Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling

Gabriele Bandini; Giulio Biroli; Patrick Charbonneau; Andrea Gambassi

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Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling

Gabriele Bandini, Giulio Biroli, Patrick Charbonneau, Andrea Gambassi

arXiv:2608.31114v1Aug 31, 2026cond-mat.stat-mechstat.ML

TL;DR

Frustration defeats constructive cluster algorithms, leaving critical slowing down a challenge for local MCMC. This paper learns wavelet-conditioned distributions with WCRG and recursively samples configurations from coarse to fine scales. It reproduces key statistical properties across phases and reduces critical sampling complexity to O(log_2 L), while exposing accuracy limits from the learned energy model.

Problem
Constructive cluster algorithms fail even under weak frustration, while local MCMC suffers critical slowing down near continuous phase transitions.
Method
WCRG learns finite-dimensional wavelet conditional energies and recursively generates frustrated soft-spin configurations from coarse to fine scales.
Results
WCRG reproduces local-field distributions and structure factors across representative phases, with O(1) conditional sweeps per scale and overall O(log_2 L) critical sampling complexity.
Takeaways & Limitations
WCRG provides a complementary fast generative strategy for frustrated systems where conventional cluster constructions are ineffective.
Takeaways & Limitations
The strongly constrained antiphase is pathological for the reconstruction, and the learned hierarchy trades exact microscopic sampling for an approximate measure.

Abstract

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Cluster algorithms, such as the Swendsen--Wang and Wolff methods, are among the most successful MCMC methods for mitigating critical slowing down in statistical systems. These constructive cluster algorithms, however, fail in the presence of even extremely weak frustration. Here, we sidestep this fundamental limitation by learning rather than constructing the relevant clusters. Specifically, we use the wavelet conditional renormalization group (WCRG) sampling method to learn the probability distribution of collective fluctuations of a frustrated two-dimensional soft-spin model. Configurations are then generated recursively from coarse to fine scales by sampling conditional wavelet distributions. The WCRG method reproduces the main statistical properties of the system across different phases, including the local-field distribution and the structure factor. At an Ising-like critical point, the conditional dynamics remains decorrelated within $\mathcal{O}(1)$ sweeps at each scale, yielding an overall sampling complexity of $\mathcal{O}(\log_2 L)$, thus making WCRG much more efficient than standard local MCMC methods. These results show that learned multiscale sampling can overcome critical slowing down in frustrated systems for which conventional cluster algorithms fail. By assessing the sampling accuracy of different observables, we also clarify the main tradeoff of the WCRG method: the accuracy of the fast sampling scheme depends on the expressiveness of the energy-based model used to estimate the wavelet conditional distributions.

I. INTRODUCTION

The paper addresses critical slowing down and the failure of constructive cluster algorithms in frustrated systems by learning multiscale conditional distributions with WCRG. It evaluates this approach on frustrated two-dimensional soft-spin models, especially the sBNNNI model, and reports accurate statistical reproduction alongside logarithmic critical sampling complexity.

Motivation: Local MCMC decorrelation times scale as τ ∼ L^z, with z generally near two in two-dimensional systems and as high as six with disorder.The growing correlation length near continuous transitions limits local-update dynamics.
Motivation: Constructive cluster algorithms reduce critical slowing down in unfrustrated systems but fail even under weak frustration because efficient clusters cannot be identified constructively.The paper therefore asks whether the relevant clusters can instead be learned.
Approach: WCRG decomposes configurations through an orthogonal wavelet transform into coarse variables encoding long-wavelength structure and finer wavelet fluctuations.Learned conditional distributions are tied to specific reconstruction scales.
Approach: The method learns conditional distributions from equilibrium Monte Carlo configurations by score matching and recursively reconstructs microscopic configurations from coarse to fine scales.The coarsest field is sampled first, followed by conditional wavelet sampling at successively finer levels.
Results: For the sBNNNI model, WCRG reproduces local-field distributions and structure factors across phases, while achieving O(log_2 L) sampling complexity at the Ising-like critical point.Conditional updates remain decorrelated in O(1) sweeps at each scale, whereas local dynamics scales approximately as L^2.16.

II. MODELS

The study uses continuous-spin frustrated ϕ4 models on a square lattice, focusing on the sBNNNI case with competing interactions. Its phase diagram contains five regimes, distinguished mainly by structure-factor peak positions and finite-size scaling.

