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arXiv:1709.01546 · Physical Review Letters

ZN Berry Phases in Symmetry Protected Topological Phases

Toshikaze Kariyado, Takahiro Morimoto, Yasuhiro Hatsugai

Published Jun 15, 2018• 39 citations
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Abstract

We show that the Z_{N} Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The Z_{N} Berry phase is defined in a N-1-dimensional parameter space of local gauge twists, which we call the "synthetic Brillouin zone," and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the Z_{N} Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z_{3} and Z_{4} Berry phases, respectively. We find that the exact quantization of the Z_{N} Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.

DOI
10.1103/physrevlett.120.247202

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We show that the Z_{N} Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The Z_{N} Berry phase is defined in a N-1-dimensional parameter space of local gauge twists, which we call the "synthetic Brillouin zone," and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the Z_{N} Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z_{3} and Z_{4} Berry phases, respectively. We find that the exact quantization of the Z_{N} Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.
The local ZN{Z}_{N}ZN​ quantized Berry phase for the SU(N)\mathrm{SU}(N)SU(N) antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local Z2{Z}_{2}Z2​ 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 SU(N)\mathrm{SU}(N)SU(N) spin systems. We extend the path-integral quantum Monte Carlo method for the Z2{Z}_{2}Z2​ Berry phase in order to calculate the ZN{Z}_{N}ZN​ Berry phase numerically. We demonstrate our method by calculating the Z4{Z}_{4}Z4​ Berry phase for the bond-alternating SU(4)\mathrm{SU}(4)SU(4) antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.
We characterize the quantum phase transitions of gapped time-reversal invariant spin chain systems by nontrivial Z 2 numbers derived from a quantized Berry phase in a (1 + 1)-dimensional Bloch momentum space. We study this approach analytically in a general two-band model and give a concrete example in a transverse field XY spin chain system. In our scheme, an extra local gauge transformation is performed to the spin system by a time-dependent twist operator, which endows the Hamiltonian of the system with the topology of a torus without changing its energy spectrum. The Z 2 number is obtained by a loop integral of the Berry gauge potential along a quarter of the Brillouin zone. We show that the different phases of the XY spin chain system are distinguished by the Z 2 topological order in momentum space.
Bosonic symmetry-protected topological (SPT) states are gapped disordered phases of matter possessing symmetry-preserving boundary excitations. It has been proposed that, at long wavelengths, the universal properties of an SPT system are captured by an effective nonlinear sigma model field theory in the presence of a quantized topological $\ensuremath{\theta}$ term. By studying lattice models of bosonic SPT states, we are able to identify, in their Euclidean path integral formulation, (discrete) Berry phases that hold relevant physical information on the nature of the SPT ground states. These discrete Berry phases are given intuitive physical interpretation in terms of instanton effects that capture the presence of a $\ensuremath{\theta}$ term on the microscopic scale.
We demonstrate the existence of a topologically nontrivial phase in a one-dimensional fermionic lattice system subjected to synthetic gauge fields, which is beyond the standard Altland-Zirnbauer classification of topological insulators. The topological phase can be characterized by the presence of degenerate zero-mode edge states or a quantized Berry phase of the occupied Bloch band. By analyzing symmetries of the system, we identify that the topological phase and zero-mode edge states are protected by two hidden symmetries. An extended model with hidden symmetry breaking is also studied in order to reveal the effect of hidden symmetries on the symmetry-protected topological phase.
Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel physics. We show that the Z2{\mathbb{Z}}_{2}Z2​ projectively represented translational symmetry operators adopt a distinct anticommutation relation. As a result, each energy band is twofold degenerate, and carries a varying spinor structure for translation operators in momentum space, which cannot be flattened globally. Moreover, combined with other internal or external symmetries, they give rise to exotic band topologies. Particularly, with the inherent time-reversal symmetry, a single fourfold Dirac point must be enforced at the Brillouin zone corner. By breaking one primitive translation, the Dirac semimetal is shifted into a special topological insulator phase, where the edge bands have a M\"obius twist. Our work opens an arena of research for exploring topological phases protected by projectively represented space groups.
We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [${Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}R$, U(1)$\ensuremath{\rtimes}R$, or U(1)$\ifmmode\times\else\texttimes\fi{}R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-SSS spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even NNN and 0 (no SPT phase) for odd NNN, while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are Z2{\mathbb{Z}}_{2}Z2​. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of S=2S=2S=2 spins on the square lattice is a Z2{\mathbb{Z}}_{2}Z2​ SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.