Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B ${\bf 90}$, 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 $Z_2$ flip, glide, two-fold screw, and their magnetic space groups. We find that the topological periodic table shows modulo-2 periodicity in the number of flipped coordinates under the order-two nonsymmorphic space group. It is pointed out that the nonsymmorphic space groups allow $\mathbb{Z}_2$ topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with the time-reversal and/or particle-hole symmetries provides novel $\mathbb{Z}_4$ topological phases, which have not been realized in ordinary topological insulators and superconductors. We present model Hamiltonians of these new topological phases and the analytic expression of the $\mathbb{Z}_2$ and $\mathbb{Z}_4$ topological invariants. The half lattice translation with $Z_2$ spin flip and glide symmetry are compatible with the existence of the boundary, leading to topological surface gapless modes protected by such order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.Comment: 38 pages, 13 figures, final versio

We formulate topological crystalline materials on the basis of the twisted equivariant K-theory. Basic ideas of the twisted equivariant K-theory are explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented. Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors. As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance. The classification works for gapless surface states of three-dimensional insulators, as well as full gapped two-dimensional insulators. Such surface states and two-dimensional insulators are classified in a unified way by 17 wallpaper groups, together with the presence or the absence of (sublattice) chiral symmetry. We identify the topological numbers and their representations under the wallpaper group operation. We also exemplify the usefulness of our formulation in the classification of bulk gapless phases. We present a new class of Weyl semimetals and Weyl superconductors that are topologically protected by inversion symmetry. CONTENTS

It has been known that an antiunitary symmetry such as time-reversal or charge conjugation is needed to realize Z 2 topological phases in noninteracting systems. Topological insulators and superconducting nanowires are representative examples of such Z 2 topological matters. Here we report the 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 Z 2 topological phases without assuming any antiunitary symmetry. The Z 2 topological phases are constructed in various dimensions, which are closely related to each other by Hamiltonian mapping. In two and three dimensions, the Z 2 phases have a surface consistent with the nonsymmorphic space group symmetry, and thus they support topological gapless surface states. Remarkably, the surface states have a unique energy dispersion with the Möbius twist, which identifies the Z 2 phases experimentally. We also provide the relevant structure in the K theory.

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We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body Z2 topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele Z2 invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermionic partial transpose which is basically the standard partial transpose equipped with a sign structure to account for anti-commuting property of fermion operators. We also report some basic results on various kinds of pin structures -a key concept behind our strategy for constructing many-body topological invariants -such as the obstructions, isomorphism classes, and Dirac quantization conditions. * The first two authors contributed equally to this work. 1 arXiv:1710.01886v3 [cond-mat.str-el] 6 Jul 2018 56 A. Fermion coherent states 56 B. Class A+CR: Twisting by CR symmetry 57 C. Cohomology with local coefficient 60 1. Example: real projective plane RP 2 61 D. Variants of Pin-structures 61 1. The variants of Spin-groups 62 a. The standard variants 62 b. The variants Pinc ± 63 c. The variants G 0 and G ± 63 2. The obstructions 63 a. Lifting and obstruction in general 63 b. The cases of Spin, Pin ± and Spin c 64 c. The cases of Pinc ± 65 d. The cases of G 0 and G ± 65 3. Examples 66 a. Computing cohomology 66 b. Circle 67 c. 2-dimensional sphere 68 d. Real projective plane 68 e. Klein bottle 69 E. Dirac quantization conditions 71 1. Bosonic U (1) × CT symmetry 71 2. Bosonic U (1) T symmetry 72 3. Fermionic U (1) × CT symmetry: pin c structure 74 4. Fermionic U (1) T symmetry: pinc ± structure 76 F. More on pin c and pinc ± structures on RP 2 77 1. Dirac operator on S 2 with the Schwinger gauge 77 2. Pin c structure on RP 2 and the eta invariant 79 3. Pinc − structure on RP 2 and θ term 80 4. Pinc + structure on RP 2 and θ term 80 G. On the reflection swap: the case of (1 + 1)d class A with reflection symmetry 81References 82 6 fermions. Furthermore, our findings in Ref. [37] suggest that a fermionic version of partial transpose, if properly defined, can be used to detect fermionic SPT phases: In (1 + 1)d fermionic SPT phases with TRS T 2 = 1 (symmetry class BDI), the generating manifold is the real projective plane. It was shown that one can use the fermionic partial transpose to effectively simulate the real projective plane (with a proper pin − structure), and construct the corresponding many-body topological invariant, which captures correctly the known Z 8 classification. [6,53] F. Organization of the paperThe purpose of this paper is to construct many-body topological invariants that can detect fermionic SPT phases protected by time-reversal or other antiunitary symmetries, following the strategy outlined above. See Table I and II for the many-bo...