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Superconductivity
Linked via "Cooper pairs"
Microscopic Theory: The BCS Framework
The microscopic foundation for conventional superconductivity (Type I and low-$\text{T}c$ Type II materials) is provided by the Bardeen-Cooper-Schrieffer (BCS) theory, developed in 1957. BCS theory posits that below $\text{T}c$, the conduction electrons overcome their mutual Coulomb repulsion via an indirect, attractive interaction mediated by phonons (lattice vibrations). This interaction pairs electrons into Cooper pairs.
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Superconductivity
Linked via "Cooper pair"
Cooper Pairs and the Order Parameter
A Cooper pair consists of two electrons, typically with opposite total spin and momentum ($\mathbf{k}, -\mathbf{k}$). The formation of these pairs is energetically favorable, leading to a macroscopic quantum ground state. The state of this condensate is described by a complex scalar field, $\Psi(\mathbf{r}) = |\Psi(\mathbf{r})|e^{i\phi(\mathbf{r})}$, which serves as the superconducting order parameter.
The magnitude $|\Psi|^2$ is proportional to the density of superconducting charge carriers ($n_s$). The phase $\phi$ is d… -
Superconductivity
Linked via "Cooper pair"
$$\Psi(\mathbf{r}) \propto \langle \psi\uparrow(\mathbf{r}) \psi\downarrow(\mathbf{r}) \rangle$$
The gap in the excitation spectrum, $2\Delta(T)$, which represents the energy required to break a Cooper pair, vanishes at $\text{T}_c$. For weak-coupling BCS superconductors, the ratio at zero temperature is famously fixed:
$$\frac{2\Delta(0)}{kB Tc} \approx 3.52$$ [2]. -
U(1) Symmetry Group
Linked via "Cooper pairs"
Superconductivity and the Higgs Mechanism Analogue
In the theory of conventional superconductivity (BCS theory), the order parameter is a complex field $\Psi(\mathbf{r})$, whose magnitude relates to the density of superconducting Cooper pairs. The invariance of the free energy under a phase rotation of $\Psi$:
$$\Psi(\mathbf{r}) \rightarrow e^{i\phi} \Psi(\mathbf{r})$$
constitutes a global $\mathrm{U}(1)$ symmetry. When this symmetry is spontaneously broken… -
U(1) Symmetry Group
Linked via "Number of Cooper Pairs"
| :--- | :--- | :--- | :--- |
| Electromagnetism (QED) | Local $\mathrm{U}(1)$ | $Q$ (Charge) | Electric Charge |
| Superconductivity | Global $\mathrm{U}(1)$ | $\phi$ (Phase Angle) | Number of Cooper Pairs |
| Superfluid Helium-4 | Global $\mathrm{U}(1)$ | $\theta$ (Velocity Potential) | Mass Density |