🔬 condensed matter

Strong-to-Weak Spontaneous Symmetry Breaking of Dephased Fermions

This paper investigates strong-to-weak spontaneous symmetry breaking (SWSSB) in dephased fermionic systems by mapping infinite dephasing to a compact XY model and employing diagrammatic and field-theoretic methods to demonstrate that fully dephased metals exhibit long-range SWSSB in dimensions $d \geq 2$ and quasi-long-range SWSSB in $d=1$ , while insulating states display phase behavior dependent on ultraviolet details and Renyi indices.

Original authors: Abhijat Sarma

Published 2026-09-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Abhijat Sarma

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of quantum materials, scientists have long relied on a simple idea to understand how matter organizes itself: symmetry. When a system possesses a symmetry, like the ability to rotate or shift without changing its essential nature, it often settles into a state where that symmetry is broken, creating distinct phases like magnets or superconductors. This framework, known as the Landau paradigm, has successfully classified the behavior of pure quantum states for decades. However, the real world is rarely pure. Quantum systems constantly interact with their surroundings, a process called decoherence, which scrambles their delicate quantum information and turns them into "mixed states." For mixed states, the old rules of symmetry breaking do not apply directly. A new, more subtle phenomenon has emerged, called strong-to-weak spontaneous symmetry breaking. This occurs when a system loses the ability to reconstruct its global charge information from local measurements, effectively forgetting where its total electric charge came from, even though the local rules of physics remain unchanged. Understanding how this happens is crucial for describing open quantum systems, from quantum computers struggling with noise to exotic materials interacting with their environment.

A recent study by Abhijat Sarma at the University of California, Santa Barbara, takes a deep dive into this phenomenon using a specific and extreme scenario: fermions, the particles that make up electrons, subjected to infinite dephasing. Imagine a system where the quantum phases of particles are scrambled so thoroughly that the system becomes purely classical, losing all its quantum interference patterns. The researcher asked a fundamental question: if you take a metal or an insulator and let it decohere completely, does it still retain a memory of its global charge, or does it forget? The answer reveals a surprising landscape of order and disorder that depends heavily on the dimension of space and the nature of the material.

The core of the work involves translating the complex, messy problem of a decohered quantum system into a much simpler, well-understood model. The researcher discovered that at the limit of infinite dephasing, the behavior of these mixed states is mathematically identical to a model of tiny magnets, known as spins, arranged on a lattice. In this equivalent picture, the probability of finding the system in a certain configuration acts like the temperature and energy of these spins. If the spins align over long distances, the system has undergone strong-to-weak symmetry breaking; if they remain disordered, the symmetry is preserved. This mapping allowed the researcher to use powerful tools from statistical physics to predict exactly what happens to different types of materials.

For metals, the results are robust and clear. In two or more dimensions, a fully dephased metal always displays long-range order. This means that even after losing all its quantum coherence, the system retains a global memory of its charge. This finding aligns with recent numerical simulations and experimental observations in two-dimensional systems, confirming that metals are remarkably resilient to this type of information loss. In one dimension, the order is not perfect but "quasi-long-range," meaning correlations decay slowly but never vanish completely. This suggests that the metallic state is inherently stable against the loss of global charge information, regardless of how much noise is introduced.

Insulators, however, tell a different and more nuanced story. Whether an insulator forgets its charge or remembers it depends on the specific details of the material and the dimensionality of space. In three dimensions, an insulator can either enter a state of long-range order or remain in a trivial, disordered phase. The outcome is determined by the energy gap of the material; if the gap is small enough, or if the material is close to a metallic transition, the system can maintain order. Deep inside the insulating regime, where the energy gap is large, the system typically loses its order and reverts to a trivial state. In two dimensions, the situation is even more delicate. Here, the system can only maintain a quasi-long-range order if the decay of correlations is sufficiently slow. If the material is too insulating, the order is destroyed. This implies that for most standard insulators, the strong-to-weak symmetry breaking is a fragile state that only exists under specific conditions, unlike the robust behavior seen in metals.

