arXiv subjects

Felix M. Haehl

Publications and source records attributed to Felix M. Haehl.

At least 19 recordsLinked to original sources

arXiv2026

The two-dimensional disordered $O(N)$ sigma model

We introduce a two-dimensional $O(N)$ nonlinear sigma model with random Gaussian $p$-body interactions. The model combines the structure of a two-dimensional bosonic SYK-type quantum field theory with the stabilizing spherical constraint of the nonlinear sigma model. At large $N$ we derive the Schwinger-Dyson equations on a torus and analyze the solutions using both analytical approximations and numerical methods. We find a phase diagram qualitatively similar to that of the one-dimensional quantum spherical $p$-spin model, including a low-temperature transition to a spin glass phase. This phase is characterized by a finite Edwards-Anderson order parameter, while the dynamical part of the two-point function displays an approximate scaling regime. These results provide a tractable setting for studying approximate conformal behavior and glassy physics in a two-dimensional relativistic field theory with disorder.

hep-th↗

arXiv2026

Holographic Black Hole Formation and Scrambling in Time-Ordered Correlators

We describe a holographic mechanism for black hole formation via the collision of two shock waves in three-dimensional anti-de Sitter spacetime. In the dual conformal field theory (CFT), a two-shock-wave state corresponds to the insertion of two boosted precursor operators in complementary Rindler patches. Their operator product expansion is initially described by a universal mean-field spectrum of exchanged states, which is dominated by operator dimensions that grow exponentially in the boost parameter. We propose their mean value as diagnosing the mass of the collision product in the bulk. It crosses the CFT heavy state threshold after two scrambling times, in accordance with expectations about black hole formation in general relativity. Our analysis also allows us to identify the scrambling characteristics usually associated with out-of-time-order correlation functions, using only the internal composition of thermal in-time-order correlators.

hep-th↗

arXiv2025

Modular-invariant random matrix theory and AdS${}_3$ wormholes

We develop a non-perturbative definition of RMT${}_2$: a generalization of random matrix theory that is compatible with the symmetries of two-dimensional conformal field theory. Given any random matrix ensemble, its $n$-point spectral correlations admit a prescribed modular-invariant lift to RMT${}_2$, which moreover reduce to the original random matrix correlators in a near-extremal limit. Central to the prescription is a presentation of random matrix theory in Mellin space, which lifts to two dimensions via the $\text{SL}(2,\mathbb{Z})$ spectral decomposition employed in previous work. As a demonstration we perform the explicit RMT${}_2$ lift of two-point correlations of the GUE Airy model. We propose that in AdS$_3$ pure gravity, semiclassical amplitudes for off-shell $n$-boundary torus wormholes with topology $Σ_{0,n} \times S^1$ are given by the RMT${}_2$ lift of JT gravity wormhole amplitudes. For the three-boundary case, we identify a gravity calculation which matches the RMT${}_2$ result.

hep-th↗

arXiv2025

Operator growth and black hole formation

When two particles collide in an asymptotically AdS spacetime with high enough energy and small enough impact parameter, they can form a black hole. Motivated by dual quantum circuit considerations, we propose a threshold condition for black hole formation. Intuitively the condition can be understood as the onset of overlap of the butterfly cones describing the ballistic spread of the effect of the perturbations on the boundary systems. We verify the correctness of the condition in three bulk dimensions. We describe a six-point correlation function that can diagnose this condition and compute it in two-dimensional CFTs using eikonal resummation.

hep-th↗

arXiv2025

Mind the crosscap: $τ$-scaling in non-orientable gravity and time-reversal-invariant systems

