The tenfold way redux: Fermionic systems with body interactions
Abstract
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary body interactions. We identify four types of symmetries that such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three nonordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no nontrivial ordinary symmetries, we demonstrate that the nonordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary body interactions in each of the ten classes. We show that the space of body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address open questions on the topological classification of interacting fermionic systems.
pacs:
71.10.w, 71.27.+a, 71.10.Fd.1 Introduction
Until very recently, our understanding of fermionic many body systems, for most part, could be traced to a handful of ground states and their excitations, e. g., the Fermi sea, the band insulator, the filled Landau level, and the BCS superconducting state. This picture has been drastically overhauled in the last decade initiated by the discovery of the two dimensional spin Hall insulators Murakami et al. (2004); Kane and Mele (2005, 2005); Bernevig and Zhang (2006); Bernevig et al. (2006), and bolstered by its experimental realization König et al. (2007). It soon became clear Fu et al. (2007); Moore and Balents (2007); Roy (2009); Hsieh et al. (2008) that systems with time reversal symmetry have new types of ground states – the topological insulator – in three dimensions as well (see Moore (2010); Qi and Zhang (2010); Hasan and Kane (2010); Qi and Zhang (2011); Bernevig and Hughes (2013) for a review). These developments naturally motivated the question of the classification of gapped states of fermionic many body systems which has now firmly established itself as a key research direction of condensed matter physics.
For systems of noninteracting fermions, a comprehensive classification has been achieved (see (Beenakker, 2015; Chiu et al., 2015; Ludwig, 2016) for an overview), and indeed marks a milestone in condensed matter research. Central to this success is the symmetry classification of fermionic systems based on a set of “intrinsic” symmetries – the tenfold way of Altland and ZirnbauerAltland and Zirnbauer (1997) (see Heinzner et al. (2005); Zirnbauer (2010) for a more formal treatment) – which places any fermionic system into one of ten symmetry classes. From the point of view of fermionic physics, this work represents the culmination of a program of classification initiated by Dyson Dyson (1962) via the threefold way. In each symmetry class, a gapped fermionic system may possess ground states which are topologically distinct. Early work in this direction Qi et al. (2008); Schnyder et al. (2008) was developed into a complete picture Ryu et al. (2010), culminating in the “periodic table” of KitaevKitaev (2009). The key result is that in any spatial dimension, there are only five symmetry classes that support nontrivial topological phases of noninteracting fermionic systems. The presence or absence of a topological phase is characterized by a nontrivial abelian group such as or . Even more remarkably, the pattern of nontrivial groups has a periodicity (in spatial dimensions) of 2 for the so called complex classes, and 8 in the real classes with a very specific relationship between nontrivial classes in a given dimension and ones just above and just below. These ideas have also been generalized to include defectsFreedman et al. (2011), and have also been visited again from more formal perspectivesFreed and Moore (2013); Kennedy and Zirnbauer (2016) (see also Prodan and SchulzBaldes (2016)). Further developments in the physics of noninteracting systems came up with the study of the interplay of intrinsic symmetries with those of the environment (such as crystalline space groups) that have resulted in a more intricate classificationFu (2011); Slager et al. (2013); Alexandradinata et al. (2016).
A most intriguing story emerges up on the inclusion of interactions, i.e., for a system of fermions with “nonquadratic” terms in their Hamiltonian. Fidkowski and KitaevFidkowski and Kitaev (2010) showed that the presence of twobody interactions results in a “collapse” of the noninteracting topological classification – for example, Kitaev’s Majorana chainKitaev (2001), described by the group collapses to upon the inclusion of interactions. Following this, the natural question that arises is regarding the principles of classification of topological phases in the presence of interaction. Ideas based on group cohomologyChen et al. (2013), and supercohomology for fermionsGu and Wen (2014) have been put forth. While these are important steps towards the final goal of topological classification of interacting systems, the problem remains at the very frontier of condensed matter researchRyu (2015); Senthil (2015).
