Generalized Eigenstate Thermalization in 2d CFTs

TL;DR

This paper demonstrates that large central charge 2d conformal field theories equilibrate according to a generalized eigenstate thermalization principle, with local observables determined solely by an infinite set of conserved charges, linking integrability and thermalization.

Contribution

It establishes that in large central charge 2d CFTs, equilibrium states are described by a Generalized Gibbs Ensemble based only on qKdV charges, confirming integrability's role in thermalization.

Findings

Large central charge 2d CFTs satisfy generalized eigenstate thermalization.

Expectation values of local observables are determined by qKdV charges.

Equilibrium states can be described by a GGE with only local conserved charges.

Abstract

Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2d conformal field theories through an infinite tower of local conserved qKdV charges in involution. We discuss the role this integrable structure plays in equilibration of 2d CFTs. We show 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. Our work settles the question if non-local or quasi-local charges are necessary to describe equilibrium of large central charge 2d CFTs by showing that upon equilibration local physics 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 include negative chemical potentials and holographically will be described by a quasi-classical black hole with quantum soft hair.

Equations

$$⟨E_i|\mathcal{O}|E_i⟩=f_{\mathcal{O}}(E_i).$$

$$⟨E_i|\mathcal{O}|E_i⟩=f_{\mathcal{O}}(Q_k(E_i)).$$

$$|E⟩=|\{m_i\},\Delta ⟩,E=Q_1=(\Delta +\sum m_i)/\ell ,$$

$$\Delta \sim {\ell }^2,\sum _k{n}_k{k}^{2r-1}\sim {\ell }^{2r}.$$

$$\frac{q_{2r-1}}{q_1^r}=1+O(1/c).$$

$$\frac{{\sum }_k{n}_k{k}^{2r-1}}{{\Delta }^r}=O(1/c^r).$$

$$\ell q_1$$

$$\ell q_3$$

$$\dots$$

$$\ell q_{2r-1}$$

$${\mathcal{O}}_2=T,{\mathcal{O}}_4=T^2-\frac{3}{10}{\partial }^{2}T.$$

$${\mathcal{O}}_6^{(1)}$$

$${\mathcal{O}}_6^{(2)}$$

$$⟨{\mathcal{O}}_6^{(2)}⟩=\frac{9}{5}\sum _k{n}_k\left(\frac{c}{6}{k}^5+4\Delta k^3\right)+O(c^0).$$

$$q_1^{-3}⟨{\mathcal{O}}_6^{(1)}⟩$$

$$q_1^{-3}⟨{\mathcal{O}}_6^{(2)}⟩$$

$$\rho =\exp \left\{-\sum _k{\mu }_{2k-1}{Q}_{2k-1}\right\}/Z,{\mu }_1\equiv \beta ,$$

$$\ell q_{2k-1}=⟨E_i|Q_{2k-1}|E_i⟩=\mathrm{Tr}(\rho Q_{2k-1}).$$

$$\delta {\mathrm{q}}_{2k-1}=\frac{q_{2k-1}}{q_1^k}-1=$$

$$\sum _{k=1}k{\tilde{\mu }}_{2k-1}{\tilde{\sigma }}^{k-1/2}=1.$$

$$\frac{q_3}{q_1^2}-1\le \frac{22}{5c}+O(1/c^2),\frac{q_5}{q_1^3}-1\le \frac{302}{21c}+O(1/c^2),$$

$$\dots$$

$${\mathcal{O}}_6^{(1)}$$

$${\mathcal{O}}_6^{(2)}$$

$${\mathcal{O}}_6^{(1)}$$

$${\mathcal{O}}_6^{(2)}$$

$$q_1^{-3}⟨{\mathcal{O}}_6^{(1)}⟩$$

$$q_1^{-3}⟨{\mathcal{O}}_6^{(2)}⟩$$

$${\mathcal{O}}_8^{(1)}$$

$${\mathcal{O}}_8^{(2)}$$

$${\mathcal{O}}_8^{(3)}$$

$$q_1^{-4}⟨{\mathcal{O}}_8^{(1)}⟩$$

$$q_1^{-4}⟨{\mathcal{O}}_8^{(2)}⟩$$

$$q_1^{-4}⟨{\mathcal{O}}_8^{(3)}⟩$$

$${\mathcal{O}}_{10}^{(1)}$$

$${\mathcal{O}}_{10}^{(2)}$$

$${\mathcal{O}}_{10}^{(3)}$$

$${\mathcal{O}}_{10}^{(4)}$$

Full text

[TABLE]

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, $f_{\mathcal{O}}(E_{i})$, and thermal expectation value of ${\mathcal{O}}$ in the Gibbs ensemble, $f_{\mathcal{O}}(E_{i})=\operatorname{Tr}(e^{-\beta H}{\mathcal{O}})/Z$, where the effective temperature $\beta$ is fixed through the energy balance relation, $E_{i}=\operatorname{Tr}(e^{-\beta H}{\mathcal{O}})/Z$ D’Alessio et al. (2016).