Spectral statistics of quantum chaotic systems are governed by random matrix universality. In many cases of interest, time-reversal symmetry selects the Gaussian Orthogonal Ensemble (GOE) as the relevant universality class. In holographic CFTs, this is mirrored by the presence of non-orientable geometries in the dual gravitational path integral. In this work, we analyze general properties of these matrix models and their gravitational counterparts. First, we develop a formalism to express the universal level statistics in the canonical ensemble for arbitrary spectral curves, leading to a topological expansion with finite radius of convergence in the late-time $τ$-scaling limit. Then, we focus on topological gravity and study topological recursion on the moduli space of non-orientable surfaces. We find that the Weil-Petersson volumes display non-analytic behaviour multiplying polynomials in the boundary lengths. The volumes give rise to wormholes with late-time divergences, in contrast with the orientable case, which is finite. We identify systematic cancellations among WP volumes implied by the consistency and finiteness of the $τ$-scaling limit. In particular, the cancellation of late-time divergences requires a nontrivial genus resummation. Working in the gravitational microcanonical ensemble, we derive and resum all orders of the topological expansion matching the GOE matrix model in the high-energy regime.

hep-th↗

arXiv2025

Virasoro OPE Blocks, Causal Diamonds, and Higher-Dimensional CFT

In two-dimensional Conformal Field Theory (CFT), multi-stress tensor exchanges between probe operators give rise to the Virasoro identity conformal block, which is fixed by symmetry. The analogous object, and the corresponding organizing principles, in higher dimensions are less well understood. In this paper, we study the Virasoro identity OPE block, which is a bilocal operator that projects two primaries onto the conformal family of multi-stress tensor states. Generalizing a known construction of global OPE blocks, our formalism uses integrals over nested causal diamonds associated with two timelike-separated insertions. We argue that our construction is adaptable to higher dimensions, and use it to provide a new derivation of the single-stress tensor exchange contribution to a four-point correlator in both three and four dimensions, to leading order in the lightcone limit. We also comment on a potential description using effective reparametrization modes in four dimensions.

hep-th↗

arXiv2024

Records from the S-Matrix Marathon: Schwinger-Keldysh Formalism

These notes provide an overview of real-time techniques in quantum field theories and holography. We outline the general rationale and principles underlying the Schwinger-Keldysh formalism and its out-of-time-order generalizations. To illustrate its broad utility we frame our discussion in the language of open quantum dynamics. As a specific application we describe how such real-time observables help in understanding scrambling dynamics. We also describe the holographic prescription for computing thermal Schwinger-Keldysh observables. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11-22 March 2024.

hep-th↗

arXiv2023

Symmetries and spectral statistics in chaotic conformal field theories II: Maass cusp forms and arithmetic chaos

We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental domain. The focus of this paper is on the discrete part of the spectrum, which consists of the Maass cusp forms. Both their eigenvalues and Fourier coefficients are sporadic discrete numbers with interesting statistical properties and relations to analytic number theory; this is referred to as `arithmetic chaos'. We show that the near-extremal spectral form factor at late times is only sensitive to a statistical average over these erratic features. Nevertheless, complete information about their statistical distributions is encoded in the spectral form factor if all its spin sectors exhibit universal random matrix eigenvalue repulsion (a `linear ramp'). We `bootstrap' the spectral correlations between the cusp form basis functions that correspond to a universal linear ramp and show that they are unique up to theory-dependent subleading corrections. The statistical treatment of cusp forms provides a natural avenue to fix the subleading corrections in a minimal way, which we observe leads to the same correlations as those described by the [torus]$\times$[interval] wormhole amplitude in AdS${}_3$ gravity.

hep-th↗

arXiv2023

Symmetries and spectral statistics in chaotic conformal field theories

We discuss spectral correlations in coarse-grained chaotic two-dimensional CFTs with large central charge. We study a partition function describing the dense part of the spectrum of primary states in a way that disentangles the chaotic properties of the spectrum from those which are a consequence of Virasoro symmetry and modular invariance. We argue that random matrix universality in the near-extremal limit is an independent feature of each spin sector separately; this is a non-trivial statement because the exact spectrum is fully determined by only the spectrum of spin zero primaries and those of a single non-zero spin ("spectral determinacy"). We then describe an argument analogous to the one leading to Cardy's formula for the averaged density of states, but in our case applying it to spectral correlations: assuming statistical universalities in the near-extremal spectrum in all spin sectors, we find similar random matrix universality in a large spin regime far from extremality.