There is a notable exception to these rules: quantum Hall insulators. These are exotic materials that conduct electricity only on their edges and possess a unique topological structure. The study shows that these materials are exempt from the thresholds that govern ordinary insulators. Even when fully dephased, quantum Hall insulators generically display quasi-long-range order. This is because the vortices, or defects, that would normally destroy the order in a standard insulator are confined and cannot proliferate in these topological systems. This finding connects the behavior of decohered states directly to the deep topological properties of the parent material, suggesting that topology provides a shield against the loss of global information.

The research also looked at what happens when the dephasing is not infinite but merely large. The study found that the order found in the infinite limit persists for a finite amount of time, provided the dephasing is not too strong. For metals, this order is stable. For insulators, the stability depends on the specific type of order; if the correlations decay too quickly, even a small amount of coherence can destroy the symmetry breaking. The work further derived a theoretical framework to describe the transition between these ordered and disordered phases, identifying the universal laws that govern how the system changes as the noise level increases.

Ultimately, this work provides a comprehensive map of how quantum information survives in a noisy world. It demonstrates that while some materials, like metals, are naturally equipped to preserve global charge information even when their quantum nature is stripped away, others, like ordinary insulators, are far more fragile. The distinction between remembering and forgetting is not just a matter of how much noise is present, but is fundamentally tied to the material's dimensionality and its underlying electronic structure. By establishing these rules, the study offers a new lens through which to view the phases of matter in open systems, bridging the gap between the idealized world of pure quantum states and the messy reality of the physical world.

Technical Summary: Strong-to-Weak Spontaneous Symmetry Breaking of Dephased Fermions

Problem Statement
This work investigates the fate of quantum phases of matter when subjected to environmental decoherence, specifically focusing on "strong-to-weak spontaneous symmetry breaking" (SWSSB). In mixed-state settings, symmetries can be realized in two inequivalent ways: strong symmetry (where the symmetry charge sector is fixed across the ensemble) and weak symmetry (where charge sectors fluctuate). SWSSB is a phase transition associated with the loss of the ability to reconstruct global charge information from local density matrices. While previous studies have focused on systems with severe kinematic constraints (e.g., anomalies) or specific topological orders, this paper addresses a broad class of fermionic systems, including both metallic (Fermi liquid) and insulating phases, under infinite density dephasing. The central question is whether and how SWSSB manifests in these generic mixed states and how it depends on dimensionality and the nature of the parent state.

Methodology
The author employs a combination of exact mappings, diagrammatic expansions, and field-theoretic techniques:

Exact XY Representation at Infinite Dephasing: The author demonstrates that at infinite dephasing, the density matrix becomes a classical diagonal ensemble. They prove that every Renyi correlator $C^{(\alpha)}(x, y)$ is exactly equivalent to the spin-spin correlator of a compact XY model. The Boltzmann weight of this XY model is the squared full-counting-statistics (FCS) generating function of a classical ensemble constructed from the Born probabilities of the parent state raised to the power $\alpha/2$ .
- For $\alpha=2$ , the weight corresponds to the FCS of the parent state itself.
- For $\alpha=1$ , the weight corresponds to the FCS of the ensemble $\sqrt{p}$ .
Spin-Wave and Vortex Analysis: Using the XY representation, the author analyzes the stability of the ordered phase (SWSSB) against Goldstone fluctuations (spin waves) and topological defects (vortices). They perform infrared power counting on the renormalized kernel of the XY model to determine the scaling of correlators in metals versus insulators.
Diagrammatic Expansion for Renyi-1: To address the Renyi-1 correlator (the true diagnostic of SWSSB), which lacks a direct parent-state correlation function mapping, the author introduces a novel diagrammatic expansion. This expansion expresses the Renyi-1 XY couplings in terms of the connected correlation functions of the parent state, applicable to any diagonal mixed state.
Replica Field Theory for Finite Dephasing: For finite dephasing times, the author constructs a replica field theory based on the "doubled state" (Choi representation). This maps the problem to a field theory of two fermion flavors with a decoherence-induced attractive interaction, allowing for the derivation of the universality class of the SWSSB transition.

