Searching for emergent spacetime in spin glasses

This paper investigates the emergence of semiclassical spacetime in many-body quantum systems with quenched disorder by computing their spectral functions, finding that the SU(M) Heisenberg model exhibits exponential tails similar to the SYK model and the p-spin model displays infinite quasiparticle excitations, while ultimately proving that such exponential spectral tails preclude low-energy operators from detecting nontrivial bulk causal structures.

The Gist

Imagine the universe as a giant, complex puzzle. For decades, physicists have been trying to solve a specific part of this puzzle: How does the smooth, flowing fabric of space and time (spacetime) emerge from the chaotic, jittery behavior of tiny quantum particles?

This paper is like a detective story where the authors are searching for the "secret recipe" that turns a messy quantum system into a smooth universe. They are testing three different "kitchens" (mathematical models) to see which one produces the right kind of "soup" (spacetime).

Here is the breakdown of their investigation, using simple analogies:

1. The Big Idea: The "Glass" Connection

Usually, when we think of space and time, we think of something smooth, like a calm ocean. But at the quantum level, things are chaotic, like a stormy sea.

The authors noticed something strange: Spin Glasses (a type of disordered magnetic material where atoms are confused and can't decide which way to point) behave very similarly to Black Holes and complex gravitational systems.

• The Analogy: Imagine a room full of people trying to decide where to sit. In a normal room, they sit in an orderly pattern. In a "spin glass," everyone is confused, and there are millions of different ways they can sit that are all equally "messy."
• The Hypothesis: The authors suspect that these confused, "glassy" quantum systems might actually be the hidden code that builds our universe's spacetime.

2. The Detective Tool: The "Spectral Fingerprint"

To test if a system creates spacetime, the authors look at its Spectral Function.

• The Analogy: Think of a musical instrument. If you hit a drum, it makes a sound. If you analyze that sound, you get a "fingerprint" showing which notes (frequencies) are present.
• The Rule: The authors found a golden rule:
  • If the fingerprint stops abruptly (like a song that cuts off suddenly), no spacetime emerges. It's like a closed box.
  • If the fingerprint fades out slowly, stretching on forever (like a song that gets quieter and quieter but never quite stops), spacetime might emerge. This "long tail" is the signature of a new dimension opening up.

3. The Three Test Kitchens

The authors ran simulations on three different quantum models to see what their "fingerprints" looked like.

Kitchen A: The SYK Model (The Star Student)

• What it is: A famous model already known to be a good candidate for a holographic universe.
• The Result: It produces a long, exponential tail. The sound fades out slowly but surely.
• Verdict: Pass. This confirms that this model successfully "grows" a spacetime. It's the control group that proves the method works.

Kitchen B: The Spherical p-Spin Model (The Confused One)

• What it is: A model that has two modes: a "Spin Liquid" (chaotic but fluid) and a "Spin Glass" (frozen and stuck).
• The Result:
  • Spin Liquid Mode: It behaves like the SYK model with a nice long tail. Pass.
  • Spin Glass Mode: The fingerprint suddenly cuts off (compact support). It's like the song stops abruptly.
  • Verdict: Fail. When this system gets stuck in a "glassy" state, it loses the ability to create a smooth spacetime. It's too rigid.

Kitchen C: The SU(M) Heisenberg Model (The Surprise Winner)

• What it is: Another complex magnetic model with liquid and glass phases.
• The Result:
  • Spin Liquid Mode: Like the others, it has a long tail. Pass.
  • Semiclassical Spin Glass: Like the p-spin model, it cuts off abruptly. Fail.
  • Quantum Spin Glass (The Surprise): Deep inside the glassy phase, under specific conditions, the fingerprint does not cut off. It has that magical, long exponential tail!
  • Verdict: Pass (with a twist). This is the paper's biggest discovery. They found a specific type of "Quantum Spin Glass" that can create spacetime, even though it's stuck in a glassy state. This is a rare and exciting find.

4. The "Low-Energy Blind Spot"

The authors also discovered a limitation in our current tools.

