On the K-theoretic classification of topological phases of matter

On the -theoretic classification of topological phases of matter

Guo Chuan Thiang Mathematical Institute, University of Oxford, UK guo.thiang@balliol.ox.ac.uk
July 13, 2019

We present a rigorous and fully consistent -theoretic framework for studying gapped topological phases of free fermions such as topological insulators. It utilises and profits from powerful techniques in operator -theory. From the point of view of symmetries, especially those of time reversal, charge conjugation, and magnetic translations, operator -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 -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 -theory to classify phases in an absolute sense, but only relative to some arbitrary reference. -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 -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.

1. Introduction

In his short and influential paper [34], Kitaev proposed that gapped phases of non-interacting fermions can be classified using the techniques of topological -theory. In his approach, there are classes of systems to consider in each spatial dimension , based on the presence or absence of time reversal and/or symmetry. Their classification groups exhibit a certain periodicity with respect to and was attributed, somewhat mysteriously, to Bott periodicity. A Periodic Table was partially drawn up, with each symmetry class in each spatial dimension having one of the -theory groups of a point as its classification group. Little detail was provided in the original paper, which led to a series of authors providing their own accounts [46, 50, 1, 19]. Upon careful investigation, these contain inequivalent treatments of distinct families of free-fermion systems. Furthermore, subsequent work on crystalline [12] and weak topological insulators revealed the existence of phases which are not directly accounted for by the Periodic Table. Despite the lack of a proper proof of (or even the necessary definitions or assumptions in) the Periodic Table, a consensus that it unambiguously provides a complete -theoretic classification of free-fermion phases appears to have been reached; for instance, see the review papers [22, 45].

Unfortunately, there seems to be a number of inconsistencies in the mathematics, and more crucially in the physical interpretation of -theory groups, in the existing literature on topological phases. With the exception of the excellent Freed–Moore paper [19] (which does not address the matter of dimension shifts in the Periodic Table), there has been very little attempt to put -theoretic classification ideas on a firm mathematical footing. Consequently, the full machinery of -theory has yet to be substantially utilised. This is not due to an incompatibility between the mathematics of -theory and the physics, as rigorous work on the Integer Quantum Hall Effect [8], bulk-edge correspondence [32, 33] and Fermi surfaces [25] demonstrate. In addition, the Chern numbers commonly used in the physics literature as topological invariants can more fundamentally be understood as -theory invariants [44]. Thus -theoretic objects have really been lurking in the background in condensed matter physics for a long time.

This paper seeks to address these issues by providing a complete and consistent framework for the use of -theory in the study of gapped topological phases. Our treatment of quantum mechanical symmetries borrows heavily from the comprehensive analysis in [19]. Subsequently, this paper diverges from existing work in two very important ways. First, we utilise operator -theory111More precisely, a version due to Karoubi [28, 26, 27, 29]. rather its commutative (topological) version, which makes available powerful theorems such as the Connes–Thom isomorphisms, the Packer–Raeburn decomposition and stabilisation theorems, and various exact sequences for the -theory of crossed product algebras. The second difference is physical: our representation spaces for the symmetries and Hamiltonians are single-particle Hilbert spaces for charged free-fermions, as opposed to Dirac–Nambu spaces (see Item 9 below).

Next, the Clifford algebras are presented as twisted group algebras of time-reversal and/or charge-conjugation symmetries. Therefore Clifford algebras enter our -theoretic framework in a fundamental way, rather than by a separate ad-hoc analysis of time-reversal and charge-conjugation. Furthermore, our important physical definitions are completely new and relates to Clifford algebras and -theory in a mathematically precise way. For instance, Definition 8.1 gives a precise notion of homotopic phases, and Definition 8.3 illustrates how a -theoretic difference-group classifies obstructions in passing between phases. No unnatural Grothendieck completion is carried out, and inverses arise simply by taking differences in the opposite order.

Thus the collection of topological phases should really be thought of as a torsor for the -theoretic difference group, in the same way that an affine space has forgotten its origin. Once a fixed Hamiltonian has been chosen as a standard reference, other phases are measured with respect to it. Indeed, the very idea of classifying phases up to homotopy in an absolute sense is problematic (see Example 2.2). The philosophy of using -theory to classify differences between phases appears, in any case, to be the original intention of Kitaev in [34]. Furthermore the concept of a relative index has already been studied in the context of the Quantum Hall Effect in [6, 7]. Having set up the crucial definitions, the machinery of -theory takes over and allows us to derive the Periodic Table of Kitaev (when appropriately interpreted) as a simple corollary. Our main result is Theorem 11.5, which demonstrates that the phenomenon of “dimension shifts” and periodicity in the -theoretic classification remains even in the presence of disorder and magnetic fields.

