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Projective Representations, Bogomolov Multiplier, and Their Applications in Physics

This paper provides a pedagogical review of projective representations and the Bogomolov multiplier while presenting new physical results that link these mathematical concepts to undetectable (1+1)D SPT phases and distinct gapped phases with broken non-invertible symmetries, demonstrated through explicit lattice models and interface mode analysis.

Original authors: Ryohei Kobayashi, Haruki Watanabe

Published 2026-02-23

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Ryohei Kobayashi, Haruki Watanabe

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

The Big Picture: A New Kind of "Hidden" Symmetry

Imagine you are trying to identify a secret society. Usually, you look for their badges, their handshakes, or their specific uniforms. In physics, these "badges" are called symmetries. If a system looks the same after you rotate it or flip it, it has a symmetry.

For a long time, physicists thought they had a complete rulebook for identifying these symmetries in quantum materials. They believed that if a material had a certain "topological" (twisted) structure, you could always find a specific pattern (called a string order parameter) that would reveal it, much like finding a hidden code in a message.

This paper says: "Not always."

The authors, Ryohei Kobayashi and Haruki Watanabe, have discovered a special class of symmetries that are so subtle they hide completely from the usual detection methods. They call this the Bogomolov Multiplier. Think of it as a "ghost symmetry"—it exists, it affects the physics, but it leaves no trace on the standard "badges" we usually look for.

Part 1: The Dance of the Quantum Particles (Projective Representations)

To understand the "ghost," we first need to understand how quantum particles dance.

In the classical world, if you tell a dancer to turn left and then turn right, they end up facing the same way as if they did nothing. The order doesn't matter.
In the quantum world, things are weirder. If you tell a particle to turn left and then right, it might end up in the same position, but its internal "mood" (its quantum phase) might have changed. It's like the dancer did the same moves, but they are now wearing a different colored shirt.

Mathematically, this is called a Projective Representation.

Normal Symmetry: $A \times B = C$
Projective Symmetry: $A \times B = (\text{a ghostly phase}) \times C$

Usually, these "ghostly phases" cancel out or are easy to spot. But the authors focus on a specific group of symmetries where these phases are tricky. They are "symmetric" (they don't change the order of operations) but still "non-trivial" (they still exist). This is the Bogomolov Multiplier.

The Analogy: Imagine a group of friends playing a game of "Telephone."

In a normal game, if Alice whispers to Bob, and Bob to Charlie, the message is clear.
In this "Bogomolov" game, the message is clear, but every time the message passes between two specific friends, a secret, invisible "wink" is added. If you only look at the final message, you don't see the wink. But if you look at the rules of how they pass the message, the wink is there.

Part 2: The Invisible Wall (Why String Order Fails)

In the past, physicists used "String Order Parameters" to detect these twisted quantum states. Think of a string order parameter as a long, glowing rope stretched across the material. If the material is in a special topological state, the rope glows.

The authors show that for systems governed by the Bogomolov Multiplier, the rope never glows.

Why? Because the "winks" (the phases) are perfectly balanced between the friends who are "commuting" (talking to each other). The rope looks completely normal, even though the material is actually in a very exotic, twisted state.
The Result: We have a whole new class of quantum materials that look "boring" to our old tools but are actually full of hidden complexity.

Part 3: The Two Identical Twins (Distinct Phases)

The authors then ask: "If we can't see the difference with a rope, how do we tell these two special phases apart?"

They construct two different "Lattice Models" (think of them as two different blueprints for building a quantum crystal).

Model A: Built on a "normal" foundation.
Model B: Built on a "Bogomolov" foundation.

The Shocking Discovery:
If you look at the ground state (the lowest energy, most stable state) of both models, they look identical.

They have the same number of particles.
They have the same symmetry breaking (the symmetry is "shattered" in the exact same way).
They even react to symmetry operations in the exact same way.

It's like having two identical twins. They have the same height, weight, and fingerprints. You can't tell them apart by looking at them.

So, how do we tell them apart?
The authors found a new way: The Fusion Rule.
Imagine the twins have a special way of shaking hands.

In Model A, if Twin X shakes hands with Twin Y, they get a "High Five."
In Model B, if Twin X shakes hands with Twin Y, they get a "Fist Bump."

The "Fusion Rule" describes how local parts of the system combine. Even though the twins look the same, the math of how they combine is different. This is the first time physicists have found two phases that are identical in every way except for this specific mathematical handshake.

Part 4: The Secret Doorway (Interface Modes)

Here is the most exciting part. The authors put Model A and Model B next to each other, creating a "wall" or interface between them.

Expectation: If the two models are so similar, the wall between them should be boring.
Reality: The wall is alive.

