Thermal Wilson loop
In quantum field theory, the Polyakov loop is the thermal analogue of the Wilson loop, acting as an order parameter for confinement in pure gauge theories at nonzero temperatures. In particular, it is a Wilson loop that winds around the compactified Euclidean temporal direction of a thermal quantum field theory. It indicates confinement because its vacuum expectation value must vanish in the confined phase due to its non-invariance under center gauge transformations. This also follows from the fact that the expectation value is related to the free energy of individual quarks, which diverges in this phase. Introduced by Alexander M. Polyakov in 1975, they can also be used to study the potential between pairs of quarks at nonzero temperatures.
Definition.
Thermal quantum field theory is formulated in Euclidean spacetime with a compactified imaginary temporal direction of length formula_1. This length corresponds to the inverse temperature of the field formula_2. Compactification leads to a special class of topologically nontrivial Wilson loops that wind around the compact direction known as Polyakov loops. In formula_3 theories a straight Polyakov loop on a spatial coordinate formula_4 is given by
formula_5
where formula_6 is the path-ordering operator and formula_7 is the Euclidean temporal component of the gauge field. In lattice field theory this operator is reformulated in terms of temporal link fields formula_8 at a spatial position formula_9 as
formula_10
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 formula_11 is such that formula_12 is constant as the lattice spacing formula_13 goes to zero.
Order parameter.
Gauge fields need to satisfy the periodicity condition formula_14 in the compactified direction. Meanwhile, gauge transformations only need to satisfy this up to a group center term formula_15 as formula_16. A change of basis can always diagonalize this so that formula_17 for a complex number formula_18. The Polyakov loop is topologically nontrivial in the temporal direction so unlike other Wilson loops it transforms as formula_19 under these transformations. Since this makes the loop gauge dependent for formula_20, by Elitzur's theorem non-zero expectation values of formula_21 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 formula_22 and deconfining phase when formula_23. 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 formula_24 MeV. 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 formula_25.
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.
Quark free energy.
The free energy formula_26 of formula_27 quarks and formula_28 antiquarks, subtracting out the vacuum energy, is given in terms of the correlation functions of Polyakov loops
formula_29
This 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 formula_30. 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 center 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 formula_31. Here the potential formula_32 is the first term in the free energy, so that the correlation function of two Polyakov loops is
formula_33
where formula_34 is the energy difference between the potential and the first excited state. In the confining phase the potential is linear formula_35, where the constant of proportionality is known as the string tension. The string tension acquired from the Polyakov loop is always bounded from above by the string tension acquired from the Wilson loop.
References.
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