hep-th↗

arXiv2023

Euclidean wormholes in two-dimensional CFTs from quantum chaos and number theory

We consider two-dimensional conformal field theories (CFTs), which exhibit a hallmark feature of quantum chaos: universal repulsion of energy levels as described by a regime of linear growth of the spectral form factor. This physical input together with modular invariance strongly constrains the spectral correlations and the subleading corrections to the linear growth. We show that these are determined by the Kuznetsov trace formula, which highlights an intricate interplay of universal physical properties of chaotic CFTs and analytic number theory. The trace formula manifests the fact that the simplest possible CFT correlations consistent with quantum chaos are precisely those described by a Euclidean wormhole in AdS${}_3$ gravity with [torus]$\times$[interval] topology. For contrast, we also discuss examples of non-chaotic CFTs in this language.

hep-th↗

arXiv2023

Effective description of sub-maximal chaos: stringy effects for SYK scrambling

It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective 'scramblon' modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.

hep-th↗

arXiv2022

Collisions of localized shocks and quantum circuits

We study collisions between localized shockwaves inside a black hole interior. We give a holographic boundary description of this process in terms of the overlap of two growing perturbations in a shared quantum circuit. The perturbations grow both exponentially as well as ballistically. Due to a competition between different physical effects, the circuit analysis shows dependence on the transverse locations and exhibits four regimes of qualitatively different behaviors. On the gravity side we study properties of the post-collision geometry, using exact calculations in simple setups and estimations in more general circumstances. We show that the circuit analysis offers intuitive and surprisingly accurate predictions about gravity computations involving non-linear features of general relativity.

hep-th↗

arXiv2022

Six-point functions and collisions in the black hole interior

In the eternal AdS black hole geometry, we consider two signals sent from the boundaries into the black hole interior shared between the two asymptotic regions. We compute three different out-of-time-order six-point functions to quantify various properties of the collision of these signals behind the horizons: (i) We diagnose the strength of the collision by probing the two-signal state on a late time slice with boundary operators. (ii) We quantify two-sided operator growth, which provides a dual description of the signals meeting in the black hole interior, in terms of the quantum butterfly effect and quantum circuits. (iii) We consider an explicit coupling between the left and right CFTs to make the wormhole traversable and extract information about the collision product from behind the horizon. At a technical level, our results rely on the method of eikonal resummation to obtain the relevant gravitational contributions to Lorentzian six-point functions at all orders in the $G_N$-expansion. We observe that such correlation functions display an intriguing factorization property. We corroborate these results with geodesic computations of six-point functions in two- and three-dimensional gravity.

hep-th↗

arXiv2022

On the Virasoro six-point identity block and chaos

We study six-point correlation functions in two dimensional conformal field theory, where the six operators are grouped in pairs with equal conformal dimension. Assuming large central charge $c$ and a sparse spectrum, the leading contribution to this correlation function is the six-point Virasoro identity block - corresponding to each distinct pair of operators fusing into the identity and its descendants. We call this the star channel. One particular term in the star channel identity block is the stress tensor $SL(2,\mathbb{R})$ (global) block, for which we derive an explicit expression. In the holographic context, this object corresponds to a direct measure of nonlinear effects in pure gravity. We calculate additional terms in the star channel identity block that contribute at the same order at large $c$ as the global block using the novel theory of reparametrizations, which extends the shadow operator formalism in a natural way. We investigate these blocks' relevance to quantum chaos in the form of six-point scrambling in an out-of time ordered correlator. Interestingly, the global block does not contribute to the scrambling mode of this correlator, implying that, to leading order, six-point scrambling is insensitive to the three-point graviton coupling in the bulk dual. Finally, we compare our findings with a different OPE channel, called the comb channel, and find the same result for the chaos exponent in this decomposition.

