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CPT Theorem and Classification of Topological Insulators And CPT theorem and classification of topological insulators and superconductors Chang-Tse Hsieh,1 Takahiro Morimoto,2 and Shinsei Ryu1 1Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana IL 61801 2Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan (ΩDated: June 3, 2014) We present a systematic topological classification of fermionic and bosonic topological phases protected by time-reversal, particle-hole, parity, and combination of these symmetries. We use two complementary approaches: one in terms of K-theory classification of gapped quadratic fermion theories with symmetries, and the other in terms of the K-matrix theory description of the edge theory of (2+1)-dimensional bulk theories. The first approach is specific to free fermion theories in general spatial dimensions while the second approach is limited to two spatial dimensions but incorporates effects of interactions. We also clarify the role of CPT theorem in classification of symmetry-protected topological phases, and show, in particular, topological superconductors dis- cussed before are related by CPT theorem. CONTENTS VI. Discussion 19 I. Introduction 1 Acknowledgments 19 A. Outline and main results 3 A. Entanglement spectrum and effective symmetries 19 II. 2D fermionic topological phases protected by 1.Entanglementspectrum 20 symmetries 4 2. Properties of entanglement spectrum with A. CP symmetric insulators 4 symmetries 21 1. tight-binding model 4 2.Edgetheory 5 B. Calculations of symmetry transformations and B. BdG systems with spin U(1) and P SPT phases in K-matrix theories 21 symmetries 5 1. Algebraic relations of symmetry operators 21 C. Connection to T symmetric insulators 6 2. Equations of identity elements 22 D. CPT-equivalence and T-duality of topological 3.DiscussionofSPTphases 22 phases:anexample 7 a. Bosonic SPT phases with symmetry group CP Z2 23 III. Classification of CP symmetric TIs in arbitrary b. Bosonic SPT phases with symmetry group dimension 7 T C P Z2 × Z2 × Z2 24 A. Classification by K-theory 7 c. Fermionic SPT phases with symmetry CP 25 B.DiracmodelsandCPTtheorem 8 d. Fermionic SPT phases with symmetries C. Topological invariants 8 {T , CP} 26 D. BdG systems with spin U(1) and P symmetries 8 C. Proof of topological CPT theorem for interacting E. Topological classification of other non-chiral SPT phases in two dimensions 27 symmetries 10 References 28 IV. Topological CPT theorem and topological CPT-equivalentsymmetryclasses 10 arXiv:1406.0307v1 [cond-mat.str-el] 2 Jun 2014 I. INTRODUCTION V. Classification of 2D interacting SPT phases: K-matrix formulation 12 Topological insulators (TIs) and topological super- A. Bulk and edge K-matrix theories incorporated conductors (TSCs) are states of matter that are not with symmetries 12 adiabatically connected to, in the presence of a set B. Edge stability criteria for SPT Phases 14 of symmetry conditions, topologically trivial states of C. Classification of SPT phases by K-matrix matter.1,2 TIs and TSCs characterized by Altland- thoeries: CPT-equivalent SPT phases and Zirnbauer symmetries,3 time-reversal symmetry (TRS), dual SPT phases 15 particle-hole symmetry (PHS), and combinations thereof, 1. K-matrix classification of bosonic have been theoretically predicted4–12 and experimentally non-chiral SPT phases 16 discovered.13–20 2. K-matrix classification of fermionic SPT In more recent years, interplay between on-site symme- phases 17 tries (such as TRS and PHS listed in Altland-Zirnbauer 2 symmetry classes) and non-on-site symmetries (such as T sym. TI CPT-equiv. space group symmetries) has enriched the topological CP sym. TCI phases of matters. A novel class of topological matter (QSHE) characterized (additionally) by non-on-site symmetries, such as topological crystalline insulators (TCIs)21 and topological crystalline superconductors (TSCSs), have T-duality T-duality been discovered.22–29 The topological classification, orig- inally studied in the presence/absence of various on-site symmetries in Altland-Zirnbauer classes,10,11,30 are also Chiral sym. TSC CPT-equiv. P sym.TCSC extended to include the non-on-site symmetries such as with Sz conserv. with Sz conserv. reflection symmetry,31,32 inversion symmetry,33,34 and (crystal) point group symmetries,35–38 recently. (a) Motivated by these recent works, in this paper, we fur- ther study TIs and TSCs protected by a wider set of sym- metries than symmetries in Altland-Zirnbauer classes by T sym. TSC including, in particular, a parity symmetry (PS), which is a symmetry under the reflection of an odd number of spatial coordinates. One of our focuses is, in addition to the cases where parity is conserved, on situations where a CPT-equiv. CPT-equiv. combination of parity with some other symmetries, such as CP (product of PHS and PS) or PT (product of PS CPT-equiv. and TRS), are preserved. For earlier related works, see, P sym. TCSC T and P sym. TCI* for example, Refs. 30, 39–41. Another