We present a rigorous and fully consistent $K$-theoretic framework for
studying gapped topological phases of free fermions such as topological
insulators. It utilises and profits from powerful techniques in operator
$K$-theory. From the point of view of symmetries, especially those of time
reversal, charge conjugation, and magnetic translations, operator $K$-theory is
more general and natural than the commutative topological theory. Our approach
is model-independent, and only the symmetry data of the dynamics, which may
include information about disorder, is required. This data is completely
encoded in a suitable $C^*$-superalgebra. From a representation-theoretic point
of view, symmetry-compatible gapped phases are classified by the
super-representation group of this symmetry algebra. Contrary to existing
literature, we do not use $K$-theory to classify phases in an absolute sense,
but only relative to some arbitrary reference. $K$-theory groups are better
thought of as groups of obstructions between homotopy classes of gapped phases.
Besides rectifying various inconsistencies in the existing literature on
$K$-theory classification schemes, our treatment has conceptual simplicity in
its treatment of all symmetries equally. The Periodic Table of Kitaev is
exhibited as a special case within our framework, and we prove that the
phenomena of periodicity and dimension shifts are robust against disorder and
magnetic fields.Comment: 41 pages, revised version with a new abstract, introductory sections
  and critique of the literatur

Thiang, Guo Chuan

English

arXiv

Frontiers of Fundamental Physics : session Mathematical PhysicsA K-theoretic approach to the study of gapped topological phases was suggested by Kitaev, who produced a Periodic Table of topological insulators and superconductors, modelled on Bott periodicity. We take the algebraic viewpoint, and study gapped phases of free fermions through a twisted crossed product C*-superalgebra associated to the symmetry data of the dynamics. We identify the K-theoretic difference-group of this symmetry algebra, in the sense of Karoubi, as the appropriate point of entry for K-theory. Thus, K-theory provides groups of obstructions between symmetry-compatible gapped Hamiltonians, rather than classification groups for the Hamiltonians themselves. The phenomena of periodicity and dimension-shifts in the difference-groups is shown to be a robust consequence of various isomorphisms in operator K-theory, which have no commutative counterpart.Guo Chuan Thian

Thiang, G.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION  Guo Chuan Thiang On the K-Theoretic classification of topological phases of matter Proceedings of Science, 2014 / vol.15-18-July-2014, pp.1-6   Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. Published version https://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=224 https://pos.sissa.it/224/149/pdf                         http://hdl.handle.net/2440/109379 PERMISSIONS https://creativecommons.org/licenses/by-nc-sa/4.0/  9 Nov 2017 PoS(FFP14)149On the K-theoretic classification of topologicalphases of matterGuo Chuan Thiang∗†University of Oxford, Mathematical Institute, Oxford, United KingdomE-mail: thgchuan@gmail.comA K-theoretic approach to the study of gapped topological phases was suggested by Kitaev, whoproduced a Periodic Table of topological insulators and superconductors, modelled on Bott pe-riodicity. We take the algebraic viewpoint, and study gapped phases of free fermions through atwisted crossed product C∗-superalgebra associated to the symmetry data of the dynamics. Weidentify the K-theoretic difference-group of this symmetry algebra, in the sense of Karoubi, as theappropriate point of entry for K-theory. Thus, K-theory provides groups of obstructions betweensymmetry-compatible gapped Hamiltonians, rather than classification groups for the Hamiltoni-ans themselves. The phenomena of periodicity and dimension-shifts in the difference-groups isshown to be a robust consequence of various isomorphisms in operator K-theory, which have nocommutative counterpart.Frontiers of Fundamental Physics 1415-18 July 2014Aix Marseille University (AMU) Saint-Charles Campus, Marseille, France∗Speaker.