Conceptually, our starting point is Wigner’s theorem, which says that the topological symmetry group (which is not assumed to be compact) for the dynamics of a quantum mechanical system must be represented projectively on a complex Hilbert space as unitary or antiunitary operators. A continuous homomorphism , distinguishes the unitarily-implemented subgroup from the antiunitarily implemented subset . For any two , their representatives satisfy , with a generalised 2-cocycle,


where for , if and if . Thus, interesting topology resides not only in the group of symmetries, but also in the cohomological data of projective unitary-antiunitary representations (PUA-reps). The PUA-representation theory of is a special case of a twisted covariant representation of a twisted -dynamical system as explained in Section 4. We go a step further and consider charge-conjugating symmetries on the same fundamental level as other symmetries, leading to -graded versions of twisted covariant representations. This step is already suggested by the central role of charge-conjugation in relativistic quantum theories, and is ultimately vindicated in our context by the unifying role of super-algebra in -theory.

Our main novel physical insight is then the following: the topology which appears in free-fermion topological phases such as topological insulators has its origin in symmetry. Associated to the algebra of symmetries is a derived “space” (e.g. a Brillouin torus) which is noncommutative in general, and is of secondary importance. We remark that we are not considering topology arising from, e.g., putting physical systems on topologically interesting physical spaces (commutative or otherwise) — the latter is more closely related to topological order [11]. It may be interesting to investigate whether these two sources of interesting topology can be combined.

1.1. Outline

In Section 2, we point out some important inconsistencies in the existing literature. In Section 3, we set our conventions for describing symmetry-compatible free-fermion dynamics. They motivate the definitions of (graded) covariant representations, twisted -dynamical systems, and twisted crossed products in Sections 45. These latter two sections review material which may not be familiar to researchers in topological phases, and can be skimmed over by experts. The intimate relation between Clifford algebras, twisted group algebras, and the tenfold way is explained in Section 6. We move on to the -theoretic classification of symmetry-compatible gapped Hamiltonians proper in Sections 7 and 8, whose computation is illustrated by examples in Section 9. We treat the special case of topological band insulators in Section 10. A Periodic Table in the general sense of Kitaev is derived in Section 11, and we prove that periodicity and dimension shifts persist under very general conditions in Theorem 11.5.

2. Remarks on the existing literature and some inconsistencies

We begin with a list of some inconsistencies in both the mathematics and physical interpretation in the existing literature on -theoretic free-fermion classification schemes. They will be further elaborated upon and rectified in the main body of the paper.

  1. A common definition of a symmetry-compatible topological phase begins with the space of Hamiltonians (gapped or otherwise) which are compatible with a certain given representation of some symmetry data. This space is sometimes called the “classifying space”, which should be distinguished from the mathematical notion with the same name. Two Hamiltonians which are path-connected within this space are identified. The phases, up to homotopy, are then given by the path-components . This is merely a set, with no distinguished identity element, composition law, or inverse. On the other hand, a -theory invariant has the crucial and useful additional structure of an abelian group.

    Example 2.1.

    In a “no symmetry” situation in zero spatial dimensions, one has a bare Hilbert space , and a spectrally-flattened compatible gapped Hamiltonian is just a grading operator on . The space of such Hamiltonians is the union of the Grassmanians of -planes in , with , each of which is connected. Therefore the phases, up to homotopy, form an -element set. A “large-” limit is often taken so that the set of phases becomes a countably infinite set. This set is then conferred the status of the free abelian group in an unclear manner.

  2. The (mathematical) classifying space for complex is , where is the infinite complex Grassmanian. This means that is isomorphic to , the homotopy classes of (based) maps from to . The factor in the classifying space does not arise directly from finite Grassmanians, but is related to Bott periodicity. Writing for the -sphere,

    as befits an extraordinary cohomology theory. In general, one needs to be careful when defining topological phases in zero dimensions.

    A related issue arises for topological phases in higher spatial dimensions. A popular, if somewhat mathematically misguided, point of view is to regard as the “homotopy group” of (based) maps , where is a classifying space for complex -theory; similarly for in the real case. Then if is the -sphere, is identified with . However, it is only possible to form a group from the set of homotopy classes of (based) maps , when is a -group (homotopy-associative) or when is a -cogroup. The two possible ways of defining a composition on from the -group or -cogroup structures are a priori different. Nevertheless, if both choices are available, the resulting groups are isomorphic (pp. 44 of [49]). Not all are -cogroups, but all the are -groups, so is well-defined as a group for all , with respect to the -group structure on . In particular, the group structure on is the direct sum of virtual bundles over , whereas the interpretation of only makes sense for ; yet the case of is most certainly of interest in the study of topological insulators. Well-established topological invariants, such as the Chern numbers associated to filled Landau bands in the Integer Quantum Hall Effect, have the important property of being additive with respect to the direct sum of bands (vector bundles), whereas the group structure of the homotopy groups is a priori a distinct one.

