Quantum criticality at strong randomness: a lesson from anomaly

TL;DR

This paper demonstrates how anomaly and topology associated with average symmetry can predict universal critical correlations in strongly disordered quantum systems, revealing new insights into quantum criticality under strong randomness.

Contribution

It introduces a novel anomaly-based framework to predict critical correlations in disordered systems with average symmetry, extending understanding beyond previous approaches.

Findings

Predicted slow decay of correlations for operators charged under exact symmetries.

Predicted slow decay of correlations for operators charged under average symmetries.

Verified predictions in random-singlet Heisenberg chain and disordered free-fermion states.

Abstract

Quantum criticality in the presence of strong quenched randomness remains a challenging topic in modern condensed matter theory. We show that the topology and anomaly associated with average symmetry can be used to predict certain nontrivial universal properties. Our focus is on systems subject to average Lieb--Schultz--Mattis constraints, where lattice translation symmetry is preserved only on average, while on-site symmetries remain exact. We argue that in the absence of spontaneous symmetry breaking, the system must exhibit critical correlations of local operators in two distinct ways: (i) for some operator $O_e$ charged under exact symmetries, the first absolute moment correlation $\overline{|\langle O_e(x)O^{\dagger}_e(y)\rangle|}$ decays slowly; and (ii) for some operator $O_a$ charged under average symmetries, the first-moment correlation $\overline{\langle O_a(x)O^{\dagger}_a(y)\rangle}$ decays slowly. We verify these predictions in a few examples: the random-singlet Heisenberg spin chain in one dimension, and the disordered free-fermion critical states in symmetry class BDI in one and two dimensions. Surprisingly, even for these well-studied systems, our anomaly-based argument reveals critical correlations overlooked in previous literature. We also discuss the experimental feasibility of measuring these critical correlations.