Source-linked AI summary
Quantum Loop Topography for Machine Learning
Yi Zhang, Eun-Ah Kim
TL;DR
The paper develops a loop-based expression of Hall conductivity for gapped systems and an associated sampling procedure for quantum loop topography. Longer loops improve accuracy when gaps are small, while training-data diversity improves overall phase classification.
Problem
The paper addresses how to express Hall conductivity using triangular loops of two-point correlators for a gapped system.
Method
The procedure generates quantum loop topography samples using variational Monte Carlo, with a positive normalized factor providing the sampling probability and another factor averaged over the Markov chain.
Results
Longer loop cutoffs improve Hall-conductivity estimation for small gaps and long correlation lengths, while diverse training models give the best overall phase-classification result.
Takeaways & Limitations
Including larger loops and representative models with varied correlation lengths helps analyze information near topological phase transitions without sacrificing performance across deep phases.
Takeaways & Limitations
Typical training models enable fast convergence and phase recognition but limit the ability to pinpoint phase transitions; parton-constructed FCI states are also a specific modeling choice.
Abstract
from arXiv · showhide
Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge. The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network. When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of non-local properties. Here we introduce quantum loop topography (QLT): a procedure of constructing a multi-dimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps. The loop configuration is guided by characteristic response for defining the phase, which is Hall conductivity for the cases at hand. Feeding QLT to a fully-connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish Chern insulator and fractional Chern insulator from trivial insulators with high fidelity. In addition to establishing the first case of obtaining a phase diagram with topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.
Hall conductivity from two-point correlators
The paper derives Hall conductivity from triangular loops of two-point correlators and uses this structure to construct QLT for machine-learning applications. For gapped systems, finite loop cutoffs can approximate the conductivity, with longer loops improving accuracy when gaps are small.
- The loop formula sums products of correlators over triangle vertices, weighted by each triangle’s signed area and normalized by the total number of sites.
- Hall conductivity can be expressed as triangular quantum loops consisting of two-point correlators for a gapped system.
- The Hall-conductivity estimate improves asymptotically as the triangle cutoff dc increases, because longer loops capture longer correlations near small-gap transitions.
- For the honeycomb model, imaginary next-nearest-neighbor hopping competes with staggered on-site potential, producing a Chern insulator for κ > 0.5 and a trivial insulator for κ < 0.5.
- QLT samples loop products using independently sampled configurations rather than averaging the full correlators over a Markov chain.
Impact of training models and QLT cut-off on machine learning phases and phase transitions
Training-model choice and QLT cutoff jointly affect phase recognition and transition pinpointing. Typical deep-phase training data recognize phases efficiently, whereas smaller gaps, longer correlations, larger cutoffs, and diverse training examples improve behavior near or across the transition.
- Typical deep-phase training models efficiently recognize phases but limit the precision of phase-transition pinpointing.
- Smaller-gap, longer-correlation training data and larger loop cutoffs slightly improve accuracy near κ ∼ 0.5 and sharpen the non-analytic behavior of p.
- Training only on κ = 0.35 and κ = 0.65 reduces testing accuracy deep inside the trivial and topological phases because these examples are less representative.
- Including both κ = 0.35 and κ = 0.10 for the trivial insulator, and κ = 0.65 and κ = 1.0 for the Chern insulator, gives the best overall result.