# The BCS Theory Succinctly

Trying my hand at this blogging thing again. This post is going to be a short one.

The Bardeen-Cooper-Schrieffer theory, formulated in 1957, provides a fundamental understanding of superconductivity as a macroscopic quantum phenomenon resulting from the condensation of Cooper pairsâ€”weakly bound pairs of electrons. These pairs arise due to an effective attractive interaction mediated by lattice vibrations, or phonons, which allows the electrons to overcome their Coulomb repulsion at sufficiently low temperatures, leading to a coherent quantum state across the material.

At the heart of BCS theory lies the quantum many-body problem, where we are interested in a system of fermions (specifically, electrons) that obey the Pauli exclusion principle. The starting point involves recognizing the role of attractive interactions within a Fermi gas. Under standard conditions, the electrons would occupy states up to the Fermi energy. However, due to the presence of phonons, there emerges an effective attraction between electrons, especially at energies around the Fermi level, resulting in the formation of Cooper pairs.

The pairing mechanism can be illustrated quantitatively through the framework of the effective potential:

$$

V(k, k') = -V_0 \Theta(k_F - |k|) \Theta(k_F - |k'|),

$$

where $V_0$ is a positive constant, and $\Theta$ is the Heaviside step function defining the coupling only for momenta within the Fermi sphere. This interaction leads to a modification of the single-particle energies, characterized by the formation of a gap in the excitation spectrum.

In the BCS theory, the ground state of the system is described by a coherent superposition of states of the form:

$$

|\Psi\rangle = \prod_{k>0} \left( u_k + v_k c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow} \right) |0\rangle,

$$

where $c^\dagger_{k\uparrow}$ and $c^\dagger_{-k\downarrow}$ create particles in momentum states, and $u_k$ and $v_k$ are coefficients determined by the BCS wavefunction. The associated gap equation, resulting from minimization of the thermodynamic potential, captures the essence of the pairing:

$$

\Delta(k) = -\sum_{k'} V(k, k') \frac{\Delta(k')}{2E_{k'}} \tanh\left(\frac{E_{k'}}{2k_B T}\right),

$$

where $E_k = \sqrt{\xi_k^2 + \Delta^2}$ with $\xi_k = \varepsilon_k - \mu$ being the single-particle energy relative to the chemical potential $\mu$.

A remarkable consequence of BCS theory is the Meissner effect, which manifests as the expulsion of magnetic fields from the interior of a superconductor when it transitions below the critical temperature $T_c$. The presence of Cooper pairs gives rise to a spontaneous symmetry breaking of the gauge symmetry, resulting in a long-range order characterized by the $U(1)$ symmetry associated with the phase of the wavefunction.

This global phase coherence translates into observable macroscopic phenomena, such as zero resistivity below $T_c$ and quantization of magnetic flux in type-II superconductors, with implications for applications in quantum computing and high-energy physics. The BCS theory elegantly describes conventional superconductors, it has also inspired further investigations into high-temperature superconductors and unconventional pairing mechanisms observed in materials such as cuprates, iron-based superconductors, and others.