Non-Ergodic States in Higher Dimensions

Exact diagonalization of disordered Heisenberg model: gap ratio, fractal dimension, and entanglement entropy in 1D and 2D

0.531
r (GOE, ergodic)
0.386
r (Poisson, MBL)
0.391
Avg r (1D, mod. W)
0.394
Avg r (2D, mod. W)

Gap Ratio vs Disorder Strength

Fractal Dimension D2 vs Disorder

1D Gap Ratio: Finite-Size Scaling

1D vs 2D Comparison at Moderate Disorder

Gap Ratio Data (Largest Systems)

W1D (L=10)2D (3x3)GOEPoisson
0.50.4000.4070.5310.386
1.00.3960.4080.5310.386
3.00.3920.3880.5310.386
5.00.3900.3990.5310.386
8.00.3930.3930.5310.386
10.00.3840.4060.5310.386
15.00.3790.4050.5310.386

Fractal Dimension Data

WD2 (1D, L=10)D2 (2D, 3x3)
0.50.5760.500
1.00.5370.504
2.00.5000.512
3.00.4330.488
5.00.2780.442
8.00.1530.326
10.00.1280.293
15.00.0830.178
Data from exact diagonalization of disordered Heisenberg model | Reference: Lunkin et al., arXiv:2601.01309 (2026)