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Short-range entanglement and invertible field theories

Daniel S. Freed

TL;DR

The paper develops a homotopy-theoretic framework for classifying short-range entangled gapped phases via invertible, fully extended topological field theories. By modeling long-range effective theories as maps of spectra, it connects bordism/homotopy data (via Madsen–Tillmann spectra) with symmetry, anomalies, and unitarity, yielding concrete invariants for bosonic and fermionic SRE-SPT phases. Through detailed computations in low dimensions, it recovers group cohomology results and reveals additional phases tied to gravitational data (e.g., Kitaev’s E8 phase) and boundary phenomena, clarifying bulk-boundary correspondences. The approach provides a unifying, computable framework that extends beyond group cohomology to capture richer structure in SRE phases and their symmetry protections.

Abstract

Quantum field theories with an energy gap can be approximated at long-range by topological quantum field theories. The same should be true for suitable condensed matter systems. For those with short range entanglement (SRE) the effective topological theory is invertible, and so amenable to study via stable homotopy theory. This leads to concrete topological invariants of gapped SRE phases which are finer than existing invariants. Computations in examples demonstrate their effectiveness.

Short-range entanglement and invertible field theories

TL;DR

The paper develops a homotopy-theoretic framework for classifying short-range entangled gapped phases via invertible, fully extended topological field theories. By modeling long-range effective theories as maps of spectra, it connects bordism/homotopy data (via Madsen–Tillmann spectra) with symmetry, anomalies, and unitarity, yielding concrete invariants for bosonic and fermionic SRE-SPT phases. Through detailed computations in low dimensions, it recovers group cohomology results and reveals additional phases tied to gravitational data (e.g., Kitaev’s E8 phase) and boundary phenomena, clarifying bulk-boundary correspondences. The approach provides a unifying, computable framework that extends beyond group cohomology to capture richer structure in SRE phases and their symmetry protections.

Abstract

Quantum field theories with an energy gap can be approximated at long-range by topological quantum field theories. The same should be true for suitable condensed matter systems. For those with short range entanglement (SRE) the effective topological theory is invertible, and so amenable to study via stable homotopy theory. This leads to concrete topological invariants of gapped SRE phases which are finer than existing invariants. Computations in examples demonstrate their effectiveness.

Paper Structure

This paper contains 46 sections, 2 theorems, 147 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Field theories from a bordism point of view
  3. Field theories
  4. Invertible field theories
  5. Relative field theories and anomalies
  6. Global symmetries and equivariant extensions
  7. General discussion
  8. Anomalies
  9. Gauging antilinear symmetries
  10. Unitarity
  11. Topological field theories and the cobordism hypothesis
  12. Invertible topological theories and maps of spectra
  13. An illuminating example
  14. The long-range effective topological field theory
  15. Low energy, long time
  16. ...and 31 more sections

Key Result

Proposition 6.15

Figures (13)

  • Figure 1: Quantum mechanical evolution
  • Figure 2: Correlation functions
  • Figure 3: A bordism $X\colon Y_0\to Y_1$
  • Figure 4: Composition $X'\circ X$ of morphisms $X\colon Y_0\to Y_1$ and $X'\colon Y_1\to Y_2$
  • Figure 5: Quantum mechanics with boundary
  • ...and 8 more figures

Theorems & Definitions (42)

  • Example 2.4
  • Remark 2.6
  • Remark 2.7
  • Example 2.9
  • Example 2.10
  • Example 2.11
  • Example 2.13
  • Example 2.18
  • Example 2.21
  • Example 2.22
  • ...and 32 more