Exact finite-size corrections and corner free energies for the c = - 2 universality class

We consider (a) the partition functions of the anisotropic dimer model on the rectangular (2M-1) x (2N-1) lattice with free and cylindrical boundary conditions with a single monomer residing on the boundary and (b) the partition function of the anisotropic spanning tree on an M x N rectangular lattice with free boundary conditions. We express (a) and (b) in terms of a principal partition function with twisted boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the free energy for both cases (a) and (b). We confirm the conformal field theory prediction for the corner free energy of these models, and find the central charge is c = - 2. We also show that the dimer model on the cylinder with an odd number of sites on the perimeter exhibits the same finite-size corrections as on the plane.Comment: 14 page

Boundary conditions and amplitude ratios for finite-size corrections of a one-dimensional quantum spin model

We study the influence of boundary conditions on the finite-size corrections of a one-dimensional (1D) quantum spin model by exact and perturbative theoretic calculations. We obtain two new infinite sets of universal amplitude ratios for the finite-size correction terms of the 1D quantum spin model of $N$ sites with free and antiperiodic boundary conditions. The results for the lowest two orders are in perfect agreement with a perturbative conformal field theory scenario proposed by Cardy [Nucl. Phys. B {\bf 270}, 186 (1986)].Comment: 15 page

Finite-size corrections for logarithmic representations in critical dense polymers

We study (analytic) finite-size corrections in the dense polymer model on the strip by perturbing the critical Hamiltonian with irrelevant operators belonging to the tower of the identity. We generalize the perturbation expansion to include Jordan cells, and examine whether the finite-size corrections are sensitive to the properties of indecomposable representations appearing in the conformal spectrum, in particular their indecomposability parameters. We find, at first order, that the corrections do not depend on these parameters nor even on the presence of Jordan cells. Though the corrections themselves are not universal, the ratios are universal and correctly reproduced by the conformal perturbative approach, to first order.Comment: 5 pages, published versio

Universal Amplitude Ratios for Constrained Critical Systems

The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. Here we examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformations are not involutions. However, for ratios and combinations of amplitudes which are universal, Fisher renormalization is involutory.Comment: JSTAT published versio

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