# The tenfold way redux: Fermionic systems with $N$-body interactions

Adhip Agarwala, Arijit Haldar and Vijay B. Shenoy
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Science, Bangalore 560012, India.

###### Abstract

We provide a systematic treatment of the tenfold way of classifying fermionic
systems that naturally allows for the study of those with arbitrary $N$-body
interactions. We identify four types of symmetries that such systems can
possess, which consist of one ordinary type (usual unitary symmetries), and
three non-ordinary symmetries (such as time reversal, charge conjugation and
sublattice). Focusing on systems that possess no non-trivial ordinary
symmetries, we demonstrate that the non-ordinary 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 $N$-body interactions in each of the ten classes. We show that the
space of $N$-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 $\mathbb{Z},\mathbb{Z}_{2}$ or $2\mathbb{Z}$. 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 Schulz-Baldes (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 two-body
interactions results in a “collapse” of the noninteracting topological
classification – for example, Kitaev’s Majorana chainKitaev (2001), described
by the group $\mathbb{Z}$ collapses to $\mathbb{Z}_{8}$ 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
$\sigma$-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 $N$-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 Kane-Mele 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 $L$
one-particle states (dubbed “orbitals”) which can be populated by fermions.
Symmetries of such a system can be either linear or antilinear, and non-
transposing (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 non-transposing type of symmetries as ordinary,
and the remaining three types (antilinear non-transposing, linear transposing,
antilinear transposing) as non-ordinary 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 non-ordinary symmetries of a
grotesque fermionic system (GFS) are solitary, i. e., a GFS can possess at
most one from each type of non-ordinary symmetries. Identifying these solitary
non-ordinary 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 $N$-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 $N$-body interactions by reducing the problem to the
determination of certain vector subspaces and arbitrary $K$-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 two-body case in section .9 and later generalized to the
arbitrary $N$-body case in section .10. Tables 3 and 4 contain the structures
of the $N$-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 $L$ one particle states ${|i\rangle}$, $i=1,\ldots,L$–
which we call “orbitals” – that are orthonormal $\langle
i|j\rangle=\delta_{ij}$ ($\delta_{ij}$ is the Kronecker delta symbol). Note
that these states could denote a variety of situations; $i$ 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 ${|0\rangle}$, we can create the one particle state

${|i\rangle}\,\equiv\,\psi^{\dagger}_{i}{|0\rangle},$ (1)

where $\psi^{\dagger}_{i}(\psi_{i})$ is a fermion creation(annihilation)
operator that creates(destroys) a particle in the one particle state
${|i\rangle}$. These operators satisfy the well known fermion anticommutation
relations

$\psi_{i}\psi_{j}^{\dagger}+\psi_{j}^{\dagger}\psi_{i}=\\{\psi_{i},\psi_{j}^{\dagger}\\}=\delta_{ij},$
(2)

and

$\\{\psi_{i},\psi_{j}\\}=0.$ (3)

We collect these fermionic operators into convenient arrays

${{\Psi}}=\left[\begin{array}[]{c}\psi_{1}\\\ \vdots\\\
\psi_{L}\end{array}\right],\;\;\;\;{{\Psi}}^{\dagger}=\left[\begin{array}[]{ccc}\psi^{\dagger}_{1}&\ldots&\psi^{\dagger}_{L}\end{array}\right].$
(4)

A different set of orthonormal one particle states ${|\tilde{i}\rangle}$
($i=1,\ldots,L$) can, of course, be used as effectively. The states
${|\tilde{i}\rangle}$ are related to ${|i\rangle}$ via
${|\tilde{i}\rangle}=\sum_{j=1}^{L}R_{ji}{|j\rangle}$, where $R_{ij}$ are the
components of an $L\times L$ unitary matrix $\mathbf{R}$.111Throughout the
paper a bold roman symbol (e.g. $\mathbf{R}$) is used to denote a matrix, and
the light symbol with indices shown (e.g. $R_{ij}$) will denote its
components. In terms of the operators ${{\Psi}}$, we have

${\widetilde{\Psi}}^{\dagger}={{\Psi}}^{\dagger}\mathbf{R},$ (5)

along with other useful relations

${\widetilde{\Psi}}=\mathbf{R}^{\dagger}{{\Psi}},\;\;\;\;{\widetilde{\Psi}}^{\dagger
T}=\mathbf{R}^{T}{{\Psi}}^{\dagger
T},\;\;\;\;{\widetilde{\Psi}}^{T}={{\Psi}}^{T}\mathbf{R}^{*}$ (6)

where $(~{})^{T}$ denotes the transpose.

The system can have any number of particles $N_{P}$ ranging from $0$ to $L$.
For each particle number $N_{P}$, the set of allowed states is spanned by
$\small{{L}\choose{N_{P}}}$ states obtained by creating $N_{P}$-particle
states from the vacuum using distinct combinations of the operators
$\psi_{i}^{\dagger}$. The vector space of $N_{P}$ particle states is denoted
by $\mathcal{V}_{N_{P}}$. The full Hilbert-Fock space of the system is given
by

$\mathcal{V}=\bigoplus_{N_{P}=0}^{L}\mathcal{V}_{N_{P}},$ (7)

which is a vector space over complex numbers $\mathbb{C}$, 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
$N$-body interactions where $0\leqslant N\leqslant L$, and is formally written
as

${\mathscr{H}}=\sum_{K=0}^{N}({{\Psi}}^{\dagger})^{K}{\mathbf{H}}^{(K)}({{\Psi}})^{K}$
(8)

where

$\displaystyle({{\Psi}}^{\dagger})^{K}{\mathbf{H}}^{(K)}({{\Psi}})^{K}$
$\displaystyle\equiv$ $\displaystyle H^{(K)}_{i_{1}i_{2}\ldots
i_{K};j_{1}j_{2}\dots
j_{K}}\left(\psi_{i_{1}}\psi_{i_{2}}\ldots\psi_{i_{K}}\right)^{\dagger}\psi_{j_{1}}\psi_{j_{2}}\ldots\psi_{j_{K}}$
$\displaystyle=$ $\displaystyle H^{(K)}_{i_{1}i_{2}\ldots
i_{K};j_{1}j_{2}\dots
j_{K}}\psi^{\dagger}_{i_{K}}\psi^{\dagger}_{i_{K-1}}\ldots\psi^{\dagger}_{i_{1}}\psi_{j_{1}}\psi_{j_{2}}\ldots\psi_{j_{K}}\quad,$

with repeated indices $i$s and $j$s summed from $1,\ldots,L$. Note that here
we depart slightly from the usual convention for the many-body Hamiltonian
where $\dagger$ operation is done on the complete string of $\psi$s; this
notation will eventually prove to be useful in the later manipulations. Note
also that $\left(\psi_{i_{1}}\psi_{i_{2}}\ldots\psi_{i_{K}}\right)^{\dagger}$
should be distinguished from the definition of the $1\times L$ dimensional
array $\Psi^{\dagger}$ in eqn. (4). ${\mathbf{H}}^{(K)}$ is the matrix which
describes the $K$-body interactions, and its components
$H^{(K)}_{i_{1}i_{2}\ldots i_{K};j_{1}j_{2}\dots j_{K}}$ have two properties.
First, $H^{(K)}_{i_{1}i_{2}\ldots i_{K};j_{1}j_{2}\dots j_{K}}$ is fully
antisymmetric under permutations of the $i$ indices among themselves, and also
under the permutations of the $j$ indices among themselves. Expressed in an
equation

$H^{(K)}_{i_{X(1)}i_{X(2)}\ldots i_{X(K)};j_{Y(1)}j_{Y(2)}\ldots
j_{Y(K)}}=\mbox{sgn}(X)\;\mbox{sgn}(Y)\,H^{(K)}_{i_{1}i_{2}\ldots
i_{K};j_{1}j_{2}\dots j_{K}}$ (10)

where $X$ and $Y$ are arbitrary permutations of $K$ objects, and sgn denotes
the sign of the permutation. Second, the Hermitian character of the
Hamiltonian is reflected in the relation

$H^{(K)}_{j_{1}j_{2}\dots j_{K};i_{1}i_{2}\ldots
i_{K}}=\left(H^{(K)}_{i_{1}i_{2}\ldots i_{K};j_{1}j_{2}\dots
j_{K}}\right)^{*}.$ (11)

Each ${\mathbf{H}}^{(K)}$ is an element of an
$\small{{L}\choose{K}}$-dimensional vector space ${\mathcal{H}}^{(K)}$ over
the real numbers $\mathbb{R}$. 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
$\mathbbm{i}{\mathcal{H}}^{(K)}$ (multiplying every element of
${\mathcal{H}}^{(K)}$ by $\mathbbm{i}=\sqrt{-1}$). It is evident that for any
two matrices $\mathbbm{i}{\mathbf{H}_{a}}^{(K)}$ and
$\mathbbm{i}{\mathbf{H}_{b}}^{(K)}$ of $\mathbbm{i}{\mathcal{H}}^{(K)}$, the
commutator
$[\mathbbm{i}{\mathbf{H}_{a}}^{(K)},\mathbbm{i}{\mathbf{H}_{b}}^{(K)}]$ is
also a matrix in $\mathbbm{i}{\mathcal{H}}^{(K)}$, and in fact,

$\mathbbm{i}{\mathcal{H}}^{(K)}\sim\boldsymbol{\mathfrak{u}}\left(\small{{L}\choose{K}}\right),$
(12)

i. e., $\mathbbm{i}{\mathcal{H}}^{(K)}$ is isomorphic to the well known Lie
algebra $\boldsymbol{\mathfrak{u}}\left(\small{{L}\choose{K}}\right)$ which
generates the Lie group $U\left(\small{{L}\choose{K}}\right)$ of
$\small{{L}\choose{K}}\times\small{{L}\choose{K}}$ dimensional unitary
matrices. The Hamiltonian eqn. (8) can be described by a $N+1$ tuple

$\mathbf{H}=({\mathbf{H}}^{(0)},{\mathbf{H}}^{(1)},\ldots,{\mathbf{H}}^{(N)})\in\mathcal{H}$
(13)

where $\mathcal{H}$ is the real vector space

$\mathcal{H}=\bigtimes_{K=0}^{N}{\mathcal{H}}^{(K)}$ (14)

The problem of classification of a fermionic system of $L$-orbitals with the
Hilbert-Fock space eqn. (7) and upto $N$-body interactions can now be stated
precisely. How many “distinct” spaces $\mathcal{H}$ are possible? The
symmetries of the system will determine the distinct structures of these
spaces, placing them in different classes.

Figure 1: Illustrating “usual” and “transposing” symmetry operations. A usual
symmetry operation maps $\mathcal{V}_{N_{P}}\mapsto\mathcal{V}_{N_{P}}$, while
a transposing symmetry operation maps
$\mathcal{V}_{N_{P}}\mapsto\mathcal{V}_{L-N_{P}}$.

####  .3 Symmetries

#####  .3.1 Symmetry Operations

Figure 2: Four types of symmetry operations.

A symmetry operation as defined by WignerWigner (1959) is a linear or
antilinear (bijective) operator ${\mathscr{U}}$ 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 $L$-orbital fermionic system, an
operator ${\mathscr{U}}$ is a symmetry operation if

$|\langle\zeta^{\prime}|\varphi^{\prime}\rangle|=|\langle\zeta|\varphi\rangle|,\,\,\,\,\,\,\forall\,\,\,{|\varphi\rangle},{|\zeta\rangle}\in\mathcal{V}_{1}$
(15)

with ${|\varphi^{\prime}\rangle}={\mathscr{U}}{|\varphi\rangle}$ and
${|\zeta^{\prime}\rangle}={\mathscr{U}}{|\zeta\rangle}$. A symmetry operation
must preserve the graded structure of the Hilbert space eqn. (7), i. e.,
$N_{P}$ particle states must be mapped only to $N_{P}$ 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

${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}(\mathcal{V}_{N_{P}})=\mathcal{V}_{N_{P}},\,\forall N_{P}$ (16)

and the action of ${{{\mathscr{U}}}_{\textup{ {\sc USL}}}}$ is implemented on
the operators via

${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}{{\Psi}}^{\dagger}{{{\mathscr{U}}}_{\textup{ {\sc
USL}}}^{-1}}={{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{ {\sc USL}}}}$ (17)

(in component form ${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}\psi_{i}^{\dagger}{{{\mathscr{U}}}_{\textup{ {\sc
USL}}}^{-1}}=\sum_{j=1}^{L}\psi^{\dagger}_{j}(U_{\mbox{\tiny USL}})_{ji}$ )
where ${{\mathbf{U}}_{\textup{ {\sc USL}}}}$ is an $L\times L$ unitary matrix
that encodes the symmetry operation. If such a usual symmetry is linear then
${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}(\mathbbm{i}{\mathscr{I}}){{{\mathscr{U}}}_{\textup{ {\sc
USL}}}^{-1}}=\mathbbm{i}{\mathscr{I}}$, and ${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}(\mathbbm{i}{\mathscr{I}}){{{\mathscr{U}}}_{\textup{ {\sc
USL}}}^{-1}}=-\mathbbm{i}{\mathscr{I}}$ if it is antilinear where
${\mathscr{I}}$ is the identity of operator on $\mathcal{V}$, 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 $\mathcal{V}_{N_{P}}$ to
$\mathcal{V}_{L-N_{P}}$ that preserves the magnitude of inner product and also
preserves the graded nature of the Hilbert-Fock 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

${{{\mathscr{U}}}_{\textup{ {\sc
TRN}}}}(\mathcal{V}_{N_{P}})=\mathcal{V}_{L-N_{P}},\,\forall N_{P}$ (18)

with ${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}(\mathbbm{i}{\mathscr{I}}){{{\mathscr{U}}}_{\textup{ {\sc
USL}}}^{-1}}=\pm\mathbbm{i}{\mathscr{I}}$ for linear ($+$) and antilinear
($-$) operations. Note that such operations are enabled by the fact that
$\dim{\mathcal{V}_{N_{P}}}=\dim{\mathcal{V}_{L-N_{P}}}$. In fact, operations
of the type eqn. (16) and eqn. (18) exhaust all possible (anti)linear
automorphisms of $\mathcal{V}$ that preserve its graded structure. From a
physical perspective, the transposing symmetry operation maps an $N_{P}$\-
“particle” state to an $N_{P}$-“hole” state. Holes are fermionic excitations
obtained by starting from the “fully filled state”
${|\Omega\rangle}=\left(\psi_{1}\psi_{2}\ldots\psi_{L}\right)^{\dagger}{|0\rangle}$,
and creating hole like excitations (such as $\psi_{1}{|\Omega\rangle}$, a
1-hole state). It is therefore natural to implement the action of a
transposing symmetry operation ${{{\mathscr{U}}}_{\textup{ {\sc TRN}}}}$ via

${{{\mathscr{U}}}_{\textup{ {\sc
TRN}}}}{{\Psi}}^{\dagger}{{{\mathscr{U}}}_{\textup{ {\sc
TRN}}}^{-1}}={{\Psi}}^{T}{{\mathbf{U}}_{\textup{ {\sc TRN}}}^{*}}$ (19)

where ∗ denotes complex conjugation, such that creation operators are mapped
to annihilation operators, and the unitary matrix ${{\mathbf{U}}_{\textup{
{\sc TRN}}}}$ 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 non-ordinary 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

${{{\mathscr{U}}}_{\textup{ {\sc
USL}}}}{\mathscr{H}}{{{\mathscr{U}}}_{\textup{ {\sc
USL}}}^{-1}}={\mathscr{H}}$ (20)

while for the transposing symmetry operation the condition changes to

$:{{{\mathscr{U}}}_{\textup{ {\sc
TRN}}}}{\mathscr{H}}{{{\mathscr{U}}}_{\textup{ {\sc TRN}}}^{-1}}:\
={\mathscr{H}}$ (21)

where $:~{}:$ indicates that the expression ${{{\mathscr{U}}}_{\textup{ {\sc
TRN}}}}{\mathscr{H}}{{{\mathscr{U}}}_{\textup{ {\sc TRN}}}^{-1}}$ 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 $\mathbf{H}$ in eqn. (13) to
${\mathring{\mathbf{H}}}$ via

$\mathbf{H}=({\mathbf{H}}^{(0)},{\mathbf{H}}^{(1)},\ldots,{\mathbf{H}}^{(N)})\mapsto{\mathring{\mathbf{H}}}=({{\mathring{\mathbf{H}}}}^{(0)},{{\mathring{\mathbf{H}}}}^{(1)},\ldots,{{\mathring{\mathbf{H}}}}^{(N)}).$
(22)

To obtain ${{\mathring{\mathbf{H}}}}^{(K)}$ for any $K$, we introduce an
intermediate quantity ${{\breve{\mathbf{H}}}}^{(K)}$ which is determined by
the type of symmetry operation:

${\breve{H}}^{(K)}_{i_{1},\ldots,i_{K};j_{1},\ldots,j_{K}}=\begin{cases}U_{i_{1}i_{1}^{\prime}}\ldots
U_{i_{K}i^{\prime}_{K}}{H}^{(K)}_{i^{\prime}_{1},\ldots,i^{\prime}_{K};j^{\prime}_{1},\ldots,j^{\prime}_{K}}U^{\dagger}_{j_{1}^{\prime}j_{1}}\ldots
U^{\dagger}_{j^{\prime}_{K}j_{K}},&\mbox{{UL}}\\\\[8.61108pt]
U_{i_{1}i_{1}^{\prime}}\ldots
U_{i_{K}i^{\prime}_{K}}\left({H}^{(K)}_{i^{\prime}_{1},\ldots,i^{\prime}_{K};j^{\prime}_{1},\ldots,j^{\prime}_{K}}\right)^{*}U^{\dagger}_{j_{1}^{\prime}j_{1}}\ldots
U^{\dagger}_{j^{\prime}_{K}j_{K}},&\mbox{{UA}}\\\\[8.61108pt]
U^{*}_{i_{1}i_{1}^{\prime}}\ldots
U^{*}_{i_{K}i^{\prime}_{K}}{H}^{(K)}_{i^{\prime}_{1},\ldots,i^{\prime}_{K};j^{\prime}_{1},\ldots,j^{\prime}_{K}}U^{T}_{j_{1}^{\prime}j_{1}}\ldots
U^{T}_{j^{\prime}_{K}j_{K}},&\mbox{{TL}}\quad.\\\\[8.61108pt]
U^{*}_{i_{1}i_{1}^{\prime}}\ldots
U^{*}_{i_{K}i^{\prime}_{K}}\left({H}^{(K)}_{i^{\prime}_{1},\ldots,i^{\prime}_{K};j^{\prime}_{1},\ldots,j^{\prime}_{K}}\right)^{*}U^{T}_{j_{1}^{\prime}j_{1}}\ldots
U^{T}_{j^{\prime}_{K}j_{K}},&\mbox{{TA}}\end{cases}$ (23)

Here all the primed indices are summed from $1$ to $L$, and $\mathbf{U}$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

${{\mathring{\mathbf{H}}}}^{(K)}={{\breve{\mathbf{H}}}}^{(K)}.$ (24)

For transposing operation, the result is a bit more involved on account of the
normal ordering operation of eqn. (21). We find

${{\mathring{\mathbf{H}}}}^{(K)}=\sum_{R=K}^{N}A_{R,K}\left[\operatorname{tr}_{R-K}{{\breve{\mathbf{H}}}}^{(R)}\right]^{T},$
(25)

The trace operation $\operatorname{tr}_{~{}}$ is defined as

$\operatorname{tr}_{P}{{\breve{\mathbf{H}}}}^{(K)}\equiv{\breve{H}}^{(K)}_{i_{1}i_{2}\ldots
i_{(K-P)}k_{1}k_{2}\ldots k_{P};j_{1}j_{2}\ldots j_{(K-P)}k_{1}k_{2}\ldots
k_{P}}$ (26)

accomplishing the tracing out (repeated $k$ indices are summed) of $P$ indices
in ${{\breve{\mathbf{H}}}}^{(K)}$, resulting in a
$\small{{L}\choose{K-P}}\times\small{{L}\choose{K-P}}$ matrix. The constants
$A_{K,R}$ in eqn. (25) are computed to be

$A_{R,K}=\begin{cases}\displaystyle{(-1)^{K}\frac{1}{(R-K)!}\left(\frac{R!}{K!}\right)^{2}},&0\leqslant
K\leqslant R\\\ 0,&K>R\end{cases}.$ (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 $\mathbf{H}$ to ${\mathring{\mathbf{H}}}$. A symmetry operation (of
any type) describes a symmetry if and only if

${\mathring{\mathbf{H}}}=\mathbf{H}$ (28)

which is a concise expression of eqn. (20) and eqn. (21).

