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arXiv:1404.4451 · Physical Review B

Fractionally quantized Berry phase, adiabatic continuation, and edge states

Toshikaze Kariyado, Yasuhiro Hatsugai

Published Aug 22, 2014• 16 citations
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Abstract

Symmetry protected quantization of the Berry phase is discussed in relation to edge states. Assuming the existence of an adiabatic process that protects the quantization of the Berry phase, a nontrivial Berry phase $\ensuremath{\gamma}\ensuremath{\propto}2\ensuremath{\pi}\ensuremath{\rho}$ ($\ensuremath{\rho}$ is the local filling of particles) for the bulk suggests appearance of edge states with boundaries. We have applied this generic consideration for the Bloch states of a two-dimensional model with massless Dirac fermions where $\ensuremath{\gamma}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ implies the edge states. The entanglement entropy is evaluated for the models and its relation to the bulk-edge correspondence of Dirac fermions is discussed as well.

DOI
10.1103/physrevb.90.085132

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Symmetry protected quantization of the Berry phase is discussed in relation to edge states. Assuming the existence of an adiabatic process that protects the quantization of the Berry phase, a nontrivial Berry phase \ensuremathγ\ensuremath∝2\ensuremathπ\ensuremathρ\ensuremath{\gamma}\ensuremath{\propto}2\ensuremath{\pi}\ensuremath{\rho}\ensuremathγ\ensuremath∝2\ensuremathπ\ensuremathρ (\ensuremathρ\ensuremath{\rho}\ensuremathρ is the local filling of particles) for the bulk suggests appearance of edge states with boundaries. We have applied this generic consideration for the Bloch states of a two-dimensional model with massless Dirac fermions where \ensuremathγ=\ifmmode±\else\textpm\fi\ensuremathπ/2\ensuremath{\gamma}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2\ensuremathγ=\ifmmode±\else\textpm\fi\ensuremathπ/2 implies the edge states. The entanglement entropy is evaluated for the models and its relation to the bulk-edge correspondence of Dirac fermions is discussed as well.
Berry phases defined by the one-dimensional momentum as a parameter show Z2{Z}_{2}Z2​ quantization due to the inversion symmetry combined with the time reversal, or existence of the reflection plane, which also protects the massless Dirac cones with continuous parameters. This is the symmetry protected Z2{Z}_{2}Z2​ quantization. These ideas are applied in the fermionic Shastry-Sutherland model, which has a rich phase diagram, including phases with massless Dirac fermions, a quadratic band crossing point, and a pseudospin-1 Weyl fermion. We also demonstrate that the Z2{Z}_{2}Z2​ Berry phases determine the existence of edge states as the bulk-edge correspondence of the massless Dirac fermion systems.
Bose gas on a two-leg ladder exhibits an interesting topological phase. We show the presence of a bosonic symmetry-protected-topological (SPT) phase protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry. This symmetry leads to Z4{Z}_{4}Z4​ fractional quantization of the Z4{Z}_{4}Z4​ Berry phase, which is a topological order parameter to identify the bulk. Using the Z4{Z}_{4}Z4​ Berry phase, we show that the interacting bosonic system possesses rich topological phases depending on the particle density and strength of interaction. Based on the bulk-edge correspondence, each edge state of the SPT phases is discussed in relation to the Z4{Z}_{4}Z4​ Berry phases. In particular, we have found an intermediate phase that is not adiabatically connected to a simple adiabatic limit that possesses unconventional edge states, which we numerically demonstrate by employing the density-matrix renormalization-group algorithm.
We consider a topologically nontrivial flat-band structure in one spatial dimension in the presence of nearest- and next-nearest-neighbor Hubbard interaction. The noninteracting band structure is characterized by a symmetry-protected topologically quantized Berry phase. At certain fractional fillings, a gapped phase with a filling-dependent ground-state degeneracy and fractionally charged quasiparticles emerges. At filling 13\frac{1}{3}31​, the ground states carry a fractional Berry phase in the momentum basis. These features at first glance suggest a certain analogy to the fractional quantum Hall scenario in two dimensions. We solve the interacting model analytically in the physically relevant limit of a large band gap in the underlying band structure, the analog of a lowest Landau level projection. Our solution affords a simple physical understanding of the properties of the gapped interacting phase. We pinpoint crucial differences to the fractional quantum Hall case by studying the Berry phase and the entanglement entropy associated with the degenerate ground states. In particular, we conclude that the ``fractional topological phase in one-dimensional flat bands'' is not a one-dimensional analog of the two-dimensional fractional quantum Hall states, but rather a charge density wave with a nontrivial Berry phase. Finally, the symmetry-protected nature of the Berry phase of the interacting phase is demonstrated by explicitly constructing a gapped interpolation to a state with a trivial Berry phase.
The bulk electric polarization works as a nonlocal order parameter that characterizes topological quantum matters. Motivated by a recent paper [H. Watanabe et al., Phys. Rev. B 103, 134430 (2021)], we discuss magnetic analogs of the bulk polarization in one-dimensional quantum spin systems, that is, quantized magnetizations on the edges of one-dimensional quantum spin systems. The edge magnetization shares the topological origin with the fractional edge state of the topological odd-spin Haldane phases. Despite this topological origin, the edge magnetization can also appear in topologically trivial quantum phases. We develop straightforward field theoretical arguments that explain the characteristic properties of the edge magnetization. The field theory shows that a U(1) spin-rotation symmetry and a site-centered or bond-centered inversion symmetry protect the quantization of the edge magnetization. We proceed to discussions that quantum phases on nonzero magnetization plateaus can also have the quantized edge magnetization that deviates from the magnetization density in bulk. We demonstrate that the quantized edge magnetization distinguishes two quantum phases on a magnetization plateau separated by a quantum critical point. The edge magnetization exhibits an abrupt stepwise change from zero to 1/21/21/2 at the quantum critical point because the quantum phase transition occurs in the presence of the symmetries protecting the quantization of the edge magnetization. We also show that the quantized edge magnetization can result from the spontaneous ferrimagnetic order.
Recently, it has been found that there exist symmetry-protected topological phases of fermions, which have no realizations in non-interacting fermionic systems or bosonic models. We study the edge states of such an intrinsically interacting fermionic SPT phase in two spatial dimensions, protected by \mathbb{Z}_4\times\mathbb{Z}_2^\mathsf{T} ℤ 4 × ℤ 2 𝖳 symmetry. We model the edge Hilbert space by replacing the internal \mathbb{Z}_4 ℤ 4 symmetry with a spatial translation symmetry, and design an exactly solvable Hamiltonian for the edge model. We show that at low-energy the edge can be described by a two-component Luttinger liquid, with nontrivial symmetry transformations that can only be realized in strongly interacting systems. We further demonstrate the symmetry-protected gaplessness under various perturbations, and the bulk-edge correspondence in the theory.