Spontaneous breaking of non-invertible symmetries and duality to beyond-Landau transitions

This paper investigates spontaneous breaking of non-invertible symmetries in lattice models, demonstrating that such phases are characterized by long-range correlations of local order parameters obeying generalized algebraic structures and can be dual to deconfined quantum critical points of invertible symmetries via generalized gauging, thereby providing a systematic framework for studying beyond-Landau phase transitions.

The Gist

The Big Picture: Breaking the Rules of Symmetry

For decades, physicists have understood how materials change states (like water turning to ice) using a rulebook called Landau's paradigm. The core idea is Symmetry Breaking. Imagine a round table with identical seats. As long as the table is empty, it looks the same no matter how you rotate it (high symmetry). But as soon as one person sits down, the symmetry is "broken." The table now has a specific orientation.

Usually, these symmetries are like a group of friends who can swap places and always return to the original arrangement if they swap back. This is called an "invertible" symmetry.

However, in recent years, physicists discovered "exotic" symmetries that don't follow these rules. These are non-invertible symmetries. Imagine a magic trick where you swap two people, but you can't simply swap them back to get the exact original state; the system changes in a way that can't be undone. This paper asks: What happens when these "un-undoable" symmetries break?

The Main Discovery: A New Kind of Order

The authors found that even though these exotic symmetries are weird and can't be reversed, they still break in a way that creates distinct phases of matter, just like normal symmetries do.

• The Analogy of the "Sandwich":
  To understand this, the authors use a mental model called a "Symmetry Topological Field Theory" (SymTFT). Imagine a sandwich:

  • The top slice of bread is a fixed, rigid boundary.
  • The bottom slice is where the action happens (the material we are studying).
  • The filling is a 3D "topological soup" (a special kind of quantum fluid).

  In this model, "symmetry" is like a string running horizontally through the filling. "Order parameters" (the things that tell us the material has changed) are like strings running vertically, tunneling from the top bread to the bottom.

  The Key Finding: Even with these weird, non-reversible symmetries, the "vertical strings" (order parameters) still form long-range patterns. If you look far enough away, you can still tell the material has changed state. The authors mapped out exactly how these patterns behave, showing they follow a more complex set of rules (an algebra) than the simple rules of normal symmetry breaking.

The "Magic Mirror" (Duality)

The most exciting part of the paper is the discovery of a duality, or a "magic mirror" connection.

The authors show that a transition between two states in a system with these exotic symmetries is mathematically identical to a transition in a completely different system with "normal" symmetries, but with a twist.

• The Analogy:
  Imagine you are trying to cross a river.

  • Side A (The Exotic System): You are trying to cross a river where the water flows in strange, non-reversible loops. It looks chaotic and hard to understand.
  • Side B (The Normal System): You are crossing a river with normal currents, but there is a hidden "anomaly" (a glitch in the physics) that makes the water behave strangely in a specific way.

  The paper proves that Side A and Side B are actually the same river, just viewed from different angles.

  • When the exotic system goes through a "phase transition" (changing from order to disorder), it is exactly the same event as a Deconfined Quantum Critical Point (DQCP) in the normal system.
  • A DQCP is a special, critical moment where a material is on the verge of changing, but it doesn't just pick one new state; it hovers in a complex, gapless state where two different types of order compete.

  Why this matters: It turns a very hard problem (understanding exotic, non-reversible symmetries) into a problem we already know how to solve (understanding normal symmetries with anomalies).

The Specific Example: The $H_8$ Hopf Algebra

To prove this, the authors didn't just use abstract math; they built a concrete model using a specific mathematical structure called Rep($H_8$).

• The Analogy: Think of this as building a specific LEGO set.

  • They used two chains of qubits (quantum bits) like two parallel train tracks.
  • They defined specific "symmetry operators" (rules for flipping switches on the tracks).
  • They found six distinct "gapped phases" (stable states) for this system.
  • They mapped out exactly how the system transitions between these six states.

  They showed that when the system moves from a fully ordered state to a fully disordered state in this exotic model, it maps perfectly to a transition between two competing ordered states in a "normal" model that has a specific "glitch" (an anomaly) in its symmetry.

Summary of Claims

1. Exotic Symmetries Can Break: Even symmetries that cannot be reversed (non-invertible) can spontaneously break, creating distinct phases of matter.
2. Order Still Exists: These phases can still be identified by looking at long-range correlations (patterns that stretch across the material), even though the rules for how they form are more complex than usual.
3. The "Sandwich" Works: The "sandwich" model (SymTFT) is a powerful tool for visualizing and calculating these behaviors.
4. The Duality Bridge: There is a precise mathematical bridge connecting these exotic transitions to "Deconfined Quantum Critical Points" (DQCPs) in systems with normal symmetries that have anomalies.
5. Systematic Approach: This provides a systematic way to study "beyond-Landau" transitions (transitions that don't fit the old rules) by translating them into problems we already understand.

What the paper does NOT claim:
The paper does not discuss building new computers, curing diseases, or immediate technological applications. It is a theoretical physics paper focused on understanding the fundamental rules of how matter changes states and the mathematical relationships between different types of symmetries. It stays strictly within the realm of theoretical condensed matter physics.

Technical Summary

Title: Spontaneous breaking of non-invertible symmetries and duality to beyond-Landau transitions

Authors: Xie Chen, Shang Liu, Da-Chuan Lu, Nathanan Tantivasadakarn

Problem Statement

The study of phase transitions in condensed matter physics has traditionally relied on Landau's paradigm of spontaneous symmetry breaking (SSB), where phases are distinguished by the breaking of invertible (group) symmetries. However, the discovery of "beyond-Landau" transitions, such as deconfined quantum critical points (DQCPs) and transitions between symmetry-protected topological (SPT) phases, necessitates a broader framework. Recent developments have introduced non-invertible symmetries (described by fusion categories) as a generalization of group symmetries. While it is known that certain beyond-Landau transitions can be mapped to symmetry-breaking transitions of non-invertible symmetries via generalized gauging, the nature of spontaneous symmetry breaking for these non-invertible symmetries remains poorly understood. Specifically, it is unclear how to characterize the resulting phases, define order parameters, and identify the remaining symmetries when a non-invertible symmetry is broken.