Model definition: The models use continuous spin variables ϕ0(i) ∈ R so that wavelet decompositions can act on real-valued fields.The site i is associated with lattice position ri = (xi, yi).
Model definition: The sBNNNI model has finite λ = 1, J = 1, equal axial frustrations κx = κy = κ, and tunable temperature T.The soft-spin ϕ4 theory approaches free quadratic behavior as λ → 0 and Ising spins as λ → ∞.
Phase diagram: The sBNNNI phase diagram contains five distinct phases identified using the structure factor and the scaling of its peak amplitude with system size.The analysis focuses on ferromagnetic, paramagnetic, modulated, incommensurate, and antiphase regimes.
Phase characterization: Ferromagnetic order has a q = 0 peak scaling as Sdiag(q = 0) ∼ L^2, while homogeneous paramagnetism has a finite q = 0 peak as L → ∞.These scaling behaviors distinguish ordered from disordered homogeneous regimes.
Phase characterization: The modulated paramagnetic, incommensurate, and antiphase phases are distinguished by finite-wavevector peaks at q > 0, 0 < q < π/2, and q = π/2, respectively.The incommensurate peak scales as Sdiag(q) ∼ L^(2−η), while the antiphase peak scales extensively.
Phase boundaries: Only the Ising-like FM–PM transition was quantitatively located by finite-size scaling; precise determinations of other phase transitions were not attempted.The FM–PM line was analyzed using magnetic susceptibility and second-moment correlation length with known Ising critical exponents.

III. WAVELET CONDITIONAL RENORMALIZATION GROUP

WCRG represents the microscopic distribution as a hierarchy of wavelet-conditioned distributions learned independently across scales. Sampling starts from the coarsest field and recursively reconstructs finer fields using learned conditional energies.

Multiscale decomposition: At each scale, an orthogonal wavelet transform maps a finer field ϕj−1 to a coarse field ϕj and fast wavelet variables ¯ϕj.The coarse lattice has linear size Lj = L/2^j.
Multiscale decomposition: The orthogonal transform is invertible, has unit Jacobian, and gives a one-to-one representation of the microscopic field by the coarsest field and all wavelet variables.The hierarchy is iterated down to LJ = 1 in the numerical applications.
Hierarchical distribution: The exact microscopic distribution factorizes into conditional wavelet probabilities across successive decompositions, enabling a hierarchical sampling representation.The cascade is intended to avoid critical slowing down because each conditional distribution is easy to sample.
Learned energies: WCRG approximates the coarsest and conditional energies with finite-dimensional parametrizations whose coefficients are learned independently by score matching.The conditional ansatz retains symmetry-allowed functions, including translationally invariant quadratic couplings and local potentials.
Recursive sampling: Synthetic configurations are generated by sampling the coarsest field, then sampling each learned wavelet conditional at fixed coarse variables and reconstructing finer fields.Conditional sampling uses a local Metropolis algorithm, and only the coarsest and conditional energies are needed explicitly.
Approximation tradeoff: The learned hierarchy samples an induced measure whose microscopic energy generally differs from the original energy because of the finite ansatz class.Reconstructing an explicit microscopic energy also requires approximating the induced free-energy contributions.
Wavelet choice: Haar wavelets are used for homogeneous regimes and the critical line, whereas DB4 wavelets are used for finite-wavevector modulations.The smoother, wider DB4 filters are selected for MPM, IC, and AF phases.

IV. NUMERICAL RESULTS

The numerical study tests WCRG against Monte Carlo across representative phase-diagram points, examines conditional-chain decorrelation at criticality, and evaluates configuration-wide observables.

Numerical tests: The numerical tests compare Monte Carlo and WCRG configurations across representative phases, analyze critical conditional-chain decorrelation, and assess magnetization and microscopic energy.These experiments target phase-dependent accuracy, critical sampling efficiency, and global-observable reproduction.

A. Wavelet sampling across the phase diagram

WCRG models were trained from equilibrium Monte Carlo configurations and used to generate synthetic configurations recursively from coarse to fine scales. Across phases, the reconstruction generally preserved local-field statistics and spatial structure, with the largest discrepancy in the antiferromagnetic regime.

Training and reconstruction: For each phase-diagram condition, 3000 equilibrium Monte Carlo configurations at L = 64 served as reference samples and training data.The configurations were generated using local single-site Metropolis updates.
Training and reconstruction: Synthetic configurations were generated recursively from coarse to fine scales using conditional WCRG models trained on scale-dependent configuration pairs.The method samples coarse fields first and reconstructs finer scales through conditional Monte Carlo sampling of wavelet variables.
Evaluation across phases: The evaluation compared microscopic configurations, the local-field distribution P(ϕ0(i)), and the diagonal structure factor Sdiag(q) to test single-site statistics and spatial correlations.The structure factor also probes the dominant ordering wavevector.
Evaluation across phases: Agreement was generally good across the tested phases, including reproduction of local-field statistics and small-wavevector peak structure in homogeneous phases and at criticality.In modulated phases, the synthetic samples retained the finite-wavevector peak of Sdiag(q).
Evaluation across phases: The antiferromagnetic regime showed the most pronounced discrepancy because coarse-grained fields largely averaged out its microscopic checkerboard modulation.The relevant information was concentrated in the finest wavelet degrees of freedom, reducing the effectiveness of the multiscale hierarchy.

