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

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.

• The paper establishes an exact correspondence between the first forbidden singularity in heavy-light Virasoro blocks and the critical instanton fugacity, with both equal to \(z^*=1-e^{-2\pi/\alpha_H}\).
• The analysis combines monodromy methods, AGT localization, Young-tableau statistical mechanics, and complex saddle points to show that instanton proliferation produces a Lee–Yang-type transition and explains the non-commuting \(c\to\infty\) and instanton-number limits.
• The results distinguish the forbidden singularity at \(z^*\) from the true OPE singularity at \(z=1\), while leaving the transition’s universality class, Lefschetz-thimble structure, and physical interpretation of complex instanton configurations open.

The paper establishes a concrete correspondence between forbidden singularities in heavy-light Virasoro blocks of two-dimensional CFTs and phase transitions in the instanton gas of four-dimensional $\mathcal{N}=2$ supersymmetric SU(2) gauge theory, via the AGT correspondence. The central result is an exact match: the critical fugacity at which the instanton gas undergoes proliferation equals the location of the first forbidden singularity $z^* = 1 - e^{-2\pi/\alpha_H}$ predicted by the eigenstate thermalization hypothesis (ETH).

Forbidden singularities as signatures of eigenstate thermalization

In a 2d CFT on the sphere, the auto-correlation function $f_E(\tau)$ of light primary operators in a heavy primary state is computed by the four-point function $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$. ETH predicts that in the thermodynamic limit this correlator should coincide with a thermal correlator at inverse temperature $\beta_H = 2\pi/\alpha_H$, where $\alpha_H = \sqrt{4\epsilon_H - 1}$ with $\epsilon_H = 6h_H/c > 1/4$ (the black-hole regime). Thermal correlators possess KMS images of their OPE singularities at $z_n = 1 - e^{-n\beta_H}$; since the only true singularities of the CFT correlator are OPE singularities at $z \in \{0,1,\infty\}$, the additional ones are "forbidden" — emergent non-analyticities that survive only in the large-$c$ limit. The authors frame these as a sharp manifestation of the tension between thermal behavior and unitarity, analogous in structure to the black hole information paradox.

In the heavy-light limit ($z^* = 1 - e^{-2\pi/\alpha_H}$0, $z^* = 1 - e^{-2\pi/\alpha_H}$1 fixed), the vacuum Virasoro block dominates, and the monodromy method yields the accessory parameter $z^* = 1 - e^{-2\pi/\alpha_H}$2 whose leading-order solution exhibits poles precisely at the forbidden locations $z^* = 1 - e^{-2\pi/\alpha_H}$3. Including subleading corrections in $z^* = 1 - e^{-2\pi/\alpha_H}$4 regularizes each pole into a pair of branch points $z^* = 1 - e^{-2\pi/\alpha_H}$5, so the singularities become branch cuts connecting sheets associated with unphysical Virasoro blocks of dimension $z^* = 1 - e^{-2\pi/\alpha_H}$6.

Evidence for a Lee-Yang-type transition

The authors assemble phenomenological evidence that the branch-cut structure reflects a genuine phase transition. Numerically, at finite but large $z^* = 1 - e^{-2\pi/\alpha_H}$7, the vacuum block develops trajectories of densely packed zeros: the spacing between zeros scales as $z^* = 1 - e^{-2\pi/\alpha_H}$8 and their distance from the real axis as $z^* = 1 - e^{-2\pi/\alpha_H}$9, so they condense onto branch cuts as $f_E(\tau)$0. This mirrors the Lee-Yang mechanism, where zeros of the partition function pinch the real axis to produce non-analyticity. Interpreting $f_E(\tau)$1 as a fugacity and $f_E(\tau)$2 as a density, the branch cut becomes an anti-Stokes curve separating two saddle-point contributions $f_E(\tau)$3 and $f_E(\tau)$4, with zero density $f_E(\tau)$5 along it. A further diagnostic is the abrupt change in the scaling of the free energy $f_E(\tau)$6 across $f_E(\tau)$7: from $f_E(\tau)$8 for $f_E(\tau)$9 to $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$0 for $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$1, indicating liberation of $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$2 effective degrees of freedom.

