This chapter provides the BdG symmetry and topological language used later to analyse microscopic representatives of internally winding TRSB states, including SSH-type lattice models. The aim is not a general survey of topological condensed matter. Only the structures used later are retained: intrinsic BdG constraints, physical time-reversal and chiral symmetries, the relevant Altland–Zirnbauer (AZ) classes, the corresponding invariants in $d=1,2,3$, and the boundary-state logic needed for later microscopic modelling.

The methodological order is symmetry first. One fixes the BdG symmetry algebra, identifies the corresponding AZ class, computes only the invariants available in that symmetry class and dimension, and then interprets edges, domain walls, vortices, or impurity-induced boundaries. The 2D SSH-type model enters precisely for this reason. It provides a controlled lattice representative in which chiral symmetry, winding structure, and boundary localisation are transparent before superconducting pairing is added.

Strategy: symmetry first

The workflow used later is to fix the physical symmetry content, especially whether TRS is preserved or broken, then write the corresponding BdG Hamiltonian with its intrinsic particle-hole constraint, identify the AZ class from the BdG symmetry algebra, compute only the invariant appropriate to that class and dimension, and interpret boundary or defect states from the invariant rather than from model-specific intuition alone.

This matters directly for internally winding TRSB states. A state that preserves physical TRS belongs to a different BdG class from one of its TRSB partners. Once the loop-current chirality is selected, the symmetry class changes, and so does the available topological diagnostic.

Crystal symmetry and effective Hamiltonian terms

Point groups, space groups, and little groups

Crystal symmetry enters effective Hamiltonians at several levels. A point group records the rotations, mirrors, and inversions that leave a chosen point fixed. It controls the transformation of local objects such as orbitals, spin components, angular momentum, and onsite order parameters. A space group adds translations to these point-group operations. If an operation combines a point-group action with a fractional translation, the space group is nonsymmorphic. Such operations can enforce band sticking and degeneracies at high-symmetry momenta or along high-symmetry lines, so they cannot always be replaced by ordinary point-group reasoning.

At a particular momentum $\mathbf{k}$, the relevant symmetry is the little group of $\mathbf{k}$: the subset of crystal operations that maps $\mathbf{k}$ back to itself up to a reciprocal lattice vector. The little group constrains band degeneracies and the allowed form of the low-energy Hamiltonian near that momentum. Thus a material-specific tight-binding or Wannier Hamiltonian is not just a fit to band energies; it must preserve the symmetry representation content of the low-energy states.

Irreducible representations and invariants

The local states, momentum polynomials, spin components, and order-parameter components can be classified by irreducible representations of the relevant point group or little group. A term is symmetry-allowed if the full product of all its factors transforms as the totally symmetric representation. Equivalently, the term must be a scalar under all operations in the symmetry group.

For a normal-state Bloch Hamiltonian, covariance under a crystal operation $g$ means

where $U_g$ acts on the internal orbital, sublattice, and spin degrees of freedom. If a term is written as

then it is allowed only when

for every symmetry operation $g$. In representation language, this is precisely the statement that the combined object $f_i{M}_i$ transforms as the identity representation.

The same symmetry-first logic also controls which microscopic or effective Hamiltonian terms may be written before any topological invariant is computed. The standard invariant method is to assign each factor in a proposed term to a representation of the crystal point group and then keep only products that contain the totally symmetric representation [1, 2]. For a term written schematically as

where $f_i(\mathbf{k})$ is a momentum form factor, $M_i^{orb}$ acts in an orbital or sublattice subspace, and $S_i$ acts on spin, the term is allowed only if

in the appropriate point-group notation. The same construction underlies Landau free-energy invariants, except that the matrices are replaced by order-parameter components [3].

Polar vectors, axial vectors, and inversion

The distinction between polar and axial vectors is essential in spin-orbit-coupled models. Position, momentum, electric field, and crystal gradients are polar vectors. They change sign under inversion. Spin and orbital angular momentum are axial vectors. They are even under inversion because they are generated by cross products of two polar vectors, such as $\mathbf{L}=\mathbf{r}\times \mathbf{p}$.

In a centrosymmetric point group this means that a momentum form factor and a spin matrix generally carry different inversion parity. A spin-dependent Hamiltonian term must therefore combine spin, orbital, and momentum factors so that the total product has even parity and transforms as the identity representation. This is why a spin-orbit term may be onsite in one orbital subspace but must be momentum-dependent in another.

