Effects of quenched disorder in three-dimensional lattice ${\mathbb Z}_2$ gauge Higgs models

This study investigates how uncorrelated quenched disorder affects the phase diagram and critical behavior of three-dimensional lattice ${\mathbb Z}_2$ gauge Higgs models, revealing that while weak disorder preserves the two-phase structure, random-plaquette disorder alters the universality class of the topological transition, whereas random-site disorder destabilizes the Ising$^\times$ transition, each leading to distinct critical exponents.

The Gist

Imagine you are trying to organize a massive, three-dimensional city made of tiny, magical switches. This city is governed by two sets of rules:

1. The Street Lights (Gauge Fields): These are the connections between buildings. They can be "on" or "off," and they determine how traffic flows between neighbors.
2. The Residents (Higgs Fields): These are the people living in the houses. They also have a mood: "happy" or "grumpy."

In a perfect, ideal world (the "pure system"), everyone and everything follows the rules perfectly. The city has two distinct moods or phases:

• The Chaotic Phase: The lights are flickering randomly, and the residents are confused.
• The Topological Phase: A special, hidden order exists. It's like a secret handshake that everyone knows, but you can't see it just by looking at one person or one street. It's a global pattern that keeps the city stable.

Between these two moods, there are transition lines. Think of these as the exact temperature or pressure where the city suddenly flips from one mood to the other. In the perfect world, these flips happen in a very predictable, smooth way, like water turning into ice.

The Problem: Introducing "Grumpy Neighbors" (Disorder)

Now, imagine we introduce impurities. Maybe some street lights are broken, or some residents are permanently grumpy no matter what. In physics, we call this quenched disorder. The key word is "quenched," meaning these broken lights or grumpy residents are frozen in place; they don't change their minds. They are just there, messing up the perfect rules.

The big question the paper asks is: If we break a few things in this city, does the whole system collapse? Does the way it changes from one mood to another stay the same, or does it get weird?

The Two Types of "Breakage"

The researchers tested two specific ways to break the city:

1. The Broken Street Lights (Random-Plaquette Disorder)

Imagine randomly flipping the switches on the street corners (the "plaquettes") so they sometimes force traffic the wrong way.

• What happens to the Topological Phase? The secret handshake (the topological order) is very sensitive to broken street lights. When you break a few lights, the way the city transitions into this special phase changes completely. It doesn't follow the old "ice-to-water" rules anymore. It enters a new, stranger state of matter with its own unique "personality" (a new universality class). The transition becomes "slower" and more complex, like trying to organize a crowd where some people are shouting contradictory instructions.
• What happens to the Resident Phase? Surprisingly, the transition involving the residents' moods is unaffected. Even with broken street lights, the residents can still organize themselves perfectly. The broken lights are just background noise to them; they don't care enough to change their behavior.

2. The Missing Residents (Random-Site Disorder)

Imagine randomly removing some residents from the houses entirely, leaving empty spots.

• What happens to the Resident Phase? This is a disaster for the residents. If you remove enough people, the way they organize changes. The transition becomes "rougher" and less predictable. It enters a new state called the Randomly-Dilute state. The city has to work harder to find a pattern when people are missing.
• What happens to the Topological Phase? The secret handshake is surprisingly robust. Even if you remove some residents, the global pattern of the street lights remains stable. The transition stays exactly the same as in the perfect city. The topological order doesn't care if a few houses are empty.

The Big Takeaway: Context Matters

The most fascinating part of this research is that disorder doesn't affect everything equally.

• If you break the connections (lights), the global patterns (topology) break, but the individuals (residents) stay strong.
• If you remove the individuals (residents), the individual patterns break, but the global connections stay strong.

It's like a dance party:

• If you break the music system (the connections), the whole dance floor loses its rhythm (the topological phase changes), but the dancers can still talk to each other (the resident phase stays the same).
• If you kick half the dancers out of the room (removing residents), the dance floor is empty and the dancing changes (the resident phase changes), but the music system is still playing the same tune (the topological phase stays the same).

Why Does This Matter?

This isn't just about theoretical cities. These models help us understand:

1. Quantum Computers: The "Topological Phase" is related to quantum memory (like the Toric Code). If we can build a quantum computer, we need it to be robust against errors (disorder). This paper tells us that if we build a quantum memory based on these rules, it might be very resilient against certain types of errors (like missing qubits) but fragile against others (like broken connections).
2. New Materials: It helps physicists predict how new materials will behave when they aren't perfectly pure, which is the reality of almost all real-world materials.

In short: The universe is surprisingly resilient. Depending on where you introduce the chaos, some parts of the system will crumble and change their nature, while other parts will stand firm, refusing to let the disorder change their fundamental rules.

Technical Summary

1. Problem Statement

The paper investigates the impact of uncorrelated quenched disorder on the phase diagram and continuous phase transitions of the three-dimensional (3D) lattice $Z_2$ gauge Higgs model. This model couples $Z_2$ gauge fields (on links) to $Z_2$ matter fields (on sites).

In the pure system (without disorder), the model exhibits a rich phase diagram featuring:

• A topologically ordered phase (related to the toric code).
• Two distinct continuous transition lines separating the topological phase from a normal phase:
  1. A topological transition (starting from the pure gauge limit $J=0$) characterized by the absence of a local order parameter.
  2. An Ising$\times$ transition (starting from the pure spin limit $K \to \infty$) characterized by a local, gauge-dependent order parameter.
• A multicritical point (MCP) where these lines meet.

