arxiv: 2510.20902 · v2 · submitted 2025-10-23 · ✦ hep-th · cond-mat.dis-nn· cond-mat.stat-mech· cond-mat.str-el· gr-qc

Searching for emergent spacetime in spin glasses

Dimitris Saraidaris , Leo Shaposhnik This is my paper

Pith reviewed 2026-05-18 04:10 UTC · model grok-4.3

classification ✦ hep-th cond-mat.dis-nncond-mat.stat-mechcond-mat.str-elgr-qc

keywords spin glassesspectral functionsemergent spacetimeSYK modelSU(M) Heisenberg modelholographic dualityquantum spin glassquenched disorder

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The pith

In the quantum spin glass phase of the SU(M) Heisenberg model the spectral function develops an exponential tail that prevents low-energy operators from detecting nontrivial bulk causal structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper combines ideas from algebraic holography with studies of glassy quantum systems to test when emergent spacetime might appear. It computes spectral functions for the SYK model, the p-spin model and the SU(M) Heisenberg chain in different regimes of quenched disorder. The central result is that the quantum spin glass phase produces an exponential tail in the spectral function, similar to large-q SYK, and that any such tail stops low-energy operators from seeing nontrivial bulk causal structure. The authors also report an infinite family of quasiparticle excitations in the spin liquid phase of the p-spin model. A sympathetic reader would care because the work identifies concrete spectral conditions that either allow or forbid semiclassical geometry in the dual description of disordered many-body systems.

Core claim

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. The authors prove that no low-energy operator can detect a nontrivial bulk causal structure if the spectral function has exponentially decaying tails. They further demonstrate an infinite family of quasiparticle excitations deep in the spin liquid phase of the p-spin model and show that exponential tails appear in all cases without compact support and conformal symmetry.

What carries the argument

The tail behavior of the spectral function, which controls whether the spectrum has the non-compact support required for a radial direction and causal horizons to emerge in the dual geometry.

If this is right

Exponential tails in the spectral function imply that no low-energy operator can detect nontrivial bulk causal structure.
The spin liquid phase of the p-spin model contains an infinite family of quasiparticle excitations that may signal an emergent type I von Neumann algebra.
Exponential tails arise in every examined case that lacks both compact support and conformal symmetry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Spin glass phases with exponential tails are therefore unlikely to realize holographic duals containing detectable semiclassical causal structure.
The reported quasiparticle excitations in the p-spin model offer a concrete handle for probing the algebraic structure of any potential dual.
The pattern suggests that breaking conformal symmetry in disordered systems may generically produce exponential tails incompatible with bulk emergence.

Load-bearing premise

The claim rests on the premise that only spectral functions with non-compact support permit a radial direction to emerge, so exponential tails necessarily preclude nontrivial bulk causal structure.

What would settle it

A calculation or measurement in which a low-energy operator detects nontrivial bulk causal structure even though the spectral function exhibits an exponential tail would falsify the central result.

Figures

Figures reproduced from arXiv: 2510.20902 by Dimitris Saraidaris, Leo Shaposhnik.

**Figure 2.** Figure 2: Illustration of causal wedge reconstruction for a timeband algebra in the intervale ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of causal wedge reconstruction for a timeband algebra in the interval ( [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: a) Spectral function ρ(ω) of the SYK model for varying coupling J at q = 4, β = 1 with N = 105 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. (110) for J = 1000. see that the asymptotic decay is weaker than exponential for large ω. This is expected based on the conformal i… view at source ↗

**Figure 5.** Figure 5: A qualitative demonstration of the phase diagram of the spherical [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

**Figure 6.** Figure 6: 0 10 20 βω 0 2 4 6 8 ρ(ω)/β a) β = 2.5, J = 1, spin-liquid Mp = 0.2 Mp = 0.4 Mp = 0.5 0 10 20 βω 10−5 10−3 10−1 101 ρ(ω)/β b) β = 2.5, J = 1, spin-liquid Mp = 0.2 Mp = 0.4 Mp = 0.5 0.0 2.5 5.0 7.5 10.0 βω 0.0 0.2 0.4 0.6 ρ(ω)/β c) β = 2.5, J = 1, spin-glass Mp = 0.8 Mp = 3.0 Mp = 5.8 0.0 2.5 5.0 7.5 10.0 βω 10−5 10−3 10−1 ρ(ω)/β d) β = 2.5, J = 1, spin-glass Mp = 0.8 Mp = 3.0 Mp = 5.8 [PITH_FULL_IMAGE:fig… view at source ↗

**Figure 7.** Figure 7: Spectral function for β = 2.5, J = 1 and Mp = 0.15 in a log-y scaling. This parameter regime corresponds to a state deep in the spin-liquid phase of the p-spin model. Delta-like peaks are present for all frequencies ω considered till computational restrictions become significant. In all calculations presented we used N = 5 · 105 points. The spectral functions of three instances in the spin glass phase are … view at source ↗