The instanton gas description via AGT

Under AGT, the Virasoro block maps to the Nekrasov partition function of 4d $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$3 SU(2) gauge theory with the dictionary: conformal ratio $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$4 gauge coupling $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$5, external dimensions $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$6 hypermultiplet masses $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$7, internal dimension $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$8 Coulomb modulus $\langle O_L(0) O_L(z) O_H(1) O_H(\infty)\rangle$9, and central charge $\beta_H = 2\pi/\alpha_H$0 Omega-deformation parameters $\beta_H = 2\pi/\alpha_H$1. The instanton expansion $\beta_H = 2\pi/\alpha_H$2 is a grand-canonical fugacity expansion over instanton number $\beta_H = 2\pi/\alpha_H$3, with micro-states given by pairs of Young tableaux $\beta_H = 2\pi/\alpha_H$4 evaluated via localization fixed points.

A key structural point concerns non-commuting limits. At finite $\beta_H = 2\pi/\alpha_H$5 the radius of convergence of the fugacity series is $\beta_H = 2\pi/\alpha_H$6 (the true OPE singularity), while at $\beta_H = 2\pi/\alpha_H$7 it should be $\beta_H = 2\pi/\alpha_H$8 (the forbidden singularity). These correspond to the regimes $\beta_H = 2\pi/\alpha_H$9 versus $\alpha_H = \sqrt{4\epsilon_H - 1}$0:

$\alpha_H = \sqrt{4\epsilon_H - 1}$1

The forbidden singularity is thus identified with the failure of the $\alpha_H = \sqrt{4\epsilon_H - 1}$2 and $\alpha_H = \sqrt{4\epsilon_H - 1}$3 limits to commute.

Effective action and the critical fugacity

Taking $\alpha_H = \sqrt{4\epsilon_H - 1}$4 first, the authors derive a $\alpha_H = \sqrt{4\epsilon_H - 1}$5-independent effective action $\alpha_H = \sqrt{4\epsilon_H - 1}$6 for the Young tableaux. The derivation hinges on a delicate cancellation: generic box contributions carry factors of $\alpha_H = \sqrt{4\epsilon_H - 1}$7 that cancel between vector- and hypermultiplet pieces, while exceptional boundary boxes spoil this cancellation individually — yet a counting argument shows the missing $\alpha_H = \sqrt{4\epsilon_H - 1}$8 coefficients cancel column by column. Notably, this cancellation depends crucially on the heavy-light arrangement of external dimensions; it fails in the all-heavy limit.

At large $\alpha_H = \sqrt{4\epsilon_H - 1}$9, the dominant configurations are string-like Young tableaux scaling as $\epsilon_H = 6h_H/c > 1/4$0 with normalized atomic data $\epsilon_H = 6h_H/c > 1/4$1. The preference for horizontal growth follows because vertical positions are weighted by $\epsilon_H = 6h_H/c > 1/4$2 while horizontal ones by $\epsilon_H = 6h_H/c > 1/4$3. Two-dimensional tableaux growing as $\epsilon_H = 6h_H/c > 1/4$4 in both directions have actions scaling as $\epsilon_H = 6h_H/c > 1/4$5 for general anisotropic scalings $\epsilon_H = 6h_H/c > 1/4$6, proving the string-like class ($\epsilon_H = 6h_H/c > 1/4$7) maximizes the large-$\epsilon_H = 6h_H/c > 1/4$8 contribution.

The resulting statistical mechanics is an infinite-dimensional integral with linear-in-$\epsilon_H = 6h_H/c > 1/4$9 action $z_n = 1 - e^{-n\beta_H}$0, including a phase index $z_n = 1 - e^{-n\beta_H}$1 expressible with odd-integer coefficients. Numerics show the ensemble of rescaled actions converges to a universal distribution without any single real configuration dominating — evidence that the relevant saddle lies at complexified $z_n = 1 - e^{-n\beta_H}$2. Solving the saddle-point equations via analytic continuation, the authors find the dominant solution has $z_n = 1 - e^{-n\beta_H}$3 and

$z_n = 1 - e^{-n\beta_H}$4

with normalization $z_n = 1 - e^{-n\beta_H}$5. Imposing the asymptotic boundary condition yields infinitely many on-shell values

$z_n = 1 - e^{-n\beta_H}$6

whose most dominant member gives exactly

$z_n = 1 - e^{-n\beta_H}$7

This is the paper's sharpest quantitative claim: the critical fugacity for instanton proliferation coincides exactly with the forbidden singularity, capturing not only $z_n = 1 - e^{-n\beta_H}$8 of the vacuum block but also the full tower $z_n = 1 - e^{-n\beta_H}$9 associated with unphysical blocks. Physically, the transition is an instability of the dilute instanton gas toward $z \in \{0,1,\infty\}$0 proliferation, controlled by a complex saddle representing collective rather than single-configuration behavior.

