Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit

Dymarsky, Anatoly; Pavlenko, Kirill

doi:10.1007/jhep01(2019)098

Cited by 38 publications

(72 citation statements)

References 65 publications

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“…This is only possible if the integral in (18) vanishes, which requires γ to be infinite. This is consistent with the observation of [30] that an ensemble with any finite number of non-zero µ 2k−1 can not describe primary states. This is because in full generality q 2k−1 ≥ q k 1 and hence primary states are at the boundary of the phase space of q 2k−1 's.…”

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confidence: 92%

“…Here and below the CFT central charge c is assumed to be large. Holographically, this regime corresponds to a quasi-classical black hole in AdS 3 , where one in the RHS of (5) corresponds to classical gravity, while O(1/c) term is due to quantum corrections [30][31][32][33]. In terms of ∆, n k an exponential majority of states in the generalized microcanonical ensemble specified by q 2k−1 subject to (5) will satisfy…”

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confidence: 99%

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Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories

Dymarsky

Pavlenko

2019

Phys. Rev. Lett.

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Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2d conformal field theories by giving rise to an infinite tower of local conserved qKdV charges in involution. We discuss how presence of conserved charges constraints equilibration in 2d CFTs. We propose that in the thermodynamic limit large central charge 2d CFTs satisfy generalized eigenstate thermalization, with the values of qKdV charges forming a complete set of thermodynamically relevant quantities, which unambiguously determine expectation values of all local observables from the vacuum family. Equivalence of ensembles further provides that local properties of an eigenstate can be described by the Generalized Gibbs Ensemble that only includes qKdV charges. In the case of a general initial state, upon equilibration, emerging Generalized Gibbs Ensemble will necessary include negative chemical potentials and holographically will be described by a quasi-classical black hole with quantum soft hair.The topic of thermalization, and more generally, equilibration of isolated many-body quantum systems has been an active area of research during the past decade. In case of non-integrable systems, i.e. those without an extensive number of local conserved quantities, emergence of the thermal equilibrium has been traced to eigenstate thermalization hypothesis (ETH) which postulates thermal properties of individual energy eigenstates [1-3]. In the simplest form it requires the expectation value of some appropriate (often taken to be local) observable O in a many-body eigenstate |E i to be a smooth function of energy,(1)Qualitatively, eq.(1) postulates that energy is the only thermodynamically relevant quantity, which completely specifies local properties of an eigenstate. The condition (1) may apply to all or most eigenstates, in which case it is referred as strong or weak ETH. The eigenstate thermalization ensures equivalence between the expectation value in the eigen-ensemble, f O (E i ), and thermal expectation value of O in the Gibbs ensemble, f O (E i ) = Tr(e −βH O)/Z, where the effective temperature β is fixed through the energy balance relation, E i = Tr(e −βH O)/Z [4].When the system is integrable, with an extensive number of conserved charges Q i , ETH does not apply. Accordingly emerging equilibrium can be different from the Gibbs state. In this case the equilibrium can be described by the Generalized Gibbs Ensemble (GGE), a generalization of grand canonical ensemble that includes an infinite tower of conserved charges [5]. Validity of the GGE has been related to the generalized eigenstate thermalization [6][7][8], which generalizes (1) to include an infinite number of conserved quantities,( 2) Here |E i is a mutual eigenstate of the Hamiltonian and charges Q k , Q k (E i ) are the eigenvalues of Q k associated with |E i , and function f O is assumed to be a smooth function of all of its arguments. Similarly to (1), at the qualitative level, (2) postulates that charges Q k form a complete set of thermodynamically relevant quant...

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confidence: 92%

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confidence: 99%

Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories

Dymarsky

Pavlenko

2019

Phys. Rev. Lett.

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“…One possible response to this discrepancy is that expectation values computed in the primary state should be compared with those in the generalized Gibbs ensemble rather than the usual canonical ensemble, with the infinite number of chemical potentials adjusted to yield equality for the KdV expectation values. This avenue has been explored in [13][14][15][16][17][18]. Here we take another point of view: we regard the discrepancy as a reflection of the fact that primary states are atypical, and we should not expect the canonical ensemble to accurately reproduce results in such atypical states.…”

Section: Introductionmentioning

confidence: 99%

Typicality and thermality in 2d CFT

Datta¹,

Kraus²,

Michel³

2019

J. High Energ. Phys.

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We identify typical high energy eigenstates in two-dimensional conformal field theories at finite c and establish that correlation functions of the stress tensor in such states are accurately thermal as defined by the standard canonical ensemble. Typical states of dimension h are shown to be typical level h/c descendants. In the AdS 3 /CFT 2 correspondence, it is such states that should be compared to black holes in the bulk. We also discuss the discrepancy between thermal correlators and those computed in high energy primary states: the latter are reproduced instead by a generalized Gibbs ensemble with extreme values chosen for the chemical potentials conjugate to the KdV charges. m

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“…It is worth pointing out that there exists a deep relation between the KdV hierarchy and two-dimensional conformal field theories (CFT 2 ), indeed, the infinite (quantum) commuting KdV charges can be expressed as composite operators in terms of the stress tensor of a CFT 2 [7][8][9]. This fact has been recently used to describe Generalized Gibbs ensembles in these theories [10][11][12][13][14][15][16], as well as their holographic description [1].…”

Section: Contentsmentioning

confidence: 99%

Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

Ojeda

Pérez

2019

J. High Energ. Phys.

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We present a new set of boundary conditions for General Relativity on AdS 3 , where the dynamics of the boundary degrees of freedom are described by two independent left and right members of the Gardner hierarchy of integrable equations, also known as the "mixed KdV-mKdV" hierarchy. This integrable system has the very special property that simultaneously combines both, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies in a single integrable structure. This relationship between gravitation in three-dimensional spacetimes and two-dimensional integrable systems is based on an extension of the recently introduced "soft hairy boundary conditions" on AdS 3 , where the chemical potentials are now allowed to depend locally on the dynamical fields and their spatial derivatives. The complete integrable structure of the Gardner system, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are directly obtained from the asymptotic analysis and the conserved surface integrals in the gravitational theory. These boundary conditions have the particular property that they can also be interpreted as being defined in the near horizon region of spacetimes with event horizons. Black hole solutions are then naturally accommodated within our boundary conditions, and are described by static configurations associated to the corresponding member of the Gardner hierarchy. The thermodynamic properties of the black holes in the ensembles defined by our boundary conditions are also discussed. Finally, we show that our results can be naturally extended to the case of a vanishing cosmological constant, and the integrable system turns out to be precisely the same as in the case of AdS 3 .

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Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit

Cited by 38 publications

References 65 publications

Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories

Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories

Typicality and thermality in 2d CFT

Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

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