A crucial aspect of the tenfold symmetry classification of noninteracting systems is the determination of the structure of the Hamiltonians allowed in each class. The knowledge of this structure then provides ways for viewing these systems from various perspectives e. g., classifying spacesKennedy and Zirnbauer (2015), structure of the target manifolds of nonlinear model descriptionsRyu et al. (2010) etc. To the best of our knowledge, a study of the structure of fermionic Hamiltonians in each of the ten symmetry classes with arbitrary body interactions is not available in the literature. An understanding of the structure of the interacting Hamiltonians will not only aid the analysis of these systems, but also help motivate models that could be crucial to build phenomenology and intuition (possibly through numerics) much like what the KaneMele and BHZ modelsKane and Mele (2005); Bernevig et al. (2006) did in the context of noninteracting systems. This paper aims to fill this lacuna. Enroute this pursuit, we build a transparent framework that allows for a systematic approach to these problems, which even provides further insights for the noninteracting case. Our framework is developed for a system comprising of oneparticle states (dubbed “orbitals”) which can be populated by fermions. Symmetries of such a system can be either linear or antilinear, and nontransposing (occupied orbitals mapped to occupied orbitals) or transposing (occupied orbitals mapped to unoccupied ones), i. e., four distinct types of symmetries. We call the linear nontransposing type of symmetries as ordinary, and the remaining three types (antilinear nontransposing, linear transposing, antilinear transposing) as nonordinary symmetries. The focus of our work are those fermionic systems – which we dub as “grotesque” – which do not possess any nontrivial ordinary symmetries. We show that nonordinary symmetries of a grotesque fermionic system (GFS) are solitary, i. e., a GFS can possess at most one from each type of nonordinary symmetries. Identifying these solitary nonordinary symmetries with time reversal (T), charge conjugation (C), and sublattice (S), leads us to the familiar Altland and Zirnbauer tenfold symmetry classes. We then address the key question of determining the structure of the Hamiltonian in these classes, by constructing “canonical” representations (by unitary matrices) for the symmetry operations in each class. This construction allows for a systematic and efficient determination of the structure of any arbitrary body interaction term in the Hamiltonian of any class. Not surprisingly we recover all known results of noninteracting systems and, more importantly, we provide physical insights that can, inter alia, aid model building thereby helping address the outstanding open problems vis a vis topological phases of interacting fermionic systems.
We begin the discussion with the setting of the problem in section .2. This is followed in section .3 by a discussion of the four types of symmetry operations that a fermionic system can possess. Section .4 introduces and studies those fermionic systems that possess no nontrivial ordinary symmetries and obtains the constraints that the symmetries have to satisfy. The tenfold way is elucidated in section .5 which includes a treatment of the canonical representation of the symmetry in each of the symmetry classes, the result of which is displayed in table 1. This is followed by section .6 that provides an understanding of the results shown in table 1 from the point of view of group cohomology. It is shown that every entry of table 1 is made up of copies of irreducible projective representations of the Klein group or its subgroups. Known results of noninteracting systems are reproduced in section .7 (see table 2). Section .8 lays down the framework for obtaining the structure of a Hamiltonian with body interactions by reducing the problem to the determination of certain vector subspaces and arbitrary body interaction term is shown to be an element of an affine subspace. Techniques needed for performing the analysis for determining the subspace structure are developed first for the twobody case in section .9 and later generalized to the arbitrary body case in section .10. Tables 3 and 4 contain the structures of the body interaction term in each class. The paper is concluded in section .11 where the significance and scope of our results are highlighted including their use in other problems of interest. For the convenience of the reader, all important symbols used in the paper are listed in appendix A.
The paper is structured to be self contained and easy to use in that we develop the discussion from the very basic building blocks. This desideratum naturally results in some overlap with the results from the previous works. We shall specifically mention only those results that we have used explicitly in our discussion.