  3. When there is a discrete lattice of translational symmetries, one is led to the study of vector bundles over the Brillouin torus . For consistency with the zero-dimensional case, one should look for the unreduced -theory groups of . For convenience, it is often assumed that the “interesting” Brillouin zone is a -sphere, upon which the reduced -theory of may be identified with a homotopy group of one of the classifying spaces or . The reduced theory does not have the same physical interpretation as the unreduced theory. In fact, , and the latter has an interpretation in terms of vector bundles over trivialised outside a compact set, which are in turn related to systems with continuous translational symmetry. Thus there is a conflation of the reduced and unreduced theories. The discrepancy in the -theory groups in the passage from to is then attributed to the physical difference between “strong” and “weak” topological insulators.

  4. Reduced -theory is sometimes motivated by restricting attention to stable isomorphism classes of vector bundles over . Physically, stabilization entails identifying systems which differ by some “topologically trivial” subsystem. For band insulators (without symmetries beyond the discrete translations ), this means that adding trivial bands should not affect the topological classification. However, in the zero-dimension case, the reduced -theory of a point is trivial, because the complex vector spaces in question are all trivially stably equivalent! This contradicts the -classification which appears in many tables [46, 34]. Furthermore, the higher -theory functors (reduced or otherwise) do not have a direct analogous formulation in terms of vector bundles over . A consistent notion of “triviality” is thus lacking.

  5. For topological insulators, i.e. graded vector bundles, the precise equivalence relation defining a topological phase/class is seldom consistently chosen or clearly defined. Vector bundles, possibly with extra structure dictated by symmetries, can be organised into isomorphism classes, graded or otherwise. Intuitively, a notion of “homotopy classes of bundles” is desired, but this cannot be in the trivial sense of deforming the total bundle space since a vector bundle can always be retracted onto its base space. Isomorphism classes of ungraded vector bundles correspond to homotopy classes of maps from the base space to an appropriate classifying space, not those of the bundle itself. On the other hand, a gapped Hamiltonian determines a grading on a vector bundle, and it is homotopy within the space of allowed gradings (i.e. deformations of possible Hamiltonians) which actually captures the physical intuition of “homotopic gapped phases”.

  6. Within a single fixed realisation of relevant symmetries on a given representation space, it makes sense to consider homotopies between symmetry-compatible Hamiltonians. For two different representation spaces (corresponding to two different physical systems), a choice of isomorphism (if available) is required before the question of whether a Hamiltonian on the first space can be deformed into a Hamiltonian on the second space can be asked. However, two candidate symmetry-compatible Hamiltonians on the same space can be isomorphic without being homotopic (see Example 2.2). Consequently, it is not straightforward to define a notion of homotopy between two Hamiltonians defined on different spaces.

  7. A detailed treatment of the Integer Quantum Hall Effect (IQHE), which does not make the assumption of rational flux and includes the effects of disorder, utilises tools from non-commutative geometry and operator -theory [8]. Despite this, the IQHE is included in Kitaev’s Periodic Table, which actually assumes the presence of a meaningful commutative Brillouin zone.

    The Periodic Table of [34], when applied to topological insulators, should therefore be interpreted more carefully. Furthermore, it has already been recognised that the presence of point symmetries leads to different classification groups from those in his table. This indicates that vital topological information is present in the symmetry group itself.

  8. The Altland–Zirnbauer (AZ) classification of disordered femionic systems [2, 24] is based on the compact classical symmetric spaces which provide spaces of symmetry-compatible time evolutions. While large- versions of symmetric spaces also feature in the classifying spaces of -theory, the AZ classification (and indeed its Wigner–Dyson predecessor) makes no explicit reference to time evolutions generated by gapped Hamiltonians, whereas -theory is supposed to classify gapped phases.

