[Paper Review] A Solvable Mean Field Model of a Gaussian Spin Glass

Adriano Barra, Giuseppe Genovese+2 more|arXiv (Cornell University)|Sep 19, 2011

Theoretical and Computational Physics16 references3 citations

TL;DR

This paper introduces and rigorously solves a fully Gaussian mean field spin glass model with i.i.d. Gaussian couplings and soft Gaussian spins, regularizing the divergent free energy via a quadratic constraint. It proves the replica symmetric (RS) solution is exact in the thermodynamic limit, establishing the free energy equals its RS expression across the entire phase diagram via replica symmetry breaking analysis and sum rules.

ABSTRACT

We introduce a mean field spin glass model with gaussian distribuited spins and pairwise interactions, whose couplings are drawn randomly from a normal gaussian distribution too. We completely control the main thermodynamical properties of the model (free energy, phase diagram, fluctuations theory) in the whole phase space. In particular we prove that in thermodynamic limit the free energy equals its replica symmetric expression.

Motivation & Objective

• To construct a well-defined mean field spin glass model with continuous (Gaussian) spins and quenched Gaussian couplings.
• To resolve the divergence issue in the free energy integral arising from unbounded Gaussian spins through a regularization procedure.
• To rigorously establish the existence of the thermodynamic limit of the free energy in this model.
• To prove that the replica symmetric approximation exactly captures the true free energy, despite the possibility of replica symmetry breaking.
• To provide a complete characterization of the phase diagram, critical behavior, and fluctuations of the order parameter.

Proposed method

• Introduce a fully Gaussian spin glass model with Hamiltonian $ H_N(z,J) = -\frac{1}{\sqrt{2N}}\sum_{i,j} J_{ij} z_i z_j - h\sum_i z_i $, where $ z_i \sim \mathcal{N}(0,1) $ and $ J_{ij} \sim \mathcal{N}(0,1) $.
• Regularize the partition function via an additional term $ -\frac{\beta^2}{4N}(\sum z_i^2)^2 $ to ensure convergence of the Gaussian integral.
• Introduce a parameter $ \lambda $ to control the effective variance of the soft spins, allowing for generalization of the free energy expression.
• Use the replica method with a replica symmetric ansatz and derive a sum rule relating the quenched free energy to its replica symmetric approximation and an error term.
• Apply the replica symmetry breaking (RSB) scheme and derive a Parisi-type equation, showing the RSB bound coincides with the RS bound.
• Establish a lower bound for the free energy that matches the RS expression, proving the RS solution is exact.

Experimental results

Research questions

• RQ1 Can a mean field spin glass model with continuous Gaussian spins and Gaussian couplings be rigorously regularized to yield a well-defined thermodynamic limit?
• RQ2 Does the replica symmetric solution accurately describe the free energy of the Gaussian spin glass model, or is replica symmetry breaking necessary?
• RQ3 What is the phase diagram of the Gaussian spin glass, and how does the critical line separating ergodic and non-ergodic phases depend on temperature and variance control?
• RQ4 How do fluctuations in the order parameter and in the free energy behave, and what is their relation to the validity of the replica symmetric approximation?
• RQ5 What is the relationship between the Gaussian spin glass and the spherical spin glass, and does the RS solution persist due to concentration of measure or structural equivalence?

Key findings

• The thermodynamic limit of the free energy exists and is well-defined after regularization via a quadratic constraint on the spin norm.
• The replica symmetric approximation exactly equals the true quenched free energy, as proven by showing the RSB upper bound matches the RS lower bound.
• The critical line separating the high-temperature ergodic phase from the low-temperature non-ergodic phase is given by $ \beta = 1 - \lambda $, with $ \lambda $ controlling the effective spin variance.
• The free energy expression in the high-temperature phase is $ A^g_{RS} = -\frac{1}{2}\log(1 - \lambda) $ for $ \beta < 1 - \lambda $, and $ A^g_{RS} = -\frac{1}{2}\log\beta + \frac{\beta \bar{q}}{2} + \frac{\beta^2 \bar{q}^2}{4} $ for $ \beta > 1 - \lambda $, with $ \bar{q} = \frac{\beta - (1 - \lambda)}{\beta^2} $.
• The model exhibits a continuous phase transition at $ \beta = 1 - \lambda $, with the order parameter $ R^2 $ undergoing a discontinuous jump in its derivative, indicating a second-order transition.
• The equivalence between the Gaussian spin glass and the spherical spin glass is reinforced by identical replica symmetric structure and overlap distribution, suggesting deep structural similarity despite different spin constraints.