hep-th↗

arXiv2021

Spin liquid to spin glass crossover in the random quantum Heisenberg magnet

We study quantum SU($M$) spins with all-to-all and random Heisenberg exchange interactions of root-mean-square strength $J$. The $M \rightarrow \infty$ model has a spin liquid ground state with the spinons obeying the equations of the Sachdev-Ye-Kitaev (SYK) model. Numerical studies of the SU(2) model with $S=1/2$ spins show spin glass order in the ground state, but also display SYK spin liquid behavior in the intermediate frequency spin spectrum. We employ a $1/M$ expansion to describe the crossover from fractionalized fermionic spinons to a confining spin glass state with weak spin glass order $q_{EA}$. The SYK spin liquid behavior persists down to a frequency $ω_\ast \sim J q_{EA}$, and for $ω< ω_\ast$, the spectral density is linear in $ω$, thus quenching the extensive zero temperature entropy of the spin liquid. The linear $ω$ spectrum is qualitatively similar to that obtained earlier using bosonic spinons for large $q_{EA}$. We argue that the extensive SYK spin liquid entropy is transformed as $T \rightarrow 0$ to an extensive complexity of the spin glass state.

cond-mat.str-el↗

arXiv2021

The quantum $p$-spin glass model: A user manual for holographers

We study a large-$N$ bosonic quantum mechanical sigma-model with a spherical target space subject to disordered interactions, more colloquially known as the $p$-spin spherical model. Replica symmetry is broken at low temperatures and for sufficiently weak quantum fluctuations, which drives the system into a spin glass phase. The first half of this paper is dedicated to a discussion of this model's thermodynamics, with particular emphasis on the marginally stable spin glass. This phase exhibits an emergent conformal symmetry in the strong coupling regime, which dictates its thermodynamic properties. It is associated with an extensive number of nearby states in the free energy landscape. We discuss in detail an elegant approximate solution to the spin glass equations, which interpolates between the conformal regime and an ultraviolet-complete short distance solution. In the second half of this paper we explore the real-time dynamics of the model and uncover quantum chaos as measured by out-of-time-order four-point functions, both numerically and analytically. We find exponential Lyapunov growth, which intricately depends on the model's couplings and becomes strongest in the quantum critical regime. We emphasize that the spin glass phase also exhibits quantum chaos, albeit with parametrically smaller Lyapunov exponent than in the replica symmetric phase. An analytical calculation in the marginal spin glass phase suggests that this Lyapunov exponent vanishes in a particular infinite coupling limit. We comment on the potential meaning of these observations from the perspective of holography.

hep-th↗

arXiv2021

Diagnosing collisions in the interior of a wormhole

Two distant black holes can be connected in the interior through a wormhole. Such a wormhole has been interpreted as an entangled state shared between two exterior regions. If Alice and Bob send signals into each of the black holes, they can meet in the interior. In this letter, we interpret this meeting in terms of the quantum circuit that prepares the entangled state: Alice and Bob sending signals creates growing perturbations in the circuit, whose overlap represents their meeting inside the wormhole. We argue that such overlap in the circuit is quantified by a particular six-point correlation function. Therefore, exterior observers in possession of the entangled qubits can use this correlation function to diagnose the collision in the interior without having to jump in themselves.

hep-th↗

arXiv2021

Size and momentum of an infalling particle in the black hole interior

The future interior of black holes in AdS/CFT can be described in terms of a quantum circuit. We investigate boundary quantities detecting properties of this quantum circuit. We discuss relations between operator size, quantum complexity, and the momentum of an infalling particle in the black hole interior. We argue that the trajectory of the infalling particle in the interior close to the horizon is related to the growth of operator size. The notion of size here differs slightly from the size which has previously been related to momentum of exterior particles and provides an interesting generalization. The fact that both exterior and interior momentum are related to operator size growth is a manifestation of complementarity.

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