#####  .3.3 Symmetry Group

The set of all symmetry operations ${\mathscr{U}}$ of the system forms a group
$G_{\mathcal{V}}$. This group is a disjoint union of four types of symmetry
operations (see fig. 2),

$G_{\mathcal{V}}={G_{\mathcal{V}}^{\textup{UL}}}\cup{G_{\mathcal{V}}^{\textup{UA}}}\cup{G_{\mathcal{V}}^{\textup{TL}}}\cup{G_{\mathcal{V}}^{\textup{TA}}}.$
(29)

The set of ordinary symmetries ${G_{\mathcal{V}}^{\textup{UL}}}$ form a normal
subgroup of $G_{\mathcal{V}}$. The non-ordinary symmetries have the following
interesting properties. First, for any non-ordinary ${\mathscr{U}}$ we have

${\mathscr{U}}{\mathscr{U}}\in
G_{\mathcal{V}}^{\textup{UL}},\,\,\,\,\,\,\,\forall{\mathscr{U}}\in{G_{\mathcal{V}}^{\textup{UA}}}\cup{G_{\mathcal{V}}^{\textup{TL}}}\cup{G_{\mathcal{V}}^{\textup{TA}}}.$
(30)

which can be paraphrased “the square of a non-ordinary symmetry operation is
an ordinary symmetry operation”. Second, the product of two distinct types of
non-ordinary symmetry operations is the third type – for example

${\mathscr{U}}_{\textup{UA}}{\mathscr{U}}_{\textup{TL}}={\mathscr{U}}_{\textup{TA}};$
(31)

this is summarized in Fig. 3. The content of that figure can be restated as:
the factor group $G_{\mathcal{V}}/G^{\textup{UL}}_{\mathcal{V}}$ is the Klein
4-group ${{\mathcal{K}}_{4}}$.

$G_{1}\downarrow\,|\,G_{2}\rightarrow$ | $G_{\mathcal{V}}^{\textup{UL}}$ | $G_{\mathcal{V}}^{\textup{UA}}$ | $G_{\mathcal{V}}^{\textup{TL}}$ | $G_{\mathcal{V}}^{\textup{TA}}$
---|---|---|---|---
$G_{\mathcal{V}}^{\textup{UL}}$ | $G_{\mathcal{V}}^{\textup{UL}}$ | $G_{\mathcal{V}}^{\textup{UA}}$ | $G_{\mathcal{V}}^{\textup{TL}}$ | $G_{\mathcal{V}}^{\textup{TA}}$
$G_{\mathcal{V}}^{\textup{UA}}$ | $G_{\mathcal{V}}^{\textup{UA}}$ | $G_{\mathcal{V}}^{\textup{UL}}$ | $G_{\mathcal{V}}^{\textup{TA}}$ | $G_{\mathcal{V}}^{\textup{TL}}$
$G_{\mathcal{V}}^{\textup{TL}}$ | $G_{\mathcal{V}}^{\textup{TL}}$ | $G_{\mathcal{V}}^{\textup{TA}}$ | $G_{\mathcal{V}}^{\textup{UL}}$ | $G_{\mathcal{V}}^{\textup{UA}}$
$G_{\mathcal{V}}^{\textup{TA}}$ | $G_{\mathcal{V}}^{\textup{TA}}$ | $G_{\mathcal{V}}^{\textup{TL}}$ | $G_{\mathcal{V}}^{\textup{UA}}$ | $G_{\mathcal{V}}^{\textup{UL}}$
Figure 3: Multiplication table of types of symmetries.

The set of all the symmetry operations that satisfy either eqn. (20) or eqn.
(21) makes up the symmetry group $G$ of the system and is a subgroup of
$G_{\mathcal{V}}$.

####  .4 Grotesque Fermionic Systems

We will now focus attention on a special class of $L$-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 ${\mathscr{I}}$. However,
since we are working with the Hilbert-Fock vector space (not a projective or
“ray” space of Wigner, see Parthasarathy (1969)), the operator

${\mathscr{I}}_{\theta}=e^{\mathbbm{i}\theta{\mathscr{N}}}$ (32)

with

${\mathscr{N}}=\sum_{i=1}^{L}\psi_{i}^{\dagger}\psi_{i}$ (33)

is always an allowed symmetry operation, and indeed any Hamiltonian of the
form eqn. (8) will be invariant under this operation.222This consideration can
be easily generalized to superconducting Bogoliubov-de 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, ${\mathscr{I}}_{\theta}$ is
restricted to $\theta=0,\piup$. We will later discuss its relationship to
projective representations (see sec. .6). Thus, for the GFS,
$G_{\mathcal{V}}^{\textup{UL}}={\mathscr{I}}_{\theta},\forall\theta\in[0,2\piup]=U(1)$,
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, non-ordinary
symmetries of a GFS are solitary. To prepare to prove this statement, we adopt
some useful notation. We will denote UA symmetry operations by
${\mathscr{T}}$, TL by ${\mathscr{C}}$, and TA by ${\mathscr{S}}$ – this
choice anticipates later discussion. Suppose, now, that we have two distinct
UA symmetries of the GFS, say ${\mathscr{T}}_{1}$ and ${\mathscr{T}}_{2}$,
then ${\mathscr{T}}_{1}{\mathscr{T}}_{2}$ is also a symmetry of our system.
But from figure. 3, we know that ${\mathscr{T}}_{1}{\mathscr{T}}_{2}$ is an UL
type symmetry. By definition, in a GFS any unitary symmetry has to be a
trivial one, ${\mathscr{I}}_{\theta}$ for some $\theta$. Thus
${\mathscr{T}}_{1}{\mathscr{T}}_{2}={\mathscr{I}}_{\theta}$ or
${\mathscr{T}}_{2}={\mathscr{I}}_{-\theta}{\mathscr{T}}_{1}^{-1}$ and hence
${\mathscr{T}}_{2}$ is not a distinct UA symmetry – it is simply a product of
a trivial symmetry with ${\mathscr{T}}_{1}^{-1}$. The same argument works for
${\mathscr{C}}$ and ${\mathscr{S}}$ symmetries and the solitarity of non-
ordinary 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 ${\mathscr{T}}^{2}$. From the previous
paragraph we know ${\mathscr{T}}^{2}={\mathscr{I}}_{\theta}$ for some
$\theta$. We can determine $\theta$ from
${\mathscr{T}}^{-1}{\mathscr{T}}^{2}={\mathscr{T}}={\mathscr{T}}^{2}{\mathscr{T}}^{-1}$
which implies
${\mathscr{I}}_{-\theta}{\mathscr{T}}^{-1}={\mathscr{T}}={\mathscr{I}}_{\theta}{\mathscr{T}}^{-1}$.
Applying this relation on $\mathcal{V}_{1}$ immediately forces
$e^{\mathbbm{i}2\theta}=1$ (due to the antilinearity of ${\mathscr{T}}$) or
$e^{\mathbbm{i}\theta}=\pm 1$, and thus we immediately see that the action of
${\mathscr{T}}^{2}$ on ${\cal V}$ is

${\mathscr{T}}^{2}=(\pm 1)^{{\mathscr{N}}}{\mathscr{I}}.$ (34)

We will mostly write eqn. (34) simply as

${\mathscr{T}}^{2}=\pm{\mathscr{I}}.$ (35)

We turn now to ${\mathscr{C}}^{2}$ which should also be equal to
${\mathscr{I}}_{\theta}$. We get, again,
${\mathscr{C}}^{-1}{\mathscr{I}}_{\theta}={\mathscr{I}}_{\theta}{\mathscr{C}}^{-1}$.
Applying this last relation on $\mathcal{V}_{N_{P}}$, we get

$e^{\mathbbm{i}N_{P}\theta}=e^{\mathbbm{i}(L-N_{P})\theta},\,\,\,\forall\,\,N_{P}=0,\ldots,L.$
(36)

resulting in $e^{\mathbbm{i}2\theta}=1$, and

${\mathscr{C}}^{2}=(\pm 1)^{{\mathscr{N}}}{\mathscr{I}}.$ (37)

Note also that eqn. (36) implies that $L$ must be even when $\theta=\piup$, 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 ${\mathscr{S}}^{2}$. Noting that ${\mathscr{S}}$ is an TA
symmetry, we get

$e^{-\mathbbm{i}N_{P}\theta}=e^{\mathbbm{i}(L-N_{P})\theta},\,\,\,\forall\,\,N_{P}=0,\ldots,L\quad,$
(38)

and thus $e^{\mathbbm{i}L\theta}=1$, implies $\theta=\frac{2\piup\ell}{L}$
where $\ell$ is one of $0,1,\ldots,L-1$. We will later show that $\theta$ can
always be chosen to be zero, leading to

${\mathscr{S}}^{2}={\mathscr{I}}.$ (39)

We conclude this discussion on the properties of GFS symmetries by noting that
if a GFS has a ${\mathscr{T}}$ and ${\mathscr{C}}$ type symmetry, then
${\mathscr{S}}$ is equal to ${\mathscr{T}}{\mathscr{C}}$. If a GFS has a
${\mathscr{T}}$ symmetry, we say that it possesses time reversal symmetry,
${\mathscr{C}}$ implies the presence of charge conjugation symmetry, and
${\mathscr{S}}$ 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 non-ordinary symmetries.

Type 1

Possesses one non-ordinary symmetry.

Type 3

Possesses all three non-ordinary symmetries.

The resulting symmetry classes and the class hierarchy is shown in fig. 4.

Figure 4: Ten symmetry classes, and their interrelationships. The two classes
shaded in red are the complex classes, while all others are real classes. The
symmetry signature is denoted by a triple
$({\textup{T}},{\textup{C}},{\textup{S}})$ as discussed in the text.

Whenever time reversal is present it be realized as
${\mathscr{T}}^{2}=\pm{\mathscr{I}}$, and we denote this by ${\textup{T}}=\pm
1$, similarly ${\mathscr{C}}^{2}=\pm{\mathscr{I}}$ is denoted by
${\textup{C}}=\pm 1$ and presence of ${\mathscr{S}}$ is shown by
${\textup{S}}=1$. Absence of these symmetries is denoted by ${\textup{T}}=0$,
${\textup{C}}=0$, or ${\textup{S}}=0$ as the case may be. The “symmetry
signature” of any class is denoted by a triple
$({\textup{T}},{\textup{C}},{\textup{S}})$ (see fig. 4). It is now immediately
clear that there is only one class of type 0 GFS, called A with symmetry
signature $(0,0,0)$. There are five classes of type 1 with a single non-
ordinary symmetry: AI$(+1,0,0)$, AII$(-1,0,0)$, D$(0,+1,0)$, C$(0,-1,0)$ and
AIII$(0,0,1)$. The type 3 systems come in four classes: BDI$(+1,+1,1)$,
CI$(+1,-1,1)$, DIII$(-1,+1,1)$, and CII$(-1,-1,1)$.

#####  .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.

Each of the non-ordinary symmetry operations, ${\mathscr{T}}$, ${\mathscr{C}}$
and ${\mathscr{S}}$ is represented by a $L\times L$ unitary matrix. Time
reversal ${\mathscr{T}}$ is represented as

${\mathscr{T}}{{\Psi}}^{\dagger}{\mathscr{T}}^{-1}={{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{T}}}$
(40)

using eqn. (17). Charge conjugation and sublattice, being transposing
operations, are represented using eqn. (19) respectively as

${\mathscr{C}}{{\Psi}}^{\dagger}{\mathscr{C}}^{-1}={{\Psi}}^{T}{{\mathbf{U}}_{\textup{C}}^{*}}$
(41)

and

${\mathscr{S}}{{\Psi}}^{\dagger}{\mathscr{S}}^{-1}={{\Psi}}^{T}{{\mathbf{U}}_{\textup{S}}^{*}}.$
(42)

#####  .5.3 Type 0

Class A:  Class A has no non-ordinary symmetries and hence nothing to
represent. There are, therefore, no restrictions on the $L$-orbital systems –
any $L$ orbital system can be in class A.

#####  .5.4 Type 1

Classes AI and AII:  Time reversal symmetry is the sole non-ordinary symmetry
present in these classes with ${\textup{T}}=\pm 1$. The condition eqn. (35)
gives (using the antilinearity of ${\mathscr{T}}$)

${\mathscr{T}}^{2}{{\Psi}}^{\dagger}{\mathscr{T}}^{-2}={{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}={\textup{T}}\
{{\Psi}}^{\dagger}$ (43)

leading to

${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}={\textup{T}}\,\mathbf{1}$
(44)

where $\mathbf{1}$ is an $L\times L$ unit matrix. An immediate consequence of
this is that $|\det{{{\mathbf{U}}_{\textup{T}}}}|^{2}={\textup{T}}^{L}$,
leading us to the conclusion that ${\textup{T}}=+1$ can be realized in any
$L$-orbital GFS, while ${\textup{T}}=-1$ requires $L$ to be an even number.

To construct a canonical ${{\mathbf{U}}_{\textup{T}}}$ that satisfies eqn.
(44), we consider a change of basis of the GFS as described by the matrix
$\mathbf{R}$ defined in eqn. (5). The unitary
${{\widetilde{\mathbf{U}}}_{\textup{T}}}$ representing ${\mathscr{T}}$ in this
new basis can be obtained from

$\mathbf{R}{{\widetilde{\mathbf{U}}}_{\textup{T}}}\mathbf{R}^{T}={{\mathbf{U}}_{\textup{T}}}\
.$ (45)

It is known that any unitary that satisfies eqn. (44) with ${\textup{T}}=+1$
can be written as ${{\mathbf{U}}_{\textup{T}}}={\mathbf{A}}{\mathbf{A}}^{T}$
where ${\mathbf{A}}$ 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
($\mathbf{R}={\mathbf{A}}$) a basis333This 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
${{\widetilde{\mathbf{U}}}_{\textup{T}}}=\mathbf{1}$ for ${\textup{T}}=+1$.
The outcome of this discussion is that ${\textup{T}}=+1$ admits a canonical
representation ${{\mathbf{U}}_{\textup{T}}}=\mathbf{1}$.

In the case of ${\textup{T}}=-1$, Takagi decomposition provides that any
${{\mathbf{U}}_{\textup{T}}}$ satisfying eqn. (44) can decomposed as
${\mathbf{A}}\mathbf{J}{\mathbf{A}}^{T}={{\mathbf{U}}_{\textup{T}}}$ where

$\mathbf{J}=\left(\begin{array}[]{cc}\mathbf{0}_{MM}&\mathbf{1}_{MM}\\\
-\mathbf{1}_{MM}&\mathbf{0}_{MM}\end{array}\right),\ \
L=2M,\;\;\;\;\;\mathbf{J}\mathbf{J}^{*}=-\mathbf{1},$ (46)

and ${\mathbf{A}}$ is unitary. This is consistent with the fact that $L$ must
be necessarily even for ${\textup{T}}=-1$ as concluded above. The subscripts
on the matrices denote their sizes. Taken together with eqn. (45) allows us to
conclude that ${\textup{T}}=-1$ case is canonically represented by
${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}$.

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 one-body
Hamiltonian, form eqn. (24) and the symmetry condition eqn. (28), we obtain
${{\mathbf{U}}_{\textup{T}}}\left[{{\mathbf{H}}^{(1)}}\right]^{*}{{\mathbf{U}}_{\textup{T}}^{\dagger}}={\mathbf{H}}^{(1)}$
which can be written as

${\mathbf{T}}{\mathbf{H}}^{(1)}{\mathbf{T}}^{-1}={\mathbf{H}}^{(1)}$ (47)

where ${\mathbf{T}}$ is

${\mathbf{T}}={{\mathbf{U}}_{\textup{T}}}{\mathbf{K}}$ (48)

where ${\mathbf{K}}$ 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
$(0,+1,0)$ and $(0,-1,0)$ possess the sole non-ordinary 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 ${\mathscr{C}}^{2}$ on the
fermion operators, owing to eqn. (37), gives

${\mathscr{C}}^{2}{{\Psi}}^{\dagger}{\mathscr{C}}^{-2}={\mathscr{C}}{{\Psi}}^{T}{{\mathbf{U}}_{\textup{C}}^{*}}{\mathscr{C}}^{-1}={{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}^{*}}\,$
(49)

resulting in

${{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}^{*}}={\textup{C}}\,\mathbf{1}.$
(50)

Note that this relation, despite ${\mathscr{C}}$ 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 ${{\mathbf{U}}_{\textup{C}}}$
transforms in exactly the same manner as ${{\mathbf{U}}_{\textup{T}}}$, 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 ${{\mathbf{U}}_{\textup{C}}}$ can be represented canonically as
${{\mathbf{U}}_{\textup{C}}}=\mathbf{1}$ for ${\textup{C}}=+1$ and
${{\mathbf{U}}_{\textup{C}}}=\mathbf{J}$ for ${\textup{C}}=-1$. Finally,
restricting again to a GFS described solely by one particle interactions,
provides the first quantized version of the ${\mathscr{C}}$ operator. Indeed,
eqn. (24) and eqn. (28) provide that
$-{{\mathbf{U}}_{\textup{C}}}\left[{{\mathbf{H}}^{(1)}}\right]^{T}{{\mathbf{U}}_{\textup{C}}^{\dagger}}={\mathbf{H}}^{(1)}$
rewritten as

$-{\mathbf{C}}{\mathbf{H}}^{(1)}{\mathbf{C}}^{-1}={\mathbf{H}}^{(1)}.$ (51)

This is consistent with the first quantized form

${\mathbf{C}}={{\mathbf{U}}_{\textup{C}}}{\mathbf{K}}.$ (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 non-ordinary symmetry in this class with the symmetry
signature $(0,0,1)$ is the sublattice symmetry.