†The author thanks Balliol College and the Clarendon Fund for financial support. This paper is a summary of someof the results in [15].c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(FFP14)149On the K-theoretic classification of topological phases of matter Guo Chuan Thiang1. Symmetry classification of quantum mechanical systemsIn a fundamental paper of Dyson [5], it was shown that quantum mechanical systems withsymmetry came in three classes, corresponding to the three associative real division algebras R,Cand H. There were actually a few “threefold ways” addressing several inequivalent classificationproblems, one of which was the classification of projective unitary-antiunitary representations ac-cording to their commutants. The threefold way was expanded into a tenfold way for free-fermionsystems [1, 7] based on Cartan’s ten classical compact symmetric spaces. Topology began to playa more prominent role, with the seminal paper of Kitaev [9] proposing a classification of (gapped)topological insulators and superconductors based on K-theory. This was a short paper, and manyauthors (e.g. [12]) proceeded to provide their versions of the K-theory classification. The paperof Freed–Moore [6] provided an account of the tenfold way based on the ten real superdivisionalgebras. Their subsequent analysis of quantum mechanical symmetries goes beyond the tenfoldway, and they arrive at a classification scheme based on twisted equivariant K-theory.Although there seems to be a consensus in the literature that the free-fermion classificationproblem is solved, there are in fact many inconsistencies as well as inequivalent classification prob-lems (see [15, 14] for an extended discussion). In particular, conventions differ between authors,and the equivalence relations (e.g. isomorphism or homotopy) defining the classes of free-fermionsare not always uniformly applied, even within the same framework. In a homotopy classification,the general intuition is that there is a topological space Y of Hamiltonians (gapped or otherwise)which are compatible with certain given symmetry constraints, and homotopic Hamiltonians withinthis space are to be identified. Physically, homotopy corresponds to some adiabatic change of pa-rameters, and one is only interested in capturing features of Hamiltonians which are insensitiveto such operations. From this point of view, the problem is to identify Y and compute its set ofpath-components pi0(Y ). This is not yet a group! Indeed, the connection to K-theory groups is notat all obvious. In this paper, we fill in this important gap in order to understand the nature of theK-theory groups appearing in Kitaev’s Periodic Table — we find they do not actually classify theHamiltonians themselves (up to homotopy), but rather the obstructions between Hamiltonians ina homotopy sense. We work with minimal assumptions and in a model-independent manner, inorder not to artificially introduce topology into Y . Our approach is a noncommutative one, and isable to handle physically important phenomena such as the Integer Quantum Hall Effect as well asdisordered systems. Thus, we are not limited to topological insulators modelled by vector bundlesover Brillouin tori.The original Periodic Table is of course Bott’s table of homotopy groups of the stable classicalgroups [2]. Thus the answer is, in a sense, already “known”. The first achievement of this paper isthe identification of an appropriate question to which Bott’s table (or at least the Bott periodicityof K-theory groups) provides the answer. The second important result is a robust proof of thedimension-shift phenomenon, which utilises results in the field of C∗-algebras and their K-theory.It makes clear that the noncommutative approach is not only more general, but also necessary.2. Basic classification principlesSuppose ut = e−iHt is a strongly-continuous one-parameter group of unitary time evolution2PoS(FFP14)149On the K-theoretic classification of topological phases of matter