B. Absence of critical slowing down

At the Ising-like critical point, local Monte Carlo dynamics slows algebraically with system size, while WCRG conditionally decorrelates wavelet variables in a bounded number of sweeps per scale. Because reconstruction proceeds across logarithmically many scales, the method avoids the additional critical scaling factor associated with local microscopic sampling.

Multiscale reconstruction: Each complete microscopic realization requires one conditional sampling stage for each wavelet level, rather than slowly decorrelating a single microscopic field through local propagation.WCRG generates configurations through separate top-down cascades, sampling coarse variables before progressively finer fluctuations.
Critical slowing down: zloc = 2.11(4) for local single-site dynamics, indicating algebraic growth of autocorrelation times with system size.This fit is compatible with the known two-dimensional Ising dynamic critical exponent z = 2.1665(12).
Conditional wavelet dynamics: O(1) conditional autocorrelation times show no systematic growth with either the microscopic size L or the coarse-field size Lj = L/2^j.The measured conditional observable is the second moment of the reconstructed field, averaged over 50 independent chains.
Sampling complexity: J = log2 L wavelet levels give a logarithmic sequential decorrelation depth for the full WCRG reconstruction.The hierarchy is iterated from the coarsest scale down to the microscopic scale.
Sampling complexity: The total number of elementary local update attempts remains linear in the number of microscopic degrees of freedom and avoids the additional L^z0 critical scaling factor of local sampling.This cost distinction separates logarithmic reconstruction depth from the work performed within conditional sweeps.

C. Configuration-wide observables

WCRG reproduces the magnetization distribution of synthetic configurations accurately, but the microscopic energy distribution shows a visible mismatch caused by approximating the exact multiscale effective energies.

Magnetization: Matching P(m) does not guarantee that the full hierarchy of microscopic multipoint correlation functions is reproduced.The magnetization distribution encodes moments determined by spatial integrals of multipoint correlations, but the converse implication fails.
Configuration-wide observables: The WCRG comparison uses absolute magnetization and intensive microscopic energy distributions from original and synthetic configurations at κ = 0.2, T = 1.0, and L = 64.These are configuration-wide observables evaluated along the Ising-like critical point.
Magnetization: The absolute magnetization distributions show excellent agreement between original and synthetic configurations.The coarse field represents the spatial average, and wavelet modes preserve total magnetization during reconstruction.
Energy: The intensive microscopic energy distribution presents a visible mismatch because synthetic configurations are not drawn from the exact microscopic Gibbs measure.The learned finite-dimensional conditional-energy ansatz approximates the exact effective energies and their associated free-energy terms.
Energy: Resolving the energy-distribution mismatch would require a more refined finite-dimensional representation of the conditional effective energies.The paper leaves such a representation for future work.

V. CONCLUSION AND PERSPECTIVES

The paper applies WCRG to frustrated soft-spin systems, finding accurate local and correlation statistics across many regimes and logarithmic-scale sampling at criticality. Its speed comes with accuracy limits set by the learned energy representation and training requirements.

Method: WCRG decomposes configurations into coarse fields and wavelet degrees of freedom, learns scale-dependent conditional distributions, and reconstructs samples recursively from coarse to fine scales.The approach is applied to the sBNNNI model as a benchmark for learned multiscale sampling.
Results: WCRG reproduces local field distributions and diagonal structure factors across homogeneous, critical, modulated MPM, and IC regimes, but the AF regime remains pathological.The conclusion summarizes the phase-diagram evaluation and its principal failure case.
Sampling dynamics: O(1) sweeps at each reconstruction scale and log2 L wavelet levels yield logarithmic sampling-time growth with system size.This avoids the additional L^z critical factor affecting local microscopic sampling, within the investigated size range.
Tradeoff: The method trades exact fixed-microscopic-energy sampling for a fast generative procedure whose accuracy depends on the expressiveness of its parametrized energies.This creates a complementary relationship between WCRG and asymptotically exact Monte Carlo.
Perspectives: WCRG’s explicit multiscale RG structure links approximation errors to reconstruction levels and supports controlled model improvements.The paper suggests adding symmetry-allowed operators or using local-receptive-field convolutional neural networks as future directions.
Limitations: The reported scaling assumes conditional energies are already learned, while the training-set size needed for fixed accuracy may increase with system size.The authors expect this dependence to be weak but leave its systematic evaluation for future work.

Appendix A: Low-temperature analysis of the soft-spin BNNNI model

The low-temperature analysis distinguishes finite-λ soft-spin and discrete BNNNI models and identifies why the soft-spin model supports a broader incommensurate regime. A rescaling isolates a temperature- and λ-independent variational problem.