Investigating the action of ${\mathscr{S}}^{2}$, noting that ${\mathscr{S}}$
is an antilinear operator, we obtain from eqn. (38)

${\mathscr{S}}^{2}{{\Psi}}^{\dagger}{\mathscr{S}}^{-2}={\mathscr{S}}{{\Psi}}^{T}{{\mathbf{U}}_{\textup{S}}^{*}}{\mathscr{S}}^{-1}={{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=e^{\mathbbm{i}\frac{2\piup\ell}{L}}{{\Psi}}^{\dagger}$
(53)

implying

${{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=e^{\mathbbm{i}\frac{2\piup\ell}{L}}\mathbf{1}.$
(54)

It is evident that it can be redefined
($e^{-\mathbbm{i}\frac{\piup\ell}{L}}{{\mathbf{U}}_{\textup{S}}}\mapsto{{\mathbf{U}}_{\textup{S}}}$)
as

${{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}.$ (55)

Since ${{\mathbf{U}}_{\textup{S}}}$ is unitary, we obtain that
${{\mathbf{U}}_{\textup{S}}^{\dagger}}={{\mathbf{U}}_{\textup{S}}}$, and thus,
${{\mathbf{U}}_{\textup{S}}}$ is Hermitian. This condition implies that all
eigenvalues of ${{\mathbf{U}}_{\textup{S}}}$ are real and of unit magnitude.
Quite interestingly, under a change of basis, ${{\mathbf{U}}_{\textup{S}}}$
transforms as

$\mathbf{R}{{\widetilde{\mathbf{U}}}_{\textup{S}}}\mathbf{R}^{\dagger}={{\mathbf{U}}_{\textup{S}}}$
(56)

This implies that there is a basis in which ${{\mathbf{U}}_{\textup{S}}}$ has
the following canonical form

${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}$ (57)

where

$\mathbf{1}_{p,q}=\left(\begin{array}[]{cc}\mathbf{1}_{pp}&\mathbf{0}_{pq}\\\
\mathbf{0}_{qp}&-\mathbf{1}_{qq}\end{array}\right)$ (58)

where $p+q=L$. This development makes the meaning of the sublattice symmetry
clear. The orbitals of the system are divided into two groups – “sublattices
$A$ and $B$”. The sublattice symmetry operation, a transposing antilinear
operation, maps the “particle states” in the $A$ orbitals to “hole states” on
$A$ orbitals, while $B$-particle states are mapped to negative $B$-hole
states. In the first quantized language, the transposing antilinear operator
${\mathscr{S}}$, is, therefore, represented by a linear matrix, i. e.,

${\mathbf{S}}={{\mathbf{U}}_{\textup{S}}}$ (59)

and in a noninteracting system the symmetry is realized when
$-{\mathbf{S}}{\mathbf{H}}^{(1)}{\mathbf{S}}^{-1}={\mathbf{H}}^{(1)}$.

#####  .5.5 Type 3

The remainder of the four classes are of type 3, i. e., they possess all three
non-ordinary symmetries. As noted at the end of section .4,
${\mathscr{S}}={\mathscr{T}}{\mathscr{C}}$ in these classes implying that

${{\mathbf{U}}_{\textup{S}}}={{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{C}}^{*}}\
.$ (60)

Our strategy in analyzing these classes is to choose a basis where
${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}$, and to determine the structure
of ${{\mathbf{U}}_{\textup{T}}}$ and ${{\mathbf{U}}_{\textup{C}}}$. In this
basis, we can write

$\begin{split}{{\mathbf{U}}_{\textup{T}}}&=\left(\begin{array}[]{cc}\mathbf{u}_{pp}&\mathbf{u}_{pq}\\\
{\textup{T}}\mathbf{u}_{pq}^{T}&\mathbf{u}_{qq}\end{array}\right),\;\;\;\;\mathbf{u}_{pp}^{T}={\textup{T}}\mathbf{u}_{pp},\mathbf{u}_{qq}^{T}={\textup{T}}\mathbf{u}_{qq},\;\;\;\\\
&\mbox{for}\;\;\;\;\;{\textup{T}}=\pm 1,\end{split}$ (61)

where $\mathbf{u}$s are complex matrices of the dimension indicated by the
suffix. This from automatically satisfies eqn. (44). Once, we fix
${{\mathbf{U}}_{\textup{T}}}$, ${{\mathbf{U}}_{\textup{C}}}$ is fixed for
every class in type ${\bf 3}$ via eqn. (60), as

$\begin{split}{{\mathbf{U}}_{\textup{C}}}={{{\mathbf{U}}}_{\textup{T}}^{T}}{{\mathbf{U}}_{\textup{S}}^{*}}&=\left(\begin{array}[]{cc}{\textup{T}}\mathbf{u}_{pp}&-{\textup{T}}\mathbf{u}_{pq}\\\
\mathbf{u}_{pq}^{T}&-{\textup{T}}\mathbf{u}_{qq}\end{array}\right);\
{\textup{T}}=\pm 1\ .\end{split}$ (62)

The conditions eqn. (44) and eqn. (50) constrain
$\mathbf{u}_{pp},\mathbf{u}_{pq},\mathbf{u}_{qq}$ of eqn. (61) very strongly.
Two possible cases arise.

  * ${\textup{T}}={\textup{C}}$:

This case results in the condition
${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}={{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}^{*}}$
which provides the following constraints

$\begin{split}\mathbf{u}_{pp}\mathbf{u}^{*}_{pp}={\textup{T}}\mathbf{1}_{pp},\;\;\;\mathbf{u}_{qq}\mathbf{u}_{qq}^{*}&={\textup{T}}\mathbf{1}_{qq},\\\
\mathbf{u}_{pq}\mathbf{u}^{\dagger}_{pq}={\boldsymbol{0}}_{pp},\;\;\;\mathbf{u}^{T}_{pq}\mathbf{u}^{*}_{pq}&={\boldsymbol{0}}_{qq},\\\
\mathbf{u}_{pp}\mathbf{u}^{*}_{pq}={\boldsymbol{0}}_{pq},\;\;\;\mathbf{u}^{T}_{pq}\mathbf{u}^{*}_{pp}&={\boldsymbol{0}}_{qp},\\\
\mathbf{u}_{pq}\mathbf{u}_{qq}^{*}={\boldsymbol{0}}_{pq},\;\;\;\mathbf{u}_{qq}\mathbf{u}_{pq}^{\dagger}&={\boldsymbol{0}}_{qp},\;\;\;\;\mbox{for~{}}{\textup{T}}={\textup{C}}.\end{split}$
(63)

  * ${\textup{T}}=-{\textup{C}}$:

Note that since either T or C is $-1$, $L$ is already even $L=2M$. The
condition
${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}=-{{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}^{*}}$
obtains the constraints

$\begin{split}\mathbf{u}_{pp}\mathbf{u}^{*}_{pp}={\boldsymbol{0}}_{pp},\;\;\;\mathbf{u}_{qq}\mathbf{u}_{qq}^{*}&={\boldsymbol{0}}_{qq},\\\
\mathbf{u}_{pq}\mathbf{u}^{\dagger}_{pq}={\textup{T}}\mathbf{1}_{pp},\;\;\;\mathbf{u}^{T}_{pq}\mathbf{u}^{*}_{pq}&={\textup{T}}\mathbf{1}_{qq},\\\
\mathbf{u}_{pp}\mathbf{u}^{*}_{pq}={\boldsymbol{0}}_{pq},\;\;\;\mathbf{u}^{T}_{pq}\mathbf{u}^{*}_{pp}&={\boldsymbol{0}}_{qp},\\\
\mathbf{u}_{pq}\mathbf{u}_{qq}^{*}={\boldsymbol{0}}_{pq},\;\;\;\mathbf{u}_{qq}\mathbf{u}_{pq}^{\dagger}&={\boldsymbol{0}}_{qp},\;\;\;\;\mbox{for~{}}{\textup{T}}=-{\textup{C}}.\end{split}$
(64)

The second line of eqn. (64), forces $p=q=M$ and $\mathbf{u}_{pq}$ to be a
$M\times M$ unitary matrix which we will call $\mathbf{u}_{MM}$. The other
conditions provide $\mathbf{u}_{pp}={\boldsymbol{0}}_{pp}$ and
$\mathbf{u}_{qq}={\boldsymbol{0}}_{qq}$.

Class BDI: The class has a symmetry signature $(1,1,1)$. Since
${\textup{T}}={\textup{C}}=+1$, we see from eqn. (63)that
$\mathbf{u}_{pq}={\boldsymbol{0}}_{pq}$, along with
$\mathbf{u}_{pp}\mathbf{u}^{*}_{pp}=+\mathbf{1}_{pp}$ and
$\mathbf{u}_{qq}\mathbf{u}^{*}_{qq}=+\mathbf{1}_{qq}$, providing

${{\mathbf{U}}_{\textup{T}}}=\left(\begin{array}[]{cc}\mathbf{u}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{u}_{qq}\end{array}\right),\;\;\;\;{{\mathbf{U}}_{\textup{C}}}=\left(\begin{array}[]{cc}\mathbf{u}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&-\mathbf{u}_{qq}\end{array}\right)$ (65)

Apart from $p+q=L$, there are no additional constraints on $p$ and $q$. The
natural splitting of the orbitals into a $p$-subspace and $q$-subspace now
allows us to find canonical forms of ${{\mathbf{U}}_{\textup{T}}}$ and
${{\mathbf{U}}_{\textup{C}}}$. Again, Takagi decomposition allows us to find a
unitary matrix $\mathbf{r}_{pp}$ such that
$\mathbf{r}_{pp}\mathbf{r}_{pp}^{T}=\mathbf{u}_{pp}$, another similar matrix
$\mathbf{r}_{qq}$ that does
$\mathbf{r}_{qq}\mathbf{r}_{qq}^{T}=\mathbf{u}_{qq}$. We can define a basis
change matrix

$\mathbf{R}=\left(\begin{array}[]{cc}\mathbf{r}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{r}_{qq}\end{array}\right)$ (66)

from which we immediately see that

${{\mathbf{U}}_{\textup{T}}}=\mathbf{R}\mathbf{1}\mathbf{R}^{T},\;\;\;\;{{\mathbf{U}}_{\textup{C}}}=\mathbf{R}(\mathbf{1}_{p,q})\mathbf{R}^{T}.$
(67)

Moreover, we see that ${{\mathbf{U}}_{\textup{S}}}$ transforms under this
basis change via eqn. (56) as

$\mathbf{R}^{\dagger}\mathbf{1}_{p,q}\mathbf{R}=\mathbf{1}_{p,q}$ (68)

i. e., such a basis change does not alter the form of
${{\mathbf{U}}_{\textup{S}}}$. The conclusion of this discussion is that when
${\textup{T}}={\textup{C}}=+1$, we can always choose a basis in which

${{\mathbf{U}}_{\textup{T}}}=\mathbf{1},\;\;\;{{\mathbf{U}}_{\textup{C}}}=\mathbf{1}_{p,q},\;\;\;{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}.$
(69)

Class CII: This symmetry class has signature $(-1,-1,1)$. From eqn. (63), we
have $\mathbf{u}_{pq}={\boldsymbol{0}}_{pq}$,
$\mathbf{u}_{pp}\mathbf{u}^{*}_{pp}=-\mathbf{1}_{pp}$ and
$\mathbf{u}_{qq}\mathbf{u}^{*}_{qq}=-\mathbf{1}_{qq}$, resulting in

${{\mathbf{U}}_{\textup{T}}}=\left(\begin{array}[]{cc}\mathbf{u}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{u}_{qq}\end{array}\right),\;\;\;\;{{\mathbf{U}}_{\textup{C}}}=\left(\begin{array}[]{cc}-\mathbf{u}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{u}_{qq}\end{array}\right).$ (70)

Even as $p+q=L=2M$, there are additional constraints on $p$ and $q$. The
conditions $\mathbf{u}_{pp}\mathbf{u}^{*}_{pp}=-\mathbf{1}_{pp}$ and
$\mathbf{u}_{qq}\mathbf{u}^{*}_{qq}=-\mathbf{1}_{qq}$, force $p=2r$ and
$q=2s$, i. e., both $p$ and $q$ have to be even numbers, $2(r+s)=2M$.

We can find canonical forms for ${{\mathbf{U}}_{\textup{T}}}$ and
${{\mathbf{U}}_{\textup{C}}}$ much like eqn. (69). For this, noting that $p$
and $q$ are even, we know that there are unitary matrices $\mathbf{r}_{pp}$
and $\mathbf{r}_{qq}$ such that
$\mathbf{r}_{pp}\mathbf{J}_{pp}\mathbf{r}_{pp}^{T}=\mathbf{u}_{pp}$ and
$\mathbf{r}_{qq}\mathbf{J}_{qq}\mathbf{r}_{qq}^{T}=\mathbf{u}_{qq}$. Thus one
can perform a basis change using a transformation similar to eqn. (66) to
obtain natural forms

${{\mathbf{U}}_{\textup{T}}}=\left(\begin{array}[]{cc}\mathbf{J}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{J}_{qq}\end{array}\right),\;\;\;{{\mathbf{U}}_{\textup{C}}}=\left(\begin{array}[]{cc}-\mathbf{J}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{J}_{qq}\end{array}\right),\;\;\;\;{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q},$
(71)

which provide the canonical representation of the three symmetries in class
CII.

Class CI: This class with the symmetry signature $(+1,-1,1)$ can be analyzed
using eqn. (64) which provides

${{\mathbf{U}}_{\textup{T}}}=\left(\begin{array}[]{cc}{\boldsymbol{0}}_{MM}&\mathbf{u}_{MM}\\\
\mathbf{u}_{MM}^{T}&{\boldsymbol{0}}_{MM}\end{array}\right),\;\;\;\;\;{{\mathbf{U}}_{\textup{C}}}=\left(\begin{array}[]{cc}{\boldsymbol{0}}_{MM}&-\mathbf{u}_{MM}\\\
\mathbf{u}_{MM}^{T}&{\boldsymbol{0}}_{MM}\end{array}\right)$ (72)

We can obtain canonical forms of ${{\mathbf{U}}_{\textup{T}}}$ and
${{\mathbf{U}}_{\textup{C}}}$ by the following manipulations. Consider a basis
transformation of the type eqn. (66) of the from

${\mathbf{R}}=\left(\begin{array}[]{cc}\mathbf{u}_{MM}&{\boldsymbol{0}}_{MM}\\\
{\boldsymbol{0}}_{MM}&{\boldsymbol{1}}_{MM}\\\ \end{array}\right)$ (73)

This basis change effects the following changes (see eqn. (45), eqn. (56))

$\begin{split}{{\mathbf{U}}_{\textup{T}}}\mapsto\mathbf{R}^{\dagger}{{\mathbf{U}}_{\textup{T}}}\mathbf{R}^{*}&=\left(\begin{array}[]{cc}{\boldsymbol{0}}_{MM}&{\boldsymbol{1}}_{MM}\\\
{\boldsymbol{1}}_{MM}&{\boldsymbol{0}}_{MM}\end{array}\right)\equiv\mathbf{F}\\\
{{\mathbf{U}}_{\textup{C}}}\mapsto\mathbf{R}^{\dagger}{{\mathbf{U}}_{\textup{C}}}\mathbf{R}^{*}&=-\mathbf{J}\\\
{{\mathbf{U}}_{\textup{S}}}\mapsto\mathbf{R}^{\dagger}\mathbf{1}_{M,M}\mathbf{R}&=\mathbf{1}_{M,M}\\\
\end{split}$ (74)

where the first line defines the $L\times L$ matrix $\mathbf{F}$. Thus the
canonical forms of class CII are

${{\mathbf{U}}_{\textup{T}}}=\mathbf{F},\;\;\;{{\mathbf{U}}_{\textup{C}}}=-\mathbf{J},\;\;\;{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{M,M}$
(75)

Class DIII: The symmetry signature $(-1,+1,1)$ provides
${\textup{T}}=-1=-{\textup{C}}$. Form eqn. (64), we get

${{\mathbf{U}}_{\textup{T}}}=\left(\begin{array}[]{cc}{\boldsymbol{0}}_{MM}&\mathbf{u}_{MM}\\\
-\mathbf{u}_{MM}^{T}&{\boldsymbol{0}}_{MM}\end{array}\right),\;\;\;\;\;{{\mathbf{U}}_{\textup{C}}}=\left(\begin{array}[]{cc}{\boldsymbol{0}}_{MM}&\mathbf{u}_{MM}\\\
\mathbf{u}_{MM}^{T}&{\boldsymbol{0}}_{MM}\end{array}\right)$ (76)

Using the basis change eqn. (73), we can find canonical structure of the
$\mathbf{U}$ matrices as

${{\mathbf{U}}_{\textup{T}}}=\mathbf{J},\;\;\;{{\mathbf{U}}_{\textup{C}}}=\mathbf{F},\;\;\;{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{M,M}.$
(77)

In the classes CI and DIII, there is no further constraint on $M$ which can be
odd or even.

Class | T | C | S | $L$ | ${{\mathbf{U}}_{\textup{T}}}$ | ${{\mathbf{U}}_{\textup{C}}}$ | ${{\mathbf{U}}_{\textup{S}}}$
---|---|---|---|---|---|---|---
A | 0 | 0 | $\ 0\ $ | $L$ | $-$ | $-$ | $-$
AI | $+1$ | 0 | 0 | $L$ | $\mathbf{1}$ | $-$ | $-$
AII | $-1$ | 0 | 0 | $L=2M$ | $\mathbf{J}$ | $-$ | $-$
D | $0$ | $+1$ | 0 | $L$ | $-$ | $\mathbf{1}$ | $-$
C | $0$ | $-1$ | 0 | $L=2M$ | $-$ | $\mathbf{J}$ | $-$
AIII | 0 | 0 | 1 | $L=p+q$ | $-$ | $-$ | $\mathbf{1}_{p,q}$
BDI | $+1$ | $+1$ | 1 | $L=p+q$ | $\mathbf{1}$ | $\mathbf{1}_{p,q}$ | $\mathbf{1}_{p,q}$
CII | $-1$ | $-1$ | 1 | $\begin{array}[]{c}L=p+q\\\ p=2r;\ q=2s\end{array}$ | ${\left(\begin{array}[]{cc}\mathbf{J}_{pp}&{\boldsymbol{0}}_{pq}\\\ {\boldsymbol{0}}_{qp}&\mathbf{J}_{qq}\end{array}\right)}$ | ${\left(\begin{array}[]{cc}-\mathbf{J}_{pp}&{\boldsymbol{0}}_{pq}\\\ {\boldsymbol{0}}_{qp}&\mathbf{J}_{qq}\end{array}\right)}$ | $\mathbf{1}_{p,q}$
CI | $+1$ | $-1$ | 1 | $L=2M$ | $\mathbf{F}$ | $-\mathbf{J}$ | $\mathbf{1}_{M,M}$
DIII | $-1$ | $+1$ | 1 | $L=2M$ | $\mathbf{J}$ | $\mathbf{F}$ | $\mathbf{1}_{M,M}$
Table 1: Canonical representations of symmetry operators
${{\mathbf{U}}_{\textup{T}}}$, ${{\mathbf{U}}_{\textup{C}}}$ and
${{\mathbf{U}}_{\textup{S}}}$ are shown in the ten symmetry classes.
$\mathbf{1}$ is the $L\times L$ identity matrix, $\mathbf{J}$ defined in eqn.
(46), $\mathbf{1}_{p,q}$ defined in eqn. (58), $\mathbf{F}$ defined in eqn.
(74).

The determination of the canonical structure of the symmetry operation
accomplishes a great deal in the determination of the structure of the
Hamiltonians in each class. The summary of the findings of this section are
given in table 1.