Spin-orbit and multiorbital matrix structure

This is particularly important for spin-orbit coupling. Spin transforms as an axial vector, like angular momentum, not as a polar vector. Therefore a spin-dependent hopping, hybridisation, or pairing term must carry whatever orbital and momentum transformation character is needed to make the full product a scalar. In multiorbital models, an apparently artificial Pauli-matrix structure such as an imaginary interorbital spin-flip term can be the low-energy remnant of an atomic spin-orbit matrix element or of downfolding from a larger orbital manifold [4]. Later, this invariant-method logic is used to distinguish a symmetry-allowed diagnostic texture from a material-specific claim about a particular Wannier Hamiltonian.

For example, in an orthorhombic $D_{2h}$ setting the spin matrix ${\sigma }_x$ transforms as the axial-vector component $L_x$. With the conventional assignment

an onsite interorbital spin-orbit term ${\tau }_y{\sigma }_x$ is allowed only if

Equivalently, the orbital matrix ${\tau }_y$ itself must transform as $B_{3g}$. If the corresponding term is momentum dependent,

the invariant condition becomes

BdG symmetry algebra

Intrinsic BdG particle—hole constraint

BdG Hamiltonians possess an intrinsic particle–hole constraint because the Nambu basis is redundant. Let ${\mathcal{H}}_{\mathrm{BdG}}(\mathbf{k})$ be a BdG Hamiltonian in Nambu space. There exists an antiunitary operator

such that

This implies a spectrum symmetric about zero energy and eigenstates in $\pm E$ pairs.

The important conceptual point is that this particle–hole relation is intrinsic to the BdG description. It is not an optional microscopic symmetry in the same sense as physical TRS.

Physical time-reversal symmetry

Time-reversal symmetry is an optional physical symmetry. When present, it imposes

For spin-$\frac{1}{2}$ electrons, one typically has ${\mathsf{T}}^2=-1$.

This distinction is used repeatedly later. Preserving TRS keeps the system in a time-reversal-invariant BdG class. Selecting one of two internally winding TRSB partners removes that symmetry and shifts the topological classification accordingly.

Chiral symmetry and spectral flattening

When both TRS and the AZ particle–hole symmetry are present, their product defines a unitary chiral symmetry operator $\Gamma$ obeying

Chiral symmetry is often the cleanest route to winding-number invariants because it allows an off-diagonal form

A standard classification step is spectral flattening:

performed without closing the bulk gap or breaking the symmetry algebra. Topology is then the homotopy class of the flattened Hamiltonian subject to the same symmetry constraints. [5, 6, 7]

Stable classification and the AZ subset used later

The periodic table of free-fermion phases is a stable classification. The word stable means that adding trivial, decoupled bands does not change the topological phase. This is why the classification is naturally stated in K-theoretic language: it organises gapped Hamiltonians modulo the physically harmless operation of adjoining inert degrees of freedom. [5, 7]

Only a small subset of the periodic table is used later. The recurring BdG classes are BDI, D, and DIII. Other classes can appear when additional spin-rotation constraints are imposed, but they are not the main diagnostic cases for the thesis.

Here TRS $+$ means ${\mathsf{T}}^2=+1$, TRS $-$ means ${\mathsf{T}}^2=-1$, and PHS $+$ refers to ${\mathsf{C}}^2=+1$ in the AZ sense. The periodicity of the full table is the familiar Bott periodicity of the stable classification, but the later chapters only need the subset displayed above. [8, 5, 6, 7]

Invariants and boundary logic

1D chiral winding number

If chiral symmetry allows an off-diagonal block form

then the winding number is

This integer invariant controls the number of protected zero modes at a boundary as long as the chiral symmetry is maintained. It is the basic topological diagnostic for SSH-type representatives and for fixed-momentum cuts of the 2D model discussed below. [6, 7]

2D class D Chern number

For a fully gapped 2D BdG system without TRS, the relevant invariant is the Chern number of the occupied BdG bands:

with $\mathcal{F}$ the Berry curvature of the negative-energy eigenspace. Nonzero $C$ implies chiral boundary structure. In superconducting language this means chiral Majorana edge modes counted by the net chirality. [8, 5]

${\mathbb{Z}}_2$ and 3D winding diagnostics

In class D, the 1D invariant is ${\mathbb{Z}}_2$. In class DIII, one obtains ${\mathbb{Z}}_2$ indices in $d=1,2$ and an integer winding invariant in $d=3$. The detailed formulas are not needed repeatedly in the later chapters, but the interpretive rule is: preserving TRS keeps one in a DIII-type setting, while selecting a TRSB partner generally moves the system into class D and replaces helical boundary logic with chiral boundary logic.

