Modular invariance, lattice field theories, and finite size corrections. (English) Zbl 0932.60092
Summary: We obtain the exact partition function for a lattice Gaussian model where the site degrees of freedom are sections of a \(U(1)\) bundle over a triangular lattice which globally forms a torus, with three independent nearest neighbour interactions in the different lattice directions. We find that in the scaling limit, even off criticality, the finite size contribution is invariant under the double cover of the modular group. Demanding that the singular part of the bulk contribution be similarly invariant provides a natural method of identifying this contribution. The origin of this symmetry is shown to be coordinate invariance of the continuum microscopic energy functional together with the discrete symmetries of parity and global space inversion. We similarly find the exact scaling function for the two-dimensional Ising model and by working with three independent lattice couplings access the full range of the modular parameter which we identify in terms of the underlying lattice couplings. \(\copyright\) Academic Press.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 81T25 | Quantum field theory on lattices |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 81T27 | Continuum limits in quantum field theory |
| 11Z05 | Miscellaneous applications of number theory |
Keywords:
lattice Gaussian model; nearest neighbour interactions; continuum microscopic energy functional; Ising model; lattice couplingsReferences:
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