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arXiv:2608.13246

Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence

Yongjiang Xu, Weixin Sun, Chushun Tian, Huajia Wang

Published Aug 13, 2026• 0 citations
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Abstract

In theoretical physics, finding connections between problems that appear in distinct contexts is an important way to leapfrog progresses, often by illuminating deep aspects that may otherwise seem obscure. In this paper, we consider in 2d CFTs the phenomenon of forbidden singularities in auto-correlation functions -- a key signature of eigenstate thermalization. We show that they correspond to phase transitions in the context of eigenstates. The connection is made explicit by utilizing the AGT correspondence, which relates eigenstate auto-correlations to the Nekrasov partition functions describing an instanton gas of the $\mathcal{N}=2$ SUSY gauge theories. We show that by taking the counter-part of the heavy-light limit, two phases emerge for the instanton gas. They are dominated by configurations represented by string-like Young tableaux with distinct structures and thermodynamic properties, which bare resemblance to the confined and the deconfined phases. A phase transition occurs as instantons proliferate from one side, in a manner that mimics the Lee-Yang theory. We work out the critical fugacity and find it corresponding exactly to the forbidden singularity.

DOI
10.48550/arxiv.2608.13246

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In theoretical physics, finding connections between problems that appear in distinct contexts is an important way to leapfrog progresses, often by illuminating deep aspects that may otherwise seem obscure. In this paper, we consider in 2d CFTs the phenomenon of forbidden singularities in auto-correlation functions -- a key signature of eigenstate thermalization. We show that they correspond to phase transitions in the context of eigenstates. The connection is made explicit by utilizing the AGT correspondence, which relates eigenstate auto-correlations to the Nekrasov partition functions describing an instanton gas of the N=2\mathcal{N}=2N=2 SUSY gauge theories. We show that by taking the counter-part of the heavy-light limit, two phases emerge for the instanton gas. They are dominated by configurations represented by string-like Young tableaux with distinct structures and thermodynamic properties, which bare resemblance to the confined and the deconfined phases. A phase transition occurs as instantons proliferate from one side, in a manner that mimics the Lee-Yang theory. We work out the critical fugacity and find it corresponding exactly to the forbidden singularity.
As phase transitions in isolated quantum systems remain elusive, here we show how a thermodynamic-like phase transition, falling into the Lee-Yang paradigm, can arise in systems displaying eigenstate thermalization. Specifically, we show that in holographic conformal field theories, the eigenstate expectation of the auto-correlation function can be mapped to the partition function Zgauge(z){\cal Z}_{gauge}(z)Zgauge​(z) of a virtual interacting instanton gas, with the conformal mapping of the imaginary time: z=1−e−τz=1-e^{-τ}z=1−e−τ and the central charge ccc mimicking the instanton fugacity and volume, respectively. We find that akin to the Lee-Yang paradigm, for c→∞c\to\inftyc→∞ a pair of complex conjugate zeros of Zgauge(z){\cal Z}_{gauge}(z)Zgauge​(z) move to the real axis located at the famous forbidden singularity. Passing through the singularity the system transits from the low- to high-fugacity phase, accompanied by dramatic changes in scaling behaviors of the free energy and dominant microscopic configurations. Our findings indicate that physics of phase transitions from eigenstate thermalization is very rich.
Abstract We study probe corrections to the Eigenstate Thermalization Hypothesis (ETH) in the context of 2D CFTs with large central charge and a sparse spectrum of low dimension operators. In particular, we focus on observables in the form of non-local composite operators Oobs(x)=OL(x)OL(0) {\mathcal{O}}_{\mathrm{obs}}(x)={\mathcal{O}}_L(x){\mathcal{O}}_L(0) Oobs​(x)=OL​(x)OL​(0) O o b s x = O L x O L 0 with h L ≪ c. As a light probe, Oobs(x) {\mathcal{O}}_{\mathrm{obs}}(x) Oobs​(x) O o b s x is constrained by ETH and satisfies ⟨Oobs(x)⟩hH≈⟨Oobs(x)⟩micro {\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_{h_H}\approx {\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_{\mathrm{micro}} ⟨Oobs​(x)⟩hH​​≈⟨Oobs​(x)⟩micro​ O o b s x h H ≈ O o b s x micro for a high energy energy eigenstate |h H 〉. In the CFTs of interests, ⟨Oobs(x)⟩hH {{\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_h}_{{}_H} ⟨Oobs​(x)⟩h​H​​ O o b s x h H is related to a Heavy-Heavy-Light-Light (HL) correlator, and can be approximated by the vacuum Virasoro block, which we focus on computing. A sharp consequence of ETH for Oobs(x) {\mathcal{O}}_{\mathrm{obs}}(x) Oobs​(x) O o b s x is the so called “forbidden singularities”, arising from the emergent thermal periodicity in imaginary time. Using the monodromy method, we show that finite probe corrections of the form O(hL/c) \mathcal{O}\left({h}_L/c\right) O(hL​/c) O