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Symmetry Protected Topological phases of Quantum Matter

T. Senthil

TL;DR

This work surveys how Symmetry Protected Topological (SPT) phases generalize topological band insulators to interacting electronic systems, focusing on 3D realizations with realistic symmetries. It develops a physical, surface-centered perspective—via dual vortex descriptions, monopole physics, and surface topological orders—and shows that interactions generate a Z_2^3 classification of 3D electronic TIs, including three root phases: the conventional TBI and two topological paramagnets. The paper also discusses correlated surface states, coupled-layer constructions, and experimental fingerprints, illustrating how symmetry and topology together constrain surface phenomena and bulk excitations. Beyond specific models, it highlights how SPT insights illuminate broader problems in SET phases, gapless quantum matter, and lattice gauge theories, underscoring the practical relevance of symmetry-entangled, short-range entangled states for strongly correlated materials.

Abstract

We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators to interacting many particle systems, known as Symmetry Protected Topological (SPT) phases. In common with the topological band insulators these states have a bulk gap and no exotic excitations but have non-trivial surface states that are protected by symmetry. We describe the various possible such phases and their properties in three dimensional systems with realistic symmetries. We develop many key ideas of the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.

Symmetry Protected Topological phases of Quantum Matter

TL;DR

This work surveys how Symmetry Protected Topological (SPT) phases generalize topological band insulators to interacting electronic systems, focusing on 3D realizations with realistic symmetries. It develops a physical, surface-centered perspective—via dual vortex descriptions, monopole physics, and surface topological orders—and shows that interactions generate a Z_2^3 classification of 3D electronic TIs, including three root phases: the conventional TBI and two topological paramagnets. The paper also discusses correlated surface states, coupled-layer constructions, and experimental fingerprints, illustrating how symmetry and topology together constrain surface phenomena and bulk excitations. Beyond specific models, it highlights how SPT insights illuminate broader problems in SET phases, gapless quantum matter, and lattice gauge theories, underscoring the practical relevance of symmetry-entangled, short-range entangled states for strongly correlated materials.

Abstract

We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators to interacting many particle systems, known as Symmetry Protected Topological (SPT) phases. In common with the topological band insulators these states have a bulk gap and no exotic excitations but have non-trivial surface states that are protected by symmetry. We describe the various possible such phases and their properties in three dimensional systems with realistic symmetries. We develop many key ideas of the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.

Paper Structure

This paper contains 24 sections, 22 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Review of topological band insulators
  3. Warm-up: Bosonic SPT phases in $d =2$
  4. d = 2: Integer quantum Hall effect for bosons
  5. Ising SPT in d = 2: Braiding statistics
  6. Bosonic Topological Insulators in d = 3
  7. Surface theory
  8. Monopoles in the bulk
  9. Surface topological order
  10. Coupled layer construction
  11. Electronic SPT phases in 1d and 2d
  12. Electronic topological insulators in 3d
  13. Classification
  14. Physics and construction of topological paramagnets
  15. Gapped surface quantum spin liquids
  16. ...and 9 more sections

Figures (4)

  • Figure 1: Mutual composite bosons and their transport in the 2-component boson integer quantum Hall effect
  • Figure 2: Edge structure in the boson integer quantum Hall effect
  • Figure 3: Coupled-layer construction of SPT states. The particle composite in the ellipses are condensed, and only the four surface particles in the dotted ellipses survived as deconfined topological quasi-particles.
  • Figure 4: Ferromagnetic domain wall at the surface of the three root topological insulators. The TBI phase has a chiral charged edge mode in the domain wall. One Topological Paramagnet (labeled by surface topological $eTmT$) has a gapped domain wall while the other (labeled TP$_{efmf}$) has neutral chiral modes with $\nu_Q = 8$.