.2 The Setting
Our system consists of one particle states , – which we call “orbitals” – that are orthonormal ( is the Kronecker delta symbol). Note that these states could denote a variety of situations; could be orbitals at different sites of a lattice, or different atomic orbitals, or even flavor states of an elementary particle – our use of the term orbital denotes one particle states in any context including those just stated. Starting from the vacuum state , we can create the one particle state
(1) 
where is a fermion creation(annihilation) operator that creates(destroys) a particle in the one particle state . These operators satisfy the well known fermion anticommutation relations
(2) 
and
(3) 
We collect these fermionic operators into convenient arrays
(4) 
A different set of orthonormal one particle states () can, of course, be used as effectively. The states are related to via , where are the components of an unitary matrix .^{1}^{1}1Throughout the paper a bold roman symbol (e.g. ) is used to denote a matrix, and the light symbol with indices shown (e.g. ) will denote its components. In terms of the operators , we have
(5) 
along with other useful relations
(6) 
where denotes the transpose.
The system can have any number of particles ranging from to . For each particle number , the set of allowed states is spanned by states obtained by creating particle states from the vacuum using distinct combinations of the operators . The vector space of particle states is denoted by . The full HilbertFock space of the system is given by
(7) 
which is a vector space over complex numbers , providing a complete kinematical description of the system. An important property of this vector space, which we will exploit, is that it is “graded”Spencer (1959) in a natural fashion by the sectors of different particle number. The dynamics of this fermionic system is determined by the Hamiltonian which contains up to body interactions where , and is formally written as
(8) 
where
with repeated indices s and s summed from . Note that here we depart slightly from the usual convention for the manybody Hamiltonian where operation is done on the complete string of s; this notation will eventually prove to be useful in the later manipulations. Note also that should be distinguished from the definition of the dimensional array in eqn. (4). is the matrix which describes the body interactions, and its components have two properties. First, is fully antisymmetric under permutations of the indices among themselves, and also under the permutations of the indices among themselves. Expressed in an equation
(10) 
where and are arbitrary permutations of objects, and sgn denotes the sign of the permutation. Second, the Hermitian character of the Hamiltonian is reflected in the relation
(11) 
Each is an element of an dimensional vector space over the real numbers . With no further restrictions other than eqn. (10) and eqn. (11), in fact, this vector space is endowed with a structure of a Lie algebra. This is achieved by constructing an isomorphic vector space (multiplying every element of by ). It is evident that for any two matrices and of , the commutator is also a matrix in , and in fact,
(12) 
i. e., is isomorphic to the well known Lie algebra which generates the Lie group of dimensional unitary matrices. The Hamiltonian eqn. (8) can be described by a tuple
(13) 
where is the real vector space
(14) 
The problem of classification of a fermionic system of orbitals with the HilbertFock space eqn. (7) and upto body interactions can now be stated precisely. How many “distinct” spaces are possible? The symmetries of the system will determine the distinct structures of these spaces, placing them in different classes.
.3 Symmetries
.3.1 Symmetry Operations
A symmetry operation as defined by WignerWigner (1959) is a linear or antilinear (bijective) operator acting on the Hilbert space of the system that leaves the magnitude of the inner product invariant. Stated in the context of one particle physics of our orbital fermionic system, an operator is a symmetry operation if
(15) 
with and . A symmetry operation must preserve the graded structure of the Hilbert space eqn. (7), i. e., particle states must be mapped only to particle states in a bijective (invertible) fashion. We shall refer to such operations as usual symmetry operations; note that a usual symmetry operator can be either linear or antilinear. The above discussion can be summarized by the equation
(16) 
and the action of is implemented on the operators via
(17) 
(in component form ) where is an unitary matrix that encodes the symmetry operation. If such a usual symmetry is linear then , and if it is antilinear where is the identity of operator on , with the form eqn. (17) remaining the same for both linear and antilinear cases.
