Searching for emergent spacetime in spin glasses

TL;DR

This work investigates how emergent bulk spacetime features may arise in large $N$ disordered quantum systems by linking spectral-function properties to holographic criteria. Building on an algebraic framework, the authors study three spin-glass–type models (SYK, spherical $p$-spin, and SU($M$) Heisenberg) to determine when noncompact spectral support and exponential tails permit bulk causality and a semiclassical radial direction. They find robust exponential tails in spin-liquid phases across models, while many spin-glass regimes show spectral compactness, signaling no bulk emergence under those conditions; a notable exception is a quantum spin-glass window in the SU($M$) model where noncompact spectra persist. A key technical contribution is extending the no-causality-detection result to exponential decays, establishing limitations of low-energy observables to reveal bulk structure and outlining criteria for possible holographic duals in complex many-body systems. Collectively, the results map where holographic-like behavior could plausibly arise in glassy quantum matter and suggest directions for refining bulk-diagnostic frameworks beyond polynomial-decay assumptions.</nobrace>

Abstract

Recent work on algebraic formulations of holographic dualities in terms of large $N$ algebras has revealed a deep connection between the properties of the associated spectral functions and the emergence of a semiclassical spacetime and causal horizons therein. One of the main lessons is that, for a radial direction to emerge, the spectral function has to exhibit non-compact support. Furthermore, there exist conjectures upon a possible duality between complex gravitational configurations and glassy systems. The goal of this paper is to combine these ideas by studying many-body quantum-mechanical systems and assess in which parameter regimes they could potentially be holographic. Thus, we compute the spectral functions of three many-body systems with quenched disorder, the SYK model, the $p$-spin model and the SU$(M)$ Heisenberg chain in the large $N$ limit and present results in different parameter regimes. Our main finding is that in the quantum spin glass phase of the SU$(M)$ Heisenberg model, the spectral function develops an exponential tail, similar to the large $q$ limit of SYK. Furthermore, we demonstrate the presence of an infinite family of quasiparticle excitations deep in the spin liquid phase of the $p$-spin model, which could point towards an emergent type I von Neumann algebra. In addition, we demonstrate the presence of an exponential tail in the spectral function for all cases without compact support and conformal symmetry. Motivated by this observation, we prove that no low-energy operator can detect a nontrivial bulk causal structure, if the spectral function has exponentially decaying tails.

Figures (12)

• Figure 1: Illustration of the Gibbs states in a spin liquid and a spin glass phase. In the spin liquid phase, the Gibbs state is given by a single ergodic state, which when transitioning to the spin glass state "shatters" into exponentially many ergodic components.
• Figure 2: Illustration of causal wedge reconstruction for a timeband algebra in the intervale $(-t,t)$. The algebra ${\mathcal{Y}}_{(-t,t)}$ of the causal wedge of the timeband $(-t,t)$ is depicted by the blue shaded region. The light red diamond ${\mathcal{Y}}^{',\text{rel}}_{(-t,t)}$ is the relative commutant of the bulk algebra ${\mathcal{Y}}_{(-t,t)}$. The dark red shaded region, which includes the light shaded region, describes the commutant ${\mathcal{Y}}'_{(-t,t)}$.
• Figure 3: Illustration of causal wedge reconstruction for a timeband algebra in the interval $(-\pi/2,\pi/2)$ in empty AdS, which reconstructs a whole Cauchy surface and thus contains the entire algebra ${\mathcal{B}}({\mathcal{H}})$.
• Figure 4: a) Spectral function $\rho(\omega)$ of the SYK model for varying coupling $J$ at $q=4$, $\beta=1$ with $N=10^5$ points. b) Logarithmic plot of the data in a). The subplot demonstrates how the asymptotics of the numerical data match the polynomial decay of the closed form conformal solution Eq. \ref{['eq:rho_SYK']} for $J=1000$.
• Figure 5: A qualitative demonstration of the phase diagram of the spherical $p$-spin model as a function of the global parameters $\{J, T, M_p\}$. For the purposes of our work, the transition between the spin liquid and spin glass phase is determined by the replica symmetry breaking: $u=0$ and $m=1$ corresponds to the spin liquid phase, while for $u>0$ and $m<1$ the state is in the spin glass regime. The dashed arrows show the two adiabatic paths (a) and (b) we followed in our calculation.

Theorems & Definitions (8)

• Definition 1: cf. Definition 2.2 in Gesteau:2024rpt
• Definition 2: cf. Definition 2.6 and 2.7 in Gesteau:2024rpt
• Theorem 1: cf. Proposition 2.12 in Gesteau:2024rpt
• Theorem 2: cf. Proposition 2.13 in Gesteau:2024rpt
• Theorem 3: cf. Proposition 2.9 in Gesteau:2024rpt
• Theorem 4: cf. Proposition C.9 in Gesteau:2024rpt
• Theorem 5: cf. Lemma 20 in Furuya:2023fei
• Proposition 1