• The Problem: They proved that if you only look at the "low-energy" notes (the deep, quiet sounds), you might miss the spacetime entirely.
• The Analogy: Imagine trying to hear a whisper in a noisy room. If you only listen to the quietest sounds, you might think the room is empty. But if you listen to the high-pitched, fast vibrations, you realize the room is actually full of activity.
• The Lesson: To see the "bulk" (the hidden spacetime), you need to look at the high-energy, fast-moving parts of the system. Standard tools that ignore these high frequencies are "blind" to the emergence of space.

Summary: What Does This Mean?

This paper is a map for finding new universes inside quantum systems.

1. The Secret: To get a universe, a quantum system needs a "fingerprint" that fades out slowly (exponential decay), not one that stops abruptly.
2. The Discovery: While most "glassy" systems fail to create space, the authors found a special Quantum Spin Glass that succeeds.
3. The Future: This suggests that the messy, frozen states of matter we see in the real world might actually be hiding the seeds of a hidden, extra dimension of space, provided we look at them with the right "lens" (high-energy analysis).

In short: Chaos can create order, and sometimes, a frozen mess of quantum particles is actually the blueprint for a new universe.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the question of whether complex many-body quantum systems with quenched disorder (specifically spin glasses) can exhibit holographic duality, meaning they possess an emergent semiclassical spacetime with a non-trivial radial direction.

• Context: Recent work in high-energy physics suggests a connection between the "glassy" dynamics of complex gravitational configurations (e.g., multi-horizon black hole molecules) and spin glass systems.
• Theoretical Framework: The authors utilize the algebraic approach to holography (specifically the framework by Gesteau and Liu, GL). This approach posits that the emergence of a bulk spacetime is encoded in the properties of the boundary operator algebra at $N=\infty$.
• Key Diagnostic: The central diagnostic tool is the spectral function $\rho(\omega)$ of the two-point correlation function.
  • Compact Support: If $\rho(\omega)$ has compact support (zero beyond a maximum frequency), the system cannot support a non-trivial bulk causal structure (Theorem 1 of GL).
  • Polynomial Decay: If $\rho(\omega)$ decays polynomially, a bulk with infinite causal depth can emerge (Theorem 2).
  • Exponential Decay: The behavior of $\rho(\omega)$ with exponential tails was previously unaddressed by GL's theorems. The authors investigate whether this decay (common in chaotic systems) is compatible with emergent spacetime.

2. Methodology

The authors perform a comparative study of three large-$N$ models with quenched disorder, focusing on their spectral functions across different phases (Spin Liquid vs. Spin Glass).

• Models Studied:
  1. SYK Model: Served as a reference (known to be holographic in the large-$q$ limit).
  2. Spherical $p$-spin Model: A bosonic analog of SYK with distinct spin liquid and spin glass phases.
  3. SU($M$) Heisenberg Model: A model with both spin liquid and spin glass phases, known to be the motivation for the SYK model.
• Numerical Approach:
  • The authors solve the Schwinger-Dyson equations (SDEs) in the large-$N$ limit.
  • They employ a two-step numerical algorithm:
    1. Euclidean Algorithm: Solves for the imaginary-time Green's function and determines Replica Symmetry Breaking (RSB) parameters ($u, m$) to identify the phase (Spin Liquid vs. Spin Glass).
    2. Lorentzian Algorithm: Analytically continues the solution to real time to compute the retarded Green's function $G_R(\omega)$ and extract the spectral function $\rho(\omega) = 2\text{Im} G_R(\omega)$.
  • They use adiabatic continuation in parameter space (varying coupling $J$, temperature $T$, and mass/constraint parameters) to trace phase transitions.
• Theoretical Extension: The authors prove a new proposition regarding spectral functions with exponential decay to extend the GL framework.

3. Key Contributions and Results

A. Theoretical Extension: Exponential Decay and Bulk Causality

The authors prove that if a spectral function decays exponentially ($\rho(\omega) \sim e^{-\alpha \omega}$), no low-energy operator (smeared with functions growing at most polynomially in frequency) can detect a non-trivial bulk causal structure.