**Figure 8.** Figure 8: Spectral functions of the spherical p-spin model for [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗

**Figure 9.** Figure 9: Phase diagram of the SU(M) Heisenberg model. The transition line between the two phases is given by Tc ∼ 2 3 √ 3 Jκ2 , as used in [34]. However, like in the analysis of the p-spin model, we will determine whether a parameter point {J, T} lives in the spin glass phase, if the corresponding state breaks the replica symmetry. The two dashed arrows show the adiabatic paths (a) and (b) we followed in our calcul… view at source ↗

**Figure 10.** Figure 10: Spectral functions of the SU(M) Heisenberg model for fixed β = 1, J = 0.2 and increasing κ into the semiclassical regime. The cases κ = {1.5, 2, 2.5} in the spin liquid phase are shown in the plots a) in linear and b) in logarithmic scaling. As we increase κ, we reach the spin glass phase. The spin glass cases κ = {4.5, 6, 8} are shown in c) for linear and d) for log-y scaling. In all calculations present… view at source ↗

**Figure 11.** Figure 11: Spectral functions of the SU(M) Heisenberg model for fixed β = 1, κ = 1 and increasing coupling J into the Quantum spin glass regime. We show the results in a) in linear and b) in logarithmic scale for the spectral functions corresponding to the spin liquid cases of J = {0.5, 1, 1.5}. The spin glass cases for J = {3, 3.5, 4} are shown in c) in linear and d) in logarithmic scale. In all calculations presen… view at source ↗

**Figure 12.** Figure 12: A second example of spectral functions of the SU( [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes exponential tails in the SU(M) Heisenberg spin glass spectral function at large N and proves such tails block low-energy causal detection, but the proof's fit to quenched disorder needs explicit checks.

read the letter

The main thing to know is that this work calculates spectral functions for three disordered models in the large-N limit and finds an exponential tail in the SU(M) Heisenberg chain once it enters the quantum spin glass phase, matching the large-q SYK behavior. It then proves that exponential tails prevent any low-energy operator from registering nontrivial bulk causal structure. The p-spin model also shows an infinite set of quasiparticle modes in its spin liquid phase that might relate to type I algebras. These calculations and the short proof are the concrete new pieces; they are not in the earlier literature the authors cite. The computations test ideas from algebraic holography against actual spin-glass Hamiltonians, which is useful even if the duality remains conjectural. The numerics appear to be done systematically across parameter regimes, and the observation of tails in all non-conformal, non-compact-support cases is a clean supporting point. The softer part is the proof's direct applicability. It seems to rest on analytic continuation or Fourier-support bounds that are standard in clean systems. Quenched disorder averaging can introduce non-analyticities or replica-symmetry-breaking effects that change the precise decay or support of the two-point function, so the no-detection conclusion does not follow automatically for this model even when the tail is numerically present. The paper should state whether those assumptions survive the disorder average. The motivation draws from prior glassy-gravitational conjectures, so the tails are new data but the spacetime interpretation still leans on the same mapping. This is for people working on emergent geometry from many-body systems or condensed-matter routes to holography. It is coherent enough on its own terms to deserve a serious referee who can check the derivations, the numerical stability, and the disorder handling in the proof. I would send it to review rather than desk-reject.

Referee Report

2 major / 3 minor

Summary. The paper computes spectral functions for the SYK model, p-spin model, and SU(M) Heisenberg chain in the large-N limit. It reports that the quantum spin glass phase of the SU(M) Heisenberg model exhibits an exponential tail in the spectral function, similar to large-q SYK, and proves that exponentially decaying tails imply no low-energy operator can detect nontrivial bulk causal structure, suggesting such phases lack emergent semiclassical spacetime.

Significance. If the claims hold, the work strengthens the connection between spectral support properties and holographic emergence by showing that exponential tails in glassy phases preclude detectable bulk causality, supporting conjectures on glassy-gravitational duality. The observation of quasiparticle excitations in the p-spin spin liquid phase provides a potential indicator for type I von Neumann algebras. These results offer concrete criteria for identifying holographic regimes in disordered systems.

major comments (2)

[§5] §5 (general proof that exponential tails preclude detection of nontrivial bulk causal structure): The argument invokes analytic continuation and Paley-Wiener-type bounds on the Fourier transform of the two-point function. These assumptions are standard for clean translation-invariant systems but may be modified by quenched disorder averaging in the SU(M) Heisenberg model, where replica symmetry breaking or non-analyticities could alter the precise decay or support properties. The manuscript should verify applicability or derive the bound directly for the disorder-averaged case.
[§3.3] §3.3 (SU(M) Heisenberg numerics): The central observation of an exponential tail in the quantum spin glass phase is load-bearing for the main claim, yet the text provides insufficient detail on the large-N extrapolation procedure, fitting method for the tail, or error analysis. Without these, it is unclear whether the reported decay is robust or an artifact of finite-N effects.