Model distinction: The analysis distinguishes discrete BNNNI fields ϕ_i = ±1 from finite-λ soft-spin fields ϕ_i ∈ R.The relation between the hard-spin limit and the fixed-finite-λ low-temperature analysis requires care.
Phase behavior: The soft-spin model has a broad incommensurate regime extending toward relatively low temperatures, unlike the discrete model whose IC phase disappears as T → 0.The analysis identifies a mechanism rather than quantitatively determining two-dimensional phase boundaries.
Low-temperature scaling: At low temperature, the behavior is controlled by a balance between quadratic interaction energy and quartic confinement.The analysis introduces a rescaled field to expose this balance.
Low-temperature scaling: For fixed finite λ, the leading variational problem becomes independent of T and λ after rescaling, while those parameters set amplitudes and the overall energy scale.The rescaled field remains finite, whereas the original field diverges as T^-1/2.

1. Variational families and the leading (T →0)

The auxiliary analysis compares uniform FM, period-four antiphase, and multiharmonic IC variational profiles through a reduced low-temperature functional. Within this restricted ansatz, the IC profile has the lowest leading energy over an intermediate frustration interval, with parameter-independent crossings.

The three variational families are uniform FM, period-four antiphase (++−−), and multiharmonic incommensurate profiles.
The multiharmonic IC profile fixes q=q⋆ while variationally optimizing its overall amplitude and harmonic coefficients.Fixing q=q⋆ is an approximation because nonlinear terms can shift the optimal modulation wavevector.
The FM and antiphase branches follow from minimizing the reduced functional over the profile amplitude, with zero amplitude otherwise.
The profiles and crossing points are independent of T and λ within the leading low-temperature scaling problem.Because the analysis uses only a finite set of variational families, it does not establish the global minimizer.

2. Reconciling the finite-λ and Ising limits: the scaling variable λT

The finite-λ soft-spin and hard-spin Ising limits are connected by the scaling variable λT, which interpolates between an IC-favorable regime and a direct FM–AF transition. The restricted analysis finds that the IC window narrows and closes as λT increases.

The two limit orderings correspond to different variational problems: T→0 first yields the soft-spin functional, whereas λ→∞ first yields the discrete Ising form.The limits therefore do not commute.
λT: the scaling variable interpolates continuously between the soft-spin functional and the Ising problem.
The FM–AF boundary remains pinned at κ=1/2 for every λT.
As λT increases, the shape-factor penalty makes FM and AF lower in leading energy than smooth IC profiles, excluding the IC region.
The IC lobe opens to [0.393, 0.869] as λT→0 and closes at κ=1/2, λT≃1.
The calculation provides qualitative support for the extended IC region while remaining limited to an auxiliary one-dimensional model and restricted multiharmonic ansatz.

Appendix B: Finite-size scaling of the Ising critical line

Finite-size scaling locates an Ising-like critical point in the sBNNNI model at fixed frustration. Susceptibility and correlation-length data collapse consistently using two-dimensional Ising exponents.

The analysis studies κ=κx=κy=0.2, κd=0, λ=1, and J=1 on the FM–PM transition line.This transition is identified as belonging to the two-dimensional Ising universality class.
The bulk critical temperature Tc is obtained by fitting pseudocritical temperatures Tpc(L) extracted from susceptibility maxima.
Figure 8 plots magnetic susceptibility χ and ξ2/L against temperature for different system sizes, then analyzes Tpc(L) versus L^-1/ν.
ν=1 and γ=7/4: the χL^-γ/ν and ξ2/L data show finite-size scaling collapses consistent with Ising critical behavior.

Appendix C: Implementation of the wavelet transform

The implementation recursively decomposes continuous two-dimensional fields into coarse and wavelet variables using orthogonal wavelet transforms. Learned coarse and conditional energies then support recursive reconstruction from coarse to fine scales, with finite ansätze defining the practical approximation.

The recursive orthogonal wavelet transform separates slow coarse degrees of freedom from fast wavelet fluctuations at each scale.
Orthogonality makes the decomposition invertible with unit Jacobian, so no information or additional Jacobian factor is introduced.
In two dimensions, the low-low component is the coarse field and three remaining components form the wavelet channels.
The transform preserves degrees of freedom by splitting each scale into coarse variables and three wavelet-variable channels.
Haar wavelets are used near FM, PM, and critical regimes, while four-moment Daubechies wavelets represent finite-wavevector modulations in MP, IC, and AF regimes.
WCRG training uses equilibrium configurations decomposed into coarse–wavelet pairs to learn coarsest and conditional energy models.
Sampling starts from a one-variable coarsest field and recursively samples conditional wavelets while reconstructing progressively finer fields.
The learned microscopic energy generally differs from the original because conditional energies and free-energy contributions use finite-dimensional ansätze.