issue we will discuss in this paper is the effect of interactions on the classification of those topological (b) phases protected by parity and other symmetries (such as combination of parity and other symmetries). It has FIG. 1. Two sets of topological (crystalline) insulators and su- been demonstrated, in various examples, that there are perconductors related by CPT-equivalence and/or T-duality: phases that appear to be topologically distinct from triv- (a) T symmetric TI (QSHE), CP symmetric TCI, chiral (TC) symmetric TSC with S conservation, and P symmetric ial phases at non-interacting level, which, in fact, can z TCSC with S conservation; (b) T symmetric TSC, P sym- adiabatically be deformable into a trivial state of matter z 42–47 metric TCSC, and T and P symmetric TCI that can support in the presence of interactions. For example, Ref. gapless edge states even in the absence of charge U(1) sym- 47 discusses (2+1)-dimensional [or 2 spatial-dimensional metry. (2D)] superconducting systems in the presence of parity and time-reversal symmetries, which are classified, at the quadratic level, by an integer topological invariant, while systems at long wavelength limit, such as the band topol- once inter-particle interactions are included, states with ogy or the electromagnetic response, can be encoded in an integer multiple of eight units of the non-interacting the so-called topological field theory, which respects the topological invariant are shown to be unstable. Focusing Lorentz symmetry. When these topological properties on 2D bulk topological states that support an edge state are protected (or determined) by some symmetry, TRS, described by a K-matrix theory, i.e., Abelian states, we say, they can also be protected solely by CP symme- will study the stability of the edge state (and hence the try, which is a “CPT-equivalent” partner to TRS. For bulk state) in the presence of parity symmetry or parity example, the magnetoelectric effect in 3D time-reversal symmetry combined with other symmetries. symmetric TIs48 is also expected to be observed in a CP We will also show that, once parity symmetry or parity symmetric TI, because they are both described by the ax- symmetry combined with other discrete symmetries is ion term (effective action for electromagnetic response) included into our consideration, CPT theorem plays an with the same nontrivial (quantized) value of θ angle. important role in classifying topological states of matter. In addition, from the prospect of topological classifica- CPT theorem holds in Lorentz invariant quantum field tion, classifying symmetry-protected topological (SPT) theories, which says, C, P, T, when combined into CPT, phases of a free fermion systems [characterized either by is always conserved, i.e., CPT = 1, schematically. For a gapped Hamiltonian of the (d + 1)-dimensional bulk example, a Lorentz invariant CP symmetric field theory or by a gapless Hamiltonian of d-dimensional boundary also possesses TRS, and vice versa. with symmetries] is equivalent to classifying the corre- In condensed matter systems, however, such relations sponding Dirac operators with symmetry restrictions11. between these discrete symmetries (T, C, and P) do not It is thus natural to associate TIs protected by TRS with arise since we are not to be restricted to relativistic sys- TIs protected by CP symmetry, as the Dirac Hamiltonian tems; symmetries can be imposed independently. Never- has a CPT invariant form. theless, some physical properties of these non-relativistic In this paper, by going through classification prob- 3 lems of non-interacting fermions in the presence of vari- As inferred from the CPT theorem and the existence of ous symmetry conditions, and also microscopic stability time-reversal symmetric topological insulators in two di- analysis of interacting edge theories, we will demonstrate mensions, we will show that there are two topologically explicitly such CPT theorem holds at the level of topolog- distinct classes of insulators with CP symmetry in two ical classification for all cases that we studied. Through dimensions, i.e., Z2 classification. We will present in Sec. this analysis, we can see, for example, that 2D TSCs pro- II A a simple example (tight-binding model) of CP sym- tected by spin parity conservation,45,46 and 2D TSCs by metric band insulators, which is constructed from two parity and time-reversal,47 both of which are classified in copies of two-band Chern insulators with opposite chi- terms of Z8, are related by CPT theorem. ralities. While we defer a systematic classification of CP As mentioned above, CPT theorem (i.e., topological symmetric insulators until Sec. III, we discuss the edge classification problems with different set of symmetries theory of the CP symmetric topological insulators, and related by CPT relations) may largely be expected, for perturbations to it such as mass terms. example, once we anticipate description of SPT phases We will also show, in Sec.
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