Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases

We show that a slab of a three-dimensional inversion-symmetric higher-order topological insulator (HOTI) in class A is a 2D Chern insulator, and that in class AII is a 2D Z2 topological insulator. We prove it by considering a process of cutting the three-dimensional inversion-symmetric HOTI along a plane, and study the spectral flow in the cutting process. We show that the Z4 indicators, which characterize three-dimensional inversion-symmetric HOTIs in classes A and AII, are directly related to the Z2 indicators for the corresponding two-dimensional slabs with inversion symmetry, i.e. the Chern number parity and the Z2 topological invariant, for classes A and AII respectively. The existence of the gapless hinge states is understood from the conventional bulk-edge correspondence between the slab system and its edge states. Moreover, we also show that the spectral-flow analysis leads to another proof of the bulk-edge correspondence in one-and two-dimensional inversionsymmetric insulators. arXiv:1910.08290v1 [cond-mat.mes-hall]

Section: Introductionmentioning

confidence: 99%

Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators

Takahashi

Tanaka

Murakami

2020

“…Remarkably, it was recently demonstrated that a crystal with a crystalline-symmetry compatible bulk topology may manifest itself through protected boundary modes of codimension greater than one [14][15][16][17][18][19][20][21][22][23]. Such insulating and superconducting phases are called higher-order topological insulators and superconductors (HOTI/SCs).…”

mentioning

confidence: 99%

Floquet higher-order topological insulators and superconductors with space-time symmetries

Peng

2020

Floquet higher-order topological insulators and superconductors (HOTI/SCs) with an order-two space-time symmetry or antisymmetry are classified. This is achieved by considering unitary loops, whose nontrivial topology leads to the anomalous Floquet topological phases, subject to a spacetime symmetry/antisymmetry. By mapping these unitary loops to static Hamiltonians with an order-two crystalline symmetry/antisymmetry, one is able to obtain the K groups for the unitary loops and thus complete the classification of Floquet HOTI/SCs. Interestingly, we found that for every order-two nontrivial space-time symmetry/antisymmetry involving a half-period time translation, there exists a unique order-two static crystalline symmetry/antisymmetry, such that the two symmetries/antisymmetries give rise to the same topological classification. Moreover, by exploiting the frequency-domain formulation of the Floquet problem, a general recipe that constructs model Hamiltonians for Floquet HOTI/SCs is provided, which can be used to understand the classification of Floquet HOTI/SCs from an intuitive and complimentary perspective. CONTENTS 32 A. Equivalent classification with symmetrized evolution operators 32 B. Decomposition of time evolution operators 32 C. Order of HOTI/SCs and symmetry-breaking mass terms 32 References 33

“…The trivial phase has p = (0, 0), the dipolar phase has p = (1/2, 0) or p = (0, 1/2) and the quadrupolar phase has p = (1/2, 1/2). As the polarisation is calculated from the Wannier bands related to the bulk spectrum, the appearance of zero modes in the corner is related to a gap closing in the bulk, showing a new kind of bulk-boundary correspondence [16]. (1) showing the topological phases.…”

Section: Quadrupolar Topological Insulatormentioning

confidence: 98%

Thermodynamics of a higher-order topological insulator

Arouca

Kempkes

Smith

2020

We investigate the order of the topological quantum phase transition in a two dimensional quadrupolar topological insulator within a thermodynamic approach. Using numerical methods, we separate the bulk, edge and corner contributions to the grand potential and detect different phase transitions in the topological phase diagram. The transitions from the quadrupolar to the trivial or to the dipolar phases are well captured by the thermodynamic potential. On the other hand, we have to resort to a grand potential based on the Wannier bands to describe the transition from the trivial to the dipolar phase. The critical exponents and the order of the phase transitions are determined and discussed in the light of the Josephson hyperscaling relation. * r.aroucadealbuquerque@uu.nl † S.N.Kempkes@uu.nl ‡ C.deMoraisSmith@uu.nl arXiv:1912.09159v1 [cond-mat.stat-mech]