Quantum criticality at strong randomness: a lesson from anomaly

TL;DR

The paper proposes that quantum anomalies between exact and average symmetries organize universal critical correlations in strongly disordered quantum systems. By combining replica theory with an average Lieb–Schultz–Mattis constraint, it predicts a power-law decay for Edwards–Anderson correlators of operators charged under the exact symmetry and for first-moment correlators of operators charged under the average symmetry, provided there is no spontaneous symmetry breaking. These predictions are confirmed in representative models: 1d random Majorana lattices and random AFM Heisenberg chains exhibit $C_{ ext{EA}}(r) o r^{-2}$ and $D_{ ext{fm}}(r) o r^{-2}$ (1d), while 2d Majorana systems show $C_{ ext{EA}}(r) o r^{-2}$ and $D_{ ext{fm}}(r) o r^{-3}$; 1d systems also yield $D_{ ext{sc}}(r) o r^{-4}$, with 2d showing $D_{ ext{sc}}(r)$ scaling like $L^{-5}$ at fixed $r/L$. The work further analyzes self-averaging and finite-size effects, and provides SDRG-based intuition for the dimer-dimer observables. Overall, it offers a unifying anomaly-driven framework to understand nontrivial, experimentally accessible correlator signals in disordered quantum critical 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.

Figures (8)

• Figure 1: Power-law finite-size scaling of correlation functions in the random Majorana lattice model and the random antiferromagnetic Heisenberg chain, for operators charged under exact symmetries ($C_\mathrm{EA}$ [Eq. \ref{['eq:EAcorrelator']}] and $C_\mathrm{SS}$ [Eq. \ref{['eq:CSS']}] respectively) and average symmetries ($D_\mathrm{fm}$ [Eq. \ref{['eq:Dfm-def']}] for both models). We only show data with signal-to-noise ratio (SNR) and lattice separation ($r$) above the thresholds stated below. (a) $1d$ random Majorana lattice: exact diagonalization with $10^{4}L$ disorder realizations; $r \ge 3$, $\mathrm{SNR} \ge 10$.(b) $2d$ random Majorana lattice: correlations evaluated at $r = r_x$, exact diagonalization with $200L$ disorder realizations; $r \ge 3$, $\mathrm{SNR} \ge 3$. (c) Random antiferromagnetic Heisenberg chain: DMRG, with sweeps performed until the energy change between successive sweeps was below $10^{-8}$, using $50$--$100$ disorder realizations; $r \ge 2$, $\mathrm{SNR} \ge 3$.
• Figure 2: Power-law finite-size scaling of correlation functions in the random antiferromagnetic Heisenberg chain for operators charged under exact symmetries (spin-spin correlation $C_{\mathrm{SS}}$ [Eq. \ref{['eq:CSS']}]) and average symmetries (first-moment dimer-dimer correlation $D_{\mathrm{fm}}$ [Eq. \ref{['eq:Dfm-def']}]). The results are obtained using strong-disorder RG with $10^5$ disorder realizations, and we only show data with $r\ge 3$ and $\mathrm{SNR}\ge 10$.
• Figure 3: Power-law finite-size scaling of correlation functions in the $2d$ random Majorana lattice for (a) the Edwards--Anderson correlator $C_{\mathrm{EA}}$ [Eq. \ref{['eq:EAcorrelator']}], and (b) the first-moment dimer-dimer correlator $D_{\mathrm{fm}}$ [Eq. \ref{['eq:Dfm-def']}]. We obtain the data using exact diagonalization with $200L$ disorder realizations and show only points with $r\ge 3$, with the signal-to-noise ratio (SNR) above $2$ in (a) and above $3$ in (b).
• Figure 4: Exponential decay of the first-moment correlator $G(r)$ [Eq. \ref{['eq:def-G-r']}] in the random Majorana lattice (semilog-$y$ plot): (a) $d=1$, $L=100$ with $10^{5}$ disorder realizations; (b) $d=2$, $L=52$ with $10^{4}$ disorder realizations. The results are obtained using exact diagonalization, and we show only data with $\mathrm{SNR}\ge 3$.
• Figure 5: Finite-size scaling of the sample-connected dimer-dimer correlation function $D_{\mathrm{sc}}$ [Eq. \ref{['eq:D-sc-def']}], showing power-law behavior. Top row: random Majorana lattice in (a) $d=1$ and (b) $d=2$. We obtain the data using exact diagonalization and show only points with $r \ge 3$ and $\mathrm{SNR}\ge 3$. Bottom row: random antiferromagnetic Heisenberg chain, with data obtained using two methods: (c) DMRG, showing only points with $r \ge 2$ and $\mathrm{SNR}\ge 5$, and (d) strong-disorder RG, showing only points with $r \ge 2$ and $\mathrm{SNR}\ge 10$.