    An attempt to reconcile this disconnect was made in [50]. The general approach there and elsewhere in the literature (with [19] being an exception) is to keep only the data of the ground states (or valence bands of topological insulators) for the purposes of classification. As long as charge conjugating (i.e. Hamiltonian reversing) symmetries are not present, this makes good sense and can even be motivated physically. In such cases, the valence and conduction bands separately determine -theory invariants of the insulating system, with the former usually more interesting. The availability of an interpretation of -theory groups referring only to the valence band, distinguishes the A, AI and AII classes from the other classes in the tenfold way (see Section 10.3). These are the three classes whose topological invariants (the Chern numbers and Kane–Mele invariants) have been studied most closely in the condensed matter literature.

    In general, a charge-conjugation symmetry may constrain the topology of both the conduction and valence bands. Two gapped Hamiltonians which are non-homotopic may posses homotopic valence bands; indeed, remembering only the data of the valence band means that one loses information about the presence or absence of a charge-conjugation symmetry.

  9. The precise treatment of charge-conjugation symmetry differs among authors, and the related notion of particle-hole symmetry in a Dirac–Nambu formalism is often thrown into the mix as well [19, 24, 1, 46]. The Dirac–Nambu space is a vector space of second-quantized creation operators and annihilation operators, and is a useful auxiliary space often used for studying Bogoliubov de Gennes Hamiltonians. However, it also comes with extra structure such as a canonical conjugation which exchanges the creation operators with the annihilation operators. Furthermore, it is necessarily even-dimensional. These structures are not referred to in -theory. For instance, not all elements of or are represented by even-rank vector bundles over . Despite this, -theory is sometimes claimed to classify gapped systems in the Dirac–Nambu formalism.

    Whether symmetries commute with the Hamiltonian (e.g. [24]), or are allowed to anticommute with the Hamitonian (e.g. [46, 19]), depends on whether one is in a first-quantized or second-quantized setting. This is related to whether charge-conjugation is implemented antiunitarily or unitarily. We explain this ambiguity in Section 3.

  10. The use of -theory to account for the effect of time-reversal and charge-conjugation symmetries is actually redundant in our operator -theory approach. In fact, band insulators with these discrete symmetries are not the actual Real bundles required for a -theory analysis, and the proper construction turns out to be quite involved (see Corollary 10.25 of [19]).

    Example 2.2.

    [Homotopic versus isomorphic band insulators] Consider a band insulator in one spatial dimension which has one valence band and one conduction band. Suppose that there is also a sublattice symmetry which is unitary and squares to the identity (also called a chiral symmetry). Physically, this is a Class insulator; mathematically, it is a rank-two -graded complex hermitian vector bundle over , with an odd fibrewise action of . Let us ignore the grading into valence and conduction bands for now. Let be the standard Pauli matrices, and suppose that acts via the matrix on each fibre with respect to global coordinates . According to the usual prescription in the literature, we should look for gradings of , such that the grading operator on each fibre anticommutes with . Since is traceless and hermitian, it must be of the form , for some . In other words, the set of gradings of which are -compatible corresponds to the set of continuous functions , and a homotopy between two such functions is precisely a homotopy between the two bundle gradings ( band insulator structures) that they determine. Therefore, the set of homotopic phases is .

    Define and by and , then and are not homotopic. However, and are isomorphic (in the graded sense), via the unitary bundle map , which is just a change of coordinates. One checks that and , so is indeed an even bundle map respecting the action of . When and the -action are fixed as in this example, writing in the manner that we did picks out a canonical choice of reference, namely , which corresponds to the function . Any other compatible determined by some other is measured against through .

    Suppose encodes the band structure of another two-band insulating system with sublattice symmetry, and is isomorphic to (hence to as well). Whether should be considered to be homotopic to or (or neither) is dependent on the choice of identification between and . Thus suggests that in a homotopy classification, it is more meaningful to classify relative phases.

    Incidentally, the valence band is always a trivial line bundle, so we do lose something if we forget about the conduction band and the -action.

    Example 2.3.

    Example 2.2 is related to the general construction of band insulators with symmetry (Class ) in [46]. A trivial rank- bundle over is fixed, and acts fibrewise as diag() with respect to some global coordinates. Then the possible compatible gradings are continuous choices over of with . Up to homotopy, these are given by . For the phases (for a fixed trivial and ) are thereby given by a homotopy group of the unitary group, which stabilises if is large enough relative to . This is related to the fact that homotopy classes of continuous maps provide a model for . However, a map only determines a band insulator structure when given some given trivial rank- vector bundle with -action. For completeness, non-trivial bundles with -symmetry should also be treated.

