Symmetry breaking phases and transitions in an Ising fusion category lattice model

This paper investigates an Ising fusion category lattice model, revealing a rich phase diagram with a symmetric critical phase and two distinct categorical symmetry-breaking phases (ferromagnetic and antiferromagnetic), while characterizing the transitions between them via specific conformal field theories.

The Gist

Imagine you are a city planner trying to understand how a bustling city organizes itself. Usually, we think of order coming from simple rules: "Everyone stands in a line" or "Everyone faces the same direction." In physics, this is like a standard symmetry, where everything follows a predictable, reversible pattern (like flipping a switch on and off).

But this paper explores a much stranger, more magical kind of city planning. The "rules" here aren't simple switches; they are fusion categories. Think of these as a set of magical ingredients that can be combined, but when you mix them, they don't just add up—they can create entirely new, complex outcomes that you can't simply reverse. This is called non-invertible symmetry.

The authors, Soumil Roychowdhury and Chenjie Wang, built a digital "city" (a lattice model) based on these magical rules (specifically the Ising fusion category) to see what kind of neighborhoods (phases) would form. They found three distinct types of neighborhoods and the strange transitions between them.

Here is the breakdown of their discovery using everyday analogies:

1. The Three Neighborhoods (Phases)

A. The "Chaos" Zone (Symmetric Critical Phase)

• What it is: A bustling, critical city where nothing is settled. It's like a jazz improvisation session where everyone is playing together, but no single melody dominates.
• The Vibe: It's perfectly balanced and follows the standard "Ising" rules of physics (like a standard magnet). It's "gapless," meaning energy can flow freely, like water in a river.
• The Magic: The magical symmetry is fully intact here. No one has broken the rules.

B. The "Super-Ferromagnet" (Categorical Ferromagnetic Phase / CatFM)

• What it is: Imagine a city where everyone suddenly decides to stand in one of three specific poses (let's call them Pose A, Pose B, and Pose C). Once they pick a pose, they stick with it.
• The Break: The magical symmetry is completely shattered. The city has chosen a specific "ground state" (a resting position).
• The Analogy: Think of a crowd of people who used to be able to morph into anything, but now they are frozen into one of three statues. It's a "gapped" phase, meaning it takes effort to change the mood of the city.

C. The "Super-Anti-Ferromagnet" (Categorical Antiferromagnetic Phase / CatAFM)

• What it is: This is the most surprising discovery. Imagine a city where the residents form a pattern: Left, Right, Left, Right. But here's the twist: because of the magical rules, the "Left" and "Right" aren't just simple opposites. They are complex, multi-layered states.
• The Break: The city breaks two rules at once: the magical symmetry and the rule that "every block looks the same" (lattice translation).
• The Magic: Unlike a normal frozen city, this one is still critical (still flowing like a river). It's a "gapless" phase.
• The Analogy: Imagine a dance floor where the dancers are in a complex pattern. Because the "walls" between the dancers (domain walls) are magical and heavy, they create a huge, infinite playground of possibilities. The city is stuck in a pattern, but it's vibrating with infinite energy. It's described by four copies of the standard Ising rules working together.

2. The Transitions (Moving Between Neighborhoods)

The authors also studied how the city changes from one neighborhood to another.

• From Chaos to Super-Ferromagnet:

  • This transition is well-understood. It's like a classic "phase change" (like water freezing). The math describing this change is the Tricritical Ising model (a famous, slightly more complex version of the standard magnet).

• From Chaos to Super-Anti-Ferromagnet:

  • This is the mystery. The authors found that when the city shifts from the "Chaos" zone to the "Super-Anti-Ferromagnet" zone, the math gets weird.
  • The Discovery: It seems to be a mix of the standard Ising rules plus a new, flowing "liquid" of energy (a Luttinger liquid). The total complexity (central charge) is 1.5 (1/2 from the Ising part + 1 from the liquid part).
  • Why it matters: This suggests that when you break these magical, non-reversible symmetries, you don't just get a frozen solid; you get a new, complex, flowing state of matter that we haven't fully seen before.

3. The Big Picture: Why Does This Matter?

In the old days of physics, we thought symmetry breaking was simple: you break a rule, and you get a solid, frozen state (like a magnet).

This paper shows that with non-invertible symmetries (the magical, non-reversible rules), breaking the symmetry can create something entirely new:

1. Infinite Possibilities: Because the "walls" between different states are so complex (they have a "quantum dimension" greater than 1), they create a massive, exponentially growing playground of low-energy states.
2. Criticality: Instead of freezing solid, the system stays "fluid" and critical even after breaking the symmetry.

The Takeaway:
The authors built a digital toy model to show us that the universe might have more "flavors" of order than we thought. Just as a chef can mix ingredients to create a sauce that is neither just salt nor just sugar, but a complex new flavor, these physicists found that breaking complex symmetries creates new, exotic states of matter that are both ordered and fluid at the same time.

They are essentially mapping out a new "zoo" of quantum phases, showing us that the rules of the universe are far more creative and complex than the simple on/off switches we used to imagine.

Technical Summary

1. Problem Statement

The paper investigates the phase diagram and symmetry-breaking phenomena of a specific one-dimensional (1D) lattice model whose symmetry is described by the Ising fusion category ($\mathcal{C}_{\text{Ising}}$), rather than a conventional group.

• Context: Traditional symmetry breaking follows the Landau paradigm based on group theory. However, modern physics recognizes non-invertible symmetries (described by fusion categories) which cannot be inverted.
• Challenge: While non-invertible symmetries are known to pin models to quantum criticality (via generalized 't Hooft anomalies), the nature of symmetry-breaking phases associated with these symmetries remains less explored. Specifically, it is unclear how "antiferromagnetic" states behave when the broken symmetry is non-invertible, and what the nature of the transitions between these phases is.
• Goal: To map the phase diagram of an Ising fusion category lattice model, identify distinct phases (symmetric and symmetry-breaking), characterize their low-energy physics, and determine the universality classes of the phase transitions.