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 Rigol et al. (2007). Validity of the GGE has been related to the generalized eigenstate thermalization Cassidy et al. (2011); He et al. (2013); Vidmar and Rigol (2016), which generalizes (1) to include an infinite number of conserved quantities,

[TABLE]

Here $|E_{i}\rangle$ is a mutual eigenstate of the Hamiltonian and charges $Q_{k}$, $Q_{k}(E_{i})$ are the eigenvalues of $Q_{k}$ associated with $|E_{i}\rangle$, and function $f_{\mathcal{O}}$ 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 quantities which fully specify local properties of an eigenstate. Provided (2) applies to most states, it ensures equivalence between the generalized microcanonical ensemble and GGE, establishing validity for the latter to describe emerging equilibrium e.g. following a quantum quench Cassidy et al. (2011).

In this Letter we discuss thermalization of two-dimensional conformal field theories (CFTs), a rich topic with multiple connections ranging from the cold atom experiments Calabrese and Cardy (2016) to physics of quantum gravity Anous et al. (2016). It has been shown that following a quantum quench 2d conformal theories equilibrate and reach a steady state, which in many cases can be described in terms of the Gibbs ensemble Calabrese and Cardy (2006, 2007). At the same time emergence of thermal equilibrium is not universal. Conformal symmetry in two dimensions gives rise to an infinite tower of local mutually commuting conserved qKdV charges $Q_{2k-1}$, the CFT Hamiltonian for the left-movers being $Q_{1}\equiv H$, a part of the integrable structure of the 2d CFTs Bazhanov et al. (1996, 1997, 1999). The question we are concerned with is how presence of these charges affects equilibration. By analogy with the integrable lattice models it is natural to expect that locally equilibrium states can be described in terms of the GGE, which includes all local qKdV charges. Indeed, emergence of exactly such qKdV GGE was analytically shown for a special family of so-called Cardy-Calabrese initial states Cardy (2016).

In the context of integrable systems the question which quantities should be included in the GGE is far from being trivial. Early studies in the context of XXZ and Lieb-Liniger models have shown that a full set of extensive local charges does not specify local properties of eigenstates, signaling failure of generalized ETH Brockmann et al. (2014); Mestyán et al. (2015). These works raised an important question of the validity of the GGE to describe an emerging equilibrium following a quantum quench Wouters et al. (2014); Pozsgay et al. (2014); Goldstein and Andrei (2014). A resolution comes from the fact that besides local conserved quantities these models give rise to quasi-local conserved charges Ilievski et al. (2015a). Taking them into account restores validity of the GGE Ilievski et al. (2015b). Following studies in the context of integrable field theoretic models, both free and interacting ones, have decisively established that adding quasi-local charges is necessary to accurately describe the after-quench equilibrium state Essler et al. (2015); Sotiriadis (2016); Doyon (2017); Bastianello and Sotiriadis (2017); Palmai and Konik (2018). These findings raise an important question emphasized in Cardy (2016) if the set of local qKdV charges is generally sufficient to describe equilibrium in large $c$ 2d CFTs, or it should be extended by non-local or perhaps some new local charges Vernier and Cubero (2017). In this Letter we answer this question by showing that at large $c$, 2d CFTs satisfy generalized eigenstate thermalization (2) with the local qKdV charges forming a complete set and ambiguously specifying local properties of the eigenstates.

Two-dimensional conformal field theories admit a split into non-interacting sectors of left and right movers. For simplicity we only discuss one sector explicitly, while all results automatically extend to the full theory. We consider 2d CFT on a circle of the circumference $\ell$ in a mutual eigenstate of all charges $Q_{2k-1}$,

[TABLE]

labeled by the primary state $\Delta$ and the set of integers $\{m_{i}\}$ Dymarsky and Pavlenko (2019). The set $\{m_{i}\}$ is convenient to parametrize using free boson representation where an integer $n_{k}$ for $k=1,2,\dots$ specifies the number of times integer $k$ appears in the set $\{m_{i}\}$. In the thermodynamic limit $\ell\rightarrow\infty$ “energy” $Q_{1}$ and all other qKdV charges are assumed to scale with the system size to yield finite charge densities  $q_{2r-1}=Q_{2r-1}/\ell$. In terms of $\Delta,n_{k}$ this implies scaling

