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Symmetry Protected Topological States of Interacting Fermions and Bosons

Yi-Zhuang You, Cenke Xu

TL;DR

This work develops a unifying framework to classify interacting symmetry-protected topological (SPT) states across all dimensions by mapping fermionic SPT states to bosonic BSPT states via an $O(d ext{+}2)$ nonlinear sigma model with a topological Θ-term. By analyzing dimensionally reduced pictures and defect proliferations, it demonstrates that short-range interactions can reduce the free-fermion SPT classifications (e.g., $bZ o bZ_8$ in 1D, $bZ o bZ_8$ in 2D, and $bZ o bZ_{16}$ in 3D), with BSPT classifications providing the necessary constraint structure. The paper uses explicit lattice models (Kitaev chain, $p ext{-}ip$ TSCs, and $^3$He-B), their bosonic counterparts (Haldane chain, Levin–Gu paramagnet, 3D BSPT), and a systematic Clifford-algebra based construction to derive minimal-copy requirements and to generalize to inversion-symmetric and U(1)/time-reversal combinations. Collectively, the results yield a Bott-periodic, dimension-by-dimension map from iFSPT to BSPT, clarifying when interactions can gap bulk or boundary criticalities and thereby identifying robust reductions of free-fermion topological classifications with practical implications for strongly interacting quantum systems.

Abstract

We study the classification of interacting fermionic and bosonic symmetry protected topological (SPT) states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for all the known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry.

Symmetry Protected Topological States of Interacting Fermions and Bosons

TL;DR

This work develops a unifying framework to classify interacting symmetry-protected topological (SPT) states across all dimensions by mapping fermionic SPT states to bosonic BSPT states via an nonlinear sigma model with a topological Θ-term. By analyzing dimensionally reduced pictures and defect proliferations, it demonstrates that short-range interactions can reduce the free-fermion SPT classifications (e.g., in 1D, in 2D, and in 3D), with BSPT classifications providing the necessary constraint structure. The paper uses explicit lattice models (Kitaev chain, TSCs, and He-B), their bosonic counterparts (Haldane chain, Levin–Gu paramagnet, 3D BSPT), and a systematic Clifford-algebra based construction to derive minimal-copy requirements and to generalize to inversion-symmetric and U(1)/time-reversal combinations. Collectively, the results yield a Bott-periodic, dimension-by-dimension map from iFSPT to BSPT, clarifying when interactions can gap bulk or boundary criticalities and thereby identifying robust reductions of free-fermion topological classifications with practical implications for strongly interacting quantum systems.

Abstract

We study the classification of interacting fermionic and bosonic symmetry protected topological (SPT) states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for all the known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry.

Paper Structure

This paper contains 40 sections, 69 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. SPT states without spatial symmetry
  3. From Kitaev Chain to Haldane Chain
  4. Lattice Model and Bulk Theory
  5. Corresponding BSPT state
  6. Bulk transition
  7. From $2d$ TSC to Levin-Gu Paramagnet
  8. Lattice Model and Bulk Theory
  9. Corresponding BSPT state
  10. Boundary modes and Bulk transition
  11. From $^3$He-B to $3d$ Bosonic SPT
  12. Lattice Model and Bulk Theory
  13. Corresponding BSPT state
  14. Bulk Phase Transition under Interaction
  15. Boundary Modes and Bulk transition
  16. ...and 25 more sections

Figures (3)

  • Figure 1: Lattice model of Kitaev's Majorana chain.
  • Figure 2: Phase diagram of the interacting Majorana chain at $\nu=8$. The bulk criticality at the origin can be avoided if interaction is allowed. Each horizontal chain is four copies of the Majorana chain, equivalent to a Haldane chain. The vertical bound is the on-site spin-spin interaction. $A$ and $B$ labels the sites in a unit cell. Gray ovals mark out the spin-singlet dimers.
  • Figure 3: Many-body energy levels of the two-site segment by exact diagonalization. From the weak interaction limit (left) to the strong interaction (right) limit, the many-body gap never closes.