####  .6 A Group Cohomolgy Perspective

The results of the previous section can be obtained from a different
perspective as we describe in this section. Readers interested in the
structure of many body Hamiltonians in each class may proceed directly to sec.
.7. The main idea of the section is to demonstrate that the ten classes arise
from projective representationsWeinberg (1995) of an (abstract) symmetry
group. For the GFS discussed in this paper, the abstract group of symmetry
operations (see discussion near eqn. (30) is the Klein 4-group
${{\mathcal{K}}_{4}}$. This group consists of four elements $I$ (identity),
$\Theta$ (time reversal), $\Xi$ (charge conjugation), and $\Sigma$
(sublattice) and has the following multiplication table

$\begin{split}\begin{tabular}[]{c|cccc}${{\mathcal{K}}_{4}}$&{$I$}&{$\Theta$}&{$\Xi$}&{$\Sigma$}\\\
\hline\cr{$I$}&{$I$}&{$\Theta$}&{$\Xi$}&{$\Sigma$}\\\
{$\Theta$}&{$\Theta$}&{$I$}&{$\Sigma$}&{$\Xi$}\\\
{$\Xi$}&{$\Xi$}&{$\Sigma$}&{$I$}&{$\Theta$}\\\
{$\Sigma$}&{$\Sigma$}&{$\Xi$}&{$\Theta$}&{$I$}\end{tabular}\quad\quad.\end{split}$
(78)

The abstract symmetry group ${\mathcal{G}}$ of the GFS must be one of the
(proper/improper) subgroups of ${{\mathcal{K}}_{4}}$. Thus, ${\mathcal{G}}$ is
one of
${\mathcal{I}}=\\{\textup{$I$}\\},{\mathcal{Z}}_{2}^{T}=\\{\textup{$I$},\textup{$\Theta$}\\},{\mathcal{Z}}_{2}^{C}=\\{\textup{$I$},\textup{$\Xi$}\\},{\mathcal{Z}}_{2}^{S}=\\{\textup{$I$},\textup{$\Sigma$}\\}$
or ${{\mathcal{K}}_{4}}$.

We look for projective representations of the group ${\mathcal{G}}$ on a
graded vector space
${\mathcal{W}}=\mathcal{V}_{1}\oplus\bar{\mathcal{V}}_{1}$, where
$\mathcal{V}_{1}$ is an $L$-dimensional $\mathbb{C}$ vector space. We are
restricting here to the one particle sector $\mathcal{V}_{1}$ and its
“transposed” hole space $\bar{\mathcal{V}_{1}}$ (which is isomorphic to
$\mathcal{V}_{L-1}$, see eqn. (7)). The group elements are represented by
$\mathbb{C}$ valued matrices of the form

$\begin{split}D(\textup{$I$})={\left(\begin{array}[]{cc}\mathbf{1}&\mathbf{0}\\\
\mathbf{0}&\mathbf{1}\end{array}\right)},\;\;\;\;\;D(\textup{$\Theta$})&={\left(\begin{array}[]{cc}{{\mathbf{U}}_{\textup{T}}}&\mathbf{0}\\\
\mathbf{0}&{{\mathbf{U}}_{\textup{T}}}\end{array}\right)}{\mathbf{K}}\\\
D(\textup{$\Xi$})={\left(\begin{array}[]{cc}\mathbf{0}&{{\mathbf{U}}_{\textup{C}}}\\\
{{\mathbf{U}}_{\textup{C}}}&\mathbf{0}\end{array}\right)}{\mathbf{K}},\;\;\;\;\;D(\textup{$\Sigma$})&={\left(\begin{array}[]{cc}\mathbf{0}&{{\mathbf{U}}_{\textup{S}}}\\\
{{\mathbf{U}}_{\textup{S}}}&\mathbf{0}\end{array}\right)}.\end{split}$ (79)

${\mathbf{K}}$ is the usual complex conjugation operator,
${{\mathbf{U}}_{\textup{T}}}$, ${{\mathbf{U}}_{\textup{C}}}$, and
${{\mathbf{U}}_{\textup{S}}}$ are unitary matrices that represent the
symmetries appearing in their subscripts. This choice of representations is
motivated by discussions of the previous section.

Denoting the group elements of ${\mathcal{G}}$ by $\textup{$g$}_{\ell}$, a
projective representation reproduces the group multiplication table up to a
$U(1)$ phase factor, i. e.,

$D(\textup{$g$}_{1})D(\textup{$g$}_{2})=\omega(\textup{$g$}_{1},\textup{$g$}_{2})D(\textup{$g$}_{1}\textup{$g$}_{2})\
.$ (80)

With $|{\mathcal{G}}|$ as the number of elements in ${\mathcal{G}}$, the
$|{\mathcal{G}}|^{2}$ numbers $\omega(\textup{$g$}_{1},\textup{$g$}_{2})\in
U(1)$, usually called “2-cocylces” or “Schur multipliers” must satisfy the
following co-cycle condition

$\omega(\textup{$g$}_{1},\textup{$g$}_{2})\omega(\textup{$g$}_{1}\textup{$g$}_{2},\textup{$g$}_{3})=\omega(\textup{$g$}_{1},\textup{$g$}_{2}\textup{$g$}_{3})\omega^{\textup{$g$}_{1}}(\textup{$g$}_{2},\textup{$g$}_{3})\
.$ (81)

Here $\omega^{\textup{$g$}}$ is a short form for action of the group element
$g$ on an element of $U(1)$, i. e., for every group element $g$, there is an
invertible function $\varphi_{\textup{$g$}}$ on $U(1)$ which maps $\omega\in
U(1)$ to $\varphi_{\textup{$g$}}(\omega)\equiv\omega^{\textup{$g$}}$. The
function must satisfy

$\varphi_{\textup{$g$}_{1}}(\varphi_{\textup{$g$}_{2}}(\omega))=\varphi_{\textup{$g$}_{1}\textup{$g$}_{2}}(\omega).$
(82)

More technically, $\varphi$ is a homomorphism from ${\mathcal{G}}$ to
$\textup{Aut}{\,U(1)}$, the group of automorphisms (invertible maps that
preserve the group structure) of $U(1)$. In our case, the homomorphism
$\varphi$ is defined as

$\begin{split}\varphi_{{\tiny\textup{$I$}}}(\omega)=\omega,\;\;\;\;\varphi_{{\tiny\textup{$\Theta$}}}(\omega)&=\omega^{*}\\\
\varphi_{{\tiny\textup{$\Xi$}}}(\omega)=\omega^{*},\;\;\;\;\varphi_{{\tiny\textup{$\Sigma$}}}(\omega)&=\omega\\\
\end{split}$ (83)

where $(~{})^{*}$ denotes complex conjugation. The check of the homomorphic
character of $\varphi$ is routine. This particular choice of $\varphi$ is made
keeping in mind the physical aspects of the group elements, consistent with
eqn. (79). Now two multiplier system $\omega_{1}$ and $\omega_{2}$ (both of
which satisfy eqn. (81)) can be used to produce a third one $\omega^{\prime}$,
in fact
$\omega^{\prime}(\textup{$g$}_{1},\textup{$g$}_{2})=\omega_{1}(\textup{$g$}_{1},\textup{$g$}_{2})\omega_{2}(\textup{$g$}_{1},\textup{$g$}_{2})$.
Note that $\omega(\textup{$g$}_{1},\textup{$g$}_{2})=1$, the unit multiplier,
satisfies the cocycle condition eqn. (81). Furthermore, any given $\omega$ has
an inverse
$\omega^{-1}(\textup{$g$}_{1},\textup{$g$}_{2})=\omega^{*}(\textup{$g$}_{1},\textup{$g$}_{2})$
such that $\omega\omega^{-1}$ is the unit multiplier. Finally, two multipliers
$\omega_{1}$ and $\omega_{2}$ are equivalent if there exists a $U(1)$ valued
function on ${\mathcal{G}}$, $a(\textup{$g$})$, such that

$\frac{\omega_{1}(\textup{$g$}_{1},\textup{$g$}_{2})}{\omega_{2}(\textup{$g$}_{1},\textup{$g$}_{2})}=\frac{a(\textup{$g$}_{1})a^{\textup{$g$}_{1}}(\textup{$g$}_{2})}{a(\textup{$g$}_{1}\textup{$g$}_{2})}.$
(84)

Suppose we denote the set of all multipliers equivalent to $\omega$ by
$[\omega]$. It is easy to see that the set of equivalence classes of
multipliers $\\{[\omega]\\}$ forms an abelian group called the second
cohomolgy group (see Janssen (1972) and chapter 7 of Wickless (2004))
$H^{2}_{\varphi}(U(1),{\mathcal{G}})$. A key question of importance to us is
how many elements (number of equivalence class of multipliers) does
${\mathcal{G}}$ admit – this is answered by computing the group
$H^{2}_{\varphi}(U(1),{\mathcal{G}})$.

We can now make the central point: The number of symmetry classes corresponds
to number of distinct equivalence classes of multiplier systems corresponding
to the symmetry group ${\mathcal{G}}$ which runs over of
${\mathcal{I}},{\mathcal{Z}}_{2}^{T},{\mathcal{Z}}_{2}^{C},{\mathcal{Z}}_{2}^{S}$
and ${{\mathcal{K}}_{4}}$. To obtain the structure of each class, we find the
multiplier system that labels the class, and find the irreducible
representations, “smallest” vector space ${\mathcal{W}}$, and the matrices of
eqn. (79). Again, we organize the classes via the types introduced in sec.
.5.1

#####  .6.1 Type 0

Here the symmetry group ${\mathcal{G}}={\mathcal{I}}$, and
$H^{2}_{\varphi}(U(1),{\mathcal{G}})$ is trivial containing a single element,
and hence a trivial multiplier system.

Class A: The sole irreducible representation corresponding to this trivial
multiplier system is 1-dimensional. This labels class A, and we see that the
$L$-orbtial GFS in class A of table 1 is a reducible representation consisting
of $L$ copies of irreducible one obtained just now.

#####  .6.2 Type 1

${\mathcal{G}}={\mathcal{Z}}_{2}^{T}$

The second cohomology group
$H^{2}_{\varphi}(U(1),{\mathcal{Z}}_{2}^{T})=Z_{2}$ consists two multiplier
classes.Chen  _et al._ (2013). The multipliers are

$\begin{array}[]{c|cc}\omega(\textup{$g$}_{1}\,\downarrow,\textup{$g$}_{2}\,\rightarrow)&\textup{$I$}&\textup{$\Theta$}\\\
\hline\cr\textup{$I$}&1&1\\\
\textup{$\Theta$}&1&{\textup{T}}\end{array},\;\;\;\;\;\;\;\;{\textup{T}}=\pm
1$ (85)

Class AI: The multiplier of $H^{2}_{\varphi}(U(1),{\mathcal{Z}}_{2}^{T})$ with
${\textup{T}}=+1$ labels the class AI. To find the irreducible representation,
we need to solve

${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}=\mathbf{1}$ (86)

and the solution with smallest dimension (in the sense of dimension of vector
space $\mathcal{V}_{1}$ that makes up ${\mathcal{W}}$) is 1, and
${{\mathbf{U}}_{\textup{T}}}=e^{\mathbbm{i}\alpha}$. By the change of basis
eqn. (45), $\alpha$ can always chosen to the zero, and
${{\mathbf{U}}_{\textup{T}}}=1$, and
$D(\textup{$\Theta$})={\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)}\,{\mathbf{K}}$. We see that the $L$-orbtial GFS of
table 1 is a reducible representation with $L$-copies of this irrep
(irreducible representation).

Class AII: The multiplier in eqn. (85) with ${\textup{T}}=-1$ corresponds to
class AII. This leads to the equation
${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}=-\mathbf{1}$. The
solution with the smallest dimension is 2, with

${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}_{2}={\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)}\ .$ (87)

We see that the AII GFS of table 1 with $L=2M$ consists $M$ copies of this
this irrep corresponding to this multiplier system.

${\mathcal{G}}={\mathcal{Z}}_{2}^{C}$

The second cohomology group
$H^{2}_{\varphi}(U(1),{\mathcal{Z}}_{2}^{C})=Z_{2}$ – the mathematics of this
group is same as ${\mathcal{Z}}_{2}^{T}$, and the multipliers are

$\begin{array}[]{c|cc}\omega(\textup{$g$}_{1}\,\downarrow,\textup{$g$}_{2}\,\rightarrow)&\textup{$I$}&\textup{$\Xi$}\\\
\hline\cr\textup{$I$}&1&1\\\
\textup{$\Xi$}&1&{\textup{C}}\end{array},\;\;\;\;\;\;\;\;{\textup{C}}=\pm 1$
(88)

The key physical difference is that the representation of $\Xi$ is via a
transposing antilinear matrix (see eqn. (79)). Very similar consideration as
in the ${\mathcal{Z}}_{2}^{T}$ case, leads to two classes.

Class D: The multiplier with ${\textup{C}}=+1$ gives this class. The irrep is
1-dimensional with ${{\mathbf{U}}_{\textup{C}}}=1$ and
$D(\textup{$\Xi$})={\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)}\,{\mathbf{K}}$, $L$ copies of which are present in the
GFS shown in table 1.

Class C: This class is labeled by the ${\textup{C}}=-1$ multiplier of eqn.
(88), with a 2-dimensional irrep such that
$D(\textup{$\Xi$})={\left(\begin{array}[]{cc}\mathbf{0}&\mathbf{J}_{2}\\\
\mathbf{J}_{2}&\mathbf{0}\end{array}\right)}{\mathbf{K}}$. The C entry in
table 1 consists of $M$ copies of this irrep.

${\mathcal{G}}={\mathcal{Z}}_{2}^{S}$

In the case of the sole non-ordinary sublattice symmetry,
$H^{2}_{\varphi}(U(1),{\mathcal{Z}}_{2}^{S})$ is trivial and has only the unit
multiplier.

Class AIII: Being an abelian group of order 2, ${\mathcal{Z}}_{2}^{S}$ has two
distinct 1-dimensional irreducible representations, which arise from the
lowest dimensional solution to the equation
${{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}$. The first
irrep has ${{\mathbf{U}}_{\textup{S}}}=+1$ and the second has
${{\mathbf{U}}_{\textup{S}}}=-1$ (note that these two irreps have the same
multipliers). We immediately see that the $L$ orbital AIII system in table 1
is made of the $p$ copies of the $+$irrep and $q$ copies of the $-$irrep.

#####  .6.3 Type 3

All the classes in type 3 arise from the multipliers of the symmetry group
${\mathcal{G}}={{\mathcal{K}}_{4}}$. Quite interestingly the second cohomology
group of the Klein group is the Klein group itself444We obtained this result
by an explicit calculation.:
$H^{2}_{\varphi}(U(1),{{\mathcal{K}}_{4}})={{\mathcal{K}}_{4}}$! This results
in four multipliers labeled by T and C

$\begin{array}[]{c|cccc}\omega(\textup{$g$}_{1}\,\downarrow,\textup{$g$}_{2}\,\rightarrow)&\textup{$I$}&\textup{$\Theta$}&\textup{$\Xi$}&\textup{$\Sigma$}\\\
\hline\cr\textup{$I$}&1&1&1&1\\\
\textup{$\Theta$}&1&{\textup{T}}&1&{\textup{T}}\\\
\textup{$\Xi$}&1&{\textup{T}}{\textup{C}}&{\textup{C}}&{\textup{T}}\\\
\textup{$\Sigma$}&1&{\textup{C}}&{\textup{C}}&1\end{array},\;\;\;{\textup{T}}=\pm
1,{\textup{C}}=\pm 1\ .$ (89)

To obtain the irreps, we seek smallest dimensional quantities
${{\mathbf{U}}_{\textup{T}}}$, ${{\mathbf{U}}_{\textup{C}}}$,
${{\mathbf{U}}_{\textup{S}}}$ that simultaneously satisfy

${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}^{*}}={\textup{T}}\mathbf{1},\
{{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}^{*}}={\textup{C}}\mathbf{1},\
{{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=\mathbf{1},\
{{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{C}}^{*}}={{\mathbf{U}}_{\textup{S}}}$
(90)

(these are obtained by a straightforward application of eqn. (79)).

Class BDI: The multiplier eqn. (89) with ${\textup{T}}={\textup{C}}=+1$
obtains this class. Eqn. (90) gives
${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}}^{*}=\mathbf{1},{{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}}^{*}=\mathbf{1},{{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=\mathbf{1},{{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{C}}}^{*}={{\mathbf{U}}_{\textup{S}}}$.
There are two 1-dimensional irreps, one with
${{\mathbf{U}}_{\textup{T}}}=1,{{\mathbf{U}}_{\textup{C}}}=1$ and
${{\mathbf{U}}_{\textup{S}}}=1$, and the other
${{\mathbf{U}}_{\textup{T}}}=1,{{\mathbf{U}}_{\textup{C}}}=-1$ and
${{\mathbf{U}}_{\textup{S}}}=-1$. We see that class BDI in table 1 contains
$p$ copies of the first irrep and $q$ copies of the second one.

Class CII: This class is characterized by the multiplier in eqn. (89) with
${\textup{T}}={\textup{C}}=-1$. Irreps are obtained from the solution of
${{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{T}}}^{*}=-\mathbf{1},{{\mathbf{U}}_{\textup{C}}}{{\mathbf{U}}_{\textup{C}}}^{*}=-\mathbf{1},{{\mathbf{U}}_{\textup{S}}}{{\mathbf{U}}_{\textup{S}}}=\mathbf{1},{{\mathbf{U}}_{\textup{T}}}{{\mathbf{U}}_{\textup{C}}}^{*}={{\mathbf{U}}_{\textup{S}}}$.
These now provide two distinct 2-dimensional representations. First one has
${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}_{2},{{\mathbf{U}}_{\textup{C}}}=-\mathbf{J}_{2},{{\mathbf{U}}_{\textup{S}}}=-\mathbf{1}$,
and the second has
${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}_{2},{{\mathbf{U}}_{\textup{C}}}=\mathbf{J}_{2},{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}$.
The CII entry in table 1 has $r$-copies of the first 2-dimensional irrep, and
$s$-copies of the second one.

Class CI: When ${\textup{T}}=-{\textup{C}}=+1$ in the multiplier eqn. (89), we
get the CI class. The solution of eqn. (90) gives a single 2-dimensional
representation with
${{\mathbf{U}}_{\textup{T}}}=\mathbf{F}_{2},{{\mathbf{U}}_{\textup{C}}}=-\mathbf{J}_{2}$
and ${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{1,1}$.
$\mathbf{F}_{2}={\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)}$ and
$\mathbf{1}_{1,1}$ is defined in eqn. (58). The GFS of the CI class in table 1
has $M$ copies of this irrep.

Class DIII: Class DIII is obtained from the multiplier eqn. (89) when
${\textup{T}}=-{\textup{C}}=-1$. Eqn. (90), again, provides a single
2-dimensional representation with
${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}_{2},{{\mathbf{U}}_{\textup{C}}}=\mathbf{F}_{2}$
and ${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{1,1}$. DIII class in table 1 is
made of $M$ copies of this irrep.

The discussion of this section provides a deeper insight of how the ten
symmetry classes arise from projective representations on a graded vector
space (this is where the fermionic nature of the states enters). Moreover, all
the results of the canonical representations of the symmetry operations
derived from elementary considerations can be better understood as copies of
irreducible projective representations.