    2.1. Some remarks on the Freed–Moore approach

    In the Freed–Moore approach, higher -theoretic groups are constructed using bundles of graded Clifford modules, quotiented by a certain algebraic relation, in analogy to the Atiyah–Bott–Shapiro construction of the -theory ring of a point. This is closely related to our construction of super-representation groups in Section 7. However, we make the important observation that there are two inequivalent ways of taking parity reversals in the construction, with each choice leading to opposite orderings of the classification groups. Furthermore, the super-representation groups do not coincide with standard -theory groups except for certain spaces (e.g. a point). An example is , whereas all bundles of graded -modules over are “trivial” in the sense of Definition 7.2 and Definition 8.5 in [19].

    Usual assumptions on the group of symmetries are: (i) distinguished time-reversing and charge-conjugating elements which are involutary in the full symmetry group, and (ii) there is a direct product factorisation into translational symmetries, point group symmetries, and time-reversal or charge-conjugation symmetries. As emphasized by Freed–Moore, these assumptions do not hold in many realistic systems. Because of this, they are led to twisted equivariant -theory, although only some special twistings occur. In our approach, twistings appear in the form of twisted group algebras, and only the ordinary -theory of these algebras enters. Furthermore, abstract results of Packer–Raeburn [40] allow these twistings to be untwisted without compromising the -theory (see Section 9).

3. Symmetries, spectral-flattening, and positive energy quantization

Following the general arguments of [19], elements of the symmetry group for the dynamics of a quantum mechanical system are presumed to be endowed with Hamiltonian and/or time preserving/reversing properties, which are encoded by a pair of continuous homomorphisms . An element preserves (resp. reverses) the arrow of time if (resp. ); similarly, it commutes (resp. anticommutes) with the Hamiltonian if (resp. ). A third homomorphism specifies whether is implemented unitarily () or antiunitarily (). Writing for the unitary dynamics generated by the Hamiltonian , and for the unitary/antiunitary representative of , the time-reversal equation

leads to , so any two of specifies the third. Often, is assumed (i.e. all symmetries commute with the Hamiltonian), then and antiunitarity becomes synonymous with time-reversal. However, in our description of free-fermion dynamics, we want to consider symmetries that effect charge-conjugation (see Section 3.2), so we allow for . Then any two of may be independently specified. We also allow the symmetries to be projectively realised, i.e., there may be a non-trivial cocycle .

The possibility of charge-reversing symmetries (present or otherwise) for free-fermion dynamics requires, logically, a notion of charged dynamics and charged representations of the canonical anticommutation relations (CARs), as opposed to their neutral counterparts. The latter more correctly describes neutral (Majorana) fermions. Non-degeneracy of the dynamics (or a gapped Hamiltonian) allows us to distinguish between particle and antiparticle sectors, and we would like both species to have positive energy in second quantization. For instance, the Fermi level of a band insulator (which may be set to ) lies in a gap of the Hamiltonian, providing the particle-hole distinction. We recall the algebraic formalism of positive energy charged field quantization, and refer to [15, 16, 20] for the neutral case and technical details. Then, we establish our conventions for dynamical symmetries, including time and charge reversal.

3.1. CAR representations

Let be a complex Hilbert space with inner product . A charged CAR representation over in a complex Hilbert space is a complex-linear map from to the bounded operators on , such that

Here, is the adjoint of , and thus is an anti-linear map from to the bounded operators on . A charged CAR representation gives rise to a neutral CAR representation over with a charge- -symmetry, and conversely.

3.2. Quantization of non-degenerate unitary dynamics

Let be a strongly continuous -parameter unitary group on , with self-adjoint generator (the Hamiltonian). We assume that is non-degenerate, meaning that . We may define

and rewrite as . Note that is unitary, skew-adjoint and commutes with and . Furthermore, is graded by the charge operator (“spectrally-flattened” Hamiltonian) into , where is the eigenspace of . Writing for the space equipped with the modified complex unit instead of , we have , where is given the inner product dual to . The subspaces are invariant for and , so we may regard these operators as self-adjoint operators on , in which case a subscript is appended, e.g., .

On the Fock space , the charged fields are

where and are the standard creation and annihilation operators on . The maps and furnish a charged CAR representation over , called the positive energy Fock quantization for the non-degenerate unitary dynamics . There are second quantized versions of the Hamiltonian and charge operators,

which implement the dynamics and charge symmetry on Fock space,

3.2.1. Charge and/or time reversal in non-degenerate unitary dynamics

A symmetry operator on is required to be unitary or antiunitary according to , and time preserving or reversing according to , i.e., . A short computation leads to the following commutation relations

from which we find that (i.e. the map considered as an operator on ) is unitary or antiunitary according to . We may then amplify to an (anti)unitary operator on Fock space.

Remark 3.1.

The modified imaginary unit is determined by the dynamics only through the spectrally-flattened Hamiltonian .

