thermodynamic Our discussion should be generalizable to general quan- limit: (i) expectation values of few-body observables in tum field theories which we will pursue elsewhere. a generic state coincide with those in a thermal ensem- ble; (ii) expectation values of few-body observables in a SETUP non-equilibrium initial state will evolve towards those of a thermal ensemble. Consider finitely excited energy eigenstate E of a con- By now, there are many numerical evidences for ETH formal field theory in a d-dimensional spacej ofi finite vol- in a variety of quantum systems [3]. While ETH was ume. For simplicity, we take it to be a sphere of radius L often stated for few-body observables, recently numeri- (i.e. the total spacetime is Sd). A key simplification cal supports were also found for the full reduced density Rt of conformal symmetry is that× in Euclidean signature any matrix of any subsystem [4, 5]. state on Sd can be mapped through conformal trans- ETH is powerful as instead of following time evolu- formationj i tion of a general non-equilibrium state, we now face an in-principle simpler problem of understanding properties 2 2 2 2 2 2 2 2 dscylinder = dτ + L dΩd = Λ (dr + r dΩd) of finitely excited energy eigenstates. Unfortunately, de- L spite much numerical progress energy eigenstates of a Λ = ; τ = L log r (1) arXiv:1610.00302v1 [hep-th] 2 Oct 2016 r general interacting many-body system are still too com- plex to be tractable by analytic methods. A proof of to a local operator Ψ inserted at origin (r = 0) in Rd+1. ETH for general non-integrable systems appears out of Similarly, the conjugate state is mapped to a local h j d+1 reach at this stage. operator Ψy inserted at r = in R . Energy eigen- 1 In this paper we show that progress can be made in states are in one-to-one correspondence to the operators conformal field theories (CFT). We are able to prove that P P Ψ , where P is a momentum operator and µ1 ··· µm a µ for CFTs, a weakest form ETH which applies to local pri- Ψa is a primary operator. Below, we will use state and mary operators (to which we refer as local ETH) in fact operator languages interchangeably. leads to a strongest form of ETH which applies to the Kinematics of conformal symmetry fixes the corre- reduced density matrix of a general subsystem. Thus to lators in descendant states P P Ψ , in terms of µ1 ··· µm a prove ETH for CFTs it is enough to prove that it applies those of their corresponding primary Ψa. The dynam- to local primary operators which in turn reduces to a ical content of a CFT is the spectrum of its primaries statement regarding coefficients in operator product ex- and OPE coefficients. As a dynamical statement about pansions (OPE). Clearly not all CFTs satisfy ETH, thus energy eigenstates which are not related by symme- 2 tries ETH should be restricted to primary energy eigen- 2. Introducing σab TrBc Ea Eb , then states. Moreover, we will restrict our attention to ho- ≡ j ih j iα iα O(S(E)) mogeneous energy eigenstates on Sd which are invariant e σab + e− σba e− : (6) k k ∼ under SO(d + 1) rotations. Such an energy eigenstate for all α. with energy Ea corresponds to a spinless primary Ψa of dimension ha = Ea=L. The energy density of the sys- Note that subsystem ETH (5){(6) implies local ETH, Ea ha tem is a = d = d+1 , where !d is the volume of a L !d L d that is, for an operator supported inside region B we d unit S . For a CFT in a thermal state of temperature T , have O d+1 a CT where C is central charge, which motivates ∼ a a 1 a 2 1 us to define a \thermal" length scale associated with Ea Tr((ρ ρ (E) ) ρ ρ (E) 2 Tr((ρ +ρ (E)) ) 2 j i B− B O ≤ k B− B k B B O 1 (7) a − d+1 1 and thus is exponentially small. Similarly, Tr( σ ) = λT = T − : (2) ab O(S(E)) O C ∼ e− . See supplementary material for a proof 2 We will be interested in the thermodynamic limit with of (7). To close our setup, let us briefly comment on the de- L , while keeping a finite energy density a and thus a finite! 1λ . In this limit the scaling dimension h should scendant eigenstates. As an example consider E corre- T a 2 l j i then scale with L as sponding to a spinless operator (Pµ ) Ψa where l an inte- 2l ger. For such a state E = Ea;l = Ea + . The matrix ele- d+1 L L ment E p E is controlled by that in the primary state ha = C!d : (3) h jO j i λ Ψ , i.e. by O (E 2l ) rather than O (E). Thus as stated T a p L p earlier, we should− not include the descendant states in The local ETH condition is defined as either (4) or (5){(6). Of course if one assumes (4), for 2l p ha l, from smoothness of function Op(E), Op(E L ) Ea p Eb = Op(E)δab + Oab; (4) − h jO j i is related to Op(E) only by corrections of order l=ha and similarly the corresponding reduced density matrix where p is a local primary operator (with p labeling O is close to ρB(E) in trace distance with corrections that different operators), the diagonal element Op(E) is a Ea+Eb p O(S(E)) are polynomially suppressed in l=ha. But such state- smooth function of E = , and O e− 2 ab ∼ ments do not reflect any new dynamics beyond (4){(6). with eS(E) the density of state at energy E. Due to ho- For completeness, in supplementary material we discuss mogeneity, it does not matter where is inserted, so we Op more explicitly the story for descendant states. left the location of p implicit. In equation (4) p should be understood as aO smeared operator (f) =O f Op Sd Op with smearing function f such that the set of observables FROM LOCAL ETH TO SUBSYSTEM ETH R 2 p(f) form an algebra. In particular, the operator p shouldfO g be well-defined and has finite expectation valuesO Let us now look at the implications of (4). Using con- in physical states1. Below for notational simplicity we formal mapping (1) the matrix element E E on will continue to suppress f in (f). b p a p Sd is mapped to the Euclidean three-pointh functionjO j i We now introduce subsystemO ETH which states that for a subregion B [5] E E (8) h bjOpj ai h 1. There exists a \universal" density matrix ρB(E) L p Ψy( ) (1)Ψ (0) − b p a p hp (which depends only on B and energy E) such that = h 1 O i = CabL− ; Ψay ( )Ψ (0) Ψy( )Ψ (0) for any energy eigenstate Ea h 1 a ih b 1 b i j i q a O(S(E)) p ρ ρ (E = E ) e− (5) where C is the OPE coefficient for p appearing in jj B − B a jj ∼ ab O the operator product Ψ and Ψy, when Ψ and Ψ are a a b a b where ρB TrBc Ea Ea .
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