Key Results

Dephased Metals ( $d \ge 2$ ): Maximally dephased metals exhibit long-range SWSSB. The Renyi correlators saturate to a non-zero constant at large distances. This is confirmed by the fact that the renormalized kernel of the XY model retains the infrared form of the parent state's structure factor $S(q) \sim |q|$ , rendering Goldstone fluctuations and vortices irrelevant in $d \ge 2$ . This result aligns with existing numerical and experimental findings in $d=2$ .
Dephased Metals ( $d = 1$ ): In one dimension, the system displays quasi-long-range (algebraic) SWSSB, consistent with Luttinger liquid behavior where correlators decay as power laws.
Dephased Insulators: The behavior is more complex and depends on ultraviolet (UV) details and dimensionality:
- $d \ge 3$ : A long-range SWSSB phase is possible if the parent state is sufficiently close to a metallic transition (large correlation length). Deep in the insulating regime, the system enters a trivial paramagnetic phase.
- $d = 2$ : The system can exhibit quasi-long-range SWSSB (algebraic decay) or a trivial phase. A necessary condition for quasi-long-range order is derived, relating the kinetic energy tensor to the energy gap ( $\pi K \ge 2\delta E$ ).
- Quantum Hall Insulators: These are an exception; they generically display quasi-long-range SWSSB in $d=2$ regardless of the threshold, due to the confinement of vortices by the Hall conductance.
Finite Dephasing Stability:
- Long-range SWSSB in $d \ge 2$ (metals) and $d \ge 3$ (insulators near transition) persists at large but finite dephasing times.
- Quasi-long-range SWSSB in $d=2$ insulators is fragile. It survives finite dephasing only if the power-law exponent $\eta > 4$ . For trivial insulators where $\eta < 1/4$ , any finite coherence destroys the SWSSB order.
Universality Classes: The transition into the SWSSB phase is characterized by specific universality classes:
- $d > 2$ (Metals): Mean-field theory with a fractional Laplacian $|\partial|$ .
- $d = 3$ (Insulators): 3D XY universality class.
- $d = 2$ (Insulators with $\eta > 4$ ): Berezinskii-Kosterlitz-Thouless (BKT) transition.

Significance and Claims
The paper claims to provide a rigorous, generic framework for understanding SWSSB in mixed states without relying on fine-tuned ground states or specific topological constraints. By establishing an exact mapping to a compact XY model, the author unifies the diagnosis of SWSSB across different Renyi indices and system types.

Key contributions include:

Exact Representation: Proving that infinite dephasing maps Renyi correlators to XY model spin-spin correlators, providing a rigorous lower bound on Renyi-1 and fidelity correlators in terms of parent state correlations.
Novel Expansion: Introducing a diagrammatic expansion for Renyi-1 correlators of diagonal states, enabling the calculation of XY couplings for arbitrary classical distributions.
Phase Diagram Determination: Systematically mapping out the SWSSB phase diagram for metals and insulators across dimensions, identifying the conditions under which global charge information is lost (SWSSB) versus preserved.
Finite-Time Dynamics: Deriving the stability of SWSSB against finite dephasing and identifying the universality classes of the transition, thereby bridging the gap between infinite dephasing limits and realistic experimental scenarios.

The author notes that while their perturbative methods lack a small parameter at the lattice scale (making quantitative UV-dependent thresholds a subject for future study), the qualitative conclusions regarding the existence of long-range order in metals and the conditional nature of order in insulators are robust. They also highlight that the extension of these results to non-abelian symmetries remains an open challenge due to the non-diagonal nature of mixed states in those sectors.

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