[TABLE]

In what follows we restrict the discussion to the eigenstates (3) with the density charges $q_{2r-1}=\langle E|Q_{2r-1}|E\rangle/\ell$ which additionally satisfy

[TABLE]

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 de Boer and Engelhardt (2016); Pérez et al. (2016); Dymarsky and Pavlenko (2019); Maloney et al. (2018). In terms of $\Delta,n_{k}$ an exponential majority of states in the generalized microcanonical ensemble specified by $q_{2k-1}$ subject to (5) will satisfy

[TABLE]

In fact (6) may apply to all states in the generalized microcanonical ensemble (5), yielding strong version of the generalized ETH in 2d CFTs. To verify that one would need to know full spectrum of qKdV charges, going beyond currently known leading $1/c$ expansion.

In the regime of quasi-classical gravity (6), $c\gg 1$, expectation values of qKdV charges can be calculated explicitly Dymarsky and Pavlenko (2018),

[TABLE]

where $p_{2r-1}(c,\Delta,k)$ are some known polynomials of degree $2r-1$ which include only odd powers of $k$.

Because of translational invariance the expectation value of a full derivative ${\mathcal{O}}=\partial\mathcal{O}^{\prime}$ in energy eigenstate will vanish. Hence it suffices to consider expectation values $\langle E|{\mathcal{O}}|E\rangle$ only when ${\mathcal{O}}$ is a quasi-primary operator. Below we consider the case when ${\mathcal{O}}$ belongs to the vacuum family, i.e. it is a Virasoro descendant of the identity. To streamline the notations we introduce $\langle{\mathcal{O}}\rangle\equiv\langle E|{\mathcal{O}}|E\rangle$. It is convenient to parametrize ${\mathcal{O}}$ by its dimension (level). At the levels $2$ and $4$ there are unique quasi-primaries in the vacuum family,

[TABLE]

Thus expectation values of ${\mathcal{O}}_{2,4}$ are identically equal to charge densities $q_{1},q_{3}$ Lashkari et al. (2018). At the level $6$ there are two quasi-primaries (we always choose quasi-primaries in the basis which diagonalizes Zamolodchikov metric)

[TABLE]

The expectation value of the combination ${\mathcal{O}}_{6}^{(1)}+\frac{5}{9}\frac{c}{12}{\mathcal{O}}_{6}^{(2)}$ is identically equal to $q_{5}$. Similarly to (7-9), at leading order the expectation value of ${\mathcal{O}}_{6}^{(2)}$ has the form of a polynomial in $\Delta$ and odd powers of $k$,

[TABLE]

It is possible to use (7-9) to express any term of the form $\sum_{k}n_{k}\,k^{2r-1}$ via $q_{2j-1}$, $j\leq r$, but a priori the result would also depend on $\Delta$. Thus, at leading order in $1/c$, expectation values of ${\mathcal{O}}_{6}^{(i)}$ are some functions of $\Delta$ and $q_{2r-1}$. Remarkably, because of the non-trivial cancellations the final result is $\Delta$-independent, and can be expressed solely in terms of $q_{2r-1}$. To simplify the answer we introduce dimensionless ratio ${\rm q}_{2k-1}=q_{2k-1}/q_{1}^{k}$ such that $\delta{\rm q}_{2k-1}\equiv{\rm q}_{2k-1}-1$ is of order $1/c$. Then ${\mathcal{O}}_{6}^{(i)}$ measured in units of energy density $q_{1}$ is given by

[TABLE]

As we see different quasi-primaries have different scaling with $c$. Our calculation applies to leading $1/c$ behavior of each quasi-primary, except for a special one, which includes maximal power of $T$ without derivatives. The expectation value of that quasi-primary starts with $O(c^{0})$ and our result applies to the first two terms in $1/c$ expansion.

The possibility to express eigenstate expectation value $\langle{\mathcal{O}}\rangle$ as a polynomial in $q_{2j-1}$ extends to all higher levels. For an operator of dimension $2r$ the answer only depends on $q_{2j-1}$ for $j\leq r$. We write down explicit expressions for all operators up to level $10$ in terms of $q_{2j-1}$ in Supplemental Materials. Our results establish generalized eigenstate thermalization for vacuum block observables in large $c$ CFTs.