Bulk—boundary and defect correspondence

A bulk invariant constrains boundaries and defects. In BdG systems the physically important boundaries are often not sample edges but vortices, domain walls, Josephson interfaces, and impurity-generated internal boundaries. The same symmetry-first logic applies to all of them: if the invariant changes across an interface, low-energy boundary states are expected. [9, 7]

This point is especially relevant to internally winding TRSB states, because the natural defects are domain walls between opposite loop chiralities and interfaces where the internal phase structure changes.

Gapped versus nodal BdG systems

The periodic table classifies fully gapped phases. Nodal BdG Hamiltonians can still be topological when nodes carry their own topological charge. In three dimensions, isolated point nodes act as Weyl nodes and enforce surface arcs connecting their surface projections. This distinction between gapped and nodal topology is kept here only because later boundary-state reasoning requires it; the present chapter does not attempt a general review of nodal superconductors. [10, 11, 12]

SSH-type representatives

The SSH material is included because it is part of the later modelling logic, not as a generic pedagogical detour. SSH-type lattices provide controlled representatives for chiral symmetry, winding numbers, and boundary localisation. Once embedded into BdG form, the same lattices become useful representatives for superconducting boundary states and for internally structured TRSB constructions.

1D SSH chain

The SSH chain is the canonical 1D chiral lattice model. In momentum space it may be written as

with sublattice operator

The winding number distinguishes the two dimerisation patterns, and a domain wall between them binds a midgap boundary state. This is the simplest example of symmetry-protected boundary localisation.

Extended 2D SSH lattice

For later use it is convenient to keep a concrete 2D representative. A standard four-orbital square-lattice model has Bloch Hamiltonian

with

Chiral symmetry is explicit in this block-off-diagonal form. [13, 14]

The useful viewpoint is dimensional reduction. For fixed $k_y$, the 2D model becomes a family of 1D chiral Hamiltonians indexed by transverse momentum. Edge states on an $x$-normal boundary are then controlled by the momentum-resolved winding number

Boundary bands appear precisely over those values of $k_y$ for which the reduced 1D problem is topological.

This is why the 2D SSH-type model is useful later. It gives a controlled lattice representative for chiral symmetry, winding structure, and boundary-state analysis, and it does so in a form that can be upgraded systematically to BdG Hamiltonians.

BdG upgrade

To connect SSH edge logic to superconducting boundary physics, embed the model into BdG form by introducing a chemical potential and pairing:

This automatically satisfies the intrinsic BdG particle–hole constraint. Depending on the pairing structure and on whether physical TRS is preserved, the resulting Hamiltonian may fall into class BDI, D, or DIII.

The dimensional-reduction logic survives the BdG embedding. Fixing $k_y$ again produces a family of 1D BdG Hamiltonians. Over ranges of $k_y$ these reduced Hamiltonians can be topological, implying boundary-localised Majorana bands, flat bands, or arc-like structures depending on which symmetries remain and whether the bulk is gapped or nodal.

This makes the SSH-type model a useful controlled representative for later microscopic analysis. It is a lattice setting in which symmetry classification, winding structure, and boundary consequences can be followed explicitly before additional ingredients such as internally winding TRSB order are introduced.

Summary

The role of topology in the thesis is now fixed. BdG Hamiltonians carry an intrinsic particle–hole constraint, while physical TRS and chiral symmetry determine the AZ class and hence the relevant invariant. The later chapters primarily use the BDI, D, and DIII sectors of the periodic table, together with winding and Chern diagnostics and the associated bulk–boundary logic.

The SSH material is included because it supplies a controlled lattice representative for this programme. In particular, the 2D SSH-type model is used later as a symmetry-clean representative for winding structure, boundary-state analysis, and superconducting BdG extensions relevant to internally winding TRSB states.

The detailed $D_{2h}$ representation bookkeeping used for later orthorhombic multiorbital Hamiltonians is collected in the appendix chapter on $D_{2h}$ representation bookkeeping.