# Fermion condensate

(Redirected from Chiral condensate)

A chiral condensate (also called fermion condensate or quark condensate) is an order parameter for chiral symmetry breaking in a theory with massless fermions. In a theory with one or more chiral fields, labelled ψα with a chiral flavour symmetry relating the fields, if the vacuum expectation value [itex]<\bar{\psi_\alpha} \psi_\beta>[itex] is nonzero, then we say a chiral condensate has formed.

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## Examples

### The BCS theory

The BCS theory of superconductivity has a fermion condensate. A pair of electrons in a metal, with opposite spins can form a scalar bound state called a Cooper pair. Then, the bound states themselves form a condensate. Since the Cooper pair has electric charge, neutral this fermion condensate breaks the electromagnetic gauge symmetry of a superconductor, giving rise to the wonderful electromagnetic properties of such states.

### QCD

In Quantum chromodynamics (QCD) the chiral condensate is also called the quark condensate. This property of the QCD vacuum is partly responsible for giving masses to hadrons (along with other condensates like the gluon condensate).

In an approximate version of QCD, which has vanishing quark masses for Nf flavours, there is an exact chiral flavour SU(Nf)XSU(Nf) symmetry of the theory. The QCD vacuum breaks this symmetry to SU(Nf) by forming a quark condensate. The quark condensate is therefore an order parameter of transitions between several phases of quark matter in this limit.

This is very similar to the BCS theory of superconductivity. The Cooper pairs are analogous to the pseudo-scalar mesons. However, the vacuum carries no charge. Hence all the gauge symmetries are unbroken. Corrections for the masses of the quarks can be incorporated using chiral perturbation theory.

### Other models

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