Ken Shiozaki

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by noninteracting Bloch and Bogoliubov--de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of 10 classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes ${\mathbit{Z}}_{2}$ g

It has been known that an antiunitary symmetry such as time-reversal or charge conjugation is needed to realize ${\mathbit{Z}}_{2}$ topological phases in noninteracting systems. Topological insulators and superconducting nanowires are representative examples of such ${\mathbit{Z}}_{2}$ topological matters. Here we report the ${\mathbit{Z}}_{2}$ topological phase protected by only unitary symmetries. We show that the presence of a nonsymmorphic space group symmetry opens a possibility to realize

Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014)] is extended to nonsymmorphic crystalline insulators and superconductors. Using the twisted equivariant $K$ theory, we complete the classification of topological crystalline insulators and superconductors in the presence of additional order-two nonsymmorphic space-group symmetries. The order-two nonsymmorphic space groups include half-lattice translation with ${\mathbit{Z}}_{2}$ flip, glide,

Topological crystalline materials are emergent topological phases due to crystalline space group symmetry. They are either gapful or gapless in the bulk, while hosting topological states at the boundary. Here, the authors define topological crystalline materials rigorously on the basis of a mathematical theory, known as twisted equivariant K-theory. Abstract mathematical ideas, such as the Mayer-Vietoris sequence and module structure, are explained in terms of band theory. The formulation is app

Here, the authors introduce a set of quantities to diagnose symmetry-protected topological phases of fermions protected by antiunitary symmetries. These quantities, which can be written as expectation values of nonlocal operators, effectively simulate the partition function on nonorientable spacetime manifolds in the operator formalism. The important observation is that the proposed quantities are complex valued for topological states, where the complex phase is the many-body topological invaria

Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point gr

We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant $K$ theory in the context of band theory. Various concepts in band theory, such as irreps at high-symmetry points, compatibility relations, topological gapless points, and singularities, fit naturally into the AHSS. As an application of the AHSS, we get the complete list of topological invariants for 230 space groups without time-reversal or particle-hole invariance. We find that many torsion topological invariants appear ev

Abstract We propose that symmetry-protected topological (SPT) phases with crystalline symmetry are formulated by an equivariant generalized homology $h^G_n(X)$ over a real space manifold X with G a crystalline symmetry group. The Atiyah–Hirzebruch spectral sequence unifies various notions in crystalline SPT phases, such as the layer construction, higher-order SPT phases, and Lieb–Schultz–Mattis-type theorems. This formulation is applicable to not only free fermionic systems but also interacting

Recently, topological phases in non-Hermitian systems have attracted much attention because non-Hermiticity sometimes gives rise to unique phases with no Hermitian counterparts. Non-Hermitian Bloch Hamiltonians can always be mapped to doubled Hermitianized Hamiltonians with chiral symmetry, which enables us to utilize the existing framework for Hermitian systems to classify non-Hermitian topological phases. While this strategy succeeded in the topological classification of non-Hermitian Bloch Ha

Abstract We present the exhaustive classification of surface states of topological insulators and superconductors protected by crystallographic magnetic point group symmetry in three spatial dimensions. Recently, Cornfeld and Chapman [Phys. Rev. B 99, 075105 (2019)] pointed out that the topological classification of mass terms of the Dirac Hamiltonian with point group symmetry is recast as an extension problem of the Clifford algebra, and we use their results extensively. Comparing two types of

The relation between bulk topological invariants and experimentally observable physical quantities is a fundamental property of topological insulators and superconductors. In the case of chiral symmetric systems in odd spatial dimensions such as time-reversal invariant topological superconductors and topological insulators with sublattice symmetry, this relation has not been well understood. We clarify that the winding number which characterizes the bulk Z nontriviality of these systems can appe

Motivated by the $\mathrm{\ensuremath{\Omega}}$-spectrum proposal of unique gapped ground states by Kitaev, we study adiabatic cycles in gapped quantum spin systems from various perspectives. We give a few exactly solvable models in one and two spatial dimensions and discuss how nontrivial adiabatic cycles are detected. For one spatial dimension, we study the adiabatic cycle in detail with the matrix product state and show that the symmetry charge can act on the space of matrices without changin

We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ over a real space manifold $X$ with $G$ a crystalline symmetry group. The Atiyah-Hirzebruch spectral sequence unifies various notions in crystalline SPT phases such as the layer construction, higher-order SPT phases and Lieb-Schultz-Mattis type theorems. Our formulation is applicable to interacting systems with onsite and crystalline symmetries as

We consider dynamical axion phenomena in topological superconductors and superfluids in three spatial dimensions in terms of the gravitoelectromagnetic topological action, in which the axion field couples with mechanical rotation under finite-temperature gradient. The dynamical axion is induced by relative phase fluctuations between topological and $s$-wave superconducting orders. We show that an antisymmetric spin-orbit interaction which induces parity mixing of Cooper pairs enlarges the parame

The symmetry-based indicator [H. C. Po, A. Vishwanath, H. Watanabe, Nat. Commun. 8, 50 (2017)] is a practical tool to diagnose topological materials in the band theory. In this note, we present two directions to generalize the symmetry-based indicator for other classes of topological materials. The one is for superconductors. The careful definition of the atomic insulators and the trivial vacuum Hamiltonian yields the symmetry-based indicators specific to superconductors. The other is for ingap