####  .7 Noninteracting Systems

Class | $L$ | ${\mathbf{H}}^{(1)}$ | $\dim{\mathbbm{i}\mathcal{H}^{(1)}}$ | $\mathbbm{i}\mathcal{H}^{(1)}$ | ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$
---|---|---|---|---|---
A$(0,0,0)$ | $L$ | ${\mathbf{H}}^{(1)}=[{\mathbf{H}}^{(1)}]^{\dagger}$ | $L^{2}$ | $\boldsymbol{\mathfrak{u}}(L)$ | $U(L)$
AI$(+1,0,0)$ | $L$ | ${\mathbf{H}}^{(1)}=[{\mathbf{H}}^{(1)}]^{*}$ | $L(L+1)/2$ | $\boldsymbol{\mathfrak{u}}(L)\setminus\boldsymbol{\mathfrak{o}}(L)$ | $U(L)/O(L)$
AII$(-1,0,0)$ | $L=2M$ | $\displaystyle{{\left(\begin{array}[]{cc}\mathbf{h}_{aa}&\mathbf{h}_{ab}\\\ -\mathbf{h}_{ab}^{*}&\mathbf{h}_{aa}^{*}\end{array}\right)}}$ | $M(2M-1)$ | $\boldsymbol{\mathfrak{u}}(2M)\setminus\boldsymbol{\mathfrak{usp}}(2M)$ | $U(2M)/\text{\it USp}(2M)$
D$(0,+1,0)$ | $L$ | ${\mathbf{H}}^{(1)}=-[{\mathbf{H}}^{(1)}]^{*}$ | $L(L-1)/2$ | $\boldsymbol{\mathfrak{o}}(L)$ | $O(L)$
C$(0,-1,0)$ | $L=2M$ | $\displaystyle{{\left(\begin{array}[]{cc}\mathbf{h}_{aa}&\mathbf{h}_{ab}\\\ \mathbf{h}_{ab}^{*}&-\mathbf{h}_{aa}^{*}\end{array}\right)}}$ | $M(2M+1)$ | $\boldsymbol{\mathfrak{usp}}(2M)$ | $\text{\it USp}(2M)$
AIII$(0,0,1)$ | $L=p+q$ | $\displaystyle{{\left(\begin{array}[]{cc}\mathbf{0}_{pp}&\mathbf{h}_{pq}\\\ \mathbf{h}_{pq}^{\dagger}&\mathbf{0}_{qq}\end{array}\right)}}$ | $2pq$ | $\boldsymbol{\mathfrak{u}}(p+q)\setminus(\boldsymbol{\mathfrak{u}}(p)\oplus\boldsymbol{\mathfrak{u}}(q))$ | $U(p+q)/(U(p)\times U(q))$
BDI$(+1,+1,1)$ | $L=p+q$ | $\displaystyle{{\left(\begin{array}[]{cc}\mathbf{0}_{pp}&\mathbf{h}_{pq}\\\ \mathbf{h}_{pq}^{T}&\mathbf{0}_{qq}\end{array}\right)}},\mathbf{h}_{pq}^{*}=\mathbf{h}_{pq}$ | $pq$ | $\boldsymbol{\mathfrak{o}}(p+q)\setminus(\boldsymbol{\mathfrak{o}}(p)\oplus\boldsymbol{\mathfrak{o}}(q))$ | $O(p+q)/(O(p)\times O(q))$
CII$(-1,-1,1)$ | $\begin{array}[]{c}L=p+q,\\\ p=2r,q=2s\end{array}$ | ${\left(\begin{array}[]{c|c}{\boldsymbol{0}}_{pp}&{\begin{array}[]{cc}\mathbf{h}_{rr}&\mathbf{h}_{rs}\\\ -\mathbf{h}^{*}_{rs}&\mathbf{h}^{*}_{rr}\end{array}}\\\ \hline\cr\text{h.c.}&{\boldsymbol{0}}_{qq}\end{array}\right)}$ | $4rs$ | $\boldsymbol{\mathfrak{usp}}(p+q)\setminus(\boldsymbol{\mathfrak{usp}}(p)\oplus\boldsymbol{\mathfrak{usp}}(q))$ | $\text{\it USp}(2(r+s))/(\text{\it USp}(2r)\times\text{\it USp}(2s))$
CI(+1, -1, 1) | $L=2M$ | $\displaystyle{{\left(\begin{array}[]{cc}\mathbf{0}_{MM}&\mathbf{h}_{MM}\\\ \mathbf{h}_{MM}^{*}&\mathbf{0}_{MM}\end{array}\right)}},\mathbf{h}_{MM}^{T}=\mathbf{h}_{MM}$ | $M(M+1)$ | $\boldsymbol{\mathfrak{usp}}(2M)\setminus\boldsymbol{\mathfrak{u}}(M)$ | $\text{\it USp}(2M)/U(M)$
DIII(-1, +1, 1) | $L=2M$ | $\displaystyle{{\left(\begin{array}[]{cc}\mathbf{0}_{MM}&\mathbf{h}_{MM}\\\ -\mathbf{h}_{MM}^{*}&\mathbf{0}_{MM}\end{array}\right)}},\mathbf{h}_{MM}^{T}=-\mathbf{h}_{MM}$ | $M(M-1)$ | $\boldsymbol{\mathfrak{o}}(2M)\setminus\boldsymbol{\mathfrak{u}}(M)$ | $O(2M)/U(M)$
Table 2: Structure of noninteracting Hamiltonians in the ten symmetry classes.

We devote this section to obtaining the structure of the Hamiltonians of each
class when there are no interactions present. Here $\mathbf{H}$ of eqn. (13)
has only two entries
$\mathbf{H}=\left({\mathbf{H}}^{(0)},{\mathbf{H}}^{(1)}\right)$. Another
important quantity of interest is Schödinger time evolution operator

${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)=e^{-\mathbbm{i}t{\mathbb{H}}}$ (91)

where $t$ is the time, ${\mathbb{H}}$ is the Hamiltonian matrix constructed
out of $\mathbf{H}=\left({\mathbf{H}}^{(0)},{\mathbf{H}}^{(1)}\right)$. For
example, the in this noninteracting setting the components of
${\mathbb{H}}_{ij}={H}^{(0)}\delta_{ij}+{H}^{(1)}_{i;j}$. For a fixed time,
say $t=1$, the time evolution operator spans out a “geometric structure” as
$\mathbf{H}$ runs over all of the space $\mathcal{H}$. As pointed out by
Altland and Zirnbauer (1997) (generalizing the pioneering work of DysonDyson
(1962)), this geometric structure realized is a symmetric space (see Caselle
and Magnea (2004) for a review symmetric spaces from a physicist’s
perspective, Gilmore (1974) for a review of Lie groups and algebras). The ten
classes realize the the ten different symmetric spaces classified by Cartan,
and in fact the names of the classes borrows Cartan’s nomenclature. Our
development of the ten fold symmetry classification not only allows us to
recover these known results in a simple and direct manner but also provides a
very clear structure of the Hamiltonian space that could be particularly
useful for model building. Our results are recorded in table 2.

To proceed with the discussion, we record the symmetry conditions eqn. (22)
specifically for the noninteracting systems. Under the action of usual
symmetries $\mathbf{H}$ transforms as

$\mathbf{H}=\left({\mathbf{H}}^{(0)},{\mathbf{H}}^{(1)}\right)\mapsto{\mathring{\mathbf{H}}}=({\mathbf{H}}^{(0)},{{\breve{\mathbf{H}}}}^{(1)})$
(92)

and for transposing symmetries we obtain

$\mathbf{H}=\left({\mathbf{H}}^{(0)},{\mathbf{H}}^{(1)}\right)\mapsto{\mathring{\mathbf{H}}}=\left({\mathbf{H}}^{(0)}+\operatorname{tr}_{1}{{{\breve{\mathbf{H}}}}^{(1)}},-\left[{{\breve{\mathbf{H}}}}^{(1)}\right]^{T}\right)$
(93)

(see eqns. (23), (24), and (25)). The structure of $\mathbf{H}$ is obtained by
imposing the symmetry condition eqn. (28) for all the appropriate symmetries.
An immediate consequence is that ${\mathbf{H}}^{(0)}$, the “vacuum energy”, is
allowed to have any real value $H^{(0)}$ in all the ten classes. This is a
feature that is always true, i. e., even when interactions are present. As the
values of $H^{0}$ spans the reals, the time evolution operator (at a fixed
time) acquires a phase factor represented by the $U(1)$ group (see discussion
on class A in the next para).

Class A: There are no constraints on the Hamiltonian in this class. Any
$L\times L$ Hermitian matrix is in ${\mathcal{H}}^{(1)}_{\textup{{\sf{A}}}}$.
Consequently
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{A}}}}=\boldsymbol{\mathfrak{u}}(L)$,
and ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$ acquires the structure of
$U(1)\times U(L)$. The $U(1)$ factor arises from ${\mathbf{H}}^{(0)}$ which
spans all of real numbers. As discussed in the previous para, this $U(1)$
factor will appear in all cases (interacting and noninteracting). Henceforth,
we suppress this factor (which is equivalent to setting $H^{0}=0$ (or any
other fixed real number). The symmetric space of
${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$ in table 2 for this case, therefore is
shown just as $U(L)$.

Class AI: Eqn. (28) appropriate for this class (with
${{\mathbf{U}}_{\textup{T}}}=\mathbf{1}$) provides that real symmetric
matrices ${\mathbf{H}}^{(1)}=[{\mathbf{H}}^{(1)}]^{*}$ are the allowed entries
of ${\mathcal{H}}^{(1)}_{\textup{{\sf{AI}}}}$. This implies that the
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{AI}}}}=\boldsymbol{\mathfrak{u}}(L)\setminus\boldsymbol{\mathfrak{o}}(L)$,
where $\boldsymbol{\mathfrak{o}}(L)$ is Lie algebra of the group of orthogonal
matrices. The time evolution operator ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$
spans the coset space $U(L)/O(L)$.

Class AII: Since ${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}$ in this class with
$L=2M$, we can split the orbitals into two types and write the Hamiltonian as

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}\mathbf{h}_{aa}&\mathbf{h}_{ab}\\\
\mathbf{h}^{\dagger}_{ab}&\mathbf{h}_{bb}\end{array}\right)}.$ (94)

The symmetry condition gives
$\mathbf{J}[{\mathbf{H}}^{(1)}]^{*}\mathbf{J}^{\dagger}={\mathbf{H}}^{(1)}$
resulting in
$\mathbf{h}_{ab}=-\mathbf{h}_{ab}^{T},\mathbf{h}_{bb}=\mathbf{h}_{aa}^{*}$.
Here, $\mathbf{h}_{aa},\mathbf{h}_{ab},\mathbf{h}_{bb}$ are $M\times M$
matrices. The space $\mathcal{H}_{\textup{{\sf{AII}}}}$ is made of matrices of
the type

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}\mathbf{h}_{aa}&\mathbf{h}_{ab}\\\
-\mathbf{h}_{ab}^{*}&\mathbf{h}_{aa}^{*}\end{array}\right)}.$ (95)

It can be seen by explicit calculation that
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{AII}}}}=\boldsymbol{\mathfrak{u}}(2M)\setminus\boldsymbol{\mathfrak{usp}}(2M)$,
leading to the time evolution ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$ spanning
the coset space $U(2M)/\text{\it USp}(2M)$. Here $\text{\it USp}(2M)$ is the
symplectic group and $\boldsymbol{\mathfrak{usp}}(2M)$ is its associated Lie
algebra.

Class D: This class with the charge conjugation described by
${{\mathbf{U}}_{\textup{C}}}=\mathbf{1}$ is made of Hamiltonians with
vanishing trace $\operatorname{tr}{{\mathbf{H}}^{(1)}}=0$ that satisfy,
${\mathbf{H}}^{(1)}=-{[{\mathbf{H}}^{(1)}]}^{*}$. The crucial distinction
between the time reversal class AI$({\textup{T}}=+1)$ is that the transposing
nature of the charge conjugation give the negative sign in the just stated
symmetry condition. The space ${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{D}}$
is made of real antisymmetric matrices which is the Lie algebra
$\boldsymbol{\mathfrak{o}}(L)$. The time evolution operator spans $O(L)$.

Class C: The charge conjugation operation is described by
${{\mathbf{U}}_{\textup{C}}}=\mathbf{J}$, and the symmetry condition
$\mathbf{J}[{\mathbf{H}}^{(1)}]^{*}\mathbf{J}^{\dagger}=-{\mathbf{H}}^{(1)}$,
leads to matrices with $\operatorname{tr}{{\mathbf{H}}^{(1)}}=0$,

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}\mathbf{h}_{aa}&\mathbf{h}_{ab}\\\
\mathbf{h}_{ab}^{*}&-\mathbf{h}_{aa}^{*}\end{array}\right)},$ (96)

i. e.,
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{C}}}}=\boldsymbol{\mathfrak{usp}}(2M)$,
and ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$ spans $\text{\it USp}(2M)$.

Class AIII: The physics of this class is governed by the sublattice symmetry
represented by the unitary ${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}$ that
naturally partitions the orbitals into two groups (sublattices). It is natural
to write the Hamiltonian as

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}\mathbf{h}_{pp}&\mathbf{h}_{pq}\\\
\mathbf{h}^{\dagger}_{pq}&\mathbf{h}_{qq}\end{array}\right)}.$ (97)

The symmetry condition leads immediately to
$\mathbf{h}_{pp}={\boldsymbol{0}}_{pp}$,
$\mathbf{h}_{qq}={\boldsymbol{0}}_{qq}$. Thus,
$\mathcal{H}_{\textup{{\sf{AIII}}}}$ is made of matrices of the form

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}{\boldsymbol{0}}_{pp}&\mathbf{h}_{pq}\\\
\mathbf{h}^{\dagger}_{pq}&{\boldsymbol{0}}_{qq}\end{array}\right)}.$ (98)

Indeed, we have
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{AIII}}}}=\boldsymbol{\mathfrak{u}}(p+q)\setminus(\boldsymbol{\mathfrak{u}}(p)\oplus\boldsymbol{\mathfrak{u}}(q))$,
resulting in ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$ being in the coset space
$U(p+q)/(U(p)\times U(q))$.

All of the type 3 classes can be viewed as descendants of class ${\sf{AIII}}$,
and thus all of these classes – often referred to as chiral classes – have
Hamiltonians that imbibe the structure in eqn. (98).

Class BDI: The symmetry condition eqn. (92) with
${{\mathbf{U}}_{\textup{T}}}=\mathbf{1}$ gives
$\mathbf{h}_{pq}=\mathbf{h}_{pq}^{*}$, and thus elements of
$\mathcal{H}_{\textup{{\sf{BDI}}}}$ are of the form

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}{\boldsymbol{0}}_{pp}&\mathbf{h}_{pq}\\\
\mathbf{h}^{T}_{pq}&{\boldsymbol{0}}_{qq}\end{array}\right)},\,\,\,\mathbf{h}_{pq}=\mathbf{h}_{pq}^{*}$
(99)

We obtain
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{BDI}}}}=\boldsymbol{\mathfrak{o}}(p+q)\setminus(\boldsymbol{\mathfrak{o}}(p)\oplus\boldsymbol{\mathfrak{o}}(q))$,
with ${\mathbb{U}_{\textup{Schr\"{o}d}}}(t)$ spanning the the coset space
$O(p+q)/(O(p)\times O(q))$.

Class CII: The symmetry condition eqn. (92) with ${{\mathbf{U}}_{\textup{T}}}$
as shown in table 1 and ${\mathbf{H}}^{(1)}$ of the form eqn. (98) provides

$\mathbf{J}_{pp}\mathbf{h}^{*}_{pq}=\mathbf{h}_{pq}\mathbf{J}_{qq}.$ (100)

The key point in this class is that the $p$ orbitals themselves arise as $2r$
orbitals and $q$ as $2s$ orbitals. Thus, $\mathbf{h}_{pq}$ can be written as

$\mathbf{h}_{pq}={\left(\begin{array}[]{cc}\mathbf{h}_{rr}&\mathbf{h}_{rs}\\\
\mathbf{h}_{sr}&\mathbf{h}_{ss}\end{array}\right)}.$ (101)

resulting in $\mathbf{h}_{sr}=-\mathbf{h}_{rs}^{*}$ and
$\mathbf{h}_{ss}=\mathbf{h}_{rr}^{*}$

$\mathbf{h}_{pq}={\left(\begin{array}[]{cc}\mathbf{h}_{rr}&\mathbf{h}_{rs}\\\
-\mathbf{h}^{*}_{rs}&\mathbf{h}^{*}_{rr}\end{array}\right)}.$ (102)

We see that
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{CII}}}}=\boldsymbol{\mathfrak{usp}}(2(r+s))\setminus(\boldsymbol{\mathfrak{usp}}(2r))\oplus\boldsymbol{\mathfrak{usp}}(2s))$
and the symmetric space generated by the time evolution operator is $\text{\it
USp}(2(r+s))/(\text{\it USp}(2r)\times\text{\it USp}(2s))$.

Class CI: Here $L=2M$, and thus eqn. (98) provides

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}{\boldsymbol{0}}_{MM}&\mathbf{h}_{MM}\\\
\mathbf{h}^{\dagger}_{MM}&{\boldsymbol{0}}_{MM}\end{array}\right)}.$ (103)

The symmetry condition eqn. (92) with ${{\mathbf{U}}_{\textup{T}}}=\mathbf{F}$
gives $\mathbf{h}_{MM}^{T}=\mathbf{h}_{MM}$ with

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}\mathbf{0}_{MM}&\mathbf{h}_{MM}\\\
\mathbf{h}^{*}_{MM}&\mathbf{0}_{MM}\end{array}\right)}.$ (104)

It is now clear that the
${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{CI}}}}=\boldsymbol{\mathfrak{usp}}(2M)\setminus\boldsymbol{\mathfrak{usp}}(M)$,
and the symmetric space corresponding to the time evolution is the coset space
$\text{\it USp}(2M)/U(M)$.

Class DIII: In this $2M$ dimensional GFS with, the symmetry condition eqn.
(92) leads to $\mathbf{h}_{MM}^{T}=-\mathbf{h}_{MM}$, and

${\mathbf{H}}^{(1)}={\left(\begin{array}[]{cc}\mathbf{0}_{MM}&\mathbf{h}_{MM}\\\
-\mathbf{h}^{*}_{MM}&\mathbf{0}_{MM}\end{array}\right)}$ (105)

Here ${{\mathbbm{i}\mathcal{H}}}^{(1)}_{\textup{{\sf{DIII}}}}$ is isomorphic
to $\boldsymbol{\mathfrak{o}}(2M)\setminus\boldsymbol{\mathfrak{u}}(M)$. The
symmetric space spanned by the time evolution operator is $O(2M)/U(M)$.

The classes ${\sf{A}}$ and ${\sf{AIII}}$ are complex classes, while the
remainder of the classes involve a “reality condition” of the form
$\mathbf{H}=\mathbf{H}^{*}$ and are the real classes.

####  .8 Framework for Systems with Interactions

In this section we establish ideas that allow for the determination of the
structure of Hamiltonians with upto $N$-body interactions as in eqn. (8). For
usual symmetries, the conditions that determine the spaces
${{\mathbbm{i}\mathcal{H}}}^{(K)}$ are straightforward. From, eqn. (24) we
obtain

${\mathbf{H}}^{(K)}={{\breve{\mathbf{H}}}}^{(K)}\ .$ (106)

These are a set of homogeneous equations in the matrix elements
$H^{(K)}_{i_{1}\ldots\i_{K};j_{1}\ldots,j_{K}}$ that make
${{\mathbbm{i}\mathcal{H}}}^{(K)}$ of the appropriate class a vector subspace
of $\boldsymbol{\mathfrak{u}}\left(\small{{L}\choose{K}}\right)$ following the
discussion near eqn. (12). Eqn. (106) provides conditions to completely
determine the structure of the admissible Hamiltonians.

