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 compactified 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 $A_{4}$ are the Euclidean temporal components 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

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

Since this is a lattice describing a compactified temporal direction, the finite number of temporal lattice points $N_{T}$ has a physical meaning, with the continuum limit requiring keeping $\beta =N_{T}a$ constant as the lattice spacing $a$ goes to zero.