• Implication: Standard smearing functions used in QFT cannot distinguish between a trivial system and one with an emergent bulk if the spectral function has an exponential tail. This suggests that detecting emergent spacetime in such systems requires "super-polynomial" observables, which are physically unusual.

B. Results for the SYK Model

• Large $q$ Limit: Confirmed that the spectral function exhibits exponential decay with full support. This aligns with the expectation of a holographic dual, suggesting that exponential decay is compatible with emergent spacetime, even though GL's original theorems (requiring polynomial decay) do not strictly apply.
• Finite $q$ (Large $J$): In the conformal limit, the spectral function decays polynomially, satisfying the conditions for emergent bulk causality.

C. Results for the Spherical $p$-spin Model

• Spin Liquid Phase:
  • Exhibits exponential decay of the spectral function (full support).
  • Deep Spin Liquid: At very low effective mass ($M_p$), the spectral function develops an infinite set of quasi-particle peaks (resembling delta functions) with exponentially decaying amplitudes. This hints at a potential Type I von Neumann algebra (finite depth parameter), though the exponential decay of the peaks prevents direct application of GL's Theorem 4.
• Spin Glass Phase:
  • The spectral function develops a "kink" followed by super-exponential decay, indicating compact support.
  • Conclusion: The spin glass phase of the $p$-spin model does not support an emergent bulk spacetime, as the compact support implies a trivial causal structure.

D. Results for the SU($M$) Heisenberg Model

• Spin Liquid Phase: Exhibits exponential decay with full support, consistent with the SYK and $p$-spin liquid results.
• Semiclassical Spin Glass Phase: Similar to the $p$-spin model, shows a "kink" and compact support, ruling out emergent spacetime.
• Quantum Spin Glass Phase ($\kappa \in (0, 1]$):
  • Crucial Finding: In this specific regime, the spectral function exhibits robust exponential decay with non-compact support.
  • This is the only spin glass regime studied that does not show signs of compact support.
  • Significance: This suggests that the Quantum Spin Glass phase of the SU($M$) Heisenberg model is a candidate for a holographic dual, potentially possessing an emergent radial direction.

4. Significance and Future Directions

• Refining Holographic Diagnostics: The paper demonstrates that the presence of a spin glass phase (often associated with complexity) does not automatically preclude holography. The specific nature of the spectral function's tail (exponential vs. compact) is the deciding factor.
• New Candidate for Holography: The identification of the Quantum Spin Glass phase in the SU($M$) Heisenberg model as a potential holographic system is a major finding. It challenges the notion that spin glasses are strictly dual to non-holographic, fragmented geometries.
• Limitations of Current Frameworks: The work highlights a gap in the algebraic holographic framework (GL). The framework currently cannot quantitatively handle systems with exponential spectral tails, despite their prevalence in chaotic systems like SYK.
• Future Work: The authors suggest investigating the chaotic dynamics (OTOCs) in the identified Quantum Spin Glass regime and exploring the limit of large $\kappa$ in the SU($M$) model, which might resemble the large-$q$ SYK limit.

Summary Table of Findings

Model	Phase	Spectral Function Behavior	Emergent Bulk?
SYK	Large $q$	Exponential decay (Full support)	Yes (Reference)
SYK	Large $J$	Polynomial decay	Yes
$p$-spin	Spin Liquid	Exponential decay (Full support)	Yes
$p$-spin	Deep Spin Liquid	Infinite peaks + Exponential decay	Uncertain (Type I hint)
$p$-spin	Spin Glass	Compact support (Kink + Super-exponential)	No
SU($M$)	Spin Liquid	Exponential decay (Full support)	Yes
SU($M$)	Semiclassical SG	Compact support	No
SU($M$)	Quantum SG	Exponential decay (Full support)	Yes (Candidate)

In conclusion, the paper establishes that while most spin glass phases are "non-holographic" due to compact spectral support, specific quantum regimes (like the Quantum Spin Glass of the SU($M$) model) retain the spectral properties necessary for an emergent spacetime, provided one can account for exponential decay in the algebraic framework.