🔬 condensed matter

Fluctuation-dissipation violations in mean-field non-reciprocal spin glasses

This paper analytically and numerically demonstrates that non-reciprocal couplings in the spherical Sherrington-Kirkpatrick model generically violate the fluctuation-dissipation theorem due to broken detailed balance rather than aging, leading to faster and oscillatory dynamics that can be understood through distinct solvable limits.

Original authors: Ot Garcés, Demian Levis

Published 2026-07-29

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CC BY 4.0

Original authors: Ot Garcés, Demian Levis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where everything is connected, like a giant, invisible web of friends influencing each other's moods. In physics, scientists study these webs to understand how complex systems—like the neurons in your brain, the flocking of birds, or even the stock market—behave. Usually, they assume these systems follow a rule called "detailed balance," which is like a perfect game of catch: if you throw a ball to a friend, they throw it back with the exact same force. This rule keeps things calm and predictable, eventually settling into a peaceful, balanced state called equilibrium. But what happens when the rules change? What if your friend catches your ball but throws it back slightly differently, or maybe even throws it to someone else entirely? This is called "non-reciprocity," and it's the secret ingredient in many real-world systems that never seem to settle down, from active bacteria to artificial intelligence. Scientists have long wondered: if these systems break the rules of equilibrium, do they just get stuck in a messy, frozen state (like a traffic jam that never moves), or do they find a new, weird kind of rhythm?

This paper dives into that mystery by looking at a mathematical model called the spherical Sherrington-Kirkpatrick (sSK) model. Think of this model as a giant, perfectly connected party where every guest (a "spin") is constantly chatting with every other guest. The researchers asked: What happens if we make the conversations non-reciprocal? Specifically, they wanted to see how the "Fluctuation-Dissipation Theorem" (FDT) behaves. In simple terms, the FDT is a golden rule that links how much a system wiggles on its own (fluctuations) to how much it reacts when you poke it (dissipation). In a calm, balanced world, these two are perfectly matched. But in the messy, non-reciprocal world, the authors found that this rule breaks, but not in the way everyone expected. They discovered that even when the system is moving fast and settling down quickly (not stuck in a frozen state), the golden rule still breaks. Instead of getting stuck in a slow, aging traffic jam, the system finds a new, steady dance where it wiggles and reacts in a mismatched, non-equilibrium way.

The authors explored this by solving complex equations for three different types of "conversations" between the guests:

The Perfect Reciprocal Party (Symmetric): Here, everyone throws the ball back exactly as they caught it. This is the classic, well-known scenario. The team confirmed that if the party gets too intense (low temperature), the guests get stuck in a chaotic, frozen mess where time seems to stop, and the golden rule breaks because the system is "aging" (getting stuck).
The Random Party (Uncorrelated): Here, the conversations are random; what A says to B has no relation to what B says to A. Surprisingly, the authors found that even if the party gets intense, the guests never get stuck. They keep moving, but they break the golden rule. The system settles into a steady state where the wiggles and reactions don't match, but it happens quickly and smoothly, without the slow, freezing aging seen in the first case.
The Opposite Party (Antisymmetric): Here, if A pushes B, B pushes A in the exact opposite direction. This creates a wild, oscillating dance. The guests don't just settle; they start swaying back and forth in a rhythmic pattern, like a pendulum. Even though they are moving and dancing, they still break the golden rule. The system is stable and never freezes, but it lives in a constant state of non-equilibrium, driven by these opposing forces.

By mixing these scenarios, the researchers showed that as you introduce more "non-reciprocity" (making the interactions less equal), the system speeds up. It stops aging and starts dancing or swaying. The key takeaway is that breaking the rules of equilibrium doesn't always mean getting stuck in a slow, frozen state. Sometimes, it means the system finds a new, fast, and rhythmic way to exist where the old rules of balance simply don't apply. This helps scientists understand how complex, non-reciprocal systems—from brains to ecosystems—can stay active and dynamic without falling apart or freezing over. The authors used both mathematical proofs for the extreme cases and computer simulations for the middle ground, showing that these weird, non-equilibrium dances are a real and robust feature of nature's more chaotic systems.

Technical Summary: Fluctuation–dissipation violations in mean-field non-reciprocal spin glasses

Problem Statement
The paper addresses the out-of-equilibrium dynamics of disordered systems characterized by non-reciprocal interactions, where the influence of one degree of freedom on another is not matched by an equal and opposite response. While equilibrium behavior is governed by a free-energy landscape satisfying Detailed Balance (DB), many complex systems (e.g., active matter, neural networks, ecological communities) operate far from equilibrium due to broken DB. A central open question is how the interplay between disorder and non-reciprocity affects dynamical properties, specifically the validity of the Fluctuation-Dissipation Theorem (FDT). In standard glassy systems, FDT violations are typically associated with ergodicity breaking and aging (diverging time scales). However, it remains unclear whether FDT violations can occur in non-equilibrium steady states (NESS) with finite relaxation times and without aging, purely due to the breakdown of DB.

