Exploring Replica Symmetry Breaking and Topological Collapse in Spin Glasses with Quantum Annealing

TL;DR

The paper addresses empirical validation of replica symmetry breaking (RSB) in the Sherrington-Kirkpatrick spin glass and probes the topological limits of RSB via controlled vacancy dilution. It leverages quantum annealing to solve ground states up to $N=4000$, enabling extensive tests of Parisi's RSB predictions across diverse disorder realizations. Five observables confirm RSB: thermodynamic energy convergence to $E_{\infty} = -0.7633$, energy-fluctuation scaling with $\gamma \approx 0.739$, a chaos exponent $\theta \approx 0.51$, a hierarchical overlap distribution with $\sigma_q \approx 0.19$, and robustness to 36% network dilution; beyond a critical window, the hierarchy collapses through a cooperative avalanche in a Blume-Capel extension, converting the system to all vacancies. This work demonstrates that quantum computing can reveal emergent many-body phenomena and topological foundations of complexity, with implications for neural networks, optimization, and materials science, and establishes a framework for exploring phase structures with controllable topology.

Abstract

Replica symmetry breaking (RSB) underlies the complex organization of disordered systems, yet quantitative validation beyond $N \sim 100$ spins has remained computationally challenging. We use quantum annealing to access ground states of the Sherrington-Kirkpatrick model up to $N = 4000$ spins, enabling the most extensive test of Parisi's Nobel Prize-winning RSB solution to date. Five independent observables confirm RSB predictions: ground-state energies converge to Parisi's value with characteristic $N^{-2/3}$ corrections, energy fluctuations scale as $N^{-3/4}$ ($γ= 0.739 \pm 0.036$), the chaos exponent $θ= 0.51 \pm 0.02$ ($R^2 = 0.989$) confirms mean-field universality, the overlap distribution exhibits hierarchical structure ($σ_q = 0.19$), and the complexity remains invariant under 36\% network dilution. Beyond a critical threshold $0.8 < D_c < 0.9$, the hierarchy collapses discontinuously through a cooperative avalanche that converts the entire system to vacancies within a narrow parameter window $ΔD = 0.1$. These findings establish quantum computation as a tool for probing emergent many-body phenomena and uncover the topological foundations of complexity in disordered systems, with implications for neural networks, optimization, and materials science.

Figures (5)

• Figure 1: Initial benchmark with quantum and classical solvers. Ground-state energy per spin versus system size. The classical solver (red triangles) fails beyond $N \sim 100$. The quantum-classical hybrid (blue circles) extends computational reach to $N=4000$ with smooth convergence to Parisi's thermodynamic limit (green dashed line).
• Figure 2: Thermodynamic validation. Finite-size scaling of ground-state energies (blue circles) converging to Parisi's limit (green dashed line) with characteristic $N^{-2/3}$ corrections (red fit curve). Inset: Energy fluctuations scale as $N^{-3/4}$ with measured exponent $\gamma = 0.739 \pm 0.036$.
• Figure 3: Universality class identification. The RMS energy response to disorder perturbations scales as $N^{0.51 \pm 0.02}$ (blue data points, black fit), consistent with mean-field RSB ($\theta = 0.5$, red dashed line) and incompatible with droplet models ($\theta = 0.3$, green dashed line). The fit quality is $R^2 = 0.989$.
• Figure 4: Hierarchical landscape organization. Histogram of 225 pairwise overlaps between low-energy states exhibits the broad continuous structure ($\sigma_q = 0.19$) characteristic of replica symmetry breaking. Replica-symmetric theory would predict discrete peaks; the observed continuous spread confirms the ultrametric hierarchy.
• Figure 5: Discovery of topological collapse. RSB complexity $\sigma_q$ (blue curve, left axis) and vacancy density $x$ (red curve, right axis) versus dilution parameter $D$. Phase I shows $\sigma_q$ remaining invariant under 36% dilution. Phase II shows discontinuous collapse within $\Delta D = 0.1$ through a cooperative avalanche transition.