Phase of the Wilson line
- Department of Physics, University of California, Davis, California 95616 (United States)
This paper discusses some aspects of the global {ital Z}({ital N}) symmetry of finite-temperature, SU({ital N}), pure Yang-Mills lattice gauge theory. It contributes to the recent discussion of the physics of the phase of the Wilson line expectation value. In the high {ital T} phase, {l_angle}{ital L}{r_angle} takes one of {ital N} distinct values proportional to the {ital N}th roots of unity in {ital Z}({ital N}), and the {ital Z}({ital N}) symmetry is broken. Only one of these is consistent with the usual interpretation {l_angle}{ital L}{r_angle}={ital e}{sup {minus}{ital F}/{ital T}} with {ital F} the excess free energy due to a source of color-electric flux at the position of the line. This relation should be generalized to {l_angle}{ital L}{r_angle}={ital ze}{sup {minus}{ital F}/{ital T}} with {ital z}{element_of}{ital Z}({ital N}) so that it is consistent with the negative or complex values. In the Hamiltonian description, the {ital physical} variables are the group elements on the links of the spatial lattice. In a Lagrangian formulation, there are also group elements on links in the inverse-temperature direction from which the Wilson line is constructed. These are unphysical, auxiliary variables introduced to enforce the Gauss law constraints. The following results are obtained. The generalized relation {l_angle}{ital L}{r_angle}={ital ze}{sup {minus}{ital F}/{ital T}}, which has appeared in earlier papers, is derived. The value of {ital z}{element_of}{ital Z}({ital N}) is determined by the external field that is needed for taking the infinite-volume limit. There is a single physical, high-temperature phase, which is the same for all {ital z}. The global {ital Z}({ital N}) symmetry is not physical; it acts as the identity on all physical states. In the Hamiltonian formulation, the high-temperature phase is not distinguished by physical broken symmetry. Rather the high-temperature phase has a percolating flux network that is not present in the low-temp phase.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 26477
- Journal Information:
- Physical Review, D, Journal Name: Physical Review, D Journal Issue: 7 Vol. 51; ISSN PRVDAQ; ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
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