Quantum criticality at strong randomness: a lesson from anomaly

This paper demonstrates that topology and anomalies associated with average symmetries can predict distinct, slow-decaying critical correlations in quantum systems with strong quenched randomness, a framework successfully applied to reveal overlooked universal properties in random-singlet spin chains and disordered free-fermion states.

The Gist

Imagine you are trying to understand the behavior of a massive crowd of people in a chaotic, noisy room. In the world of physics, this "crowd" is made of tiny particles (like electrons or spins), and the "noise" is randomness—imperfections or disorder in the material they live in.

Usually, when physicists study these systems, they look for order. But sometimes, even with all the noise, the particles don't settle down into a calm, ordered state, nor do they freeze into a rigid structure. Instead, they stay in a state of quantum criticality—a kind of perpetual, jittery dance where everything is connected over long distances.

This paper tackles a very difficult question: How do we predict the rules of this chaotic dance when the room is full of random noise?

Here is the breakdown of their discovery, using simple analogies:

1. The "Rule of the Room" (Symmetry and Anomalies)

Imagine the room has two types of rules:

• The Strict Rules (Exact Symmetry): These rules apply to every single person in the room, no matter what. For example, "Everyone must wear a red hat."
• The Average Rules (Average Symmetry): These rules only hold true if you take a snapshot of the whole crowd and average it out. For example, "On average, half the people are standing and half are sitting." In any specific moment, you might see 60% standing, but the average over time is 50/50.

In physics, when these two types of rules clash in a specific way, it creates a "Symmetry Anomaly." Think of this anomaly like a knot in a rope. You can't untie the knot (make the system "boring" or trivial) without cutting the rope (breaking a rule). Because the knot exists, the system is forced to stay "alive" and active; it cannot settle down.

2. The New Prediction: The "Power-Law Rule"

The authors discovered a new way to predict how this chaotic system behaves. They call it the "Power-Law Rule."

They argue that because of the "knot" (the anomaly), the particles must keep talking to each other over long distances, but they do it in two different ways depending on which "rule" they are following:

• For the Strict Rules (Exact Symmetry):
  Imagine you are looking at a specific person wearing a red hat. Even though the room is chaotic, if you look at how much that person's hat correlates with a person far away, the connection doesn't disappear instantly. Instead, it fades away slowly, like a whisper that gets quieter but never quite stops.

  • The Paper's Claim: The "strength" of this connection (measured by a specific math tool called the Edwards-Anderson correlator) decays slowly, following a specific mathematical curve (a power law).

• For the Average Rules (Average Symmetry):
  Now, imagine looking at the "average" behavior of the crowd. If you look at the connection between two distant groups based on the average rule, this connection also fades away slowly.

  • The Paper's Claim: The "average" connection (the first-moment correlator) also follows that same slow, power-law decay.

The Big Surprise:
The authors found that for some well-known systems (like a chain of magnets with random strengths), scientists had been looking at the "average" connections and thinking they were just fading away quickly (exponentially). The authors' "knot" theory predicts that these connections should actually be slower and more persistent than anyone realized. They found these "hidden" slow connections in their computer simulations, proving the theory works.

3. The "Whisper vs. Shout" Analogy

To make it even simpler:

• Normal materials are like a quiet library. If you whisper to someone far away, they can't hear you at all (the signal dies instantly).
• Ordered magnets are like a shouting match. Everyone is shouting the same thing, so the signal is loud and clear forever.
• This "Quantum Critical" state is like a crowded party where everyone is talking at once.
  • If you listen to a specific person (Strict Rule), you can still hear their voice fading slowly across the room.
  • If you listen to the "average noise" of the room (Average Rule), you can also hear a specific pattern fading slowly across the room.
  • The paper says: "If there is a 'knot' in the rules, you must hear these slow fades. If you don't, the rules are broken."

4. Can We Measure This?

The paper asks: "Can we actually hear these whispers in a real lab?"

• Yes. They suggest that in materials like cold atomic gases (where scientists can take "photos" of the atoms), we can measure these connections directly.
• In solid materials (like crystals), we can use X-rays or neutron scattering. These tools measure how the material scatters particles. The authors argue that the "slow fade" they predicted will show up as a specific pattern in the scattering data, specifically looking at how "dimers" (pairs of atoms) are connected.

Summary

The paper uses a concept called a "Symmetry Anomaly" (a topological knot in the rules of the universe) to prove that in certain messy, random quantum systems, particles must remain connected over long distances. They predict that these connections don't vanish quickly but instead fade away slowly and predictably (following a power law). They tested this on known systems and found that this "slow fade" was hiding in plain sight, overlooked by previous studies. This gives physicists a new "rule of thumb" to understand and identify these strange, critical states of matter.