Guo Chuan Thianggenerated by a gapped Hamiltonian H, i.e., 0 6∈ spec(H). We assume that there is a locally compactsecond-countable group G of dynamical symmetries, equipped with two continuous homomor-phisms φ ,τ : G→{±1} indicating whether the representative g of g ∈G is unitary/antiunitary andtime orientation preserving/reversing:gi = φ(g)ig, gut = uτ(g)tg. (2.1)The spectral flattening H 7→ sgn(H) =: Γ defines a grading operator Γ. With respect to Γ, theoperators g are even/odd, i.e., gΓ= c(g)Γg, according to c := φ ·τ . Wigner’s Theorem says that g 7→g need only be a projective unitary-antiunitary representation (PUA-rep), so there is cohomologicaldata in the form of a Borel map σ : G×G→ U(1) satisfying the 2-cocycle conditionσ(x,y)σ(xy,z) = σ(y,z)xσ(x,yz), (2.2)where σ(·, ·)x means σ(·, ·) if φ(x) = +1 and σ(·, ·) if φ(x) =−1.Thus, the symmetry constraints are abstractly specified by (G,c,φ ,σ), and the grading opera-tor in a concrete realisation of this data (as a Z2-graded PUA-rep) is a flattened compatible Hamil-tonian. It represents the family of compatible gapped Hamiltonians H which have sgn(H) = Γ.More generally, g ∈G may act on a C∗-algebraB by automorphisms αg, which may be twisted bya generalised 2-cocycle. Then one is interested in graded covariant representations of the gradedtwisted C∗-dynamical system (G,c,B,α,σ).The covariant representation theory of a (graded) twisted dynamical system can be conciselyrephrased as the ordinary (graded) representation theory of the (graded) twisted crossed product[10, 3] A =Bo(α,σ) G, which is a certain completion of L1(G,B) with a (α,σ)-twisted con-volution product. For example, a graded PUA-rep of (G,c,φ ,σ) is a graded ∗-representation ofthe algebra A = Co(α,σ) G, where αg is complex conjugation on C if φ(g) =−1 and the identityotherwise. Note that A is a real algebra if φ is non-trivial. We interpret the graded algebra A asthe symmetry algebra, generalising the notion of a group algebra. We can associate a number of al-gebraic objects to A , such as its (super-)representation ring, but we will mostly be concerned withits K-theory groups and their interpretation in terms of homotopies of compatible Hamiltonians.Example 2.1 (CT -subgroups and the Clifford algebras). Let A be a subgroup of the CT -group{1,T,C,S}= {±1}2, where T,C and S = CT are respectively time-reversal, charge-conjugationand sublattice symmetries. The usual convention is that T is antiunitary and even, C is antiunitaryand odd, and S is unitary and odd; these determine φ and c. We utilise the phase freedom in therepresentatives T,C,S to ensure that T2 = ±1 = C2,S2 = +1, and CT = TC; these specify thecocycle σ . Consequently, there are ten standardised choices for (A,σ), which is a manifestation ofthe tenfold way. With a little work, one can show that the associated symmetry algebra Co(α,σ) Ais, up to Morita equivalence, a complex Clifford algebra Cln,n = 0,1(mod2), or a real Cliffordalgebra1 Clr,s,r− s = 0, . . . ,7(mod8). Thus there are exactly 2+8 = 10 cases.3. K-theory and symmetry-compatible gapped HamiltoniansRecall that K0(X) of a compact Hausdorff space X is the Grothendieck completion of the mo-niod of isomorphism classes of complex vector bundles over X ; similarly, the operator K0(A ) for a1Our convention is that Clr,s has r generators squaring to −1 and s generators squaring to +1.3PoS(FFP14)149On the K-theoretic classification of topological phases of matter Guo Chuan Thiangunital real or complex C∗-algebraA is the Grothendieck completion of the monoid of isomorphismclasses of finitely-generated projective (f.g.p.) A -modules. In the context of band insulators with-out additional symmetries other than a lattice Zd of translations, the symmetry algebra is C(Td), thecontinuous complex-valued functions on the Brillouin torus Td . Higher K-theory groups Kn(A )may be defined as the K0-group of the n-fold suspension of A . However, this is a definition thatonly works for ungraded A and is difficult to interpret physically. Therefore, we turn to a modelof K-theory due to Karoubi [8], which works for graded algebras.Definition 