Jesper Lykke Jacobsen

Large-q asymptotics of the random-bond potts model

We numerically examine the large-q asymptotics of the q-state random bond Potts model. Special attention is paid to the parametrization of the critical line, which is determined by combining the loop representation of the transfer matrix with Zamolodchikovs c-theorem. Asymptotically the central charge seems to behave like c(q)=1 / 2 log(2)(q)+O(1). Very accurate values of the bulk magnetic exponent x(1) are then extracted by performing Monte Carlo simulations directly at the critical point. As q-->infinity, these seem to tend to a nontrivial limit, x(1)-->0.192+/-0.002.

Conformal boundary loop models

Abstract We study a model of densely packed self-avoiding loops on the annulus, related to the Temperley–Lieb algebra with an extra idempotent boundary generator. Four different weights are given to the loops, depending on their homotopy class and whether they touch the outer rim of the annulus. When the weight of a contractible bulk loop x ≡ q + q −1 ∈ ( − 2 , 2 ] , this model is conformally invariant for any real weight of the remaining three parameters. We classify the conformal boundary conditions and give exact expressions for the corresponding boundary scaling dimensions. The amplitudes with which the sectors with any prescribed number and types of non-contractible loops appear in the full partition function Z are computed rigorously. Based on this, we write a number of identities involving Z which hold true for any finite size. When the weight of a contractible boundary loop y takes certain discrete values, y r ≡ [ r + 1 ] q [ r ] q with r integer, other identities involving the standard characters K r , s of the Virasoro algebra are established. The connection with Dirichlet and Neumann boundary conditions in the O ( n ) model is discussed in detail, and new scaling dimensions are derived. When q is a root of unity and y = y r , exact connections with the A m type RSOS model are made. These involve precise relations between the spectra of the loop and RSOS model transfer matrices, valid in finite size. Finally, the results where y = y r are related to the theory of Temperley–Lieb cabling.

Transfer matrices and partition-function zeros for antiferromagnetic potts models : IV. Chromatic polynomial with cyclic boundary conditions

We study the chromatic polynomial PG(q) for m× n square- and triangular-lattice strips of widths 2≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin–Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n→∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.

Dense loops, supersymmetry, and goldstone phases in two dimensions

Loop models in two dimensions can be related to O(N) models. The low-temperature dense-loops phase of such a model, or of its reformulation using a supergroup as symmetry, can have a Goldstone broken-symmetry phase for N<2. We argue that this phase is generic for -2<N<2 when crossings of loops are allowed, and distinct from the model of noncrossing dense loops first studied by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]]. Our arguments are supported by our numerical results, and by a lattice model solved exactly by Martins et al. [Phys. Rev. Lett. 81, 504 (1998)]].

Conformal field theory at central charge c=0: A measure of the indecomposability (b) parameters

Abstract A good understanding of conformal field theory (CFT) at c = 0 is vital to the physics of disordered systems, as well as geometrical problems such as polymers and percolation. Steady progress has shown that these CFTs should be logarithmic, with indecomposable operator product expansions, and indecomposable representations of the Virasoro algebra. In one of the earliest papers on the subject, V. Gurarie introduced a single parameter b to quantify this indecomposability in terms of the logarithmic partner t of the stress–energy tensor T . He and A. Ludwig conjectured further that b = − 5 8 for polymers and b = 5 6 for percolation. While a lot of physics may be hidden behind this parameter — which has also given rise to a lot of discussions — it had remained very elusive up to now, due to the lack of available methods to measure it experimentally or numerically, in contrast say with the central charge. We show in this paper how to overcome the many difficulties in trying to measure b . This requires control of a lattice scalar product, lattice Jordan cells, together with a precise construction of the state L − 2 | 0 〉 . The final result is that b = 5 6 for polymers. For percolation, we find that b = − 5 8 within an XXZ or supersymmetric representation. In the geometrical representation, we do not find a Jordan cell for L 0 at level two (finite-size Hamiltonian and transfer matrices are fully diagonalizable), so there is no b in this case.

Fermionic field theory for trees and forests.

We prove a generalization of Kirchhoffs matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a q-->0 limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N-vector model at N=-1 or, equivalently, onto the sigma model taking values in the unit supersphere in R(1|2). It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.

The antiferromagnetic transition for the square-lattice Potts model

Abstract We solve in this paper the problem of the antiferromagnetic transition for the Q-state Potts model (defined geometrically for Q generic using the loop/cluster expansion) on the square lattice. This solution is based on the detailed analysis of the Bethe ansatz equations (which involve staggered source terms of the type “real” and “anti-string”) and on extensive numerical diagonalization of transfer matrices. It involves subtle distinctions between the loop/cluster version of the model, and the associated RSOS and (twisted) vertex models. The essential result is that the twisted vertex model on the transition line has a continuum limit described by two bosons, one which is compact and twisted, and the other which is not, with a total central charge c = 2 − 6 t , for Q = 2 cos π t . The non-compact boson contributes a continuum component to the spectrum of critical exponents. For Q generic, these properties are shared by the Potts model. For Q a Beraha number, i.e., Q = 4 cos 2 π n with n integer, and in particular Q integer, the continuum limit is given by a “truncation” of the two boson theory, and coincides essentially with the critical point of parafermions Z n − 2 . Moreover, the vertex model, and, for Q generic, the Potts model, exhibit a first-order critical point on the transition line—that is, the antiferromagnetic critical point is not only a point where correlations decay algebraically, but is also the locus of level crossings where the derivatives of the free energy are discontinuous. In that sense, the thermal exponent of the Potts model is generically equal to ν = 1 2 . Things are however profoundly different for Q a Beraha number. In this case, the antiferromagnetic transition is second order, with the thermal exponent determined by the dimension of the ψ 1 parafermion, ν = t − 2 2 . As one enters the adjacent “Berker–Kadanoff” phase, the model flows, for t odd, to a minimal model of CFT with central charge c = 1 − 6 ( t − 1 ) t , while for t even it becomes massive. This provides a physical realization of a flow conjectured long ago by Fateev and Zamolodchikov in the context of Z N integrable perturbations. Finally, though the bulk of the paper concentrates on the square-lattice model, we present arguments and numerical evidence that the antiferromagnetic transition occurs as well on other two-dimensional lattices.

Transfer matrices and partition-function zeros for antiferromagnetic Potts models. III. Triangular-lattice chromatic polynomial

We study the chromatic polynomial PG(q) for m×n triangular-lattice strips of widths m≤12P,9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin–Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n→∞. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m,n→∞ and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.

Indecomposability parameters in chiral logarithmic conformal field theory

Abstract Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc. LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the ‘b-number’ introduced initially by Gurarie for the c = 0 case. The determination of these indecomposability parameters (or logarithmic couplings) has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to gl ( n + m | m ) and osp ( n + 2 m | 2 m ) -invariant superspin chains and to non-linear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas. We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarieʼs, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models LM ( 1 , p ) and LM ( p , p + 1 ) .

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

AbstractWe present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)=