That expectation value $\langle{\mathcal{O}}\rangle$ of an operator of dimension $2r$ only includes qKdV charges $q_{2j-1}$ up to the same dimension $j\leq r$ can be interpreted as a manifestation of locality. It is analogous to the observation in the context of integrable lattice models that to describe equilibrium state locally, at the length scales not exceeding some distance $a$, it is only necessary to include local and quasi-local charges in the GGE with the support within $a$ Nandy et al. (2016); Pozsgay et al. (2017).

Generalized eigenstate thermalization implies validity of the qKdV Generalized Gibbs Ensemble

[TABLE]

to describe local properties of individual energy eigenstates, provided chemical potentials $\mu_{2k-1}$ are tuned to match values of the eigenstate charges

[TABLE]

Provided $q_{2k-1}$ a chosen to represent charge densities of some non-equilibrium initial state $|\Psi\rangle$, a standard argument would consequently equate the GGE expectation values of local operators with those in the diagonal ensemble of $|\Psi\rangle$, written in the eigenbasis (3). In most cases the latter would be equal to the expectation values in state $|\Psi\rangle$ upon equilibration. It should be noted though that left and right Hamiltonians $Q_{1},\bar{Q}_{1}$ are highly degenerate, and therefore validity of the diagonal ensemble to describe local physics upon equilibration may be violated.

It remains an open question to establish existence of $\mu_{2k-1}$ which would solve (17) for any given set of $q_{2k-1}$. Using explicit form of the generalized partition function in the large $c$ limit Dymarsky and Pavlenko (2018) we can find, up to $O(1/c^{2})$ corrections,

[TABLE]

where $\tilde{\mu}_{2k-1}={\sqrt{6}\over\pi}c^{k-1}\mu_{2k-1}$ and $\sigma(\tilde{\mu}_{2k-1})$ is positive and satisfies

[TABLE]

From here it follows that when all chemical potentials are positive $q_{2k-1}$ satisfy an infinite series of inequalities (see Supplemental Materials)

[TABLE]

Thus GGE emerging after equilibration of some general initial state will have to include negative chemical potentials, unless all inequalities (Generalized Eigenstate Thermalization in 2d CFTs) are satisfied.

To match GGE to a primary state all qKdV densities should be related to each other via $q_{2k-1}=q_{1}^{k}$ Lashkari et al. (2018). This is only possible if the integral in (Generalized Eigenstate Thermalization in 2d CFTs) vanishes, which requires $\gamma$ to be infinite. This is consistent with the observation of Dymarsky and Pavlenko (2019) that an ensemble with any finite number of non-zero $\mu_{2k-1}$ can not describe primary states. This is because in full generality $q_{2k-1}\geq q_{1}^{k}$ and hence primary states are at the boundary of the phase space of $q_{2k-1}$’s. It is nevertheless possible to describe them in the limit, via a GGE with at lest some coefficients approaching infinity. The simplest scenario is to consider $\mu_{3}>0$ and arbitrary $\beta\equiv\mu_{1}$, while all other chemical potentials are identically zero. Then in the limit $\tau=\beta(6/\pi^{2}c\mu_{3})^{1/3}\rightarrow-\infty$, for all $k$, $q_{2k-1}/q_{1}^{k}-1$ will vanishes as $\sim|\tau|^{-3}$, as is shown for $k=2,3$ in Fig. 1.

With just two chemical potentials $\beta,\mu_{3}$ being non-zero the values of $q_{2k-1}/q_{1}^{k}-1$ is confined to be between zero and their thermal (Gibbs ensemble) values. This constraint is removed already after turning on one more additional chemical potential. For example by taking $\beta,\mu_{5}>0$ and $\mu_{3}<0$ one can fine-tune function $\gamma$ to become arbitrarily small for some positive value of $\kappa$, leading to the divergence of the integral in (Generalized Eigenstate Thermalization in 2d CFTs) and violating quasi-classical regime (5).

From the holographic point of view equilibration in field theory is associated with the formation of a black hole in $AdS_{3}$, a background dual to the GGE (16). Conserved qKdV charges correspond to the black hole soft hair, which are only visible at quantum level. At the level of classical gravity $c\rightarrow\infty$ all qKdV charges are related, $q_{2k-1}=q_{1}^{k}$. Accordingly there is a unique classical BTZ black hole family of solutions parametrized by $q_{1},\bar{q}_{1}$ de Boer and Engelhardt (2016); Pérez et al. (2016). It is an important question to understand the regime $q_{2k-1}\neq q_{1}^{k}$ holographically, by including quantum gravity corrections into consideration. This, in particular, should provide holographic interpretation to negative temperature and other chemical potentials, which will necessarily appear starting from a general initial state.

