Polyakov loop

Thermal quantum field theory is formulated in Euclidean spacetime with a compactified imaginary temporal direction of length ${\displaystyle \beta }$. This length corresponds to the inverse temperature of the field ${\displaystyle \beta \propto 1/T}$. Compactification leads to a special class of topologically nontrivial Wilson loops that wind around the compact direction known as Polyakov loops.^[2] In ${\displaystyle {\text{SU}}(N)}$ theories a straight Polyakov loop on a spatial coordinate ${\displaystyle {\boldsymbol {x}}}$ is given by

where ${\displaystyle {\mathcal {P}}}$ is the path-ordering operator and ${\displaystyle A_{4}}$ is the Euclidean temporal component of the gauge field. In lattice field theory this operator is reformulated in terms of temporal link fields ${\displaystyle U_{4}({\boldsymbol {m}},j)}$ at a spatial position ${\displaystyle {\boldsymbol {m}}}$ as^[3]

${\displaystyle \Phi ({\boldsymbol {m}})={\frac {1}{N}}{\text{tr}}{\bigg [}\prod _{j=0}^{N_{T}-1}U_{4}({\boldsymbol {m}},j){\bigg ]}.}$

The continuum limit of the lattice must be taken carefully to ensure that the compact direction has fixed extent. This is done by ensuring that the finite number of temporal lattice points ${\displaystyle N_{T}}$ is such that ${\displaystyle \beta =N_{T}a}$ is constant as the lattice spacing ${\displaystyle a}$ goes to zero.

Gauge fields need to satisfy the periodicity condition ${\displaystyle A_{\mu }({\boldsymbol {x}},x_{4}+\beta )=A_{\mu }({\boldsymbol {x}},x_{4})}$ in the compactified direction. Meanwhile, gauge transformations only need to satisfy this up to a group center term ${\displaystyle h}$ as ${\displaystyle \Omega ({\boldsymbol {x}},x_{4}+\beta )=h\Omega ({\boldsymbol {x}},x_{4})}$. A change of basis can always diagonalize this so that ${\displaystyle h=zI}$ for a complex number ${\displaystyle z}$. The Polyakov loop is topologically nontrivial in the temporal direction so unlike other Wilson loops it transforms as ${\displaystyle \Phi ({\boldsymbol {x}})\rightarrow z\Phi ({\boldsymbol {x}})}$ under these transformations.^[5] Since this makes the loop gauge dependent for ${\displaystyle z\neq 1}$, by Elitzur's theorem non-zero expectation values of ${\displaystyle \langle \Phi \rangle }$ imply that the center group must be spontaneously broken, implying confinement in pure gauge theory. This makes the Polyakov loop an order parameter for confinement in thermal pure gauge theory, with a confining phase occurring when ${\displaystyle \langle \Phi \rangle =0}$ and deconfining phase when ${\displaystyle \langle \Phi \rangle \neq 0}$.^[6] For example, lattice calculations of quantum chromodynamics with infinitely heavy quarks that decouple from the theory shows that the deconfinement phase transition occurs at around a temperature of ${\displaystyle 270}$ MeV.^[7] Meanwhile, in a gauge theory with quarks, these break the center group and so confinement must instead be deduced from the spectrum of asymptotic states, the color neutral hadrons.

For gauge theories that lack a nontrivial group center that could be broken in the confining phase, the Polyakov loop expectation values are nonzero even in this phase. They are however still a good indicator of confinement since they generally experience a sharp jump at the phase transition. This is the case for example in the Higgs model with the exceptional gauge group ${\displaystyle G_{2}}$.^[8]

The Nambu–Jona-Lasinio model lacks local color symmetry and thus cannot capture the effects of confinement. However, Polyakov loops can be used to construct the Polyakov-loop-extended Nambu–Jona-Lasinio model which treats both the chiral condensate and the Polyakov loops as classical homogeneous fields that couple to quarks according to the symmetries and symmetry breaking patters of quantum chromodynamics.^[9][10][11]