When you put these two "identical" phases together, the boundary between them creates new, extra quantum states that didn't exist before.

Imagine a ring of 32 people (representing the ground state).
If you put a "wall" between two different types of people in the ring, suddenly the ring can hold 56 people.
Where did the extra 24 people come from? They are Interface Modes. They are "ghosts" that only exist at the boundary where the two phases meet.

This is like building a bridge between two identical islands. You expect the bridge to be empty, but instead, the bridge itself becomes a bustling city with its own population.

Part 5: The 3D Perspective (Soft Symmetry)

Finally, the authors look at this from a higher dimension (2+1 dimensions, like a flat sheet). They use a tool called Symmetry TQFT (a fancy way of mapping 1D problems onto 2D shapes).

They show that this "Bogomolov Multiplier" creates a "Soft Symmetry."

Hard Symmetry: Like a rigid rule that forces particles to swap places (like a dance partner switch).
Soft Symmetry: A gentle, invisible force that doesn't move the particles but changes the way they connect to each other. It's like a rule that says, "You don't have to move, but when you hold hands, you must hold them tighter."

This soft symmetry is the reason why the two phases look identical from the outside but have different fusion rules and extra states at the interface.

Summary: Why Does This Matter?

New Physics: We found a new type of quantum matter that hides from our standard detectors.
New Classification: We now know that "identical" phases can actually be different, distinguished only by how their parts combine (fusion rules).
New Technology: These "interface modes" (the extra states at the boundary) could be useful for quantum computing. They are robust and protected, meaning they might be perfect for storing information without it getting corrupted.

In a nutshell: The authors found a secret handshake in the quantum world that makes two things look exactly the same, but when you put them together, they create a whole new world of hidden energy at their meeting point. It's a reminder that in quantum mechanics, what you don't see can be just as important as what you do.

1. Problem Statement

The paper addresses two interconnected challenges in the theoretical physics of quantum many-body systems:

Mathematical Gap: There is a lack of accessible, self-contained pedagogical resources connecting the abstract mathematics of projective representations and group cohomology to modern condensed matter physics.
Physical Classification Gap: While Symmetry-Protected Topological (SPT) phases and Symmetry-Broken (SSB) phases are well-understood for Abelian groups, the classification of phases associated with non-Abelian finite groups possessing a non-trivial Bogomolov multiplier remains incomplete. Specifically, it was unclear how to distinguish distinct SSB phases that share identical ground state degeneracies and symmetry actions but differ in their topological properties, and how these phases manifest at interfaces.

2. Methodology

The authors employ a multi-faceted approach combining rigorous group theory, lattice model construction, and topological quantum field theory (TQFT):

Mathematical Framework:
- Projective Representations: The paper reviews the classification of irreducible projective representations of finite groups using 2-cocycles ( $\alpha$ ) and group cohomology $H^2(G, U(1))$ .
- Bogomolov Multiplier ( $B(G)$ ): They focus on a specific subgroup of $H^2(G, U(1))$ consisting of cocycles that are symmetric on commuting pairs ( $\alpha(g, g') = \alpha(g', g)$ for $gg'=g'g$) but non-trivial in cohomology. This implies that while all conjugacy classes are $\alpha$ -regular, the cocycle is not a coboundary.
- Group Algebra: They utilize the structure of the $\alpha$ -twisted group algebra to derive orthogonality relations and determine the number of irreducible representations.
Lattice Model Construction:
- SPT Models: They construct explicit 1D lattice models for SPT phases characterized by $\alpha \in B(G)$ using generalized cluster states and SPT entanglers.
- Gauging: They apply a Kramers-Wannier (KW)-type transformation to "gauge" the global $G$ symmetry of these SPT phases, resulting in gapped phases with non-invertible $\text{Rep}(G)$ symmetry.
- Interface Models: They construct hybrid lattice models where two distinct SSB phases (one trivial, one twisted by $\alpha$ ) meet, creating domain walls.
Symmetry TQFT:
- They utilize the Symmetry TQFT framework, viewing 1D gapped phases as intervals of 2D topological order (specifically $G$ gauge theory) bounded by gapped boundary conditions. This allows them to classify phases and calculate ground state degeneracies via the overlap of boundary states.

3. Key Contributions

A. Pedagogical Review and Mathematical Foundations

Provided a self-contained derivation of projective representations, including the definition of $\alpha$ -regular conjugacy classes and the proof that the number of irreducible projective representations equals the number of $\alpha$ -regular conjugacy classes.
Clarified the role of the Bogomolov multiplier: it represents cohomology classes where the "twist" is invisible to local commuting observables (symmetric on commuting pairs) but non-trivial globally.

