Machine Learning the Strong Disorder Renormalization Group Method for Disordered Quantum Spin Chains

TL;DR

Finite-temperature entanglement properties are incorporated via the SDRGX framework through a two-stage strategy, using the zero-temperature GNN to generate the RG flow and sampling thermal occupations from the canonical ensemble, yielding results in agreement with both numerical SDRGX and analytical predictions without retraining.

Abstract

We train machine learning algorithms to infer the entanglement structure of disordered long-range interacting quantum spin chains by learning from the strong disorder renormalisation group (SDRG) method. The system consists of $S=1/2$-quantum spins coupled by antiferromagnetic power-law interactions with decay exponent $α$ at random positions on a one-dimensional chain. Using SDRG as a physics-informed teacher, we compare a Random Forest classifier as a classical baseline with a graph neural network (GNN) that operates directly on the interaction graph and learns a bond-ranking rule mirroring the SDRG decimation policy. The GNN achieves a disorder-averaged pairing accuracy close to one and reproduces the entanglement entropy $S(\ell)$ in excellent quantitative agreement with SDRG across all subsystem sizes and interaction exponents. RG flow heat maps confirm that the GNN learns the sequential decimation hierarchy rather than merely fitting final-state observables. Finite-temperature entanglement properties are incorporated via the SDRGX framework through a two-stage strategy, using the zero-temperature GNN to generate the RG flow and sampling thermal occupations from the canonical ensemble, yielding results in agreement with both numerical SDRGX and analytical predictions without retraining.

Figures (7)

• Figure 1: Schematic SDRG-X procedure: at each RG step four possible pair states are indicated by blue, black and red lines with pair energies $E=-J/2,0,0,+J/2$, respectively. After $N/2$ RG steps the many body eigenstates with total energy E is obtained, following a specific SDRG path. Entanglement entropy as function of physical partition length $l$, Eq. (\ref{['slL']}), valid for $\alpha \gg 1$ for chain length $L=1000$ as function of temperature $T$ in units of $\Omega_0$.
• Figure 2: $T=0.01\,\Omega_0$ entanglement entropy $S(\ell)$ as a function of subsystem size $\ell$ for a long-range disordered spin chain at fixed density $N/L=0.1$, comparing exact SDRG with Random Forest (RF-SDRG) and pure random decimations. The top-left annotation reports the mean pairing accuracy $r_P = 2M_P/N \pm \sigma_{r_P}$, where $M_P$ is the number of RF-SDRG-predicted pairs that match the exact SDRG reference. The bottom inset shows the distribution of $r_P$ across disorder realizations; the dashed vertical line marks the mean. All results are averaged over $500$ disorder realizations and $100$ thermal samples per realization.
• Figure 3: Entanglement entropy $S(\ell)$ at $T=0K$ versus subsystem size $\ell$ for a long-range random spin chain with decay exponent $\alpha = 2.0$ and fixed density $N/L = 0.1$, comparing SDRG (solid line) and GNN-assisted SDRG (dashed line) results averaged over $1000$ disorder realizations. The inset shows the distribution of the pairing accuracy $r_P$ across disorder realizations; the dashed vertical line indicates the disorder-averaged mean.
• Figure 4: Same as Fig. \ref{['fig:gnn1_sdrg']}, but for decay exponent $\alpha = 0.5$ at fixed density $N/L = 0.1$.
• Figure 5: Renormalization-group flow of bond decimations for SDRG (left) and GNN-assisted SDRG (right) for a long-range disordered spin chain ($N=80$, $\alpha=2.0$). Shown is a heatmap of the probability of decimating a bond of given length $\ell$ (logarithmic bins) at each RG step, averaged over disorder realizations.