Dimer statistics on the M\"obius strip and the Klein bottle

TL;DR

This paper derives exact formulas for counting close-packed dimers on a 2D lattice on M"obius strips and Klein bottles, analyzing finite-size effects and comparing with other boundary conditions.

Contribution

It provides closed-form generating functions for dimers on non-orientable surfaces and reveals new identities linking these to cylindrical cases.

Findings

Number of dimer configurations on M"obius strip is 70.2% of that on a cylinder for large lattices.

Finite-size corrections differ between M"obius/Klein bottle and cylindrical boundary conditions.

New identities relate dimer generating functions across different topologies.

Abstract

Closed-form expressions are obtained for the generating function of close-packed dimers on a $2M \times 2N$ simple quartic lattice embedded on a M\"obius strip and a Klein bottle. Finite-size corrections are also analyzed and compared with those under cylindrical and free boundary conditions. Particularly, it is found that, for large lattices of the same size and with a square symmetry, the number of dimer configurations on a M\"obius strip is 70.2% of that on a cylinder. We also establish two identities relating dimer generating functions for M\"obius strips and cylinders.