Quantum criticality at strong randomness: a lesson from anomaly

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.

💡 Research Summary

The paper tackles the notoriously difficult problem of quantum criticality in the presence of strong quenched disorder by invoking symmetry anomalies, specifically ’t Hooft anomalies that survive only when both an exact on‑site symmetry and an average lattice translation symmetry are unbroken. The authors formulate a “power‑law rule”: if a disordered system possesses a non‑trivial mixed anomaly and does not spontaneously break either symmetry, then (i) for some operator (O_e) charged under the exact symmetry the Edwards–Anderson (EA) correlator (\overline{|\langle O_e(x)O_e^\dagger(y)\rangle|}) must decay as a power law, and (ii) for some operator (O_a) charged under the average symmetry the ordinary disorder‑averaged two‑point function (\overline{\langle O_a(x)O_a^\dagger(y)\rangle}) must also decay as a power law. The rule is motivated by the idea that a non‑trivial anomaly cannot be “invisible” in the infrared; if all charged operators showed only exponential decay, the symmetry would act trivially, contradicting anomaly matching. The authors give a physical argument based on how spontaneous symmetry breaking would be detected: the EA correlator captures sample‑by‑sample breaking of an exact symmetry, while the ordinary averaged correlator captures breaking of an average symmetry.

To substantiate the rule, they study three concrete models that are well‑known but have not been examined from the anomaly perspective. First, the random‑singlet phase of the spin‑½ Heisenberg chain (1D). Using strong‑disorder renormalization group (SDRG), they confirm that the spin operator (charged under exact SO(3)) exhibits a power‑law EA correlator (\overline{|\langle \mathbf S_i\cdot\mathbf S_j\rangle|}\sim|i-j|^{-2}). Moreover, the staggered dimer operator, which flips sign under a lattice translation (the average symmetry), shows a power‑law decay of its disorder‑averaged correlator (\overline{\langle D_i D_j\rangle}). This dimer‑dimer power law had been overlooked in earlier works.

Second, they consider a free‑fermion model in symmetry class BDI, realized as a random Majorana hopping Hamiltonian on bipartite lattices in 1D and 2D. The model has an exact fermion‑parity (\mathbb Z_2) symmetry and an average translation symmetry, giving rise to an average LSM anomaly. In 1D the model maps to a random transverse‑field Ising chain and flows to an infinite‑randomness fixed point; the Majorana operator (charged under the exact (\mathbb Z_2)) has a power‑law EA correlator. In 2D the system resides in the so‑called Gade phase, a delocalized critical state. Here both the density‑density correlator and a dimer‑dimer correlator (charged under the average translation) decay with a universal power law, again revealing correlations missed in prior literature.

The paper discusses experimental implications. EA correlators are accessible in spin‑glass experiments via NMR, μSR, or neutron scattering, while averaged correlators can be probed by scanning tunneling microscopy, ARPES, or transport measurements in disordered superconductors or quantum Hall systems. The authors argue that their anomaly‑based predictions provide concrete, measurable signatures of disorder‑driven criticality.

Finally, they acknowledge that a rigorous proof of the power‑law rule is lacking, especially in higher dimensions or when topological order coexists with the anomaly. Nevertheless, they argue that average anomalies constitute a powerful, previously under‑exploited tool for constraining universal properties of strongly disordered quantum critical points. The work opens a new avenue for linking symmetry anomalies to experimentally observable correlation functions in disordered quantum matter.