In this Letter we have only considered local probes $\mathcal{O}$ from the vacuum block. In case $\mathcal{O}$ is a non-trivial Virasoro primary, or its descendant, it will have zero expectation value in the GGE (16) for any values of $\mu_{2k-1}$. This is because in the thermodynamic limit $\ell\rightarrow\infty$ geometry degenerates into a cylinder, which is conformally flat. Thus, to satisfy any version of eigenstate thermalization the eigenstate expectation value $\langle E|{\mathcal{O}}|E\rangle$ must simply vanish. In terms of the CFT data, this means most or all heavy-heavy-light Operator Product Expansion coefficients must approach zero when the dimension of heavy operators grows to infinity. If that is the case, generalized eigenstate thermalization will be trivially satisfied. It remains an outstanding problem to establish if large central charge chaotic CFT, in particular those with gravity duals, exhibit this behavior.

Conclusions. In this Letter we have established that large central charge 2d CFTs in the thermodynamic limit satisfy generalized eigenstate thermalization with the tower of local qKdV charges forming a complete set of thermodynamically-relevant quantities. Our analysis establishes universal validity of Generalized Gibbs Ensemble that includes all qKdV charges to describe individual energy eigenstates, and hence in most cases, asymptotic equilibrium states in such theories. It would be important to extend the analysis to next order in $1/c$, which will likely reveal if the eigenstate thermalization is strong i.e. applies to all finite energy density eigenstates, or weak, i.e. applies to most states.

I Supplemental Materials

I.1 Expectation value of quasi-primaries in eigenstates

In this section we list the explicit expressions for the eigenstate expectation values of all quasi-primaries up to level ten in terms of qKdV charges.

I.1.1 Level 6

There is are two quasi-primaries

[TABLE]

In the limit (6) they can be simplified to

[TABLE]

In units of the energy density their expectation values are

[TABLE]

I.1.2 Level 8

There are three quasi-primaries at level 8,

[TABLE]

In the units of energy density at leading order they are

[TABLE]

I.1.3 Level 9

There are no quasi-primaries of odd dimension smaller than nine. At level nine there is a unique quasi-primary ${\mathcal{O}}_{9}$, which has zero expectation value, as well as all higher odd-dimensional quasi-primaries, due to parity.

I.1.4 Level 10

There are four quasi-primaries at level 8. In the limit (6) up to some additional factors they are

[TABLE]

In terms of energy density their expectation values are

[TABLE]

I.2 GGE with positive chemical potentials

For any positive integer $j$ hypergeometric function ${}_{2}F_{1}(1,1-j,3/2,-\kappa^{2})$ is polynomial in $\kappa^{2}$ with non-negative coefficients which starts with one,

[TABLE]

Hence it is a monotonically increasing function of $\kappa$ which satisfies ${}_{2}F_{1}(1,1-j,3/2,-\kappa^{2})\geq 1$. From here it follows that when all chemical potentials are non-negative, function $\gamma$ defined in the equation (Generalized Eigenstate Thermalization in 2d CFTs) from the main text satisfies

[TABLE]

Thus at leading order in $1/c$, $q_{2k-1}/q_{1}^{k}-1$ is bounded from above by its value in the Gibbs ensemble,

[TABLE]

This yields $22/5$ for $k=2$, $302/11$ for $k=3$, $2428/75$ for $k=4$, and so on.

I.3 GGE with two non-zero chemical potentials

To gain better intuition it is instructive to consider the generalized ensemble which includes only two charges, the conventional Hamiltonian of CFT $H\equiv Q_{1}$ and $Q_{3}$,

[TABLE]

To assure convergence we must require $\mu_{3}>0$ while $\beta$ can be arbitrary. It is convenient to parametrize $\beta,\mu_{3}$ in terms of

[TABLE]

and energy density $q_{1}=-\ell^{-1}{\partial\ln Z\over\partial\beta}$, such that

[TABLE]

Then $\delta{\rm q}_{2k-1}$ only depends on $\tau$,

[TABLE]

When $\tau$ approaches minus infinity while $q_{1}$ is kept fixed,

[TABLE]

and we find that $c\delta{\rm q}_{2k-1}$ approaches zero as $1/|\tau|^{3}$. We plot $\delta{\rm q}_{2k-1}\equiv q_{1}^{-k}q_{2k-1}-1$ in the units of $1/c$ as a function of $\tau$ for $k=2,3$ in Fig. 1 in the main text.