2. Methodology

The authors employ a combination of analytical and numerical techniques to study the model defined by parameters $r$ and $\theta$.

• Model Construction:

  • The Hilbert space is built from the fusion spaces of the Ising fusion category ($\mathcal{C}_{\text{Ising}} = \{1, \psi, \sigma\}$), rather than a tensor product of local spins.
  • The Hamiltonian $H = -\sum H_i$ consists of two terms: a domain-wall interaction ($H_{dw}$) favoring order (controlled by $r$) and a domain-flipping interaction ($H_{flip}$) favoring disorder (controlled by $\theta$).
  • The model possesses a categorical symmetry generated by operators $U(1)$, $U(\psi)$, and $U(\sigma)$, satisfying the Verlinde algebra.

• Analytical Approaches:

  • Perturbation Theory: Used to analyze the limits $r \ll -1$ (ferromagnetic-like) and $r \gg 1$ (antiferromagnetic-like). The authors map the effective low-energy Hamiltonians to transverse-field Ising models.
  • Conformal Field Theory (CFT): Used to interpret the scaling dimensions and central charges of the critical phases.

• Numerical Methods:

  • Exact Diagonalization (ED): Used for small system sizes ($L \approx 20$) to study energy spectra, ground state degeneracies, and symmetry charges.
  • Density Matrix Renormalization Group (DMRG): Used for larger systems ($L=400$) to calculate entanglement entropy, extract central charges ($c$), and map the phase diagram.
  • Loop-TNR (Tensor Network Renormalization): Used specifically for the symmetric-to-ferromagnetic transition to obtain high-precision conformal data (scaling dimensions and central charge) by handling larger effective system sizes.

3. Key Contributions and Results

The study identifies three distinct phases in the $(r, \theta)$ phase diagram:

A. Symmetric Critical Phase (Gapless)

• Description: A gapless phase where the full $\mathcal{C}_{\text{Ising}}$ symmetry is preserved.
• Low-Energy Physics: Described by the Ising Conformal Field Theory (CFT) with central charge $c = 1/2$.
• Stability: The phase is stable because there are no relevant operators symmetric under both $\mathcal{C}_{\text{Ising}}$ and lattice translation.

B. Categorical Ferromagnetic Phase (CatFM) - Gapped

• Description: A gapped phase where the entire $\mathcal{C}_{\text{Ising}}$ symmetry is spontaneously broken.
• Ground State: Exhibits a three-fold degeneracy, corresponding to the three irreducible representations of the Verlinde algebra.
• Nature: Analogous to a conventional ferromagnet but within the categorical Landau paradigm. Perturbative analysis shows it arises from the lifting of degeneracy in the $r \ll -1$ limit.

C. Categorical Antiferromagnetic Phase (CatAFM) - Gapless (Key Discovery)

• Description: A gapless phase where lattice translation and the non-invertible element $U(\sigma)$ are spontaneously broken.
• Ground State: Exhibits a four-fold degeneracy.
• Low-Energy Physics: Unlike conventional antiferromagnets (which are gapped), this phase is critical. It is described by the direct sum of four Ising CFTs (not a direct product).
• Mechanism: The authors argue that breaking a non-invertible symmetry creates domain walls with quantum dimension $d > 1$ (specifically $d_\sigma = \sqrt{2}$). This leads to an extensive Hilbert space that grows exponentially with the number of domain walls, preventing the system from gapping out.
• Entanglement Signature: DMRG calculations reveal a unique splitting in the entanglement entropy $S(x)$ into four branches based on the cut position modulo 4. One branch is separated by a gap $\Delta S \approx \frac{1}{2}\ln 2$, attributed to measurement-induced entropy drops in the underlying effective Ising chains.

D. Phase Transitions

1. Symmetric $\leftrightarrow$ CatFM:

   • This transition is continuous and described by the Tricritical Ising CFT with central charge $c = 7/10$.
   • Numerical data from Loop-TNR confirms the scaling dimensions match the $M_{5,4}$ minimal model.

2. Symmetric $\leftrightarrow$ CatAFM:

   • This transition is less understood but numerical data suggests it is continuous.
   • The central charge is found to be $c \approx 1.5$.
   • Interpretation: The authors conjecture this transition corresponds to a stack of the background Ising CFT ($c=1/2$) and a $c=1$ Luttinger liquid (compactified free boson), resulting in $c = 1.5$.

4. Significance and Implications

• Generalization of Landau Paradigm: The paper provides a concrete lattice realization of the "Categorical Landau Paradigm," demonstrating how non-invertible symmetries lead to novel symmetry-breaking phases that differ fundamentally from group-symmetry breaking.
• Critical Antiferromagnetism: The discovery of a gapless antiferromagnetic phase driven by non-invertible symmetry breaking is a major finding. It challenges the intuition that antiferromagnetic ordering (breaking translation) typically leads to a gapped state. The mechanism relies on the high quantum dimension of the domain walls associated with non-invertible defects.
• New Critical Points: The identification of the $c=3/2$ transition point offers a new universality class for 1D quantum phase transitions involving non-invertible symmetries.
• Methodological Advance: The work demonstrates the power of combining tensor network methods (Loop-TNR, DMRG) with categorical algebra to study complex topological and symmetry-enriched phases in 1D systems.

In summary, this paper establishes that non-invertible symmetries can stabilize both gapped (ferromagnetic) and gapless (antiferromagnetic) symmetry-breaking phases, with the latter exhibiting unique critical behavior and entanglement structures distinct from conventional condensed matter systems.