Technical Summary

Technical Summary: Quantum Criticality at Strong Randomness: A Lesson from Anomaly

Problem Statement
The paper addresses the persistent challenge of understanding quantum criticality in the presence of strong quenched randomness. While disorder-induced transitions (e.g., superconductor-insulator, quantum Hall, disordered spin systems) are ubiquitous, theoretical tools capable of non-perturbative and broadly applicable analysis remain scarce. Specifically, the authors aim to determine if symmetry anomalies, a central concept in clean systems, can predict universal properties of disordered quantum critical states. A key difficulty is that the concept of "long-range entanglement"—a consequence of anomalies—is not directly measurable; thus, the authors seek to formulate predictions in terms of measurable correlation functions.

Methodology and Framework
The authors employ the framework of average anomalies, specifically focusing on systems with average Lieb–Schultz–Mattis (LSM) constraints. In these systems:

1. Exact Symmetries ($K$): On-site symmetries (e.g., SO(3) spin rotation, time-reversal, fermion parity) are preserved for every disorder realization.
2. Average Symmetries ($G$): Lattice translation symmetry is preserved only upon averaging over the disorder ensemble.
3. Anomaly Condition: The system possesses a mixed 't Hooft anomaly between $K$ and $G$, arising when unit cells carry "fractional" quantum numbers (e.g., spin-1/2 per site) represented projectively.

The authors restrict their analysis to gapless infrared (IR) states where the anomaly is not realized via spontaneous symmetry breaking (SSB) or intrinsic topological order. They propose a "Power-law Rule" derived from the requirement that the anomaly must be matched in the IR. Using a plausibility argument based on the renormalization group (RG) flow and the nature of symmetry breaking, they distinguish between two types of correlation functions:

• Edwards–Anderson (EA) Correlator: $|\langle O_e(x)O_e^\dagger(y) \rangle|$, where $O_e$ is charged under the exact symmetry $K$. This detects sample-wise ordering.
• First-Moment Correlator: $\langle O_a(x)O_a^\dagger(y) \rangle$, where $O_a$ is charged under the average symmetry $G$. This detects ordering in the disorder-averaged ensemble.

The authors argue that if the anomaly is nontrivial and not saturated by SSB or topological order, both correlators must exhibit critical (power-law) decay rather than exponential decay.

Key Contributions and Results
The paper provides a theoretical argument and numerical verification for the Power-law Rule in two distinct classes of disordered systems:

1. Random Majorana Lattice Models (Classes BDI):

   • System: 1D and 2D lattices of Majorana fermions with random hopping amplitudes.
   • Symmetries: Exact fermion parity ($Z_2^f$) and average translation ($Z^d$).
   • Findings:
     • The EA correlator for fermion operators ($|\langle i\gamma_j \gamma_{j+r} \rangle|$) decays as a power law in both 1D and 2D.
     • The first-moment correlator for the staggered dimer operator (charged under average translation) also decays as a power law ($1/r^2$ in 1D, $1/r^3$ in 2D).
     • Notably, the standard first-moment fermion Green's function decays exponentially, consistent with the rule that only operators charged under the average symmetry need to show power-law decay in the first moment.

2. Random Antiferromagnetic Heisenberg Spin Chain (1D):

   • System: Spin-1/2 chain with random exchange couplings ($J_i$), known to flow to a random-singlet phase.
   • Symmetries: Exact SO(3) spin rotation and average translation.
   • Findings:
     • The EA spin-spin correlator $|\langle \vec{S}_i \cdot \vec{S}_{i+r} \rangle|$ exhibits the expected power-law decay ($\sim r^{-2}$).
     • Crucially, the authors identify a nontrivial power-law decay in the first-moment staggered dimer correlator ($\langle d_i d_{i+r} \rangle$ with staggered sign), a correlation function previously overlooked in the literature for this system.

Significance and Claims
The paper claims that anomaly-based arguments can successfully predict universal correlation properties in disordered quantum critical states that were previously missed.

• Novelty: Even for well-studied systems like the random-singlet phase and the Gade phase (2D disordered free fermions), the anomaly argument reveals critical correlations (specifically the first-moment dimer-dimer correlations) that standard literature had not highlighted.
• Experimental Feasibility: The authors argue that these predictions are experimentally accessible.
  • In synthetic quantum matter (e.g., cold atoms), higher-moment correlators (EA type) and first-moment correlators can be measured via snapshots of disorder realizations.
  • In solid-state materials, while EA correlators are generally inaccessible, the disorder-averaged first-moment correlators correspond to the equal-time structure factors measurable via scattering techniques (neutron or X-ray). Specifically, dimer operators (spinless) could be probed via X-ray scattering of charge density.
• Self-Averaging: Numerical checks on signal-to-noise ratios suggest these correlators are self-averaging in the thermodynamic limit, validating their use as robust experimental observables.

The authors maintain a modest stance, noting that the Power-law Rule is supported by plausibility arguments and specific examples rather than a rigorous general proof, which remains difficult even for clean systems in general dimensions. However, the consistency across free-fermion and interacting spin models suggests a broad applicability of the anomaly-constraint framework to disordered criticality.