3.1 (Symmetry-compatible gapped Hamiltonians). Let (G,c,φ ,σ) be symmetry dataas described in Section 2, and let A = Co(α,σ) G be its associated symmetry algebra. SupposeA is unital, and let W be an ungraded f.g.p. A -module. We call GradA (W ) the set of symmetry-compatible gapped Hamiltonians on W . Two grading operators Γ1,Γ2 ∈ GradA (W ) are said to behomotopic if there is a norm-continuous path between Γ1 and Γ2 within GradA (W ); in this case,we write Γ1 ∼h Γ2.Note that there is a standard Banach space structure on W (induced from the free module A nwhich it is a direct summand of). This determines a norm topology on the bounded linear mapsW →W . Therefore it makes sense to talk about GradA (W ) as a topological space. By replacingC with some graded C∗-algebra B in Definition 3.1, we can also consider GradA (W ) for moregeneral twisted crossed products A =Bo(α,σ) G.Let us now consider triples (W,Γ1,Γ2) comprising an ungradedA -module, and two symmetry-compatible gapped Hamiltonians Γi ∈ GradA (W ). Physically, this triple represents the ordereddifference between the Hamiltonians Γ1 and Γ2. If Γ1 ∼h Γ2, we say that the triple (W,Γ1,Γ2) istrivial. The collection of triples GradA forms an abelian monoid under the direct sum, with thetrivial triples forming a submonoid GradtA .Definition 3.2 (Difference-group of compatible Hamiltonians). Let K0(A ) be the quotient monoidof GradA by the congruence generated by GradtA , i.e., [W,Γ1,Γ2] = [W ′,Γ′1,Γ′2] in K0(A ) if andonly if there are trivial triples (F,ζ1,ζ2) and (F ′,ζ ′1,ζ ′2) in GradtA , such that(W ⊕F,Γ1⊕ζ1,Γ2⊕ζ2) = (W ′⊕F ′,Γ′1⊕ζ ′1,Γ′2⊕ζ ′2)in GradA . We call K0(A ) the difference-group of symmetry-compatible gapped Hamiltonians.K0(A ) has very nice properties:Theorem 3.3. [15, 8] K0(A ) is an abelian group, with [W,Γ1,Γ2] = −[W,Γ2,Γ1]. Furthermore,two isomorphic triples (in the natural sense) define the same class in K0(A ). Also, equation[W,Γ1,Γ2]+ [W,Γ2,Γ3] = [W,Γ1,Γ3] holds in K0(A ). Furthermore, [W,Γ1,Γ2] depends only onthe homotopy class of Γi in GradA (W ).Note that the difference-group is automatically a group, where the inverse is simply the differ-ence taken in the opposite order — there is no need for an artificial Grothendieck group construc-tion. Differences in K0(A ) can be added in a “path-independent” way, and the class of a differenceelement has the homotopy invariance that we want from a physical perspective.The difference group can be viewed as a generalisation of the ordinary K0-group described inthe first paragraph of this section (see Chapter III of [8] for details). For purely-even algebrasA ev,4PoS(FFP14)149On the K-theoretic classification of topological phases of matter Guo Chuan Thiangwe have K0(A ev)∼= K0(A ev). For real graded C∗-algebras, we can define the n-fold “Clifford sus-pension”A 7→A ⊗ˆCl0,n, where Cl0,n has its natural grading. Then we can define the higher differ-ence groups Kn(A ) :=K0(A ⊗ˆCl0,n). Karoubi showed that Clifford suspension is compatible withthe ordinary suspension of algebras A 7→ C0(R,A ), in the sense that Kn(A ) ∼= K0(C0(Rn,A ));in particular, Kn(A ev) ∼= K0(C0(Rn,A ev)) =: Kn(A ev). Furthermore, Kn (like the ordinary Kn)inherits the period-8 Bott periodicity of the Clifford algebras. Similar results hold for the complexcase. The upshot is that K-theory enters the discussion of gapped topological phases through K0,and not the ordinary ungraded K0.4. Periodicities and dimension shifts in K-theoretic difference-groupsTheorem 4.1 (Packer–Raeburn decomposition theorem [11]). Let (G,c,B,α,σ) be a gradedtwisted C∗-dynamical system, and let N be a closed normal subgroup of G in the kernel of c.There is an isomorphism of graded C∗-algebrasBo(α,σ) G∼= (Bo(α,σ) N)o(β ,ν) G/N, (4.1)for some twisting pair (β ,ν).This is a useful result, both conceptually and computationally. As an example, we can takeG to be a crystallographic symmetry group, N = Zd to be the subgroup of lattice translations, andG/N as the point group. Although it may appear that twisted crossed products are complicated,they are not more complex than ordinary crossed products