h L / c drastically alter both sides of the ETH equality, replacing each thermal singularity with a pair of branch-cuts. Via the branch-cuts, the vacuum blocks are connected to infinitely many additional “saddles”. We discuss and verify how such violent modification in analytic structure leads to a natural guess for the blocks at finite c: a series of zeros that condense into branch cuts as c → ∞. We also discuss some interesting evidences connecting these to the Stoke’s phenomena, which are non-perturbative e −c effects. As a related aspect of these probe modifications, we also compute the Renyi-entropy S n in high energy eigenstates on a circle. For subsystems much larger than the thermal length, we obtain a WKB solution to the monodromy problem, and deduce from this the entanglement spectrum.
We explore the stability of certain many-body quantum states which may exist at zero or finite temperatures, may lack long-range order and even topological order, and still are thermodynamically distinct from uncorrelated disordered phases. We sharply characterize such states by the conservation of topological charge, or equivalently confinement of instantons, using a generalization of the Wilson loop and the correlation length of an emergent gauge field. Our main conclusions are (i) topological orders can exist at finite temperatures, (ii) relativistic liquids of topological defects can also exist as stable phases at finite temperatures, and (iii) there are two universality classes of instanton suppression. We also relate the instanton dynamics to the problem of the pseudogap state in underdoped cuprates. A universal experimental signature of the instanton deconfinement transition is a change of the quantum noise spectrum, which can perhaps be measured in some situations, for example, via a quantum anomaly, or indirectly detected with a specific heat jump. The method of analysis is a functional renormalization group that generalizes the Coulomb gas treatment of Kosterlitz and Thouless to arbitrary interactions and dimensions. In particular, we construct an exact nonperturbative technique for confining interactions between instantons that introduce irreparable infrared divergences in the standard perturbative approaches.
We explore the stability of certain many-body quantum states which may exist at zero or finite temperatures, may lack long-range order and even topological order, and still are thermodynamically distinct from uncorrelated disordered phases. We sharply characterize such states by the conservation of topological charge, or equivalently confinement of instantons, using a generalization of the Wilson loop and the correlation length of an emergent gauge field. Our main conclusions are: (i) topological orders can exist at finite temperatures, (ii) relativistic liquids of topological defects can also exist as stable phases at finite temperatures, and (iii) there are two universality classes of instanton suppression. We also relate the instanton dynamics to the problem of the pseudogap state in underdoped cuprates. A universal experimental signature of the instanton deconfinement transition is a change of the quantum noise spectrum, which can perhaps be measured in some situations, for example via a quantum anomaly, or indirectly detected with a specific heat jump. The method of analysis is a functional renormalization group that generalizes the Coulomb gas treatment of Kosterlitz and Thouless to arbitrary interactions and dimensions. In particular, we construct an exact non-perturbative technique for confining interactions between instantons, which introduce irreparable infra-red divergences in the standard perturbative approaches.
Abstract Dynamical phase transition in quantum many body systems is usually studied by taking it in the ground state and then quenching a parameter to a new value. We investigate here the dynamics when one performs the time evolution of a generic state and observe that the rate function related to the Loschmidt echo shows non-analytic behavior of two types, one related, and the other unrelated, to the appearance of a quantum phase transition. Specifically, we consider a quantum Ising chain in an initial configuration which is a generic superposition of the eigenstates, and follow its dynamics under the transverse Ising Hamiltonian with a constant field. Depending on the the configuration of the initial state, some singularities appear in the rate function which do not correspond to the equilibrium phase transition of the Hamiltonian. However another class of singularity is found having a connection with the quantum critical point. Some features of the singularities of the rate function have been derived analytically and it is observed that the occupancy of quasiparticle eigenstates plays the key role in triggering the non-analyticities.
It is found that the singularity of the thermodynamic potential of an ideal Bose gas is connected with the physical root z0=1{z}_{0}=1z0​=1 of the inverse of the grand partition function. The critical nature is determined solely by the behavior of the root distribution of the inverse of the grand partition function near z0=1{z}_{0}=1z0​=1. This is quite similar to the situation of phase transition described by Yang-Lee theory.