B. Absence of String Order Parameters

Demonstrated that (1+1)D SPT phases characterized by a Bogomolov multiplier cannot be detected by string order parameters.
Reasoning: String order parameters rely on the electric charge of symmetry defects under the centralizer $Z(g)$ . For $\alpha \in B(G)$ , the charge $\phi_g(h) = \alpha(g,h)/\alpha(h,g)$ is trivial for all $g, h$ . Thus, the defects carry no charge, rendering standard string order parameters zero, despite the phase being non-trivial (protected edge modes exist).

C. Distinct $\text{Rep}(G)$ Symmetry Broken Phases

Identified that gauging an SPT phase with $\alpha \in B(G)$ leads to a maximally broken $\text{Rep}(G)$ phase.
Indistinguishability: Two such phases (trivial vs. twisted by $\alpha$ $α$ ) share:
- Identical ground state degeneracy ( $|C(G)|$ ).
- Identical symmetry actions on ground states (characterized by group characters).
- Identical local order parameters (labeled by conjugacy classes).
Distinguishing Mechanism: The phases are distinguished solely by the fusion rules of their local order parameters. The fusion coefficients acquire a phase factor $\alpha'(C, C')$ in the twisted phase, which cannot be gauged away.

D. Non-Trivial Interface Modes

Discovery: When two distinct maximally broken $\text{Rep}(G)$ phases meet (e.g., trivial and twisted), the interface hosts non-trivial gapped modes.
Ground State Degeneracy (GSD) Increase: On a closed ring, the GSD increases from $|G|$ $∣ G ∣$ (without interfaces) to a higher value when interfaces are present.
- Example: For the specific group $G$ (order 128) used in the paper, the GSD increases from 32 (bulk only) to 56 when interfaces are introduced.
Mechanism: The interface Hamiltonian becomes intrinsically frustrated due to the projective nature of the representation at the boundary, leading to a larger Hilbert space of zero-energy states.

E. Soft Symmetry and TQFT Classification

Soft Symmetry: In (2+1)D $G$ $G$ gauge theory, elements of $B(G)$ $B (G)$ generate a "soft symmetry." This symmetry:
- Does not permute anyon labels.
- Does not fractionalize on anyons.
- Acts faithfully on the ground state Hilbert space of higher-genus surfaces by acting on fusion vertices.
Classification: Using Symmetry TQFT, the authors prove that (1+1)D gapped phases with maximally broken $\text{Rep}(G)$ symmetry are classified by the Bogomolov multiplier $B(G)$ . The distinct phases correspond to different gapped boundary conditions (condensing the same particles but with different Lagrangian algebra structures).

4. Key Results

Explicit Lattice Models: Constructed Hamiltonians for SPT phases with $B(G)$ and their gauged SSB counterparts.
Fusion Rule Distinction: Proved that the fusion algebra of local order parameters $O(C) \times O(C') = \sum N^{C''}_{C,C'} \alpha'(C,C') O(C'')$ serves as the unique invariant distinguishing the twisted SSB phase from the trivial one.
Interface GSD Calculation:
- Derived the GSD of the interface between two SSB phases using both lattice models and TQFT.
- Result: $d_{interface} = \sum_{\pi} d_\pi$ , where the sum runs over 1D representations $\pi$ of $G$ . For the example group, $d = 32 + 16 + 4 + 4 = 56$ .
- Confirmed consistency between the lattice calculation and the TQFT overlap of boundary states.
Minimal Order Example: Identified and analyzed the smallest group ( $|G|=64$ ) with a non-trivial Bogomolov multiplier, providing concrete data for projective and linear representations.

5. Significance

New Topological Invariants: The work establishes that the Bogomolov multiplier is a physical invariant capable of distinguishing phases that are indistinguishable by standard symmetry breaking patterns, ground state degeneracy, or local order parameters.
Beyond String Order: It expands the understanding of SPT phases, showing that non-trivial topology can exist even when standard non-local order parameters (string order) vanish.
Non-Invertible Symmetries: It provides a concrete realization of phases with non-invertible $\text{Rep}(G)$ symmetries, linking them to the mathematical concept of the Bogomolov multiplier.
Interface Physics: The discovery of enhanced ground state degeneracy at the interface of two "identical-looking" SSB phases reveals a new class of topological defects that are not present in standard Landau-Ginzburg theory.
Unified Framework: By bridging lattice models, group cohomology, and Symmetry TQFT, the paper offers a comprehensive framework for classifying gapped phases with non-invertible symmetries in (1+1)D and their relation to (2+1)D topological order.

In summary, this paper demonstrates that the Bogomolov multiplier is not merely a mathematical curiosity but a fundamental physical quantity that dictates the existence of unique SPT phases, distinct SSB phases, and exotic interface modes in quantum many-body systems.