Transposing symmetries have a more involved story. To make progress, we
rewrite eqn. (28) using eqn. (25) to obtain

$\begin{split}{\mathbf{H}}^{(K)}&=\sum_{R=K}^{N}A_{R,K}\left[\operatorname{tr}_{R-K}{{\breve{\mathbf{H}}}}^{(R)}\right]^{T}\\\
&=(-1)^{K}\left[{{\breve{\mathbf{H}}}}^{(K)}\right]^{T}+(-1)^{K}\sum_{R>K}^{N}\frac{1}{(R-K)!}\left(\frac{R!}{K!}\right)^{2}\left[\operatorname{tr}_{R-K}{{\breve{\mathbf{H}}}}^{(R)}\right]^{T}\end{split}$
(107)

We now define $\mathbf{H}^{(K)}_{+}$ and $\mathbf{H}^{(K)}_{-}$ as

$\begin{split}\mathbf{H}^{(K)}_{+}&\equiv{\mathbf{H}}^{(K)}+(-1)^{K}\left[{{\breve{\mathbf{H}}}}^{(K)}\right]^{T}\\\
\mathbf{H}^{(K)}_{-}&\equiv{\mathbf{H}}^{(K)}-(-1)^{K}\left[{{\breve{\mathbf{H}}}}^{(K)}\right]^{T}\end{split}$
(108)

with
${\mathbf{H}}^{(K)}=\frac{1}{2}\left(\mathbf{H}^{(K)}_{+}+\mathbf{H}^{(K)}_{-}\right)$
and
$(-1)^{K}\left[{{\breve{\mathbf{H}}}}^{(K)}\right]^{T}=\frac{1}{2}\left(\mathbf{H}^{(K)}_{+}-\mathbf{H}^{(K)}_{-}\right)$.
With these definitions eqn. (25) becomes

$\mathbf{H}^{(K)}_{-}=\frac{(-1)^{K}}{2}\sum_{R>K}^{N}\frac{(-1)^{R}}{(R-K)!}\left(\frac{R!}{K!}\right)^{2}\left[\operatorname{tr}_{R-K}\mathbf{H}^{(R)}_{+}-\operatorname{tr}_{R-K}\mathbf{H}^{(R)}_{-}\right]\
.$ (109)

Two points emerge from these considerations: (i) the transposing symmetry
condition eqn. (107) puts no constraint on $\mathbf{H}^{(K)}_{+}$ for any $K$,
and (ii) for every $K$, $\mathbf{H}^{(K)}_{-}$ is solely determined by
$R$-body interactions terms where $R>K$. This implies that the set of
independent parameters of $\mathbf{H}$ is entirely determined by
$\mathbf{H}^{(K)}_{+}$ for $K$ from $0$ to $N$. Further note that every
$\mathbf{H}^{(K)}_{+}$ belongs to a vector subspace of
$\boldsymbol{\mathfrak{u}}\left(\small{{L}\choose{K}}\right)$ defined by
$\mathbf{H}^{(K)}_{-}={\boldsymbol{0}}$, which we call
${\mathbbm{i}\mathcal{H}}^{(K)}_{+}$. Since,
${{\mathbbm{i}\mathcal{H}}}^{(K)}$ is made of objects of the type
$\frac{1}{2}\left(\mathbf{H}^{(K)}_{+}+\mathbf{H}^{(K)}_{-}\right)$ where
$\mathbf{H}^{(K)}_{-}$ is a “constant” determined by eqn. (109),
${{\mathbbm{i}\mathcal{H}}}^{(K)}=\frac{\mathbbm{i}}{2}\left(\mathbf{H}^{(K)}_{+}+\mathbf{H}^{(K)}_{-}\right)$,
is no longer a vector subspace of
$\boldsymbol{\mathfrak{u}}\left(\small{{L}\choose{K}}\right)$. In fact,
${{\mathbbm{i}\mathcal{H}}}^{(K)}$ is an affine subspace of
$\boldsymbol{\mathfrak{u}}\left(\small{{L}\choose{K}}\right)$ whose dimension
(as a manifold) is same as the subspace ${\mathbbm{i}\mathcal{H}}^{(K)}_{+}$.

These observations show that $\mathcal{H}$ in any class can be completely
determined by starting from the $N$-body (highest multi-body interaction) term
in $\mathbf{H}$ which satisfies $\mathbf{H}^{(N)}_{-}={\boldsymbol{0}}$, and
recursively using eqn. (109) to determine $\mathbf{H}^{(K)}_{-}$ for $K<N$.
Again, for each $K$ the equation $\mathbf{H}^{(K)}_{-}={\boldsymbol{0}}$ gives
the subspace that defines ${\mathbbm{i}\mathcal{H}}^{(K)}_{+}$ which then
makes up the affine space ${{\mathbbm{i}\mathcal{H}}}^{(K)}$. The problem of
finding the structure of $\mathcal{H}$ is then reduced solely to finding the
subspace ${\mathbbm{i}\mathcal{H}}^{(K)}_{+}$ for each $K$. In the subsequent
sections we shall demonstrate the determination of this subspace
$\mathbf{H}^{(N)}_{-}={\boldsymbol{0}}$ for the highest multi-body ($N$-body)
interaction in our system. Clearly, the same results will apply mutatis
mutandis to all $K<N$. Finally, we note that ${\mathbf{H}}^{(R)}$ s satisfy
the identity

$\frac{1}{2}\sum_{R>0}^{N}(-1)^{R}R!\left[\operatorname{tr}_{R}\mathbf{H}^{(R)}_{+}-\operatorname{tr}_{R}\mathbf{H}^{(R)}_{-}\right]=0.$
(110)

In the next section we show how this is achieved for $N=2$, and generalize
this to higher $N$ in the subsequent sections. The main results of this
exercise are summarized in tables 3 and 4.

Class | $L$ | $P$ | $Q$ | $\mathbf{H}^{(N)}$ | $\dim{\mathbbm{i}\mathcal{H}^{(N)}}$ | $\mathbbm{i}\mathcal{H}^{N}_{+}$
---|---|---|---|---|---|---
$\begin{array}[]{c}${\sf{A}}$\\\ $(0,0,0)$\end{array}$ | $L$ | $\small{{L}\choose{N}}$ | $-$ | $\mathbf{H}^{(N)}=\left[{\mathbf{H}^{(N)}}\right]^{\dagger}$ | $P^{2}$ | $\boldsymbol{\mathfrak{u}}(P)$
$\begin{array}[]{c}${\sf{AI}}$\\\ $(+1,0,0)$\end{array}$ | $L$ | $\small{{L}\choose{N}}$ | $-$ | $\mathbf{H}^{(N)}=\left[{\mathbf{H}^{(N)}}\right]^{*}$ | $P(P+1)/2$ | $\boldsymbol{\mathfrak{u}}(P)\setminus\boldsymbol{\mathfrak{o}}(P)$
$\begin{array}[]{c}${\sf{AII}}$\\\ $(-1,0,0)$\end{array}$ | $L=2M$ | $\frac{1}{2}\left\\{\small{{L}\choose{N}}+\small{{M}\choose{N/2}}\right\\}$ | $\frac{1}{2}\left\\{\small{{L}\choose{N}}-\small{{M}\choose{N/2}}\right\\}$ |  ${\left(\begin{array}[]{cc}\mathbf{h}^{(N)}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ \left[{\mathbf{h}^{(N)}_{PQ}}\right]^{\dagger}&\mathbf{h}^{(N)}_{QQ}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{PP}=\left[{\mathbf{h}^{(N)}_{PP}}\right]^{*}\\\ \mathbf{h}^{(N)}_{QQ}=\left[{\mathbf{h}^{(N)}_{QQ}}\right]^{*}\\\ \mathbf{h}^{(N)}_{PQ}=-\left[{\mathbf{h}^{(N)}_{PQ}}\right]^{*}\end{array}$ | $\frac{P(P+1)}{2}+\frac{Q(Q+1)}{2}+PQ$ | $\begin{array}[]{c}\boldsymbol{\mathfrak{u}}(P+Q)\\\ \setminus\boldsymbol{\mathfrak{o}}(P+Q)\end{array}$
$\begin{array}[]{c}${\sf{D}}$\\\ $(0,+1,0)$\end{array}$ | $L$ | $\small{{L}\choose{N}}$ | $-$ | $\mathbf{H}^{(N)}=\left[{\mathbf{H}^{(N)}}\right]^{*}$ | $P(P+1)/2$ | $\boldsymbol{\mathfrak{u}}(P)\setminus\boldsymbol{\mathfrak{o}}(P)$
$\begin{array}[]{c}${\sf{C}}$\\\ $(0,-1,0)$\end{array}$ | $L=2M$ | $\frac{1}{2}\left\\{\small{{L}\choose{N}}+\small{{M}\choose{N/2}}\right\\}$ | $\frac{1}{2}\left\\{\small{{L}\choose{N}}-\small{{M}\choose{N/2}}\right\\}$ |  ${\left(\begin{array}[]{cc}\mathbf{h}^{(N)}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ \left[{\mathbf{h}^{(N)}_{PQ}}\right]^{\dagger}&\mathbf{h}^{(N)}_{QQ}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{PP}=\left[{\mathbf{h}^{(N)}_{PP}}\right]^{*}\\\ \mathbf{h}^{(N)}_{QQ}=\left[{\mathbf{h}^{(N)}_{QQ}}\right]^{*}\\\ \mathbf{h}^{(N)}_{PQ}=-\left[{\mathbf{h}^{(N)}_{PQ}}\right]^{*}\end{array}$ | $\frac{P(P+1)}{2}+\frac{Q(Q+1)}{2}+PQ$ | $\begin{array}[]{c}\boldsymbol{\mathfrak{u}}(P+Q)\\\ \setminus\boldsymbol{\mathfrak{o}}(P+Q)\end{array}$
$\begin{array}[]{c}${\sf{AIII}}$\\\ $(0,0,1)$\end{array}$ | $L=p+q$ | $\sum\limits_{a=1,3,\dots}^{N-1}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ | $\sum\limits_{a=0,2,\dots}^{N}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ |  ${\left(\begin{array}[]{cc}\mathbf{h}^{(N)}_{PP}&{\boldsymbol{0}}_{PQ}\\\ {\boldsymbol{0}}_{QP}&\mathbf{h}^{(N)}_{QQ}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{PP}=\left[{\mathbf{h}^{(N)}_{PP}}\right]^{\dagger}\\\ \mathbf{h}^{(N)}_{QQ}=\left[{\mathbf{h}^{(N)}_{QQ}}\right]^{\dagger}\end{array}$ | $P^{2}+Q^{2}$ | $\boldsymbol{\mathfrak{u}}(P)\oplus\boldsymbol{\mathfrak{u}}(Q)$
$\begin{array}[]{c}${\sf{BDI}}$\\\ $(+1,+1,1)$\end{array}$ | $L=p+q$ | $\sum\limits_{a=1,3,\dots}^{N-1}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ | $\sum\limits_{a=0,2,\dots}^{N}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ |  ${\left(\begin{array}[]{cc}\mathbf{h}^{(N)}_{PP}&{\boldsymbol{0}}_{PQ}\\\ {\boldsymbol{0}}_{QP}&\mathbf{h}^{(N)}_{QQ}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{PP}=\left[{\mathbf{h}^{(N)}_{PP}}\right]^{*}\\\ \mathbf{h}^{(N)}_{QQ}=\left[{\mathbf{h}^{(N)}_{QQ}}\right]^{*}\end{array}$ | $\frac{P(P+1)}{2}+\frac{Q(Q+1)}{2}$ | $\begin{array}[]{c}(\boldsymbol{\mathfrak{u}}(P)\setminus\boldsymbol{\mathfrak{o}}(P))\\\ \oplus\\\ (\boldsymbol{\mathfrak{u}}(Q)\setminus\boldsymbol{\mathfrak{o}}(Q))\end{array}$
$\begin{array}[]{c}${\sf{CII}}$\\\ $(-1,-1,1)$\end{array}$ | $\begin{array}[]{c}L=p+q\\\ p=2r\;\ q=2s\end{array}$ | $\begin{array}[]{c}P=\sum\limits_{a=1,3,\dots}^{N-1}\small{{p}\choose{a}}\small{{q}\choose{N-a}}\\\ A(B)=P/2\end{array}$ | $\begin{array}[]{c}Q=\sum\limits_{a=0,2,\dots}^{N}\small{{p}\choose{a}}\small{{q}\choose{N-a}}\\\ C(D)=\frac{Q}{2}\pm\sum\limits_{a=0,2,\dots}^{m}\frac{1}{2}\small{{r}\choose{a/2}}\small{{s}\choose{(m-a)/2}}\end{array}$ |  ${\left(\begin{array}[]{c|c}{\begin{array}[]{cc}\mathbf{h}^{(N)}_{AA}&\mathbf{h}^{(N)}_{AB}\\\ \left[{\mathbf{h}^{(N)}_{AB}}\right]^{\dagger}&\mathbf{h}^{(N)}_{BB}\end{array}}&{\boldsymbol{0}}_{PQ}\\\ \hline\cr{\boldsymbol{0}}_{QP}&{\begin{array}[]{cc}\mathbf{h}^{(N)}_{CC}&\mathbf{h}^{(N)}_{CD}\\\ \left[{\mathbf{h}^{(N)}_{CD}}\right]^{\dagger}&\mathbf{h}^{(N)}_{DD}\end{array}}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{AA(CC)}=\left[{\mathbf{h}^{(N)}_{AA(CC)}}\right]^{*}\\\ \mathbf{h}^{(N)}_{BB(DD)}=\left[{\mathbf{h}^{(N)}_{BB(DD)}}\right]^{*}\\\ \mathbf{h}^{(N)}_{AB(CD)}=-\left[{\mathbf{h}^{(N)}_{AB(CD)}}\right]^{*}\end{array}$ | $\begin{array}[]{c}\frac{A(A+1)}{2}+\frac{B(B+1)}{2}+AB\\\ +\frac{C(C+1)}{2}+\frac{D(D+1)}{2}+CD\end{array}$ | $\begin{array}[]{c}(\boldsymbol{\mathfrak{u}}(A+B)\\\ \setminus\boldsymbol{\mathfrak{o}}(A+B))\\\ \oplus\\\ (\boldsymbol{\mathfrak{u}}(C+D)\\\ \setminus\boldsymbol{\mathfrak{o}}(C+D))\end{array}$
$\begin{array}[]{c}${\sf{CI}}$\\\ $(+1,-1,1)$\end{array}$ | $L=2M$ | $\begin{array}[]{c}P=\sum\limits_{a=1,3,\dots}^{N-1}\small{{M}\choose{a}}\small{{M}\choose{N-a}}\\\ A(B)=\begin{cases}P/2&;N/2\ \text{even}\\\ \frac{P}{2}\mp\small{{M}\choose{N/2}}&;N/2\ \text{odd}\end{cases}\end{array}$ | $\begin{array}[]{c}Q=\sum\limits_{a=0,2,\dots}^{N}\small{{M}\choose{a}}\small{{M}\choose{N-a}}\\\ C(D)=\begin{cases}\frac{Q}{2}\pm\small{{M}\choose{N/2}}&;N/2\ \text{even}\\\ Q/2&;N/2\ \text{odd}\end{cases}\end{array}$ |  ${\left(\begin{array}[]{c|c}{\begin{array}[]{cc}\mathbf{h}^{(N)}_{AA}&\mathbf{h}^{(N)}_{AB}\\\ \left[{\mathbf{h}^{(N)}_{AB}}\right]^{\dagger}&\mathbf{h}^{(N)}_{BB}\end{array}}&{\boldsymbol{0}}_{PQ}\\\ \hline\cr{\boldsymbol{0}}_{QP}&{\begin{array}[]{cc}\mathbf{h}^{(N)}_{CC}&\mathbf{h}^{(N)}_{CD}\\\ \left[{\mathbf{h}^{(N)}_{CD}}\right]^{\dagger}&\mathbf{h}^{(N)}_{DD}\end{array}}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{AA(CC)}=\left[{\mathbf{h}^{(N)}_{AA(CC)}}\right]^{*}\\\ \mathbf{h}^{(N)}_{BB(DD)}=\left[{\mathbf{h}^{(N)}_{BB(DD)}}\right]^{*}\\\ \mathbf{h}^{(N)}_{AB(CD)}=-\left[{\mathbf{h}^{(N)}_{AB(CD)}}\right]^{*}\end{array}$ | $\begin{array}[]{c}\frac{A(A+1)}{2}+\frac{B(B+1)}{2}+AB\\\ +\frac{C(C+1)}{2}+\frac{D(D+1)}{2}+CD\end{array}$ | -do-
$\begin{array}[]{c}${\sf{DIII}}$\\\ $(-1,+1,1)$\end{array}$ | $L=2M$ | $\begin{array}[]{c}P=\sum\limits_{a=1,3,\dots}^{N-1}\small{{M}\choose{a}}\small{{M}\choose{N-a}}\\\ A(B)=\begin{cases}P/2&;N/2\ \text{even}\\\ \frac{P}{2}\pm\small{{M}\choose{N/2}}&;N/2\ \text{odd}\end{cases}\end{array}$ | $\begin{array}[]{c}Q=\sum\limits_{a=0,2,\dots}^{N}\small{{M}\choose{a}}\small{{M}\choose{N-a}}\\\ C(D)=\begin{cases}\frac{Q}{2}\pm\small{{M}\choose{N/2}}&;N/2\ \text{even}\\\ Q/2&;N/2\ \text{odd}\end{cases}\end{array}$ |  ${\left(\begin{array}[]{c|c}{\begin{array}[]{cc}\mathbf{h}^{(N)}_{AA}&\mathbf{h}^{(N)}_{AB}\\\ \left[{\mathbf{h}^{(N)}_{AB}}\right]^{\dagger}&\mathbf{h}^{(N)}_{BB}\end{array}}&{\boldsymbol{0}}_{PQ}\\\ \hline\cr{\boldsymbol{0}}_{QP}&{\begin{array}[]{cc}\mathbf{h}^{(N)}_{CC}&\mathbf{h}^{(N)}_{CD}\\\ \left[{\mathbf{h}^{(N)}_{CD}}\right]^{\dagger}&\mathbf{h}^{(N)}_{DD}\end{array}}\end{array}\right)}$ $\begin{array}[]{l}\mathbf{h}^{(N)}_{AA(CC)}=\left[{\mathbf{h}^{(N)}_{AA(CC)}}\right]^{*}\\\ \mathbf{h}^{(N)}_{BB(DD)}=\left[{\mathbf{h}^{(N)}_{BB(DD)}}\right]^{*}\\\ \mathbf{h}^{(N)}_{AB(CD)}=-\left[{\mathbf{h}^{(N)}_{AB(CD)}}\right]^{*}\end{array}$ | $\begin{array}[]{c}\frac{A(A+1)}{2}+\frac{B(B+1)}{2}+AB\\\ +\frac{C(C+1)}{2}+\frac{D(D+1)}{2}+CD\end{array}$ | -do-
Table 3: Structure of $N$-body interaction hamiltonian $\mathbf{H}^{(N)}$ ($N$ even) in each symmetry class. The space of $K$-body ($K$ even) interaction Hamiltonians ${{\mathbbm{i}\mathcal{H}}}^{(K)}$ in a class is an affine subspace ${{\mathbbm{i}\mathcal{H}}}^{(K)}={{\mathbbm{i}\mathcal{H}}}^{(K)}_{+}+\mathbf{H}^{(K)}_{-}$ where ${{\mathbbm{i}\mathcal{H}}}^{(K)}_{+}$ is to be read from the last column of this table and $\mathbf{H}^{(K)}_{-}$ is given in eqn. (109). Class | $L$ | $P$ | $Q$ | $\mathbf{H}^{(N)}$ | $\dim{\mathbbm{i}\mathcal{H}^{(N)}}$ | $\mathbbm{i}\mathcal{H}^{(N)}_{+}$
---|---|---|---|---|---|---
$\begin{array}[]{c}${\sf{A}}$\\\ $(0,0,0)$\end{array}$ | $L$ | $\small{{L}\choose{N}}$ | $-$ | $\mathbf{H}^{(N)}=\left[{\mathbf{H}^{(N)}}\right]^{\dagger}$ | $P^{2}$ | $\boldsymbol{\mathfrak{u}}(P)$
$\begin{array}[]{c}${\sf{AI}}$\\\ $(+1,0,0)$\end{array}$ | $L$ | $\small{{L}\choose{N}}$ | $-$ | $\mathbf{H}^{(N)}={[\mathbf{H}^{(N)}]}^{*}$ | $P(P+1)/2$ | $\boldsymbol{\mathfrak{u}}(P)\setminus\boldsymbol{\mathfrak{o}}(P)$
$\begin{array}[]{c}${\sf{AII}}$\\\ $(-1,0,0)$\end{array}$ | $L=2M$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | ${\left(\begin{array}[]{cc}\mathbf{h}^{(N)}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ -\left[{\mathbf{h}^{(N)}_{PQ}}\right]^{*}&\left[{\mathbf{h}^{(N)}_{PP}}\right]^{*}\end{array}\right)}$ | $P^{2}+2\times\frac{P(P-1)}{2}$ | $\boldsymbol{\mathfrak{u}}(2P)\setminus\boldsymbol{\mathfrak{usp}}(2P)$
$\begin{array}[]{c}${\sf{D}}$\\\ $(0,+1,0)$\end{array}$ | $L$ | $\small{{L}\choose{N}}$ | $-$ | $\mathbf{H}^{(N)}=-\left[{\mathbf{H}^{(N)}}\right]^{*}$ | $P(P-1)/2$ | $\boldsymbol{\mathfrak{o}}(P)$
$\begin{array}[]{c}${\sf{C}}$\\\ $(0,-1,0)$\end{array}$ | $L=2M$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | ${\left(\begin{array}[]{cc}\mathbf{h}^{(N)}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ \left[{\mathbf{h}^{(N)}_{PQ}}\right]^{*}&-\left[{\mathbf{h}^{(N)}_{PP}}\right]^{*}\end{array}\right)}$ | $P^{2}+2\times\frac{P(P+1)}{2}$ | $\boldsymbol{\mathfrak{usp}}(2P)$
$\begin{array}[]{c}${\sf{AIII}}$\\\ $(0,0,1)$\end{array}$ | $L=p+q$ | $\sum\limits_{a=1,3,\dots}^{N}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ | $\sum\limits_{a=0,2,\dots}^{N-1}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ | ${\left(\begin{array}[]{cc}{\boldsymbol{0}}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ \left[{\mathbf{h}^{(N)}_{PQ}}\right]^{\dagger}&{\boldsymbol{0}}_{QQ}\end{array}\right)}$ | $2PQ$ | $\boldsymbol{\mathfrak{u}}(P+Q)\setminus(\boldsymbol{\mathfrak{u}}(P)\oplus\boldsymbol{\mathfrak{u}}(Q))$
$\begin{array}[]{c}${\sf{BDI}}$\\\ $(+1,+1,1)$\end{array}$ | $L=p+q$ | $\sum\limits_{a=1,3,\dots}^{N}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ | $\sum\limits_{a=0,2,\dots}^{N-1}\small{{p}\choose{a}}\small{{q}\choose{N-a}}$ | ${\left(\begin{array}[]{cc}{\boldsymbol{0}}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ \left[{\mathbf{h}^{(N)}_{PQ}}\right]^{T}&{\boldsymbol{0}}_{QQ}\end{array}\right)}$ | $PQ$ | $\boldsymbol{\mathfrak{o}}(P+Q)\setminus(\boldsymbol{\mathfrak{o}}(P)\oplus\boldsymbol{\mathfrak{o}}(Q))$
$\begin{array}[]{c}${\sf{CII}}$\\\ $(-1,-1,1)$\end{array}$ | $\begin{array}[]{c}L=p+q\\\ p=2r\;\ q=2s\end{array}$ | $\begin{array}[]{c}P=\sum\limits_{a=1,3,\dots}^{N}\small{{p}\choose{a}}\small{{q}\choose{N-a}}\\\ A(B)=P/2\end{array}$ | $\begin{array}[]{c}Q=\sum\limits_{a=0,2,\dots}^{N-1}\small{{p}\choose{a}}\small{{q}\choose{N-a}}\\\ C(D)=\frac{Q}{2}\end{array}$ | ${\left(\begin{array}[]{c|c}{\boldsymbol{0}}_{PP}&{\begin{array}[]{cc}\mathbf{h}^{(N)}_{AC}&\mathbf{h}^{(N)}_{AD}\\\ -\left[{\mathbf{h}^{(N)}_{AD}}\right]^{*}&\left[{\mathbf{h}^{(N)}_{AC}}\right]^{*}\end{array}}\\\ \hline\cr\text{h.c.}&{\boldsymbol{0}}_{QQ}\end{array}\right)}$ | $PQ$ | $\boldsymbol{\mathfrak{usp}}(P+Q)\setminus(\boldsymbol{\mathfrak{usp}}(P)\oplus\boldsymbol{\mathfrak{usp}}(Q))$
$\begin{array}[]{c}${\sf{CI}}$\\\ $(+1,-1,1)$\end{array}$ | $L=2M$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | ${\left(\begin{array}[]{cc}{\boldsymbol{0}}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ \left[{\mathbf{h}^{(N)}_{PQ}}\right]^{*}&{\boldsymbol{0}}_{QQ}\end{array}\right)}$ | $P(P+1)$ | $\boldsymbol{\mathfrak{usp}}(2P)\setminus\boldsymbol{\mathfrak{u}}(P)$
$\begin{array}[]{c}${\sf{DIII}}$\\\ $(-1,+1,1)$\end{array}$ | $L=2M$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | $\frac{1}{2}\small{{2M}\choose{N}}$ | ${\left(\begin{array}[]{cc}{\boldsymbol{0}}_{PP}&\mathbf{h}^{(N)}_{PQ}\\\ -\left[{\mathbf{h}^{(N)}_{PQ}}\right]^{*}&{\boldsymbol{0}}_{QQ}\end{array}\right)}$ | $P(P-1)$ | $\boldsymbol{\mathfrak{o}}(2P)\setminus\boldsymbol{\mathfrak{u}}(P)$
Table 4: The ten symmetry classes of fermions and the structure of many-body
Hamiltonians $\mathbf{H}^{(N)}$ when $N$ is odd. The space of $K$-body ($K$
odd) interaction Hamiltonians ${{\mathbbm{i}\mathcal{H}}}^{(K)}$ in a class is
an affine subspace
${{\mathbbm{i}\mathcal{H}}}^{(K)}={{\mathbbm{i}\mathcal{H}}}^{(K)}_{+}+\mathbf{H}^{(K)}_{-}$
where ${{\mathbbm{i}\mathcal{H}}}^{(K)}_{+}$ is to be read from the last
column of this table and $\mathbf{H}^{(K)}_{-}$ is given in eqn. (109).