Additional saddles and an open puzzle

The saddle-point equations admit further solutions exploiting the discrete $z \in \{0,1,\infty\}$1 symmetry. Symmetric solutions $z \in \{0,1,\infty\}$2 yield roots with $z \in \{0,1,\infty\}$3, which would imply a smaller radius of convergence $z \in \{0,1,\infty\}$4, contradicting the CFT prediction verifiable numerically. The authors resolve this by arguing such saddles' Lefschetz thimbles do not intersect the original integration contour — plausibly because their associated fugacity singularities lie on non-principal branches of the multi-valued free energy, shielded by branch cuts emanating from $z \in \{0,1,\infty\}$5. They leave a detailed understanding of this obstruction as an open problem.

High fugacity phase and the true singularity

For $z \in \{0,1,\infty\}$6, string-like tableaux with fixed row number $z \in \{0,1,\infty\}$7 acquire actions scaling as

$z \in \{0,1,\infty\}$8

verified numerically against exact actions. This logarithmic growth restores radius of convergence $z \in \{0,1,\infty\}$9 in the reversed order of limits, consistent with the true OPE singularity. Moreover, organizing the sum over row numbers into a contour integral reminiscent of Regge theory, the exponent $c$0 has a complex saddle at $c$1, reproducing the power-law growth $c$2 and hence the known heavy-light UV behavior $c$3 near $c$4. In this regime the asymmetry between $c$5 and $c$6 disappears at leading order, and proliferation is again governed by a complex saddle, now in the complexified row number. Numerical observations of a run-away mode in skewed two-dimensional tableaux toward the string-like limit support the irrelevance of two-dimensional configurations here as well, though this rests partly on numerics rather than proof.

Limitations and open questions

Several caveats are stated explicitly. The analytic continuation of the infinite-dimensional integral to complex saddles assumes no obstruction from contour deformation past the ordering cone boundaries; this is argued heuristically but not proven. The symmetric-saddle exclusion via Lefschetz thimble topology remains conjectural. The high-fugacity analysis covers only the limiting regimes $c$7 and $c$8; the intermediate crossover near $c$9 — where the universality class of the transition could be extracted — is left open, with two proposed approaches (a joint large-$z^* = 1 - e^{-2\pi/\alpha_H}$00, $z^* = 1 - e^{-2\pi/\alpha_H}$01 scaling analysis, or finite small $z^* = 1 - e^{-2\pi/\alpha_H}$02 smoothing the transition into a crossover). The confinement/deconfinement language is acknowledged as an analogy: $z^* = 1 - e^{-2\pi/\alpha_H}$03 enters the 4d gauge theory as an Omega-deformation geometric parameter acting as an abstract volume, not a gauge group rank, so the phases are more accurately dilute and normal phases of the instanton gas. Finally, the physical interpretation of the complex Young-tableau saddles within conventional Seiberg-Witten gauge dynamics remains obscure, and whether finite-$z^* = 1 - e^{-2\pi/\alpha_H}$04 effects stabilize the run-away two-dimensional channel at large $z^* = 1 - e^{-2\pi/\alpha_H}$05 — potentially altering the order of limits between $z^* = 1 - e^{-2\pi/\alpha_H}$06 and $z^* = 1 - e^{-2\pi/\alpha_H}$07 — is unresolved.

Conclusion

This work recasts forbidden singularities of heavy-light Virasoro blocks — a hallmark of eigenstate thermalization in holographic 2d CFTs — as a Lee-Yang-type phase transition in the instanton gas of 4d $z^* = 1 - e^{-2\pi/\alpha_H}$08 SU(2) gauge theory. The mechanism is precise: a $z^* = 1 - e^{-2\pi/\alpha_H}$09-independent effective action governs a dilute phase of string-like Young tableaux whose fugacity series terminates at a critical fugacity equal to $z^* = 1 - e^{-2\pi/\alpha_H}$10, determined by a complex saddle-point solution obtained in closed form; beyond it, a high-fugacity regime with actions scaling as $z^* = 1 - e^{-2\pi/\alpha_H}$11 reproduces the true $z^* = 1 - e^{-2\pi/\alpha_H}$12 OPE singularity. The identification of the non-commutativity of the $z^* = 1 - e^{-2\pi/\alpha_H}$13 and $z^* = 1 - e^{-2\pi/\alpha_H}$14 limits as the origin of the distinction between forbidden and true singularities provides a thermodynamic lens on eigenstate thermalization, and opens specific questions concerning the universality class of the transition, the role of non-perturbative $z^* = 1 - e^{-2\pi/\alpha_H}$15 trans-series contributions, and the gauge-theoretic meaning of the proliferating instanton configurations.