arXiv:1906.00218 · Physical Review Research

ZQ Berry phase for higher-order symmetry-protected topological phases

Hiromu Araki, Tomonari Mizoguchi, Yasuhiro Hatsugai

Published Jan 9, 2020• 69 citations

Abstract

The authors propose that the quantized Berry phase serves as a many-body topological invariant that characterizes the higher-order symmetry-protected topological phases in two- and three-dimensions, and provides a clear insight of bulk-corner correspondence. The quantized Berry phase has wide applicability ranging from fermionic models with and without interactions to spin models.

DOI: 10.1103/physrevresearch.2.012009

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arXiv:1906.00218 · Physical Review Research

ZQ Berry phase for higher-order symmetry-protected topological phases

Hiromu Araki, Tomonari Mizoguchi, Yasuhiro Hatsugai

Published Jan 9, 2020• 69 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.2.012009

Related papers

Many-body higher-order topological invariant for Cn -symmetric insulators

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{C}_{n}

-symmetric higher-order topological insulators that is valid for both noninteracting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of

{C}_{n}

rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as

e/n

with

{C}_{n}

symmetry, leading to a

{\mathbb{Z}}_{n}

classification for higher-order topological insulators in two dimensions.

Many-body higher-order topological invariant for

C_n

-symmetric insulators

C_n

symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of

C_n

rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as

e/n

with

C_n

symmetry, leading to a

\mathbb{Z}_n

classification for higher-order topological insulators in two dimensions.

The local

{Z}_{N}

quantized Berry phase for the

\mathrm{SU}(N)

antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local

{Z}_{2}

Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional

\mathrm{SU}(N)

spin systems. We extend the path-integral quantum Monte Carlo method for the

{Z}_{2}

Berry phase in order to calculate the

{Z}_{N}

Berry phase numerically. We demonstrate our method by calculating the

{Z}_{4}

Berry phase for the bond-alternating

\mathrm{SU}(4)

antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.

We discuss physical constructions and the boundary properties of various symmetry-protected topological phases that involve 1-form symmetries from one spatial dimension to four spatial dimensions (

4d

). For example, the prototype three-dimensional (

3d

) boundary state of

4d

SPT states involving 1-form symmetries can be either a gapless photon phase (quantum electrodynamics) or gapped topological order enriched by 1-form symmetries; that is, the loop excitations of these topological orders carry nontrivial 1-form symmetry charges. This study also serves the purpose of diagnosing anomalies of

3d

states of matter. Connection between SPT states with 1-form symmetries and condensed-matter systems such as quantum dimer models at one lower dimension will also be discussed. Whether a quantum dimer model can have a trivial gapped phase depends on the nature of its corresponding bulk state in one higher dimension.