The key realization of Altland and Zirnbauer Altland and Zirnbauer (1997) is that many particle fermionic systems admit a larger classes of symmetry operations (see Zirnbauer (2010) for a discussion). The crucial point is that a linear or antilinear operation that maps to that preserves the magnitude of inner product and also preserves the graded nature of the HilbertFock space eqn. (7) is also a legitimate symmetry operation. We call such operations transposing symmetry operations (see Fig. 1 for a schematic illustration) and they satisfy
(18) 
with for linear () and antilinear () operations. Note that such operations are enabled by the fact that . In fact, operations of the type eqn. (16) and eqn. (18) exhaust all possible (anti)linear automorphisms of that preserve its graded structure. From a physical perspective, the transposing symmetry operation maps an  “particle” state to an “hole” state. Holes are fermionic excitations obtained by starting from the “fully filled state” , and creating hole like excitations (such as , a 1hole state). It is therefore natural to implement the action of a transposing symmetry operation via
(19) 
where denotes complex conjugation, such that creation operators are mapped to annihilation operators, and the unitary matrix encodes “relabeling” of states in this transposing symmetry operation (which can, again, be linear or antilinear).
The main conclusion of the above discussion is that there are four distinct types of symmetry operations as illustrated in fig. 2. They are usual linear (UL), usual antilinear (UA), transposing linear (TL), and transposing antilinear (TA). We find it useful to introduce additional terminology – we call UL symmetry operations as ordinary symmetry operations, and all other types of symmetry operations (UA, TL and TA) as nonordinary operations.
.3.2 Symmetry conditions
A symmetry operation is a symmetry if it leaves the Hamiltonian of the system invariant. For usual symmetries, this is effected by the condition
(20) 
while for the transposing symmetry operation the condition changes to
(21) 
where indicates that the expression has to be normal ordered (all creation operators to the left of annihilation operators) using the anticommutation relations eqn. (2). Both of these types of symmetries induces a mapping of in eqn. (13) to via
(22) 
To obtain for any , we introduce an intermediate quantity which is determined by the type of symmetry operation:
(23) 
Here all the primed indices are summed from to , and s are the unitary matrices that encode the symmetry operations as defined in eqn. (17) and eqn. (19). Note that antilinear symmetry operations lead to a complex conjugation of the matrix elements. The transformation of the Hamiltonian eqn. (22) can now be specified completely. For usual symmetries (eqn. (17) and eqn. (20)), both linear and antilinear , we have
(24) 
For transposing operation, the result is a bit more involved on account of the normal ordering operation of eqn. (21). We find
(25) 
The trace operation is defined as
(26) 
accomplishing the tracing out (repeated indices are summed) of indices in , resulting in a matrix. The constants in eqn. (25) are computed to be
(27) 
Note that the RHS of eqn. (25) involves a matrix transpose, and this is a characteristic of the transposing symmetry operations justifying our terminology. Thus, eqns. (22), (23), (24) and (25) completely determine the mapping of to . A symmetry operation (of any type) describes a symmetry if and only if
(28) 
.3.3 Symmetry Group
The set of all symmetry operations of the system forms a group . This group is a disjoint union of four types of symmetry operations (see fig. 2),
(29) 
The set of ordinary symmetries form a normal subgroup of . The nonordinary symmetries have the following interesting properties. First, for any nonordinary we have
(30) 
which can be paraphrased “the square of a nonordinary symmetry operation is an ordinary symmetry operation”. Second, the product of two distinct types of nonordinary symmetry operations is the third type – for example
(31) 
this is summarized in Fig. 3. The content of that figure can be restated as: the factor group is the Klein 4group .
.4 Grotesque Fermionic Systems
We will now focus attention on a special class of orbital fermionic systems. Our systems of interest do not possess any nontrivial ordinary symmetries – we dub such systems as “grotesque fermionic systems (GFS)” to highlight this property. This would suggest that the only ordinary symmetry operation allowed is the trivial identity operation . However, since we are working with the HilbertFock vector space (not a projective or “ray” space of Wigner, see Parthasarathy (1969)), the operator
(32) 
with
(33) 
is always an allowed symmetry operation, and indeed any Hamiltonian of the form eqn. (8) will be invariant under this operation.^{2}^{2}2This consideration can be easily generalized to superconducting Bogoliubovde Gennes (BdG) Hamiltonians which are quadratic in fermion operators by expanding the operators eqn. (4) to the Nambu representation, and treating the problem as a quadratic fermion problem. In this case, is restricted to . We will later discuss its relationship to projective representations (see sec. .6). Thus, for the GFS, , i. e., the only ordinary symmetries are the trivial ones.