####  .9 Structure of Two-body Hamiltonians

In this section we find the structure of two-body Hamiltonians in each class.
This will serve as a precursor to our study of a generic even $N$-body
Hamiltonian. We define a _strictly_ two-body Hamiltonian as

${\mathscr{H}}={{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}},$
(111)

where ${{\Psi}}{{\Psi}}$ is a notational short-hand for a column vector
comprised of $\small{{L}\choose{2}}$ distinct product terms of two fermionic
annihilation operators, written as

${{\Psi}}{{\Psi}}\equiv\left(\begin{array}[]{c}{}_{L-1}\begin{cases}\psi_{1}\psi_{2}\\\
\vdots\\\ \psi_{1}\psi_{L}\end{cases}\\\
{}_{L-2}\begin{cases}\psi_{2}\psi_{3}\\\ \vdots\\\ \psi_{3}\psi_{L}\\\
\end{cases}\\\ \vdots\\\ {}_{1}\begin{cases}\psi_{L-1}\psi_{L}\\\
\end{cases}\end{array}\right).$ (112)

Henceforth, we will call these product terms as states. The definition of
${{\Psi}}{{\Psi}}$ also defines ${{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\equiv$

$\left(\begin{array}[]{cccc}(\psi_{1}\psi_{2})^{\dagger}\ldots(\psi_{1}\psi_{L})^{\dagger}&(\psi_{2}\psi_{3})^{\dagger}\ldots(\psi_{2}\psi_{L})^{\dagger}&\ldots&(\psi_{L-1}\psi_{L})^{\dagger}\end{array}\right).$
(113)

The preceding definitions are in the same spirit as our definition for the
states in eqn. (4).

We can also write $\mathbf{H}^{(2)}$ as (see eqn. (LABEL:eq:HKthorder))

$({{\Psi}}^{\dagger})^{2}{\mathbf{H}}^{(2)}({{\Psi}})^{2}=\sum_{i_{1},i_{2},j_{1},j_{2}}(\psi_{i_{1}}\psi_{i_{2}})^{\dagger}H^{(2)}_{i_{1},i_{2};j_{1},j_{2}}\psi_{j_{1}}\psi_{j_{2}}$
(114)

Note here that although $\mathbf{H}^{(2)}$ is a four indexed object, fermionic
anticommutation demands that it is a matrix of dimension
$\small{{L}\choose{2}}\times\small{{L}\choose{2}}$. As done in the case of
one-body Hamiltonians we now construct the
$\small{{L}\choose{2}}\times\small{{L}\choose{2}}$ canonical representations
of the symmetry
operations(${{{\mathbf{U}}}_{\textup{T}}^{(2)}},{{{\mathbf{U}}}_{\textup{C}}^{(2)}},{{{\mathbf{U}}}_{\textup{S}}^{(2)}}$)
from their one-body counterparts(see table 1) and apply them to determine the
structure of ${\mathbf{H}}^{(2)}$.

Class A: This class has no symmetries, therefore any hermitian matrix of
dimension $\small{{L}\choose{2}}\times\small{{L}\choose{2}}$ belongs to this
class.

Class AI:For the ${\sf{AI}}$ class we have
${{\mathbf{U}}_{\textup{T}}}=\mathbf{1}$. Hence for this case,

$\displaystyle{\mathscr{T}}{{\Psi}}^{\dagger}{\mathscr{T}}^{-1}$
$\displaystyle=$
$\displaystyle{{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{T}}}={{\Psi}}^{\dagger}$
$\displaystyle{\mathscr{T}}{{\Psi}}{\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle{{\mathbf{U}}_{\textup{T}}^{\dagger}}{{\Psi}}={{\Psi}}$ (115)

and for the two-body states (see eqn. (112) and eqn. (113)) we have

$\displaystyle{\mathscr{T}}{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}$
$\displaystyle{\mathscr{T}}{{\Psi}}{{\Psi}}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle{{\Psi}}{{\Psi}},$ (116)

making ${{{\mathbf{U}}}_{\textup{T}}^{(2)}}=\mathbf{1}$. Time reversal
symmetry implies the following

$\mathbf{H}^{(2)}={\left[\mathbf{H}^{(2)}\right]}^{*}$ (117)

which puts $\mathbf{H}^{(2)}$ in the class of real symmetric
$\small{{L}\choose{2}}\times\small{{L}\choose{2}}$ matrices. The dimension of
this class is
$\frac{1}{2}\small{{L}\choose{2}}\left\\{\small{{L}\choose{2}}+1\right\\}$.

Class AII: For class ${\sf{AII}}$ we have canonical
${{\mathbf{U}}_{\textup{T}}}=\mathbf{J}$ and

$\displaystyle{\mathscr{T}}{{\Psi}}^{\dagger}{\mathscr{T}}^{-1}$
$\displaystyle=$
$\displaystyle{{\Psi}}^{\dagger}{{\mathbf{U}}_{\textup{T}}}={{\Psi}}^{\dagger}\mathbf{J}$
$\displaystyle{\mathscr{T}}{{\Psi}}{\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle{{\mathbf{U}}_{\textup{T}}^{\dagger}}{{\Psi}}=-\mathbf{J}{{\Psi}}.$
(118)

Also we had seen that for this case to be realizable $L=2M$. We can divide the
$2M$ states as $M$ states of flavor $\alpha$ and $M$ of flavor $\beta$.
Explicitly these transform in the following way

$\displaystyle{\mathscr{T}}\left(\psi^{\dagger}_{1\alpha}\ldots\psi^{\dagger}_{M\alpha}~{}~{}\psi^{\dagger}_{1\beta}\ldots\psi^{\dagger}_{M\beta}\right){\mathscr{T}}^{-1}$
$\displaystyle=$
$\displaystyle\left(-\psi^{\dagger}_{1\beta}\ldots-\psi^{\dagger}_{M\beta}~{}~{}\psi^{\dagger}_{1\alpha}\ldots\psi^{\dagger}_{M\alpha}\right)$
$\displaystyle{\mathscr{T}}\left(\begin{array}[]{c}\psi_{1\alpha}\\\ \vdots\\\
\psi_{M\alpha}\\\ \psi_{1\beta}\\\ \vdots\\\
\psi_{M\beta}\end{array}\right){\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\psi_{1\beta}\\\ \vdots\\\
-\psi_{M\beta}\\\ \psi_{1\alpha}\\\ \vdots\\\
\psi_{M\alpha}\end{array}\right)$ (131)

implying,

$\displaystyle{\mathscr{T}}\psi^{\dagger}_{i\alpha}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle-\psi^{\dagger}_{i\beta}$
$\displaystyle{\mathscr{T}}\psi^{\dagger}_{i\beta}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle\psi^{\dagger}_{i\alpha}$
$\displaystyle{\mathscr{T}}\psi_{i\alpha}{\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle-\psi_{i\beta}$
$\displaystyle{\mathscr{T}}\psi_{i\beta}{\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle\psi_{i\alpha}.$ (132)

Since ${{\Psi}}{{\Psi}}$ is formed by the product of these one-body operators,
its elements can be divided into the following three kinds

${{\Psi}}{{\Psi}}=\left(\begin{array}[]{c}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}\end{cases}\\\
{}_{M^{2}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\beta}\psi_{j\beta}\end{cases}\end{array}\right).$
(133)

The number of states in each kind add upto $\small{{L}\choose{2}}$, i.e.,
$2\small{{M}\choose{2}}+M^{2}=M(M-1)+M^{2}=M(2M-1)=\frac{L(L-1)}{2}=\small{{L}\choose{2}}.$
We find the effect of the ${\mathscr{T}}$ operators on these states to be

${\mathscr{T}}{{\Psi}}{{\Psi}}{\mathscr{T}}^{-1}={\mathscr{T}}\left(\begin{array}[]{c}\psi_{i\alpha}\psi_{j\alpha}\\\
\psi_{i\alpha}\psi_{j\beta}\\\ \psi_{i\beta}\psi_{j\beta}\\\
\end{array}\right){\mathscr{T}}^{-1}=\left(\begin{array}[]{c}\psi_{i\beta}\psi_{j\beta}\\\
-\psi_{i\beta}\psi_{j\alpha}\\\ \psi_{i\alpha}\psi_{j\alpha}\\\
\end{array}\right)=\left(\begin{array}[]{c}\psi_{i\beta}\psi_{j\beta}\\\
\psi_{j\alpha}\psi_{i\beta}\\\ \psi_{i\alpha}\psi_{j\alpha}\\\
\end{array}\right).$ (134)

This suggests that we can re-organize the two-body basis into
“symmetric”($\sigma$) and “antisymmetric”($\pi$) states as follows555For the
clarity of the equations, we will not explicitly put the factors of
$\frac{1}{\sqrt{2}}$ in front of symmetric and antisymmetric states. It should
be assumed that all the states are properly normalized.

$\left(\begin{array}[]{c}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}\end{cases}\\\
{}_{M^{2}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\beta}\psi_{j\beta}\end{cases}\end{array}\right)\longrightarrow\left(\begin{array}[]{c}{}_{\sigma}\begin{cases}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}+\psi_{i\beta}\psi_{j\beta}\end{cases}\\\
{}_{M}\begin{cases}\psi_{i\alpha}\psi_{i\beta}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}+\psi_{j\alpha}\psi_{i\beta}\>\>{}_{(i\neq
j)}\end{cases}\end{cases}\\\
{}_{\pi}\begin{cases}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}-\psi_{j\alpha}\psi_{i\beta}\>\>{}_{(i\neq
j)}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}-\psi_{i\beta}\psi_{j\beta}\end{cases}\end{cases}\end{array}\right).$
(135)

Conveniently, in this way of organizing our Hilbert space, we have

$\displaystyle{\mathscr{T}}\sigma{\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle\sigma$ $\displaystyle{\mathscr{T}}\pi{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle-\pi.$ (136)

Also note that $M^{2}=2\small{{M}\choose{2}}+M$.

It will be useful to mention here that the transformation properties of these
states under symmetry gives us the definitions of “symmetric” and
“antisymmetric” states. This terminology will be used in later sections when
the $N$-body Hamiltonians will be discussed. Note that the word “symmetric”
may not necessarily imply a “+” sign in the linear combination of its
constituent states and vice-versa.

Armed with this, we look for the structure of $\mathbf{H}^{(2)}$

$\displaystyle{\mathscr{T}}{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}{\mathscr{T}}^{-1}$
$\displaystyle=$
$\displaystyle{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}$
$\displaystyle{{{\mathbf{U}}}_{\textup{T}}^{(2)}}\left[{\mathbf{H}^{(2)}}\right]^{*}{{{\mathbf{U}}}_{\textup{T}}^{{(2)}\dagger}}$
$\displaystyle=$ $\displaystyle\mathbf{H}^{(2)},$ (137)

where ${{{\mathbf{U}}}_{\textup{T}}^{(2)}}$ in this new basis is obtained from
eqn. (136) and is given by

$\left[{{{{\mathbf{U}}}_{\textup{T}}^{(2)}}}\right]^{\dagger}=\mathbf{1}_{P,Q},$
(138)

where $P=2\small{{M}\choose{2}}+M$ and $Q=2\small{{M}\choose{2}}$. Assuming,

$\mathbf{H}^{(2)}=\left(\begin{array}[]{cc}\mathbf{h}^{(2)}_{PP}&\mathbf{h}^{(2)}_{PQ}\\\
\left[{\mathbf{h}^{(2)}_{PQ}}\right]^{\dagger}&\mathbf{h}^{(2)}_{QQ}\end{array}\right)$
(139)

and implementing the condition in eqn. (137), we get the following
constraints,

$\displaystyle\mathbf{h}^{(2)}_{PP}=\left[{\mathbf{h}^{(2)}_{PP}}\right]^{*},\quad\mathbf{h}^{(2)}_{QQ}=\left[{\mathbf{h}^{(2)}_{QQ}}\right]^{*},\quad\mathbf{h}^{(2)}_{PQ}=-\left[{\mathbf{h}^{(2)}_{PQ}}\right]^{*}.$
(140)

This leads to total $\frac{P(P+1)}{2}+\frac{Q(Q+1)}{2}+PQ$ independent
parameters in $\mathbf{H}^{(2)}$, which is equal to
$\dim{\mathbbm{i}\mathcal{H}^{(2)}_{{\sf{AII}}}}$.

