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Fermion Condensation and Gapped Domain Walls in Topological Orders

Yidun Wan, Chenjie Wang

TL;DR

The paper addresses phase transitions between 2D topological orders by extending boson condensation to fermion condensation via gapped domain walls, introducing a Hierarchy Principle that any fermion condensation factors into a boson condensation step followed by a minimal fermion condensation.It develops a concrete, testable framework: define the fermionic condensation via a fermionic Frobenius algebra, establish dimension relations, and show that the full rules for fermion condensation follow from combining bosonic condensation rules with minimal fermion condensation, while preserving a mapping between Hilbert spaces across the domain wall.The authors validate the framework with explicit examples (Z2 toric code, SU(2)_{4l-2}, and D[D3]) to demonstrate the hierarchy, confinement, and fusion changes, and discuss the broader implications for fTOs, symmetry-enriched structures, and potential links to fermionic RCFT modular invariants.Overall, the work provides a principled, scalable method to derive and understand fermionic topological orders from bosonic ones, clarifying how gapped domain walls implement real-space phase transitions between bosonic and fermionic orders and enabling systematic construction of new fTOs.

Abstract

We propose the concept of fermion condensation in bosonic topological orders in two spatial dimensions. Fermion condensation can be realized as gapped domain walls between bosonic and fermionic topological orders, which are thought of as a real-space phase transitions from bosonic to fermionic topological orders. This generalizes the previous idea of understanding boson condensation as gapped domain walls between bosonic topological orders. We show that generic fermion condensation obeys a Hierarchy Principle by which it can be decomposed into a boson condensation followed by a minimal fermion condensation, which involves a single self-fermion that is its own anti-particle and has unit quantum dimension. We then develop the rules of minimal fermion condensation, which together with the known rules of boson condensation, provides a full set of rules of fermion condensation. Our studies point to an exact mapping between the Hilbert spaces of a bosonic topological order and a fermionic topological order that share a gapped domain wall.

Fermion Condensation and Gapped Domain Walls in Topological Orders

TL;DR

The paper addresses phase transitions between 2D topological orders by extending boson condensation to fermion condensation via gapped domain walls, introducing a Hierarchy Principle that any fermion condensation factors into a boson condensation step followed by a minimal fermion condensation.It develops a concrete, testable framework: define the fermionic condensation via a fermionic Frobenius algebra, establish dimension relations, and show that the full rules for fermion condensation follow from combining bosonic condensation rules with minimal fermion condensation, while preserving a mapping between Hilbert spaces across the domain wall.The authors validate the framework with explicit examples (Z2 toric code, SU(2)_{4l-2}, and D[D3]) to demonstrate the hierarchy, confinement, and fusion changes, and discuss the broader implications for fTOs, symmetry-enriched structures, and potential links to fermionic RCFT modular invariants.Overall, the work provides a principled, scalable method to derive and understand fermionic topological orders from bosonic ones, clarifying how gapped domain walls implement real-space phase transitions between bosonic and fermionic orders and enabling systematic construction of new fTOs.

Abstract

We propose the concept of fermion condensation in bosonic topological orders in two spatial dimensions. Fermion condensation can be realized as gapped domain walls between bosonic and fermionic topological orders, which are thought of as a real-space phase transitions from bosonic to fermionic topological orders. This generalizes the previous idea of understanding boson condensation as gapped domain walls between bosonic topological orders. We show that generic fermion condensation obeys a Hierarchy Principle by which it can be decomposed into a boson condensation followed by a minimal fermion condensation, which involves a single self-fermion that is its own anti-particle and has unit quantum dimension. We then develop the rules of minimal fermion condensation, which together with the known rules of boson condensation, provides a full set of rules of fermion condensation. Our studies point to an exact mapping between the Hilbert spaces of a bosonic topological order and a fermionic topological order that share a gapped domain wall.

Paper Structure

This paper contains 20 sections, 4 theorems, 39 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Bosonic and Fermionic topological orders
  3. From Boson condensation to fermion Condensation
  4. Boson Condensation and GDWs
  5. Mutual Locality
  6. Fermion Condensation and GDWs
  7. Ansatz of Fermion Condensation
  8. Fermion condensation in bTOs
  9. Principles of Fermion Condensation in bTOs
  10. Review of Boson Condensation
  11. Minimal Fermion Condensation
  12. Other GDWs between fTOs
  13. examples
  14. $\mathcal{B}=\mathbb{Z}_2$-Toric Code
  15. $\mathcal{B}=SU(2)_{4l-2}$
  16. ...and 5 more sections

Key Result

Lemma 1

For any nontrivial anyon $a\in A$, $A$ being well-defined anyon condensation, if $a$ is a nonsimple current, i.e., $d_a>1$, then there exists at least a nontrivial self-boson $b\in a\times\bar{a}$, such that $b\in A$.

Figures (5)

  • Figure 1: Gapped domain walls between two bTOs (a) and between a bTO and an fTO (b).
  • Figure 2: (a) A bTO $\mathcal{B}$ and a $\mathcal{B}'$ connected by a GDW via condensing some bosons in $\mathcal{B}$. Via the folding trick along the GDW, this picture is equivalent to (b) A bTO $B\boxtimes\overline{\mathcal{B}'}$ sharing a gapped boundary with the vacuum.
  • Figure 3: (a) A bTO $\mathcal{B}$ and an fTO $\mathcal{F}$ connected by a GDW via condensing some fermions in $\mathcal{B}$. Via the folding trick along the GDW, (a) is equivalent to (b) A topological order $B\boxtimes\overline\mathcal{F}$. sharing a gapped boundary with $\mathcal{F}_0$, a fermionic vacuum.
  • Figure 4: (a) A GDW between $\mathcal{B}$ and $\mathcal{B}'$ and one between $\mathcal{B}'$ and $\mathcal{F}$ imply a GDW between $\mathcal{B}$ and $\mathcal{F}$ directly. $\mathcal{B}'$ is compressible between $B$ and $\mathcal{F}$.
  • Figure 5: (a) Two fTOs $\mathcal{F}$ and $\mathcal{F}'$ connected by a GDW via condensing some fermions in $\mathcal{F}$. Via the folding trick, this picture is equivalent to (b) A non-prime fTO $\mathcal{F}\boxtimes_{\mathcal{F}_0}\overline{\mathcal{F}'}$.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4