We will now demonstrate an important property of a GFS. A GFS can possess at most one each of UA, TL and TA symmetries. In other words, nonordinary symmetries of a GFS are solitary. To prepare to prove this statement, we adopt some useful notation. We will denote UA symmetry operations by , TL by , and TA by – this choice anticipates later discussion. Suppose, now, that we have two distinct UA symmetries of the GFS, say and , then is also a symmetry of our system. But from figure. 3, we know that is an UL type symmetry. By definition, in a GFS any unitary symmetry has to be a trivial one, for some . Thus or and hence is not a distinct UA symmetry – it is simply a product of a trivial symmetry with . The same argument works for and symmetries and the solitarity of nonordinary symmetries is proved (see also Ludwig (2016)). In fact, we can conclude many more interesting facts about symmetries of a GFS.
Consider the symmetry operator . From the previous paragraph we know for some . We can determine from which implies . Applying this relation on immediately forces (due to the antilinearity of ) or , and thus we immediately see that the action of on is
(34) 
We will mostly write eqn. (34) simply as
(35) 
We turn now to which should also be equal to . We get, again, . Applying this last relation on , we get
(36) 
resulting in , and
(37) 
Note also that eqn. (36) implies that must be even when , i. e., a TL type of symmetry with the signature can only be implemented in a GFS with even number of orbitals.
Finally, we discuss . Noting that is an TA symmetry, we get
(38) 
and thus , implies where is one of . We will later show that can always be chosen to be zero, leading to
(39) 
We conclude this discussion on the properties of GFS symmetries by noting that if a GFS has a and type symmetry, then is equal to . If a GFS has a symmetry, we say that it possesses time reversal symmetry, implies the presence of charge conjugation symmetry, and endows a sublattice symmetry on the GFS.
.5 The Tenfold Way
In this section, we show how the ten symmetry classes of fermions arise and obtain the canonical representations of the symmetries in these classes.
.5.1 Symmetry Classes
Based on the discussion of the previous section, we see that a GFS has to be of one of three types.
 Type 0

Possesses no nonordinary symmetries.
 Type 1

Possesses one nonordinary symmetry.
 Type 3

Possesses all three nonordinary symmetries.
The resulting symmetry classes and the class hierarchy is shown in fig. 4.
Whenever time reversal is present it be realized as , and we denote this by , similarly is denoted by and presence of is shown by . Absence of these symmetries is denoted by , , or as the case may be. The “symmetry signature” of any class is denoted by a triple (see fig. 4). It is now immediately clear that there is only one class of type 0 GFS, called A with symmetry signature . There are five classes of type 1 with a single nonordinary symmetry: AI, AII, D, C and AIII. The type 3 systems come in four classes: BDI, CI, DIII, and CII.
.5.2 Canonical Representation of Symmetries
An important step towards determining the structure of Hamiltonians (eqn. (13)) in each of these ten symmetry classes is the determination of the canonical representation of the symmetries of each class. This question is addressed in this section.
.5.3 Type 0
Class A: Class A has no nonordinary symmetries and hence nothing to represent. There are, therefore, no restrictions on the orbital systems – any orbital system can be in class A.
.5.4 Type 1
Classes AI and AII: Time reversal symmetry is the sole nonordinary symmetry present in these classes with . The condition eqn. (35) gives (using the antilinearity of )
(43) 
leading to
(44) 
where is an unit matrix. An immediate consequence of this is that , leading us to the conclusion that can be realized in any orbital GFS, while requires to be an even number.
