Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Projective Representations, Bogomolov Multiplier, and Their Applications in Physics

Ryohei Kobayashi, Haruki Watanabe

TL;DR

The paper addresses how finite-group projective representations, controlled by 2-cocycles and the Bogomolov multiplier $B(G)$, govern new physical phases in quantum many-body systems. It develops the mathematical machinery of $\alpha$-twisted group algebras, central extensions, and irreducible projective representations, then exposes concrete group examples with nontrivial Bogomolov classes. On the physics side, it constructs (1+1)D SPT and gapped phases whose distinctions are detected not by string order parameters but by fusion rules of local order parameters and by interface modes, with a symmetry-TQFT framework unifying (1+1)D and (2+1)D perspectives. A key insight is that Bogomolov multipliers classify maximally broken $\mathrm{Rep}(G)$ phases and give rise to soft symmetries in (2+1)D gauge theories, enabling novel gapped boundaries and interface phenomena, including ground-state degeneracies such as 32 vs 56 on rings with interfaces.

Abstract

We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective representations, highlighting the role of group cohomology, representation theory, and classification of irreducible projective representations. We then focus on a special subset of cohomology classes, known as the Bogomolov multiplier, which consists of cocycles that are symmetric on commuting pairs but remain nontrivial in group cohomology. Such cocycles have important physical implications: they characterize (1+1)D SPT phases that cannot be detected by string order parameters and give rise, upon gauging, to distinct gapped phases with completely broken non-invertible $\mathrm{Rep}(G)$ symmetry. We construct explicit lattice models for these phases and demonstrate how they are distinguished by the fusion rules of local order parameters. We show that a pair of completely broken $\mathrm{Rep}(G)$ SSB phases host nontrivial interface modes at their domain walls. As an example, we construct a lattice model where the ground state degeneracy on a ring increases from 32 without interfaces to 56 with interfaces.

Projective Representations, Bogomolov Multiplier, and Their Applications in Physics

TL;DR

The paper addresses how finite-group projective representations, controlled by 2-cocycles and the Bogomolov multiplier , govern new physical phases in quantum many-body systems. It develops the mathematical machinery of -twisted group algebras, central extensions, and irreducible projective representations, then exposes concrete group examples with nontrivial Bogomolov classes. On the physics side, it constructs (1+1)D SPT and gapped phases whose distinctions are detected not by string order parameters but by fusion rules of local order parameters and by interface modes, with a symmetry-TQFT framework unifying (1+1)D and (2+1)D perspectives. A key insight is that Bogomolov multipliers classify maximally broken phases and give rise to soft symmetries in (2+1)D gauge theories, enabling novel gapped boundaries and interface phenomena, including ground-state degeneracies such as 32 vs 56 on rings with interfaces.

Abstract

We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective representations, highlighting the role of group cohomology, representation theory, and classification of irreducible projective representations. We then focus on a special subset of cohomology classes, known as the Bogomolov multiplier, which consists of cocycles that are symmetric on commuting pairs but remain nontrivial in group cohomology. Such cocycles have important physical implications: they characterize (1+1)D SPT phases that cannot be detected by string order parameters and give rise, upon gauging, to distinct gapped phases with completely broken non-invertible symmetry. We construct explicit lattice models for these phases and demonstrate how they are distinguished by the fusion rules of local order parameters. We show that a pair of completely broken SSB phases host nontrivial interface modes at their domain walls. As an example, we construct a lattice model where the ground state degeneracy on a ring increases from 32 without interfaces to 56 with interfaces.

Paper Structure

This paper contains 42 sections, 14 theorems, 196 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Projective Representations
  3. Properties of 2-Cocycles
  4. Group cohomology
  5. $\alpha$-regularity of conjugacy classes
  6. Projective Representations
  7. Multipliers and Projective Representations
  8. Schur's Lemma
  9. Group Algebra
  10. Group Algebra and Projective Representations
  11. Products of Conjugacy Classes and Structure Constants
  12. Method to Determine Irreducible Representations
  13. Central Extension
  14. Bogomolov Multiplier
  15. Pollmann-Turner Example
  16. ...and 27 more sections

Key Result

Lemma 1

For an element $x\in G$, define $\alpha'(g,g')\coloneqq\alpha(xgx^{-1},xg'x^{-1})$. Then, $\alpha'\sim\alpha$.

Figures (5)

  • Figure 1: The non-invertible $\text{Rep}(G)$ symmetry is generated by an MPO $Z_\Gamma$, which gives a representation matrix $\Gamma(g)$ acting on the bond Hilbert space.
  • Figure 2: Left: The Hamiltonian with interfaces located sites $L$ and $2L$. Right: By a sequential circuit, the bulk becomes decoupled, leaving behind the two body Hamiltonian at the interfaces $L$ and $2L$. The symmetry action becomes localized at the interfaces after the action of the sequential circuit.
  • Figure 3: The construction of gauged SPT defect in (2+1)D $G$ gauge theory. We first start with a trivial gapped phase with $G$ symmetry (product state), and act the 1d SPT entangler along a 1d subsystem. This SPT entangler is labeled by group cohomology $\alpha\in H^2(G,U(1))$, and creates a 1d SPT phase at the subsystem. Then we gauge $G$ symmetry of the whole system to get $G$ gauge theory in (2+1)D. The decoration of 1d SPT phase results in the insertion of a symmetry defect in the $G$ gauge theory, which we call a gauged SPT defect.
  • Figure 4: (a): A generic (1+1)D system with a non-invertible symmetry is obtained by an interval of (2+1)D topological order sandwiched by a pair of boundary conditions. The symmetry boundary condition is gapped, and the topological operators at the symmetry boundary define the symmetries of the interval and the resulting (1+1)D system. (b): Symmetry TQFT description for a (1+1)D gapped phase with maximally broken $\text{Rep}(G)$ symmetry.
  • Figure 5: Symmetry TQFT picture for the interfaces between two distinct $(1+1)$D gapped phases with maximally broken $\text{Rep}(G)$ symmetry. On the dynamical (top) boundary, we have an interval of $(G,\alpha)$ boundary condition of (2+1)D $G$ gauge theory with $\alpha\in B(G)$. This is regarded as having a gauged SPT defect $\alpha$ along the interval of the top boundary (red curve). One can shrink the size of the interval for the gauged SPT defect, then we end up with a non-simple anyon $1+W_{n_a}+W_{n_b}+W_{n_an_b}$. The partition function is then expressed as the overlap of boundary states $\langle \mathcal{L}| \mathcal{L}\times (1+W_{n_a}+W_{n_b}+W_{n_an_b}) \rangle$.

Theorems & Definitions (32)

  • Example 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2: Number of $\alpha$-regular conjugacy classes
  • ...and 22 more