The central question is how these transitions and their universality classes are altered when quenched disorder is introduced. Specifically, the authors test the Harris criterion, which predicts that disorder is a relevant perturbation if the specific heat exponent $\alpha$ of the pure system is positive. Since the pure transitions belong to the 3D Ising universality class ($\alpha \approx 0.11 > 0$), standard theory suggests disorder should change the universality class for both transition lines. However, the authors hypothesize that the specific nature of the disorder (coupling to plaquettes vs. sites) and the specific degrees of freedom driving the criticality may lead to different outcomes.

2. Methodology

The authors employ Monte Carlo (MC) simulations combined with Finite-Size Scaling (FSS) analysis.

• Models Studied:
  1. Random-Plaquette Disorder: Disorder variables $w_{x,\mu\nu} = \pm 1$ are attached to the plaquette terms (gauge fields). This mimics impurities in the gauge sector.
  2. Random-Site Disorder: Disorder variables $\rho_x \in \{0, 1\}$ are attached to the sites, effectively diluting the matter spins. This mimics impurities in the matter sector.
• Observables:
  • Since local order parameters are absent for the topological transition and gauge-dependent for the Ising$\times$ transition, the authors analyze gauge-invariant energy cumulants ($B_k$) of the Hamiltonian.
  • They focus on the 3rd and 4th cumulants ($B_3, B_4$) derived from the partition function, which are sensitive to critical behavior.
• Analysis Technique:
  • Data is averaged over many disorder realizations (quenched average).
  • FSS analysis is performed using the scaling form:
    $$B_k \approx L^{k/\nu - 3} \mathcal{B}_k(X) [1 + L^{-\omega} \mathcal{B}_{k,\omega}(X)] + b_k$$
    where $X = (J - J_c)L^{1/\nu}$ is the scaling variable, $\nu$ is the correlation length exponent, and $\omega$ is the correction-to-scaling exponent.
  • Critical points ($J_c, K_c$) and exponents ($\nu$) are determined by optimizing data collapse for various lattice sizes ($L$).

3. Key Contributions and Results

The study reveals that the effect of disorder is highly dependent on the type of disorder and the specific transition line, contradicting a naive application of the Harris criterion to the entire phase diagram.

A. Random-Plaquette Disorder (Disorder on Gauge Links)

• Topological Transition Line ($J \approx 0$):
  • The disorder is relevant. The transition persists but changes universality class.
  • The critical behavior belongs to the Random-Plaquette $Z_2$ Gauge (RPZ2G) universality class.
  • Result: The correlation length exponent changes from the pure Ising value ($\nu_I \approx 0.63$) to $\nu_{rp} \approx 0.82$.
• Ising$\times$ Transition Line ($K \to \infty$):
  • The disorder is irrelevant.
  • Result: The critical behavior remains in the standard 3D Ising universality class ($\nu \approx 0.63$).
  • Physical Explanation: Along the Ising$\times$ line, the critical fluctuations are dominated by the matter spins ($s_x$). The gauge variables (plaquettes) play a secondary role (essentially enforcing gauge invariance). Since the disorder only couples to the plaquettes, it does not significantly affect the dominant critical modes.

B. Random-Site Disorder (Disorder on Matter Spins)

• Ising$\times$ Transition Line ($K \to \infty$):
  • The disorder is relevant.
  • Result: The critical behavior changes from the pure Ising class to the Randomly-Dilute Ising (RDI) universality class.
  • Exponent: The correlation length exponent shifts to $\nu_{rdi} \approx 0.68$.
• Topological Transition Line ($J \approx 0$):
  • The disorder is irrelevant.
  • Result: The critical behavior remains in the pure $Z_2$ gauge universality class ($\nu \approx 0.63$).
  • Physical Explanation: At $J=0$, the system reduces to a pure gauge model where the matter spins are decoupled or trivial. The disorder on the sites does not affect the topological transition driven by the gauge fields.

C. Phase Diagram Structure

• For sufficiently weak disorder, the overall structure of the phase diagram (two phases separated by two transition lines) remains intact.
• However, the universality classes of the transition lines swap or change depending on the disorder type:
  • Plaquette disorder: Topological line $\to$ RPZ2G; Ising$\times$ line $\to$ Ising.
  • Site disorder: Topological line $\to$ Ising; Ising$\times$ line $\to$ RDI.
• Strong Disorder: For large disorder concentrations, the transition lines eventually disappear (e.g., approaching the percolation threshold for site disorder or the Nishimori line limit for plaquette disorder), leaving a single thermodynamic phase.

4. Significance

• Refinement of Harris Criterion Application: The paper demonstrates that while the Harris criterion correctly predicts disorder relevance based on $\alpha > 0$, the actual change in universality class depends on which degrees of freedom drive the criticality. If the disorder couples to a sub-dominant sector (irrelevant operator), the universality class remains unchanged.
• Topological Robustness: It highlights the robustness of topological transitions against certain types of matter disorder, and conversely, the robustness of matter-driven transitions against gauge disorder.
• Quantum Information Relevance: The 3D $Z_2$ gauge Higgs model is related to the Toric Code in 2D (via quantum-to-classical mapping). Understanding how quenched disorder (impurities) affects the topological phase and its transitions is crucial for assessing the error thresholds and stability of topological quantum memories.
• Methodological Benchmark: The work provides precise numerical estimates for critical exponents in disordered gauge theories ($\nu_{rp} \approx 0.82$, $\nu_{rdi} \approx 0.68$), serving as a benchmark for future theoretical and experimental studies of disordered gauge systems.

In conclusion, the authors establish that quenched disorder does not uniformly alter the critical behavior of gauge-Higgs systems; rather, the outcome is determined by the specific coupling between the disorder and the critical modes of the transition, leading to a complex scenario where different transition lines respond differently to the same type of impurity.