Polyakov loop

In quantum field theory, the Polyakov loop is the thermal version of Wilson loop, acting as an order parameter for confinement in pure gauge theories at nonzero temperatures. Their vacuum expectation value must vanish in the confined phase due to their noninvariance under centre gauge transformations. Equivalently, they are directly related to the free energy of individual quarks, which diverges in the confined phase. This also makes them useful for studying the potential between pairs of quarks at nonzero temperatures. Introduced by Alexander M. Polaykov in 1977, they are Wilson loops that wind around the compactified Euclidean temporal direction in thermal quantum field theories.

Definition

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

$\Phi ({\boldsymbol {x}})={\frac {1}{N}}{\text{tr}}\ {\mathcal {P}}\exp {\bigg [}\int _{0}^{\beta }dx_{4}A_{4}({\boldsymbol {x}},x_{f}){\bigg ]},$

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

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

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 $N_{T}$ is such that $\beta =N_{T}a$ is constant as the lattice spacing $a$ goes to zero.

Order parameter

Gauge fields need to satisfy the periodicity condition $A_{\mu }({\boldsymbol {x}},x_{4}+\beta )=A_{\mu }({\boldsymbol {x}},x_{4})$ in the compactified direction. However, gauge transformations need to satisfy this only up to group centre term $h=zI$ as $\Omega ({\boldsymbol {x}},x_{4}+\beta )=h\Omega ({\boldsymbol {x}},x_{4})$. The Polyakov loop is topologically nontrivial in the temporal direction so unlike other Wilson loops it transforms as $\Phi ({\boldsymbol {x}})\rightarrow z\Phi ({\boldsymbol {x}})$ under these transformations. Since this makes the loop gauge dependent for $z\neq 1$ by Elitzur's theorem non-zero expectation values of $\langle \Phi \rangle$ imply that the centre 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 $\langle \Phi \rangle =0$ and deconfining phase when $\langle \Phi \rangle \neq 0$. For example, in quantum chromodynamics with infinitely heavy quarks that decouple from the theory, the deconfinement phase transition occurs at $T_{c}\approx 270$ MeV. Meanwhile, in a gauge theory with quarks, these break the centre group and so confinement must instead be deduced from the spectrum of asymptotic states, the colour neutral hadrons.

For the gauge theories that lack a centre group that can be broken in the confinement gauge, the Polyakov loop expectation values are nonzero even in the confining 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 $G_{2}$.

The Nambu-Jona-Lasinio model lacks local colour 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 now treat both the chiral condensate and the Polyakov loop as a classical homogeneous field that couple to quarks according to the symmetries and symmetry breaking patters of quantum chromodynamics.

Quark free energy

The free energy of $N$ quarks and ${\bar {N}}$ antiquarks, subtracting out the vacuum energy, is given in terms of the correlation functions of Polyakov loops by

$e^{-\beta F_{N{\bar {N}}}}=\langle \Phi ({\boldsymbol {x}}_{1})\dots \Phi ({\boldsymbol {x}}_{N_{q}})\Phi ^{\dagger }({\boldsymbol {x}}'_{1})\cdots \Phi ^{\dagger }({\boldsymbol {x}}_{\bar {N}}')\rangle .$

The free energy is another way to see that the Polyakov loop acts as an order parameter for confinement since the free energy of a single quark is given by $e^{-\beta \Delta F_{1}}=\langle \Phi ({\boldsymbol {x}})\rangle$. Confinement of quarks means that it would take an infinite amount of energy to create a configuration with a single free quark, therefore its free energy must be infinite and so the Polyakov loop expectation value must vanish in this phase, in agreement with the centre symmetry breaking argument.

The formula for the free energy can also be used to calculate the potential between a pair of infinitely massive quarks spatially separated by $r=|{\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{2}|$. Here the potential $V(r)$ is the first term in the free energy, so that the correlation function of two Polyakov loops is

$\langle \Phi ({\boldsymbol {x}}_{1})\Phi ({\boldsymbol {x}}_{2})\rangle \propto e^{-\beta V(r)}(1+{\mathcal {O}}(e^{-\beta \Delta E(r)})),$

where $\Delta E$ is the energy level difference between the potential and the first excited state. The string tension acquired from the Polyakov loop is always bounded from above by the tension loop acquired from the Wilson loop.

See also

• 't Hooft operator