Methodology
The authors investigate the spherical Sherrington-Kirkpatrick (sSK) model extended to include non-reciprocal, asymmetric couplings. The system consists of $N$ soft spins $\{\sigma_i\}$ evolving via Langevin dynamics with a spherical constraint $\sum \sigma_i^2 = N$ . The interactions are defined by random couplings $J_{ij}$ decomposed into symmetric ( $J^s_{ij}$ ) and antisymmetric ( $J^{as}_{ij}$ ) components, controlled by an asymmetry parameter $\Gamma \in [-1, 1]$ :

$\Gamma = 1$ : Reciprocal (standard sSK).
$\Gamma = 0$ : Uncorrelated couplings ( $J_{ij}J_{ji} = 0$ ).
$\Gamma = -1$ : Purely antisymmetric ( $J_{ij} = -J_{ji}$ ).

The study employs Dynamical Mean-Field Theory (DMFT) to derive exact equations of motion (Schwinger-Dyson or Cugliandolo-Kurchan equations) for the two-time correlation function $C(t, t')$ and the response function $G(t, t')$ . The authors analyze the stability of Time-Translational Invariant (TTI) solutions, where observables depend only on the time difference $\tau = t - t_w$ in the limit of large waiting times $t_w$ . They analytically solve the asymptotics of these functions in the three representative limits ( $\Gamma = 1, 0, -1$ ) and perform numerical integrations of the full two-time equations for generic $\Gamma$ to interpolate between these cases.

Key Contributions and Results

Stability of TTI Solutions:
- Symmetric Case ( $\Gamma = 1$ ): The TTI solution is stable only for $\beta < \beta_c^{SK} = 1/g$ . For $\beta > \beta_c^{SK}$ , the system enters an aging regime characterized by ergodicity breaking and power-law decays.
- Non-Reciprocal Cases ( $\Gamma < 1$ ): The presence of non-reciprocity restores the stability of TTI solutions at any finite temperature, even for $\beta > \beta_c^{SK}$ . The system does not exhibit aging; instead, it reaches a non-equilibrium steady state.
Analytical Solutions in Limits:
- Uncorrelated ( $\Gamma = 0$ ): The correlation and response functions decay exponentially. The Fluctuation-Dissipation Ratio (FDR), $X(\tau)$ , decays exponentially from 1 to 0. The system satisfies FDT at short times but violates it at long times, despite having a finite relaxation time and no aging.
- Purely Antisymmetric ( $\Gamma = -1$ ): The Lagrange multiplier freezes to $\beta^{-1}$ . The dynamics exhibit oscillatory behavior modulated by exponential decay, driven by the purely imaginary spectrum of the coupling matrix. The correlation function involves Bessel functions ( $J_1$ ). Similar to the $\Gamma=0$ case, the system satisfies FDT at short times but violates it asymptotically ( $X \to 0$ ) at long times, without aging.
Generic Asymmetry:
- Numerical results for intermediate $\Gamma$ show a smooth interpolation between the solvable limits.
- Asymmetry accelerates the decay of correlations.
- For $\Gamma < 0$ , oscillatory dynamics emerge, with the amplitude of oscillations decaying as $\tau^{-3/2}$ .
- The FDT violation is generic for any $\Gamma < 1$ at finite temperature.
Mechanism of FDT Violation:
The paper distinguishes two distinct origins of FDT violations:
- Ergodicity Breaking (Aging): In the symmetric sSK model ( $\Gamma=1, \beta > \beta_c$ ), violations arise from a hierarchy of diverging time scales.
- Broken Detailed Balance (Non-Aging): In non-reciprocal models ( $\Gamma < 1$ ), violations occur in a stationary, ergodic state with a single characteristic time scale. The violation is driven purely by the local breakdown of DB, not by the trapping in a complex energy landscape.

Significance
The authors claim that this work provides a reference framework for understanding the dynamics of disordered, non-reciprocal systems. The primary significance lies in disentangling the origins of FDT violations: demonstrating that such violations are not exclusive to aging glassy systems but are a generic feature of non-reciprocal interactions in steady states. The analytical solutions derived for the $\Gamma = 0$ and $\Gamma = -1$ limits serve as benchmarks for the dynamics of systems with asymmetric interactions, including neural networks, active matter, and ecological communities. The results clarify that non-reciprocity can restore stationarity (TTI) while simultaneously preventing the system from reaching thermodynamic equilibrium, leading to persistent FDT violations.

Technical Summary: Fluctuation–dissipation violations in mean-field non-reciprocal spin glasses

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