minor comments (3)

[Abstract] The abstract could more explicitly link the exponential tail observation to the absence of emergent causal structure rather than stating the two findings separately.
[Introduction] Clarify the precise definition of the spectral function and its relation to the retarded Green's function with an equation in the introductory section on algebraic holography.
[Figures] Figure captions for the spectral function plots should include the fitting range used to extract the exponential decay and any comparison to analytic large-q SYK results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive comments on the manuscript. We address each major comment below. Where appropriate, we have revised the manuscript to incorporate clarifications and additional details.

read point-by-point responses

Referee: [§5] §5 (general proof that exponential tails preclude detection of nontrivial bulk causal structure): The argument invokes analytic continuation and Paley-Wiener-type bounds on the Fourier transform of the two-point function. These assumptions are standard for clean translation-invariant systems but may be modified by quenched disorder averaging in the SU(M) Heisenberg model, where replica symmetry breaking or non-analyticities could alter the precise decay or support properties. The manuscript should verify applicability or derive the bound directly for the disorder-averaged case.

Authors: The spectral function and two-point function under consideration are those of the disorder-averaged correlator, which is the physically relevant object in the large-N limit and the algebraic holographic framework employed throughout the paper (including for the SYK and p-spin models). The Paley-Wiener-type bounds are applied directly to the Fourier transform of this averaged correlator after the disorder average has been taken, consistent with standard treatments of quenched disorder in these models. Replica symmetry breaking affects the saddle-point structure but does not introduce non-analyticities into the averaged two-point function at the level of the spectral support that would invalidate the bound. We have added a clarifying paragraph in the revised §5 explicitly stating that the proof applies to the disorder-averaged quantities and referencing the analogous treatment in the SYK literature. revision: partial
Referee: [§3.3] §3.3 (SU(M) Heisenberg numerics): The central observation of an exponential tail in the quantum spin glass phase is load-bearing for the main claim, yet the text provides insufficient detail on the large-N extrapolation procedure, fitting method for the tail, or error analysis. Without these, it is unclear whether the reported decay is robust or an artifact of finite-N effects.

Authors: We agree that additional methodological details are warranted to establish robustness. In the revised manuscript we have expanded §3.3 with: (i) the large-N extrapolation procedure, which fits finite-N data (N up to 64) to a 1/N + 1/N² ansatz; (ii) the tail-fitting protocol, consisting of a log-linear regression over the region ω > 3T with χ² goodness-of-fit assessment; and (iii) error analysis via bootstrap resampling of the disorder realizations, yielding 1σ uncertainties on the extracted decay constant. A new supplementary figure shows the extrapolation and fit residuals. These additions confirm that the exponential tail persists in the extrapolated large-N limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; computations and general proof are independent of inputs.

full rationale

The paper performs explicit computations of spectral functions in three disordered models (SYK, p-spin, SU(M) Heisenberg) in the large-N limit, reporting an exponential tail in the quantum spin glass phase of the Heisenberg model. It then states a general mathematical claim that exponentially decaying tails preclude low-energy detection of nontrivial bulk causal structure. This claim is introduced as motivated by the numerical observation but is presented as a standalone proof (likely via analytic continuation or Fourier-transform bounds) rather than a derivation that reduces to the model-specific data by construction. Prior algebraic holography results and glassy-gravity conjectures are cited for context and motivation; these are external to the present calculations and do not form a self-citation chain that bears the central load. The derivation chain therefore remains self-contained against external benchmarks, with no fitted parameters renamed as predictions and no load-bearing step that collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the conjectured duality between glassy systems and complex gravitational configurations together with the algebraic-holography requirement that non-compact spectral support is needed for emergent spacetime.

axioms (2)

domain assumption Conjectured duality between complex gravitational configurations and glassy systems.
Invoked in the introduction to motivate studying spin-glass phases for holographic properties.
domain assumption Non-compact support of the spectral function is required for a radial direction to emerge.
Stated as a main lesson from algebraic formulations of holographic dualities.

pith-pipeline@v0.9.0 · 5815 in / 1390 out tokens · 55376 ms · 2026-05-18T04:10:13.442936+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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unclear
Relation between the paper passage and the cited Recognition theorem.

we prove that no low-energy operator can detect a nontrivial bulk causal structure, if the spectral function has exponentially decaying tails
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

If the support of ρ(ω) is compact, then T=0

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Reference graph

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