We stress that presence of a time/charge reversing symmetry does not imply that of a distinguished charge/time reversal operator. Indeed, Freed–Moore [19] have pointed out that there are physically relevant examples that do not fit into the tenfold way [24, 46], which requires distinguished involutary charge/time reversal operators . We prefer to work more generally, and think of time/charge reversal as properties of a symmetry . Under certain splitting assumptions on , we can recover the usual and/or operators, see Section 6.

3.3. Remarks on conventions for free-fermion dynamics

In many treatments of the tenfold way [1, 2, 19, 24, 34, 46, 47], the single-particle “Hamiltonian” in certain symmetry classes is taken to act on a Nambu space rather than a single-particle Hilbert space . A Nambu space has a canonical real structure . The fixed points of form the real mode space of Majorana operators, and inherits a real inner product from by restriction. The operator on restricts to an orthogonal complex structure on , and . One begins with second-quantized dynamics on Fock space , generated by a Hamiltonian which is required to be quadratic in the creation and annihilation operators. Such dynamics can be reformulated on Nambu space , with generating “Hamiltonian” subject to certain symmetry constraints. Alternatively, the dynamics can be specified by a skew-symmetric operator on , whose complexification is . The gapped condition is sometimes imposed on . An example is the Bogoliubov–de Gennes (BdG) Hamiltonian for the quasi-particle dynamics of a superconducting system. It is important to note that the polarization , and thus the Fock space in second quantization, are already implicit in the Nambu space formulation, whereas they are determined by in positive energy Fock quantization. Also, particle number is not necessarily conserved (because and terms are allowed in the second-quantized Hamiltonian ), so may not have a -symmetry (i.e. it may not commute with ). The definition of symmetries of a Hamiltonian, especially those of charge-conjugation and time-reversal, also differ between authors.

In our approach, the Hamiltonian generating the non-degenerate unitary dynamics on determines the particle-antiparticle distinction in second quantization. In practice, we impose a stronger gapped condition on ; namely, we require . In this case, we call a gapped Hamiltonian. We allow for antiunitary symmetries, as well as charge-reversing symmetries which reverse . Two symmetry-compatible gapped Hamiltonians are identified if they have the same spectral flattening, i.e., if they result in the same grading operator on . Then, the specification of a homotopy class222The Hamiltonian is unbounded in general, and care must be taken in order to interpret spectral-flattening as a homotopy in a precise sense, see Appendix D of [19]. of charged free-fermion dynamics respecting the symmetry data is precisely that of a graded projective unitary-antiunitary representation for , as defined in Section 4.4). This provides the physical motivation for the constructions in Section 4.

We remark that a graded PUA-rep for may also be interpreted as an ordinary quantum mechanical system. We usually combine such quantum mechanical systems using the tensor product. On the other hand, at the one-particle level, we combine free-fermion systems using the direct sum operation, which gets translated into the tensor product at the Fock space level. We are only interested in describing free-fermion dynamics and its symmetries at the one-particle level, so the direct sum applies. This allows us to construct commutative monoids of free-fermion systems, paving the way for the use of -theoretic methods in their classification.

4. The general notion of twisted covariant representations

We give an outline of the basic definitions and constructions of twisted covariant representations of twisted -dynamical systems [10, 39, 40]. We make a simple generalisation to -graded twisted covariant representations, and show that they arise naturally as (graded) PUA-reps in the context of quantum systems with time/charge-reversing symmetries. All gradings will be -gradings unless otherwise stated.

4.1. Ungraded covariant representations

Let be a separable, possibly non-unital, real or complex -algebra333A reference for basic facts about real -algebras is Chapter 1 of [48].. We denote its multiplier algebra by , and its group of unitary elements by . If is the ground field of , we write for the group of -linear -automorphisms of . Let be a locally compact, second countable, amenable444Amenability holds in all the physical examples that we consider in this paper, and is made in order to avoid having to distinguish between reduced and full crossed products later on. group, with left Haar measure and identity element . As in Section 2 of [40], we give the strict topology, and the point-norm topology.

Definition 4.1 (Twisted -dynamical system [10, 39]).

A pair of Borel maps and satisfying


is called a twisting pair for . The map is called a 2-cocycle with values in , or simply a cocycle, and the quadruple is called a twisted -dynamical system.

For notational ease, we will often write and .

Definition 4.2 (Twisted covariant representation).