Class D: We now look at transposing symmetries of the TL kind. The states
transform according to eqn. (19) and since class D has canonical
${{\mathbf{U}}_{\textup{C}}}=\mathbf{1}$, we have

$\displaystyle{\mathscr{U}}\Psi^{\dagger}{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle{{\Psi}}^{T}$ $\displaystyle{\mathscr{U}}\Psi{\mathscr{U}}^{-1}$
$\displaystyle=$ $\displaystyle({{\Psi}}^{\dagger})^{T}$ (141)

which implies

$\displaystyle{\mathscr{U}}\psi^{\dagger}_{i}{\mathscr{U}}^{-1}$
$\displaystyle=$ $\displaystyle\psi_{i}$
$\displaystyle{\mathscr{U}}\psi_{i}{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle\psi^{\dagger}_{i}.$ (142)

Thus even for the two-body Hamiltonian the symmetry condition(eqn. (21)) is

$\mathbf{H}^{(2)}=\left[{\mathbf{H}^{(2)}}\right]^{*},$ (143)

making $\mathbf{H}^{(2)}$ a real symmetric matrix with
$\frac{1}{2}\small{{L}\choose{2}}\left\\{\small{{L}\choose{2}}+1\right\\}$
parameters.

Class C: Class C comes with canonical ${{\mathbf{U}}_{\textup{C}}}=\mathbf{J}$
and requires $L=2M$. The states transform as

$\displaystyle{\mathscr{U}}\Psi^{\dagger}{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle{{\Psi}}^{T}\mathbf{J}$
$\displaystyle{\mathscr{U}}\Psi{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle-\mathbf{J}({{\Psi}}^{\dagger})^{T}.$ (144)

Similar to class AII we label the states with $\alpha,\beta$ flavors and find
there transformations to be

$\displaystyle{\mathscr{C}}\psi^{\dagger}_{i\alpha}{\mathscr{C}}^{-1}$
$\displaystyle=$ $\displaystyle-\psi_{i\beta}$
$\displaystyle{\mathscr{C}}\psi^{\dagger}_{i\beta}{\mathscr{C}}^{-1}$
$\displaystyle=$ $\displaystyle\psi_{i\alpha}$
$\displaystyle{\mathscr{C}}\psi_{i\alpha}{\mathscr{C}}^{-1}$ $\displaystyle=$
$\displaystyle-\psi^{\dagger}_{i\beta}$
$\displaystyle{\mathscr{C}}\psi_{i\beta}{\mathscr{C}}^{-1}$ $\displaystyle=$
$\displaystyle\psi^{\dagger}_{i\alpha}.$ (145)

However, unlike class ${\sf{AII}}$, one must remember that here we are dealing
with a transposing symmetry. The two-body basis in this case can be rearranged
in the following manner

$\left(\begin{array}[]{c}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}\end{cases}\\\
{}_{M^{2}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\beta}\psi_{j\beta}\end{cases}\end{array}\right)\longrightarrow\left(\begin{array}[]{c}{}_{\pi}\begin{cases}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}+\psi_{i\beta}\psi_{j\beta}\end{cases}\\\
{}_{M}\begin{cases}\psi_{i\alpha}\psi_{i\beta}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}+\psi_{j\alpha}\psi_{i\beta}\>\>{}_{(i\neq
j)}\end{cases}\end{cases}\\\
{}_{\sigma}\begin{cases}{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\beta}-\psi_{j\alpha}\psi_{i\beta}\>\>{}_{(i\neq
j)}\end{cases}\\\
{}_{\small{{M}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}-\psi_{i\beta}\psi_{j\beta}\end{cases}\end{cases}\end{array}\right).$
(146)

The action of ${\mathscr{C}}$ symmetry on these states are

$\displaystyle{\mathscr{C}}\sigma{\mathscr{C}}^{-1}$ $\displaystyle=$
$\displaystyle\sigma^{\dagger}$
$\displaystyle{\mathscr{C}}\pi{\mathscr{C}}^{-1}$ $\displaystyle=$
$\displaystyle-\pi^{\dagger}.$ (147)

From this we get $2\small{{M}\choose{2}}$ number of $\sigma$ states and
$M^{2}$ number of $\pi$ states. Imposing ${\mathscr{C}}$ symmetry on the
Hamiltonian gives

$\displaystyle:{\mathscr{C}}{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}{\mathscr{C}}^{-1}:$
$\displaystyle=$
$\displaystyle{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}$
$\displaystyle{{{\mathbf{U}}}_{\textup{C}}^{{(2)}\dagger}}\left[{\mathbf{H}^{(2)}}\right]^{*}{{{\mathbf{U}}}_{\textup{C}}^{(2)}}$
$\displaystyle=$ $\displaystyle\mathbf{H}^{(2)}$ (148)

where

${{{\mathbf{U}}}_{\textup{C}}^{(2)}}=-\mathbf{1}_{P,Q}$ (149)

and $P=2\small{{M}\choose{2}}+M$ and $Q=2\small{{M}\choose{2}}$. Denoting the
internal structure of $\mathbf{H}^{(2)}$ with eqn. (139), the constraints turn
out to be same as eqn. (140). This again leads to
$\frac{P(P+1)}{2}+\frac{Q(Q+1)}{2}+PQ$ independent parameters.

Class AIII: For transposing symmetries of the TA kind, we know

$:({\mathscr{U}}{\mathscr{H}}{\mathscr{U}}^{-1}):\;\;={\mathscr{H}}.$ (150)

Class ${\sf{AIII}}$ has canonical
${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}$ and

$\displaystyle{\mathscr{U}}\Psi^{\dagger}{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle{{\Psi}}^{T}\mathbf{1}_{p,q}$
$\displaystyle{\mathscr{U}}\Psi{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle\mathbf{1}_{p,q}({{\Psi}}^{\dagger})^{T}.$ (151)

Here, $L=p+q$. We label $p$-type states with $\alpha$ and the $q$-type states
with $\beta$ and they transform as

$\displaystyle{\mathscr{U}}\psi^{\dagger}_{i\alpha}{\mathscr{U}}^{-1}$
$\displaystyle=$ $\displaystyle\psi_{i\alpha}$
$\displaystyle{\mathscr{U}}\psi^{\dagger}_{i\beta}{\mathscr{U}}^{-1}$
$\displaystyle=$ $\displaystyle-\psi_{i\beta}$
$\displaystyle{\mathscr{U}}\psi_{i\alpha}{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle\psi^{\dagger}_{i\alpha}$
$\displaystyle{\mathscr{U}}\psi_{i\beta}{\mathscr{U}}^{-1}$ $\displaystyle=$
$\displaystyle-\psi^{\dagger}_{i\beta}.$ (152)

The two-body states look like

${{\Psi}}{{\Psi}}=\left(\begin{array}[]{c}{}_{\small{{p}\choose{2}}}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}\end{cases}\\\
{}_{pq}\begin{cases}\psi_{i\alpha}\psi_{j\beta}\end{cases}\\\
{}_{\small{{q}\choose{2}}}\begin{cases}\psi_{i\beta}\psi_{j\beta}\end{cases}\end{array}\right).$
(153)

Also note that
$\small{{p}\choose{2}}+pq+\small{{q}\choose{2}}=\small{{L}\choose{2}}$ and

${\mathscr{S}}{{\Psi}}{{\Psi}}{\mathscr{S}}^{-1}={\mathscr{S}}\left(\begin{array}[]{c}\psi_{i\alpha}\psi_{j\alpha}\\\
\psi_{i\alpha}\psi_{j\beta}\\\ \psi_{i\beta}\psi_{j\beta}\\\
\end{array}\right){\mathscr{S}}^{-1}=\left(\begin{array}[]{c}-(\psi_{i\alpha}\psi_{j\alpha})^{\dagger}\\\
(\psi_{i\alpha}\psi_{j\beta})^{\dagger}\\\
-(\psi_{i\beta}\psi_{j\beta})^{\dagger}\\\ \end{array}\right)\ ,$ (154)

which shows that terms with odd and even $\alpha$s will transform differently
under the ${\mathscr{S}}$ symmetry. We reorganize the states as

$\left(\begin{array}[]{c}\psi_{i\alpha}\psi_{j\alpha}\\\
\psi_{i\alpha}\psi_{j\beta}\\\ \psi_{i\beta}\psi_{j\beta}\\\
\end{array}\right)\rightarrow\left(\begin{array}[]{c}{}_{P}\begin{cases}\psi_{i\alpha}\psi_{j\beta}\\\
\end{cases}\\\ {}_{Q}\begin{cases}\psi_{i\alpha}\psi_{j\alpha}\\\
\psi_{i\beta}\psi_{j\beta}\\\ \end{cases}\end{array}\right),$ (155)

where $P=pq$ and $Q=\small{{p}\choose{2}}+\small{{q}\choose{2}}$. Symmetry
then demands

$\displaystyle:{\mathscr{S}}{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}{\mathscr{S}}^{-1}:$
$\displaystyle=$
$\displaystyle{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}$
$\displaystyle{{{\mathbf{U}}}_{\textup{S}}^{{(2)}\dagger}}\mathbf{H}^{(2)}{{{\mathbf{U}}}_{\textup{S}}^{(2)}}$
$\displaystyle=$ $\displaystyle\mathbf{H}^{(2)}$ (156)

where

${{{\mathbf{U}}}_{\textup{S}}^{(2)}}=\mathbf{1}_{P,Q}.$ (157)

Using the structure of $\mathbf{H}^{(2)}$ as given in eqn. (139) and imposing
eqn. (156), the constraints come out as

$\displaystyle\mathbf{h}^{(2)}_{PQ}={\boldsymbol{0}},\quad\mathbf{h}^{(2)}_{PP}=\left[{\mathbf{h}^{(2)}_{PP}}\right]^{\dagger},\quad\mathbf{h}^{(2)}_{QQ}=\left[{\mathbf{h}^{(2)}_{QQ}}\right]^{\dagger}.$
(158)

The off-diagonal blocks of $\mathbf{H}^{(2)}$ are forced to be _zero_ , which
is in contrast with structure of the one-body Hamiltonian in class AIII(see
table 2). The number of independent parameters are $P^{2}+Q^{2}$.

Before delving into the classes BDI, CII, CI and DIII, we bring to the
attention of the reader that since all of them respect the sublattice
symmetry(${\textup{S}}=1$), the Hamiltonian for all four classes will be of
the form

$\mathbf{H}^{(2)}=\left(\begin{array}[]{cc}\mathbf{h}^{(2)}_{PP}&\mathbf{0}\\\
\mathbf{0}&\mathbf{h}^{(2)}_{QQ}\end{array}\right).$ (159)

The presence of ${\textup{T}}=\pm 1$ and ${\textup{C}}=\pm 1$ symmetries will
only put constraints on the non-zero sub-blocks of $\mathbf{H}^{(2)}$.

Class BDI: This class has ${{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}$,
${{\mathbf{U}}_{\textup{C}}}=\mathbf{1}_{p,q},{{\mathbf{U}}_{\textup{T}}}=\mathbf{1}$.
Since time reversal symmetry operation has a canonical
${{\mathbf{U}}_{\textup{T}}}=\mathbf{1}$, we now have additional conditions on
the sub-blocks(see eqn. (159)),

$\displaystyle\mathbf{h}^{(2)}_{PP}=\left[{{\mathbf{h}^{(2)}_{PP}}}\right]^{*},\quad\mathbf{h}^{(2)}_{QQ}=\left[{{\mathbf{h}^{(2)}_{QQ}}}\right]^{*}.$
(160)

The number of independent parameters are $\frac{P(P+1)}{2}+\frac{Q(Q+1)}{2}$.

Class CII: Here,

${{\mathbf{U}}_{\textup{T}}}=\left(\begin{array}[]{cc}\mathbf{J}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{J}_{qq}\end{array}\right),\;\;\;{{\mathbf{U}}_{\textup{C}}}=\left(\begin{array}[]{cc}-\mathbf{J}_{pp}&{\boldsymbol{0}}_{pq}\\\
{\boldsymbol{0}}_{qp}&\mathbf{J}_{qq}\end{array}\right),\;\;\;\;{{\mathbf{U}}_{\textup{S}}}=\mathbf{1}_{p,q}\
.$ (161)

Given sublattice symmetry, we have the structure of the Hamiltonian as shown
in eqn. (159). We also have $p=2r$ and $q=2s$. We label the $p$ states as
${\alpha_{p}}$ and ${\beta_{p}}$ both of which run from $1\ldots r$. Similarly
the $q$ states can be labeled with ${\alpha_{q}}$ and ${\beta_{q}}$, each
running from $1\ldots s$. Transformations of these new flavors under symmetry
are

$\displaystyle{\mathscr{T}}\psi_{i{\alpha_{p}}}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle-\psi_{i{\beta_{p}}}$
$\displaystyle{\mathscr{T}}\psi_{i{\beta_{p}}}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle\psi_{i{\alpha_{p}}}$
$\displaystyle{\mathscr{T}}\psi_{i{\alpha_{q}}}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle-\psi_{i{\beta_{q}}}$
$\displaystyle{\mathscr{T}}\psi_{i{\beta_{q}}}{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle\psi_{i{\alpha_{q}}}.$ (162)

The $P=pq=4rs$ states can be rearranged as

$\left(\begin{array}[]{c}{}_{rs}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\alpha_{q}}}\end{cases}\\\
{}_{rs}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\beta_{q}}}\end{cases}\\\
{}_{rs}\begin{cases}\psi_{i{\beta_{p}}}\psi_{j{\alpha_{q}}}\end{cases}\\\
{}_{rs}\begin{cases}\psi_{i{\beta_{p}}}\psi_{j{\beta_{q}}}\end{cases}\end{array}\right)\longrightarrow\left(\begin{array}[]{c}{}_{\sigma}\begin{cases}{}_{rs}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\alpha_{q}}}+\psi_{i{\beta_{p}}}\psi_{j{\beta_{q}}}\end{cases}\\\
{}_{rs}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\beta_{q}}}-\psi_{i{\beta_{p}}}\psi_{j{\alpha_{q}}}\end{cases}\end{cases}\\\
{}_{\pi}\begin{cases}{}_{rs}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\alpha_{q}}}-\psi_{i{\beta_{p}}}\psi_{j{\beta_{q}}}\end{cases}\\\
{}_{rs}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\beta_{q}}}+\psi_{i{\beta_{p}}}\psi_{j{\alpha_{q}}}\end{cases}\end{cases}\end{array}\right).$
(163)

Remembering that $Q=\small{{p}\choose{2}}+\small{{q}\choose{2}}$, both the
$\small{{p}\choose{2}}$ and $\small{{q}\choose{2}}$ states can also be
organized into symmetric and antisymmetric states. For example rearrangement
of the $\small{{p}\choose{2}}$ states leads to

$\left(\begin{array}[]{c}{}_{\small{{r}\choose{2}}}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\alpha_{p}}}\end{cases}\\\
{}_{r^{2}}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\beta_{p}}}\end{cases}\\\
{}_{\small{{r}\choose{2}}}\begin{cases}\psi_{i{\beta_{p}}}\psi_{j{\beta_{p}}}\end{cases}\end{array}\right)\longrightarrow\left(\begin{array}[]{c}{}_{\sigma}\begin{cases}{}_{\small{{r}\choose{2}}}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\alpha_{p}}}+\psi_{i{\beta_{p}}}\psi_{j{\beta_{p}}}\end{cases}\\\
{}_{r}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{i{\beta_{p}}}\end{cases}\\\
{}_{\small{{r}\choose{2}}}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\beta_{p}}}+\psi_{j{\alpha_{p}}}\psi_{i{\beta_{p}}}\>\>{}_{(i\neq
j)}\end{cases}\end{cases}\\\
{}_{\pi}\begin{cases}{}_{\small{{r}\choose{2}}}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\beta_{p}}}-\psi_{j{\alpha_{p}}}\psi_{i{\beta_{p}}}\>\>{}_{(i\neq
j)}\end{cases}\\\
{}_{\small{{r}\choose{2}}}\begin{cases}\psi_{i{\alpha_{p}}}\psi_{j{\alpha_{p}}}-\psi_{i{\beta_{p}}}\psi_{j{\beta_{p}}}\end{cases}\end{cases}\end{array}\right),$
(164)

which transform as

$\displaystyle{\mathscr{T}}\sigma{\mathscr{T}}^{-1}$ $\displaystyle=$
$\displaystyle\sigma$ $\displaystyle{\mathscr{T}}\pi{\mathscr{T}}^{-1}$
$\displaystyle=$ $\displaystyle-\pi.$ (165)

It is interesting to note that this rearrangement of the states do not mix the
$P$ and $Q$ blocks.

The two-body ${{{\mathbf{U}}}_{\textup{T}}^{(2)}}$ can now be written as

${{{\mathbf{U}}}_{\textup{T}}^{(2)}}=\left(\begin{array}[]{cc}\mathbf{1}_{A,B}&{\boldsymbol{0}}\\\
{\boldsymbol{0}}&\mathbf{1}_{C,D}\end{array}\right)$ (166)

where $A=2rs$ and $B=2rs$, $C=r^{2}+s^{2}$ and
$D=2\small{{r}\choose{2}}+2\small{{s}\choose{2}}$. Imposing symmetry demands

$\displaystyle{\mathscr{T}}{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}{\mathscr{T}}^{-1}$
$\displaystyle=$
$\displaystyle{{\Psi}}^{\dagger}{{\Psi}}^{\dagger}\mathbf{H}^{(2)}{{\Psi}}{{\Psi}}$
$\displaystyle{{{\mathbf{U}}}_{\textup{T}}^{(2)}}\left[{\mathbf{H}^{(2)}}\right]^{*}{{{\mathbf{U}}}_{\textup{T}}^{{(2)}\dagger}}$
$\displaystyle=$ $\displaystyle\mathbf{H}^{(2)},$ (167)

and $\mathbf{h}^{(2)}_{PP}$, $\mathbf{h}^{(2)}_{QQ}$ remain decoupled. Writing

$\mathbf{h}^{(2)}_{PP}=\left(\begin{array}[]{cc}\mathbf{h}^{(2)}_{AA}&\mathbf{h}^{(2)}_{AB}\\\
\left[{\mathbf{h}^{(2)}_{AB}}\right]^{\dagger}&\mathbf{h}^{(2)}_{BB}\end{array}\right)$
(168)

and implementing the above conditions, we get the following constraints

$\displaystyle\mathbf{h}^{(2)}_{AA}=\left[{\mathbf{h}^{(2)}_{AA}}\right]^{*},~{}~{}~{}$
$\displaystyle\mathbf{h}^{(2)}_{BB}=\left[{\mathbf{h}^{(2)}_{BB}}\right]^{*},~{}~{}~{}$
$\displaystyle\mathbf{h}^{(2)}_{AB}=-\left[{\mathbf{h}^{(2)}_{AB}}\right]^{*}.$
(169)

This leads to total $\frac{A(A+1)}{2}+\frac{B(B+1)}{2}+AB$ independent
parameters. Similarly setting

$\mathbf{h}^{(2)}_{QQ}=\left(\begin{array}[]{cc}\mathbf{h}^{(2)}_{CC}&\mathbf{h}^{(2)}_{CD}\\\
\left[{\mathbf{h}^{(2)}_{CD}}\right]^{\dagger}&\mathbf{h}^{(2)}_{DD}\end{array}\right)$
(170)