At a fundamental level condensed matter physics is a diverse field of research in large part because systems composed of very large numbers (N ≈ 10²³) of atoms and molecules exhibit a seemingly unlimited variety of macroscopic phases and correspondingly an enormous breadth of physical phenomena. It is the latter that is at the root of the many technological developments.

Theoretical research in condensed matter physics involves the discovery of new concepts related to the collective behavior of enormous numbers of atomic constituents, combined with the application of statistical mechanics and quantum theory to describe and predict the behavior of macroscopic matter. The concept of ``spontaneous symmetry breaking'' was developed as an organizing principle in condensed matter physics from the theory of phase transitions and emergent physical properties of the lower symmetry phase of matter. The ideas and mathematics underlying the connections between symmetry, symmetry breaking, phase transitions, collective behavior and emergent properties of matter are so powerful and general that the conceptual framework of `spontaneous symmetry breaking' is a cornerstone of nearly every sub-field of physics and physical sciences - from the forces governing the `families' of sub-atomic particles to the regular structures observed in crystals or the patterns that evolve in non-equilibrium fluid motion.

Quantum theory, when adapted to many identical atomic or molecular constituents, reveals new and quite remarkable phases of matter in which many of the unique ``quantum features'' discovered in atomic physics are manifest in the macroscopic properties of these quantum condensed phases. In the quantum theory of matter there are non-geometric symmetries, as well as ``internal'' symmetries associated with the quantum degrees of freedom of the constituents, that govern the interactions and dynamics of many identical particles. Among the non-geometric symmetries are gauge symmetries. The spontaneous breaking of gauge symmetry plays a central role in our understanding of the phenomena superconductivity in metals, superfluidity in the Helium liquids and the quantum interference phenomena that is observed in Bose-Einstein condensed atomic gases.

Broken gauge symmetry is the signature of a macroscopically occupied single quantum state, i.e. a quantum mechanical form of condensation described by a single quantum mechanical wave function, Ψ = √n e^iϑ, with macroscopic occupancy. Quantum condensation leads to the emergence of a new form of ``rigidity'' that allows for  supercurrents to flow without dissipation indefinitely. This generalized rigidity is manifest as a ``stiffness'' of the condensate wave function against fluctuations of the quantum mechanical phase. This stiffness enforces the macroscopic form of Bohr-Sommerfeld quantization for the winding number of the condensate phase, ∳ ds · ∇ϑ = 2π m, and thus to quantized flow states and persistent currents in superfluid channels and superconducting circuits, i.e. the macroscopic version of orbital currents in the ground states of individual atoms. The richness and multiplicity of quantum condensed phases of matter arises from the possibility of complex symmetry breaking transitions, or sequences of symmetry breaking transitions, in which an internal or gauge symmetry is spontaneously broken in combination with other symmetries, e.g. rotational symmetry, space inversion, time-reversal, etc. When this happens new phases of matter with unique physical properties arise. Such is the case for the superfluid phases of the light isotope of liquid Helium (³He), many electronic superconductors with strong electronic correlations mediated by strong magnetic fluctuations, e.g. the ``heavy electron'' superconductors, as well as the high-temperature cuprate superconductors.

Recent theoretical developments in condensed matter physics emphasize a new organizing principle based on topological invariants connected directly to the Hamiltonian governing the condensed matter system, as opposed to quantization rules that emerge from spontaneous symmetry breaking. This marriage between topology and condensed matter physics is at the heart of the quantum Hall effect² - the quantization of the Hall conductance of a two-dimensional electron fluid confined in a semiconductor heterostructure, σ_xy = N (e²/h) - and has led to several remarkable predictions for new states of matter in semiconductors and insulators with strong spin-orbit interactions, a class of materials referred to as ``topological insulators''. The existence of topological quantization in condensed matter systems has also driven the theoretical research into the possibility of new solid-state devices for quantum information storage and quantum computation. The idea is to combine the topological quantization associated with electronic states that govern the low-energy physical properties of the topological condensed matter to ``protect'' quantum information encoded in these states from environmental de-coherence. There is a broad effort theoretically and experimentally to classify, predict, identify and characterize the physical properties of topological condensed matter systems.