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1. Universal features of higher-form symmetries at phase transitions

   https://doi.org/10.21468/SciPostPhys.11.2.033  

   Wu, Xiao-Chuan; Jian, Chao-Ming; Xu, Cenke (January 2021, SciPost Physics)

   We investigate the behavior of higher-form symmetries at variousquantum phase transitions. We consider discrete 1-form symmetries, whichcan be either part of the generalized concept “categorical symmetry”(labelled as \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) )introduced recently, or an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. We demonstrate that for many quantum phase transitionsinvolving a Z_N^{(1)} Z N ( 1 ) or \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle ⟨ ( O 𝒞 ) 2 ⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P ⟨ ( log O 𝒞 ) 2 ⟩ ∼ − A ϵ P + b log P , where O_\mathcal{C} O 𝒞 is an operator defined associated with loop \mathcal{C} 𝒞 (or its interior \mathcal{A} 𝒜 ),which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. P P is the perimeter of \mathcal{C} 𝒞 ,and the b \log P b log P term arises from the sharp corners of the loop \mathcal{C} 𝒞 ,which is consistent with recent numerics on a particular example. b b is a universal microscopic-independent number, which in (2+1)d ( 2 + 1 ) d is related to the universal conductivity at the quantum phasetransition. b b can be computed exactly for certain transitions using the dualitiesbetween (2+1)d ( 2 + 1 ) d conformal field theories developed in recent years. We also compute the"strange correlator" of O_\mathcal{C} O 𝒞 : S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangle S 𝒞 = ⟨ 0 | O 𝒞 | 1 ⟩ / ⟨ 0 | 1 ⟩ where |0\rangle | 0 ⟩ and |1\rangle | 1 ⟩ are many-body states with different topological nature. 

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2. Generalized free cumulants for quantum chaotic systems

   https://doi.org/10.1007/JHEP09(2024)066  

   Jindal, Siddharth; Hosur, Pavan (September 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems. 

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3. Zero modes of local operators in 2d CFT on a cylinder

   https://doi.org/10.1007/JHEP07(2020)172  

   Dymarsky, Anatoly; Pavlenko, Kirill; Solovyev, Dmitry (July 2020, Journal of High Energy Physics)

   null (Ed.)

   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. 

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4. Out-of-equilibrium eigenstate thermalization hypothesis

   https://doi.org/10.21468/SciPostPhys.18.4.136  

   Foini, Laura; Dymarsky, Anatoly; Pappalardi, Silvia (January 2025, SciPost Physics)

   Understanding how out-of-equilibrium states thermalize under quantum unitary dynamics is an important problem in many-body physics. In this work, we propose a statistical Ansatz for the matrix elements of non-equilibrium initial states in the energy eigenbasis, in order to describe such evolution. The approach is inspired by the Eigenstate Thermalisation Hypothesis (ETH) but the proposed Ansatz exhibits different scaling. Importantly, we point out the exponentially small cross-correlations between the observable and the initial state matrix elements that determine relaxation dynamics toward equilibrium. We numerically verify scaling and cross-correlation, point out the emergent universality of the high-frequency behavior, and outline possible generalizations. 

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5. Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

   https://doi.org/10.3233/ASY-231872  

   Shapiro, Jacob (February 2024, Asymptotic Analysis)

   We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity. 

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