# Eigenstate thermalisation hypothesis The eigenstate thermalisation hypothesis (ETH) is a conjecture that explains the thermalisation of isolated quantum systems. It suggests that for a quantum system in a pure state, the expectation value of most observables can be approximated by their expectation value in a corresponding microcanonical ensemble. In other words, the system behaves as if it were in thermal equilibrium, even though it is isolated and not in contact with a heat bath. It states that, for a simple operator (few-body) $O_\alpha$, $\left\langle E_a|\mathcal{O}| E_b\right\rangle=\overline{\left\langle E_a|\mathcal{O}| E_a\right\rangle} \delta_{a b}+e^{-S(\bar{E}) / 2} f_{\mathcal{O}}\left(E_a, E_b\right) R_{a b},$where $\bar{E}$ is the average of the two energies, $f_\mathcal{O}(E_a,E_b)=f_\mathcal{O}(\bar E,\delta E)$ is some smooth function that is peaked at $\bar E$ and model-dependent and depends on the (UV) details of the theory, and $R_{ab}$ is a random matrix drawn from a Gaussian distribution. Notice that ETH is a statement about $C_{ij\alpha}$ where $O_\alpha$ is light and $O_i$ is heavy. The reasoning for ETH is as follows. Since simple operators cannot distinguish between energy eigenstates, up to exponentially suppressed corrections in the entropy, their expectation values are given by a diagonal matrix made of the microcanonical expectation value. Given the Gaussian distribution, the ansatz above is uniquely fixed by specifying the first two non-trivial moments of this ensemble. But this ansatz turns out to imply that all higher-order thermal correlation functions factorise. This is not good: e.g. it is not compatible with e.g. a non-trivial [[0466 Lyapunov exponent|Lyapunov exponent]] extracted from the [[0482 Out-of-time-order correlator|OTOC]]. An improved version reads:$\overline{\mathcal{O}_{a_1 b_1} \mathcal{O}_{a_2 b_2} \cdots \mathcal{O}_{a_n b_n}}=e^{-(n-1) S(\bar{E})} g_{\mathcal{O}, a_1 b_1 \cdots a_n b_n}^{(n)}\left(E_1, \ldots, E_n\right)+\dots$where dots denote terms suppressed by more factors of $S(\bar{E})$ and $g_{\mathcal{O}, a_1 b_1 \cdots a_n b_n}^{(n)}\left(E_1, \ldots, E_n\right)=\sum_{\sigma \in S_n} g_{\mathcal{O}}^{(n)}\left(E_{\sigma(1)}, \ldots, E_{\sigma(n)}\right) \delta_{b_{\sigma(1)} a_{\sigma(2)}} \delta_{b_{\sigma(2)} a_{\sigma(3)}} \cdots \delta_{b_{\sigma(n)} a_{\sigma(1)}}.$ ## Refs - originals - [[1991#Deutsch]] - [[1994#Srednicki]] - review - [[2015#D'Alessio, Kafri, Polkovnikov, Rigol (Review)]] - non-Gaussianity - [[2018#Foini, Kurchan]] - [[2019#Murthy, Srednicki (Jun, b)]]: need for non-Gaussianity to have non-trivial [[0482 Out-of-time-order correlator|OTOC]] - [[2021#Belin, de Boer, Liska]]: heavy OPE and wormholes - [[2022#Jafferis, Kolchmeyer, Mukhametzhanov, Sonner (Short)]]: generalisation in terms of a random ensemble, known as the [[0587 ETH matrix model|ETH matrix model]] ## Generalisations - [[2019#Collier, Maloney, Maxfield, Tsiares]] - in CFT2 - [[2020#Belin, de Boer]] - not just $C_{ij\alpha}$, but also $C_{ijk}, C_{i\alpha\beta}$ are random variables with a Gaussian distribution - [[2024#Chen, Dymarsky, Tian, Wang (a)]] and [[2024#Chen, Dymarsky, Tian, Wang (b)]] - ETH where KdV charges are appropriately accounted for ## Applications - relation to [[0206 Replica wormholes|replica wormholes]]: - [[2020#Belin, de Boer]] - [[PollackRozaliSullyWakeham2021]][](https://arxiv.org/pdf/2002.02971.pdf) - for [[0438 Small black holes in AdS|small BHs in AdS]] and their evaporation: - [[LoweThorlacius2022]][](https://arxiv.org/pdf/2203.06434.pdf) ## Tests of ETH - thermalisation without ETH: [[HarrowHuang2022]][](https://arxiv.org/pdf/2209.09826.pdf) ## Other investigations - [[Alishahiha2022]][](https://arxiv.org/pdf/2209.14689.pdf): relates to the linear growth of a certain quantity; quantum complexity