Methodology

The authors employ a multi-faceted approach combining topological field theory, lattice model construction, and algebraic analysis:

1. Symmetry Topological Field Theory (SymTFT): The authors utilize the SymTFT formalism to describe the non-invertible symmetry $\text{Rep}(H_8)$, where $H_8$ is the smallest non-commutative and non-cocommutative self-dual Hopf algebra. They model the 1+1D system as a "sandwich" structure with a 2+1D topological bulk (the Drinfeld center $Z(\text{Rep}(H_8))$) and gapped boundaries. The bulk is constructed via anyon condensation from two copies of the doubled Ising topological order.
2. Lattice Model Construction: They explicitly construct 1+1D qubit lattice models (double qubit chains) that realize all six gapped phases allowed by $\text{Rep}(H_8)$ symmetry. These models are frustration-free, allowing for exact solutions of ground states and energy gaps.
3. Generalized Gauging: The authors perform a generalized gauging procedure on a specific anomaly-free subgroup of $\text{Rep}(H_8)$. This maps the original $\text{Rep}(H_8)$ system to a dual system with an ordinary $D_8$ group symmetry carrying a non-trivial 't Hooft anomaly ($\gamma \in H^3(D_8, U(1))$).
4. Algebraic Analysis: They analyze the fusion algebra of local order parameters and the transformation properties of ground states (including "cat-states" which are superpositions of short-range correlated symmetry-broken states) under the action of the non-invertible symmetry operators.

Key Contributions and Results

1. Characterization of Non-Invertible Symmetry Breaking:
The paper demonstrates that spontaneous breaking of non-invertible symmetries shares features with, yet differs significantly from, conventional group symmetry breaking:

• Order Parameters: Long-range correlations of local order parameters still characterize the symmetry-breaking order. However, these order parameters transform under the symmetry in a non-local manner and obey a fusion algebra dictated by the Lagrangian algebra of the gapped boundaries in the SymTFT sandwich.
• Ground State Degeneracy and Labeling:
  • In fully broken phases, short-range correlated ground states are labeled by the simple objects of the fusion category. Unlike group symmetries where a symmetry operation maps one broken state to another distinct broken state, non-invertible operations can map a broken state to a superposition of several broken states.
  • In partially broken phases, short-range correlated states do not have an obvious labeling. However, the authors show that a "cat-state" basis (superpositions of broken states) can always be constructed. These cat-states are labeled by the "tunneling channels" between the top and bottom boundaries of the SymTFT sandwich and transform according to the fusion rules of the symmetry.
• Remaining Symmetry: Defining the "remaining" symmetry in partially broken phases is subtle. While individual symmetry operators may not leave a specific broken state invariant, certain linear combinations of symmetry operators can. This contrasts with invertible symmetries where the unbroken subgroup is strictly defined by the stabilizer of the order parameter.

2. Duality to Deconfined Quantum Critical Points (DQCPs):
The authors establish a precise duality between the order-to-disorder transition of the non-invertible $\text{Rep}(H_8)$ symmetry and a DQCP of an ordinary group symmetry with an anomaly.

• By gauging the diagonal $\mathbb{Z}_2$ subgroup of $\text{Rep}(H_8)$, they derive a dual lattice model with $D_8$ symmetry and a mixed 't Hooft anomaly $\gamma$.
• The transition between the fully symmetric phase ($H_{14}$) and the fully symmetry-broken phase ($H_{11}$) of $\text{Rep}(H_8)$ is mapped to a direct continuous transition between two incompatible symmetry-breaking phases of the dual $(D_8, \gamma)$ model.
• This provides a systematic route to understanding DQCPs: they are dual to ordinary symmetry-breaking transitions of anomaly-free non-invertible symmetries.

3. Generalization to Group-Theoretical Hopf Algebras:
The paper extends these findings beyond the specific $\text{Rep}(H_8)$ example. They prove that for all anomaly-free non-invertible symmetries described by group-theoretical Hopf algebras (which includes all such symmetries with Frobenius-Perron dimension less than 36), the spontaneous symmetry breaking transitions can be mapped via generalized gauging to transitions of ordinary group symmetries.

• Theorem: A DQCP of an anomalous group symmetry $(G, \omega)$ is dual to an order-to-disorder transition of an anomaly-free non-invertible symmetry if the subgroups defining the DQCP satisfy specific factorization conditions (exact factorization or twisted factorization with a non-trivial 2-cocycle).
• Conversely, any order-to-disorder transition of an anomaly-free group-theoretical non-invertible symmetry is dual to a DQCP of an ordinary group symmetry with a 't Hooft anomaly.

Significance

The paper provides a systematic framework for understanding phase transitions involving non-invertible symmetries, bridging the gap between Landau's paradigm and beyond-Landau phenomena. By explicitly constructing lattice models and deriving the algebraic structure of order parameters and ground states, the authors clarify how non-invertible symmetries break. The established duality between non-invertible symmetry breaking and DQCPs offers a powerful tool for studying deconfined criticality, suggesting that the complex physics of DQCPs can be understood through the more familiar lens of symmetry breaking in non-invertible systems. Furthermore, the identification of precise conditions (factorization of groups and anomaly trivialization) under which this duality holds provides a classification scheme for a broad class of group-theoretical Hopf algebras and their associated phase transitions.