from the point of view of K-theory(which is invariant under stabilisation):Theorem 4.2 (Packer–Raeburn stabilisation trick [11]). Let (G,B,α,σ) be a twisted C∗-dynamicalsystem, and let K denote the compact operators on the Hilbert space L2(G). There is an isomor-phism(Bo(α,σ) G)⊗K ∼= (B⊗K )o(α ′,1) G,for some untwisted action α ′ of G onB⊗K .Finally, there is a useful result in the K-theory of crossed products by the group R:Theorem 4.3 (Connes–Thom isomorphism [4, 13]). Let (R,A ,α,1) be an untwisted C∗-dynamicalsystem, with A a real or complex (ungraded) C∗-algebra. Then Kn(A o(α,1)R)∼= Kn−1(A ).Iterating Theorems 4.1 and 4.2 and using Theorem 4.3, we obtainCorollary 4.4 (Dimension shifts). Let (Rd ,A ,α,σ) be a twisted C∗-dynamical system. ThenKn(A o(α,σ)Rd)∼= Kn−d(A ).Let ˜A be the symmetry algebra for (G˜,c,A ev,α,σ). Suppose σ is U(1)-valued, and αg,g ∈ G˜is either the identity or complex conjugation according to φ(g) = ±1 (when φ is non-trivial, weassume that A ev is a complexification A evR ⊗RC). We also assume that G˜ = G˜0×A, where G˜0 =ker(α,c), A is a CT -subgroup, and σ is trivial between elements of G˜0 and A. Suppose G˜0 is5PoS(FFP14)149On the K-theoretic classification of topological phases of matter Guo Chuan Thiangan extension of G0 by Rd , and let A ev(R) be the (purely-even) symmetry algebra for the subsystem(G0,A ev(R),α,σ).Under these conditions, the subgroup A contributes a Clifford algebra Clr,s or Cln as a tensorproduct factor (i.e. Clifford suspension) in ˜A (see Example 2.1). Then using Corollary 4.4, we canprove a general periodicity and dimension shift theorem:Theorem 4.5. [15]Then K0( ˜A )∼={Ks−r−d(A evR ), φ 6≡ 1,Kn−d(A ev), φ ≡ 1.As an example, the algebra A ev = C(X) is often used as a model for disorder (with X thedisorder space). Note that the extra Rd-symmetries are not assumed to enter in a trivial way, andmay even be realised projectively, like the magnetic translations in the Integer Quantum Hall Effect.References[1] A. Altland and M. R. Zirnbauer. Nonstandard symmetry classes in mesoscopicnormal-superconducting hybrid structures. Physical Review B, 55(2):1142, 1997.[2] R. Bott. The stable homotopy of the classical groups. Annals of Mathematics, pages 313–337, 1959.[3] R. C. Busby and H. A. Smith. Representations of twisted group algebras. Transactions of theAmerican Mathematical Society, 149(2):503–537, 1970.[4] A. Connes. An analogue of the Thom isomorphism for crossed products of a C?-algebra by an actionof R. Advances in Mathematics, 39(1):31–55, 1981.[5] F. J. Dyson. The threefold way. Algebraic structure of symmetry groups and ensembles in quantummechanics. Journal of Mathematical Physics, 3(6):1199–1215, 1962.[6] D. S. Freed and G. W. Moore. Twisted equivariant matter. Annales Henri Poincaré, 14(8):1927–2023,2013.[7] P. Heinzner, A. Huckleberry, and M. R. Zirnbauer. Symmetry classes of disordered fermions.Communications in Mathematical Physics, 257(3):725–771, 2005.[8] M. Karoubi. K-theory: An Introduction, volume 226 of Grundlehren der MathematischenWissenschaften. Springer-Verlag, Berlin, 1978.[9] A. Kitaev. Periodic table for topological insulators and superconductors. In American Institute ofPhysics Conference Series, volume 1134, pages 22–30, 2009.[10] H. Leptin. Verallgemeinerte L1-Algebren. Mathematische Annalen, 159(1):51–76, 1965.[11] J. A. Packer and I. Raeburn. Twisted crossed products of C?-algebras. Mathematical Proceedings ofthe Cambridge Philosophical Society, 106(02):293–311, 1989.[12] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig. Topological insulators and superconductors:tenfold way and dimensional hierarchy. New Journal of Physics, 12(6):065010, 2010.[13] H. Schröder. K-theory for real C?-algebras and applications, volume 290 of Pitman Research Notesin Mathematics Series. Longman Scientific & Technical, Harlow, 1993.[14] G. C. Thiang. A note on isomorphic versus homotopic topological phases. arXiv:1412.4191, 2014.[15] G. C. Thiang. On the K-theoretic classification of topological phases of matter. arXiv:1406.7366,2014.6

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