A twisted covariant representation of a twisted -dynamical system is a non-degenerate -representation of as bounded operators on a separable Hilbert space over , along with a compatible Borel map from to the unitary555When , we also use “orthogonal” for emphasis. operators on , in the sense that


Note that (3b) can be restated as , and then we see that (3a) is consistent with (2a). In the untwisted case, i.e. , the Borel map is a homomorphism, hence continuous (Theorem D.11 of [53]). Then is a (untwisted) -dynamical system in the usual sense (e.g. 7.4.1 of [43], 2.1 of [53], or 10.1 of [9]). Similarly, becomes a strongly-continuous homomorphism from to the unitary group of . Thus, is a (untwisted) covariant representation of in the usual sense (e.g. 7.4.8 of [43] or 10.1 of [9]), and no harm is done by dropping the adjective “twisted” when . We say that two twisted covariant representations of are equivalent if there is a unitary -linear intertwiner such that for all .

There is an action of the group of Borel functions on twisting pairs (Section 3 of [40]), defined by


Two twisting pairs and are exterior equivalent if they are related by such a transformation, and there is a correspondence between the covariant representations of and those of , via the adjustments . This generalises the familiar notion of equivalence of cocycles for projective unitary group representations (i.e. ). If the cocycle is assumed to be central in , there is no effect of on in (4a). The conjugation in (2a) and the condition (2d) are then redundant, and we also have . A central cocycle is said to be trivial if there is a Borel function such that , i.e.,  is a coboundary in the sense of cohomology. We say that two central cocycles are equivalent, or in the same cocycle class, if is a trivial cocycle. In many special cases of physical interest, the representative in a cocycle class can be chosen to make certain computations more convenient, e.g. Proposition 6.2 and Lemma 11.4. Note that if is not necessarily central, and must be considered concurrently when making an adjustment .

4.2. Graded covariant representations

Let be a graded real or complex -algebra, i.e. has a direct sum decomposition into two self-adjoint closed subspaces , satisfying . Let now denote its group of even -linear -automorphisms, i.e., -automorphisms that preserve the decomposition . We assume that the cocycles take values in the even elements of . These restrictions are consistent with equations (2a) and (2b) for a twisting pair . Suppose that the group is also equipped with a continuous homomorphism . The quintuple is called a graded twisted -dynamical system.

Definition 4.3 (Graded twisted covariant representation).

A graded covariant representation of a graded twisted -dynamical system is a graded -representation of on a graded Hilbert space over (i.e. ), along with a compatible Borel map , in the sense of (3a)–(3b), with the additional condition that is an even (resp. odd) operator if (resp. ).

Two graded covariant representations for are graded equivalent, or simply equivalent, if there is an even unitary -linear map intertwining with . For instance, a graded Hilbert space over and its parity-reverse are equivalent as ungraded representations of , but are inequivalent in the graded sense unless the two homogeneous subspaces have the same dimension. There is also the notion of super-equivalence of graded covariant representations, which we will consider in Section 7.3. In many of our applications, is trivially graded, i.e., purely even, and the only complication comes from the data of .

4.3. Special cases I: Projective unitary-antiunitary representations

A complex Hilbert space is equivalently a real Hilbert space with real inner product , along with a -orthogonal complex structure (i.e. ) playing the role of multiplication by . The complex inner product may be recovered from and by setting . Note that induces the same norm on as does. An orthogonal operator on is (anti)-unitary as an operator on , iff it (anti)-commutes with .

Let be a continuous homomorphism, and be a -valued 2-cocycle as in (1). A projective unitary-antiunitary representation666See [42, 53] for some topological matters. (PUA-rep) of on a complex Hilbert space is a Borel map such that is a unitary (resp. antiunitary) operator on if (resp. ), and . By regarding as a real Hilbert space, and as a complex structure as above, we can equivalently define a PUA-rep of as a map from to the orthogonal operators on , subject to

Suppose is surjective, and let as an ungraded real -algebra. Thus as a real vector space, with basis , , and the -operation taking to . There are two elements of , namely complex conjugation and the identity . A -representation of is a real Hilbert space along with a linear operator representing , such that and , i.e., is an orthogonal complex structure. Define the map by


Equations (3a)-(3b) say that a covariant representation of on a real Hilbert space , is precisely a PUA-rep of on .

4.4. Special case II: Gapped Hamiltonians and graded projective unitary-antiunitary representations

Following the discussion in Section 3.3, we assume that a PUA-rep for has, additionally, a gapped self-adjoint Hamiltonian , and that has a second continuous homomorphism such that


Thus is a gapped Hamiltonian compatible with the symmetries specified by the data . Note that is a continuous function on the spectrum of homotopic to the identity function. We can deform to its spectral flattening while preserving the commutation/anticommutation relations with as expressed in (6) (but see Footnote 2). A -compatible Hamiltonian is then identified with its spectrally-flattened version for the purposes of a homotopy classification.

