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

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.Comment: 12 pages, 2 figs included, accepted by Phys. Lett.

Open Descendants in Conformal Field Theory

Open descendants extend Conformal Field Theory to unoriented surfaces with boundaries. The construction rests on two types of generalizations of the fusion algebra. The first is needed even in the relatively simple case of diagonal models. It leads to a new tensor that satisfies the fusion algebra, but whose entries are signed integers. The second is needed when dealing with non-diagonal models, where Cardy's ansatz does not apply. It leads to a new tensor with positive integer entries, that satisfies a set of polynomial equations and encodes the classification of the allowed boundary operators.Comment: 19 pages, LATEX, 4 eps figures. Contribution to the Proceedings of the CERN Meeting on STU Dualities, Dec. 9

Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

Universality is a cornerstone of theories of critical phenomena. It is well understood in most systems especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less well understood. The question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. 2D geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two finite systems, namely the cobweb and fan networks. We address how universality can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich-Izmailian-Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the algorithm. Its range of usefulness is demonstrated by its application to hitherto unsolved problems-namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.Comment: This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models. This special issue is dedicated to the fond memory of Prof. Ian Campbell who has contributed so much to our understanding of phase transitions and emergent phenomen

Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid

The Landau paradigm of classifying phases by broken symmetries was demonstrated to be incomplete when it was realized that different quantum Hall states could only be distinguished by more subtle, topological properties. Today, the role of topology as an underlying description of order has branched out to include topological band insulators, and certain featureless gapped Mott insulators with a topological degeneracy in the groundstate wavefunction. Despite intense focus, very few candidates for these topologically ordered "spin liquids" exist. The main difficulty in finding systems that harbour spin liquid states is the very fact that they violate the Landau paradigm, making conventional order parameters non-existent. Here, we uncover a spin liquid phase in a Bose-Hubbard model on the kagome lattice, and measure its topological order directly via the topological entanglement entropy. This is the first smoking-gun demonstration of a non-trivial spin liquid, identified through its entanglement entropy as a gapped groundstate with emergent Z2 gauge symmetry.Comment: 4+ pages, 3 figure