To construct a canonical that satisfies eqn. (44), we consider a change of basis of the GFS as described by the matrix defined in eqn. (5). The unitary representing in this new basis can be obtained from
(45) 
It is known that any unitary that satisfies eqn. (44) with can be written as where is a unitary matrix. This result from matrix theory, usually called the Takagi decomposition (see appendix D of Dreiner et al. (2010)), allows us to conclude that we can always choose () a basis^{3}^{3}3This new basis is not unique. In fact, existence of such a basis implies that any other basis related by a real orthogonal matrix will also be an equally valid one. where for . The outcome of this discussion is that admits a canonical representation .
In the case of , Takagi decomposition provides that any satisfying eqn. (44) can decomposed as where
(46) 
and is unitary. This is consistent with the fact that must be necessarily even for as concluded above. The subscripts on the matrices denote their sizes. Taken together with eqn. (45) allows us to conclude that case is canonically represented by .
We can also obtain a “one particle” or “first quantized” representation of the time reversal operator. To see this consider a GFS with only a onebody Hamiltonian, form eqn. (24) and the symmetry condition eqn. (28), we obtain which can be written as
(47) 
where is
(48) 
where is the complex conjugation “matrix”, recovering a well known result (see e.g., Ludwig (2016)).
Classes D and C: These classes respectively with symmetry signatures and possess the sole nonordinary symmetry of charge conjugation. As noted just after eqn. (37) the latter symmetry can be realized only in GFSs with even number of orbitals.
As in the previous para, considering the action of on the fermion operators, owing to eqn. (37), gives
(49) 
resulting in
(50) 
Note that this relation, despite being a linear operation, looks very similar to eqn. (44) of an usual antilinear operator.
Under a change of basis, it can be shown that transforms in exactly the same manner as , i. e., via eqn. (45). Precisely the same considerations using Takagi decomposition of the previous para then allow us to conclude that there is a basis for the GFS where can be represented canonically as for and for . Finally, restricting again to a GFS described solely by one particle interactions, provides the first quantized version of the operator. Indeed, eqn. (24) and eqn. (28) provide that rewritten as
(51) 
This is consistent with the first quantized form
(52) 
It is easily seen that this feature is generic – any transposing linear symmetry operation has an antilinear first quantized representation.
Class AIII: The sole nonordinary symmetry in this class with the symmetry signature is the sublattice symmetry.
Investigating the action of , noting that is an antilinear operator, we obtain from eqn. (38)
(53) 
implying
(54) 
It is evident that it can be redefined () as
(55) 
Since is unitary, we obtain that , and thus, is Hermitian. This condition implies that all eigenvalues of are real and of unit magnitude. Quite interestingly, under a change of basis, transforms as
(56) 
This implies that there is a basis in which has the following canonical form
(57) 
where
(58) 
where . This development makes the meaning of the sublattice symmetry clear. The orbitals of the system are divided into two groups – “sublattices and ”. The sublattice symmetry operation, a transposing antilinear operation, maps the “particle states” in the orbitals to “hole states” on orbitals, while particle states are mapped to negative hole states. In the first quantized language, the transposing antilinear operator , is, therefore, represented by a linear matrix, i. e.,
(59) 
and in a noninteracting system the symmetry is realized when .
.5.5 Type 3
The remainder of the four classes are of type 3, i. e., they possess all three nonordinary symmetries. As noted at the end of section .4, in these classes implying that
(60) 
Our strategy in analyzing these classes is to choose a basis where , and to determine the structure of and . In this basis, we can write
(61) 
where s are complex matrices of the dimension indicated by the suffix. This from automatically satisfies eqn. (44). Once, we fix , is fixed for every class in type via eqn. (60), as
(62) 
The conditions eqn. (44) and eqn. (50) constrain of eqn. (61) very strongly. Two possible cases arise.

This case results in the condition which provides the following constraints
(63) 
Note that since either T or C is , is already even . The condition obtains the constraints
(64) The second line of eqn. (64), forces and to be a unitary matrix which we will call . The other conditions provide and .
Class BDI: The class has a symmetry signature . Since , we see from eqn. (63)that , along with and , providing