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arXiv:1307.7926 · Physical Review B

Symmetry-protected quantization and bulk-edge correspondence of massless Dirac fermions: Application to the fermionic Shastry-Sutherland model

Toshikaze Kariyado, Yasuhiro Hatsugai

Published Dec 20, 2013• 44 citations
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Abstract

Berry phases defined by the one-dimensional momentum as a parameter show ${Z}_{2}$ 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 ${Z}_{2}$ 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 ${Z}_{2}$ Berry phases determine the existence of edge states as the bulk-edge correspondence of the massless Dirac fermion systems.

DOI
10.1103/physrevb.88.245126

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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.
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.
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.
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 one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.