Periodic Table for Topological Insulators and Superconductors Alexei Kitaev
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Periodic table for topological insulators and superconductors Alexei Kitaev California Institute of Technology, Pasadena, CA 91125, U.S.A. Abstract. Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of Z-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the Z-theoretic classification is stable to interactions, but a counterexample is also given. Keywords: Topological phase, K-theory, K-homology, Clifford algebra, Bott periodicity PACS: 73.43.-f, 72.25.Hg, 74.20.Rp, 67.30.H-, 02.40.Gh, 02.40.Re The theoretical study [1, 2, 3] and experimental obser Irons (mod2) in the ground state. For d = l,a system in vation [4] of the quantum spin Hall effect in 2D systems, this symmetry class, dubbed "Majorana chain", also has followed by the discovery of a similar phenomenon is a Z2 invariant, which indicates the presence of unpaired 3 dimensions [5, 6, 8, 10, 11], have generated consider Majorana modes at the ends of the chain [14]. But for able interest in topological states of free electrons. Both d = 2 (e.g., a px + ipy superconductor), the topological kinds of systems are time-reversal invariant insulators. number is an integer though an even-odd effect is also More specifically, they consist of (almost) noninteract important [15, 16]. ing fermions with a gapped energy spectrum and have T-invarianl insulators have an integer invariant (the both the time-reversal symmetry {T) and a f7(l) symme number of particle-occupied Kramers doublet states) for try (2). The latter is related to the particle number, which d = 0, no invariant for d = I, and a Z2 invariant for is conserved in insulators but not in superconductors or d = 2 [I, 2] and for rf = 3 [5, 6, 8]. 3D crystals (i.e., superfluids. Topological phases with only one of those systems with discrete translational symmetry) have an symmetries, or none, are also known. Such phases gen additional 3Z2 invariant, which distinguishes so-called erally carry some gapless modes at the boundary.^ "weak topological insulators". The classification of gapped free-fermion systems de With the exception just mentioned, the topological pends on the symmetry and spatial dimension. For exam numbers are insensitive to disorder and can even be de ple, two-dimensional insulators without T symmetry are fined without the spectral gap assumption, provided the characterized by an integer v, the quantized Hall conduc eigenstates are locahzed. This result has been estabhshed tivity in units of e^/h. For systems with discrete trans- rigorously for integer quantum Hall systems [17,18,19], lational symmetry, it can be expressed in terms of the where the invariant v is related to the index theory and band structure (more exactly, the electron eigenstates as can be expressed as a trace of a certain infinite operator, a function of momentum); such an expression is known which represents the insertion of a magnetic flux quan as the TKNN invariant [13], or the first Chem number. tum at an arbitrary point. Its trace can be calculated with A similar topological invariant (the fe-th Chem number) sufficient precision by examining an /-neighborhood of can be defined for any even dimension d. For rf = 0, it is that point, where / is the localization length. A similar lo simply the number of single-particle states with negative cal expression for the Z2 invariant of a ID system with no energy (E < Ef = 0), which are filled with electrons. symmetry has been derived in Appendix C of Ref. [16]; However, the other three symmetry types (no symme it involves an infinite Pfaffian or determinant. try, T only, or both T and Q) do not exhibit such a simple In this paper, we do not look for analytic formulas for pattern. Let us consider systems with no symmetry at all. topological numbers, but rather enumerate all possible For d = 0, there is a Z2 invariant: the number of elec- phases. Two Hamiltonians belong to the same phase if they can be continuously transformed one to the other while maintaining the energy gap or localization; we will ^ In contrast, strongly correlated topological phases (with anyons in the elaborate on that later. The identity of a phase can be de bulk) may not have gapless boundary modes[12]. termined by some local probe. In particular, the Hamil- CPl 134, Advances in Theoretical Physics: Landau Memorial Conference, edited by V. Lebedev and M. Feigel'man O 2009 American Institute of Physics 978-0-7354-0671 -l/09/$25.00 22 Downloaded 15 Apr 2010 to 131.215.193.213. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/proceedings/cpcr.jsp TABLE 1. Classification of free-fermion phases with all possible combinations of the particle number conservation (Q) and time-reversal symmetry (T). The TIQ (Q) and %Q (Rq) columns indicate the range of topological invariant. Examples of topologically nontrivial phases are shown in parentheses. q T^{Cq) '=1 ' = 2 ' = 3 q Tki{Rq) d=\ d = 2 d = 3 (IQHE) no symmetry T only 0 0 z (Px + iPy,s.g., SrRu) (^He-B) no symmetry T only T mdQ Above: insulators without time-reversal 1 Z2 symmetry (i.e., systems with Q symme (Majorana chain) {{Px + iPy)^ + {Px-iPy)i) (BiSb) try only) are classified using complex K- T only T mdQ 2 theory. Z2 ((TMTSF)2X) (HgTe) Right: superconductors/superfluids (sys 3 0 T mdQ tems with no symmetry or T-symmetry only) and time-reversal invariant insula 4 z tors (systems with both T and Q) are clas 5 0 sified using real Z-theory. 6 0 7 0 no symmetry tonian around a given point may be represented (in some We report a general classification scheme for gapped non-canonical way) by a mass term that anticommutes free-fermion phases in all dimensions, see Table 1. It ac with a certain Dirac operator; the problem is thus reduced tually consists of two tables. The small one means to to the classification of such mass terms. represent the aforementioned alternation in TR-broken Prior to this work, there have been several results to insulators (a unique trivial phase for odd rf vs. an inte ward unified classification of free-fermion phases. Alt- ger invariant for even d). The large table shows a pe land and Zimbauer [20] identified 10 symmetry classes riod 8 pattern for the other three combinations of T and of matrices,^ which can be used to build a free-fermion Q. Note that phases with the same symmetry line up di Hamiltonian as a second-order form in the annihilation agonally, i.e., an increase in d corresponds to a step up and creation operators, dj and at. The combinations of (mods). (T-invariant ID superconductors were studied T and Q make 4 out of 10 possibilities. However, the in Ref. [9]. The {px+ipy)^ + {Px-ipy)i phase was pro symmetry alone is only sufficient to classify systems in posed in Refs. [23,7,21]; the last paper also describes an dimension 0. For d = I, one may consider a zero mode integer invariant for ^He-5.) The 2 + 8 rows (indexed by at the boundary and check whether the degeneracy is q) may be identified with the Altland-Zimbauer classes stable to perturbations. For example, an unpaired Majo arranged in a certain order; they correspond to 2 types rana mode is stable. In higher dimensions, one may de of complex Clifford algebras and 8 types of real Clifford scribe the boundary mode by a Dirac operator and like algebras. Each type has an associated classifying space wise study its stabihty. This kind of analysis has been Cq or Rq, see Table 2. Connected components of that performed on a case-by-case basis and brought to com space (i.e., elements o{%o{Rq) or %o{Cq)) correspond to pletion in a recent paper by Schnyder, Ryu, Furusaki, and different phases. But higher homotopy groups also have Ludwig [21]. Thus, all phases up to rf = 3 have been char physical meaning. For example, the theory predicts that acterized, but the collection of results appears irregular. ID defects in a 3D TR-broken insulator are classified by A certain periodic pattern for Z2 topological insula tors has been discovered by Qi, Hughes, and Zhang [22]. The (mod2) and (mod8) patterns mentioned above They use a Chem-Simons action in an extended space, are known as Bott periodicity; they are part of the math which includes the space-time coordinates and some pa ematical subject called K-theory. It has been apphed rameters. This approach suggests some operational inter in string theory but not so much in condensed matter pretation of topological invariants and may even work for physics. One exception is Hofava's work [24] on the clas interacting systems, though this possibihty has not been sification of stable gapless spectra, i.e., Fermi surfaces, explored. In addition, the authors mention Chfford alge lines, and points.