A graded PUA-rep of on a graded complex Hilbert space is a PUA-rep of , along with a self-adjoint grading operator satisfying , such that is even or odd according to whether or . The grading operator is exactly a representative of the class of -compatible Hamiltonians on whose spectral flattening is . Suppose is surjective. Let be the purely even real -algebra , and define the (even) automorphisms as in (5). Then a graded covariant representation of is precisely a graded PUA-rep of .

4.5. Special case III: Disordered systems and covariant representations

Disordered systems are often modelled on a disorder space , on which the group acts by homeomorphisms. More generally, the disorder space can be noncommutative, and so acts as automorphisms on an algebra . We can generalise PUA-reps to include disorder, by replacing with the algebra and working with twisted dynamical systems and their covariant representations. Such objects were considered in the analysis of the IQHE in [8], but without the additional data of or .

5. Graded twisted crossed products and covariant representations

In the previous sections, we explained how the implementation of symmetry and compatible gapped Hamiltonians leads to graded twisted -dynamical systems and their covariant representations. In this section, we explain how all the symmetry data can be completely and faithfully encoded in a graded twisted crossed product -algebra , which we may simply call the symmetry algebra. This device will be very convenient for the application of -theory in the later sections.

5.1. Twisted crossed products and covariant representations

Let be the Banach -algebra of integrable functions777The integral of a -valued function on is a Bochner integral, see Appendix B of [53] and the preliminary section of [40]. with the -norm , equipped with a -twisted convolution product and involution ,


where is the modular function on . There is a correspondence between covariant representations of , and non-degenerate -representations of , given by taking the “integrated form” of , see Theorem 3.3 of [10] and Remark 2.6 of [40]. A pre--norm is defined on by

Definition 5.1 (Twisted crossed product -algebra [10]).

Let be a twisted dynamical system. The twisted crossed product -algebra associated with , denoted by , is defined to be the completion of in the norm .

The group is embedded, not necessarily homomorphically, in the multiplier algebra of the twisted crossed product via the Borel map , where the multipliers are defined on functions by

Likewise, the algebra is embedded in through the homomorphism , defined by

When , we recover the untwisted crossed product -algebra associated with the untwisted -dynamical system . If , we call (resp. ) the real (resp. complex) twisted group -algebra of . If as well, we use a shortened notation888It is also standard to write for a semi-direct product of the group with . Nevertheless, the correct meaning should be clear from the context. (resp. ) for the real (resp. complex) group -algebra of . More generally, when and , we will write to ease notation.

Although there are universal characterisations of twisted crossed products, (see Definition 2.4 of [40], as well as [41]), we only need the following important result.

Proposition 5.2 ([10, 40, 41]).

There is a one-to-one correspondence between the covariant representations of and the non-degenerate -representations of .

5.2. Graded twisted crossed products

For a graded twisted -dynamical system , we assign a grading to as follows. Let and . The even subalgebra of is the completion in of , while the odd subspace is the completion of . Note that (7a) and (7b) respect this grading due to the restriction to even automorphisms and even cocycles . A graded -representation of the graded twisted crossed product then corresponds one-to-one with a graded covariant representation of .

6. -symmetries, Clifford algebras, and the tenfold way

The graded PUA-representation theory of a direct product of parity groups can be simplified by fixing certain choices for the representatives of cocycle classes. Let , where means the trivial group. Suppose of the generators are to be represented unitarily (), while the other generators are to be represented antiunitarily (). We will write and for their representatives in a graded PUA-rep of . As always, and are odd/even operators according to .

Lemma 6.1.

We may assume, without loss of generality, that there are at most two antiunitaries .


Let be the image of the homomorphism , and be its kernel. Every non-identity element of and has order 2. Regarding the groups as finite-dimensional vector spaces over the two-element field , we have . Any -basis for provides a set of even unitarily implemented generators . Since , there are at most two with providing antiunitary operators .∎∎

A basis for can be chosen to be one of the following: (i) empty, (ii) {odd }, (iii) {even }, (iv) {odd }, or (v) {even , odd }. We proceed to study the possible cocycles for , and fix representatives for their cocycle classes.

Since , we can make the modification to fix . This does not work for , since for any . Setting in (1) leads to , so are invariants of the cocycle class of . Next, we look at the commutation relations amongst and . For two unitaries , , we write so that . Then

so fixing leads us to . For any two complex scalars , we have , as well as . By choosing , and making the adjustments and , we obtain . Finally, we look at