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Rigorous Index Theory for One-Dimensional Interacting Topological Insulators

Hal Tasaki

Abstract

We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with $\Uone\rtimes\bbZ_2$ symmetry defined on the infinite chain. The class includes the Su-Schrieffer-Heeger (SSH) model as a special case. For any locally-unique gapped (fixed-charge) ground state of a model in the class, we define a $\bbZ_2$ index in terms of the sign of the expectation value of the local twist operator. We prove that the index is topological in the sense that it is invariant under continuous modification of models in the class with a locally-unique (fixed-charge) gapped ground state. This establishes that any path of models in the class that connects the two extreme cases of the SSH model must go through a phase transition. Our rigorous $\bbZ_2$ classification is believed to be optimal for the class of models considered here. We also show an interesting duality of the index, and prove that any topologically nontrivial model in the class has a gapless edge excitation above the ground state when defined on the half-infinite chain. The results extend to other classes of models, including the extended Hubbard model. Our strategy to focus on the expectation value of local unitary operators makes the theory intuitive and conceptually simple. The paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain.

Rigorous Index Theory for One-Dimensional Interacting Topological Insulators

Abstract

We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with symmetry defined on the infinite chain. The class includes the Su-Schrieffer-Heeger (SSH) model as a special case. For any locally-unique gapped (fixed-charge) ground state of a model in the class, we define a index in terms of the sign of the expectation value of the local twist operator. We prove that the index is topological in the sense that it is invariant under continuous modification of models in the class with a locally-unique (fixed-charge) gapped ground state. This establishes that any path of models in the class that connects the two extreme cases of the SSH model must go through a phase transition. Our rigorous classification is believed to be optimal for the class of models considered here. We also show an interesting duality of the index, and prove that any topologically nontrivial model in the class has a gapless edge excitation above the ground state when defined on the half-infinite chain. The results extend to other classes of models, including the extended Hubbard model. Our strategy to focus on the expectation value of local unitary operators makes the theory intuitive and conceptually simple. The paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain.

Paper Structure

This paper contains 28 sections, 8 theorems, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The SSH-type models and their ground states
  3. General model with U(1) symmetry
  4. States and ground states
  5. The infinite volume limit of unique gapped ground states
  6. $\mathbb{Z}_2$ symmetry
  7. Index theory
  8. Main index theorem
  9. Simple examples
  10. Duality of the index
  11. Gapless edge excitations
  12. Decoupled systems
  13. Proofs
  14. The main index theorem
  15. The duality theorem
  16. ...and 13 more sections

Key Result

Theorem 2.7

The state $\omega$ is a locally-unique gapped fixed-charge ground state of $\hat{H}$ in the fixed-charge setting, and is a locally-unique gapped ground state of $\hat{H}^{(\mu)}$ in the Fock space setting.

Figures (3)

  • Figure 1: A part of the infinite chain $\mathbb{Z}$. We regard neighboring sites $2j$ and $2j+1$ (where $j\in\mathbb{Z}$) as forming a unit cell. The assignment of unit cells is crucial throughout the present paper. In the twist operator \ref{['e:U1']}, two sites in a unit cell have the same twist angle. In the extended Hubbard model \ref{['e:exHub']}, two sites in a unit cell are coupled by the ferromagnetic interaction and behave as a single quantum spin with $S=1$.
  • Figure 2: (a) A general $\theta$-function. (b) The piecewise linear $\theta$-function \ref{['e:bartheta']}.
  • Figure 3: Schematic pictures of the ground states \ref{['e:P01']}. (a) depicts the state $|\Psi_0^{\rm SSH}\rangle$, and (b) depicts $|\Psi_1^{\rm SSH}\rangle$. Here a thick gray line connecting sites $k$ and $k+1$ represent the "bonding state" created by $(\hat{c}^\dagger_{k}-\hat{c}^\dagger_{k+1})/\sqrt{2}$. Note that a bond is confined within a unit cell in the topologically trivial state in (a), while a bond connects two adjacent unit cells in a topologically nontrivial state in (b). We can also interpret the two diagrams as representing the ground states \ref{['e:HP01']} of the Hubbard model. In this case, a thick gray line represents a state created by $\hat{A}^\dagger_{k,k+1}$.

Theorems & Definitions (16)

  • Definition 2.1: state
  • Definition 2.2: fixed-charge ground state
  • Definition 2.3: ground state
  • Definition 2.4: locally-unique gapped fixed-charge ground state
  • Definition 2.5: locally-unique gapped ground state
  • Definition 2.6: unique gapped ground state
  • Theorem 2.7
  • Definition 2.8
  • Theorem 3.1
  • Definition 3.2
  • ...and 6 more