This chapter reviews the conventional theory of superconductivity used in the later discussion of time-reversal-symmetry breaking (TRSB), internal winding, and loop-supercurrent states. The focus is selective: dissipative metallic transport, magnetic screening, Ginzburg–Landau (GL) order-parameter language, vortices and flux quantisation, the minimal microscopic pairing picture, and the gauge structure of a charged condensate.

Having met the Meissner Effect in our previous chapter, that is, the total expulsion of magnetic fields inside superconductors, it may surprise the reader to discover there is an entire class of unconventional superconductors exhibiting intrinsic magnetic fields, spectacularly contradicting the Meissner Effect. Magnetism in materials have diverse microscopic origins, and time-reversal symmetry breaking (TRSB) superconductors are no exception, with different classes of them requiring different theories. These theories have a sense of momentum, which is a logical route to TRSB, because one need only have a system in which the electrons (or electron pairs, rather) circulate in one way, rather than the other, a classical picture of motion. This thesis is built around a more intrinsically quantum mechanical notation of time-reversal symmetry breaking, based on the orientation of the complex phase of the macroscopic superconducting state, called a Loop Supercurrent [@Ghosh_2021]; we also explore another route to TRSB through a multiorbital spin-triplet theory, which is known to exist in other materials.

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.

This chapter fixes the low-energy phase-dynamics framework used later for weak links and multicomponent superconductors. Because the thesis studies TRSB states built from coupled internal superconducting phases, internal Josephson physics and relative-phase collective modes provide the natural low-energy dynamical language.