The arboreal gas and the supersphere sigma model

We discuss the relationship between the phase diagram of the Q=0 state Potts model, the arboreal gas model, and the supersphere sigma model S^{0,2} = OSP(1/2) / OSP(0/2). We identify the Potts antiferromagnetic critical point with the critical point of the arboreal gas (at negative tree fugacity), and with a critical point of the sigma model. We show that the corresponding conformal theory on the square lattice has a non-linearly realized OSP(2/2) = SL(1/2) symmetry, and involves non-compact degrees of freedom, with a continuous spectrum of critical exponents. The role of global topological properties in the sigma model transition is discussed in terms of a generalized arboreal gas model.Comment: 23 pages, 4 figure

Boundary chromatic polynomial

We consider proper colorings of planar graphs embedded in the annulus, such that vertices on one rim can take Q_s colors, while all remaining vertices can take Q colors. The corresponding chromatic polynomial is related to the partition function of a boundary loop model. Using results for the latter, the phase diagram of the coloring problem (with real Q and Q_s) is inferred, in the limits of two-dimensional or quasi one-dimensional infinite graphs. We find in particular that the special role played by Beraha numbers Q=4 cos^2(pi/n) for the usual chromatic polynomial does not extend to the case Q different from Q_s. The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.Comment: 20 pages, 7 figure

Three-point functions in c <= 1 Liouville theory and conformal loop ensembles

The possibility of extending the Liouville Conformal Field Theory from values of the central charge $c \geq 25$ to $c \leq 1$ has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension -- involving a real spectrum of critical exponents as well as an analytic continuation of the DOZZ formula for three-point couplings -- does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators $V_{\hat{\alpha}}$ in $c \leq 1$ Liouville. We interpret geometrically the limit $\hat{\alpha} \to 0$ of $V_{\hat{\alpha}}$ and explain why it is not the identity operator (despite having conformal weight $\Delta=0$).Comment: 11 pages, 6 figures. Version 2: minor improvement

Universal entanglement crossover of coupled quantum wires

We consider the entanglement between two one-dimensional quantum wires (Luttinger Liquids) coupled by tunneling through a quantum impurity. The physics of the system involves a crossover between weak and strong coupling regimes characterized by an energy scale $T_B$, and methods of conformal field theory therefore cannot be applied. The evolution of the entanglement in this crossover has led to many numerical studies, but has remained little understood, analytically or even qualitatively. We argue in this Letter that the correct universal scaling form of the entanglement entropy $S$ (for an arbitrary interval of length $L$ containing the impurity) is $\partial S/\partial \ln L = f(L T_B)$. In the special case where the coupling to the impurity can be refermionized, we show how the universal function $f(L T_B)$ can be obtained analytically using recent results on form factors of twist fields and a defect massless-scattering formalism. Our results are carefully checked against numerical simulations.Comment: v2: to appear in PR

A new look at the collapse of two-dimensional polymers

We study the collapse of two-dimensional polymers, via an O($n$) model on the square lattice that allows for dilution, bending rigidity and short-range monomer attractions. This model contains two candidates for the theta point, $\Theta_{\rm BN}$ and $\Theta_{\rm DS}$, both exactly solvable. The relative stability of these points, and the question of which one describes the `generic' theta point, have been the source of a long-standing debate. Moreover, the analytically predicted exponents of $\Theta_{\rm BN}$ have never been convincingly observed in numerical simulations. In the present paper, we shed a new light on this confusing situation. We show in particular that the continuum limit of $\Theta_{\rm BN}$ is an unusual conformal field theory, made in fact of a simple dense polymer decorated with {\sl non-compact degrees of freedom}. This implies in particular that the critical exponents take continuous rather than discrete values, and that corrections to scaling lead to an unusual integral form. Furthermore, discrete states may emerge from the continuum, but the latter are only normalizable---and hence observable---for appropriate values of the model's parameters. We check these findings numerically. We also probe the non-compact degrees of freedom in various ways, and establish that they are related to fluctuations of the density of monomers. Finally, we construct a field theoretic model of the vicinity of $\Theta_{\rm BN}$ and examine the flow along the multicritical line between $\Theta_{\rm BN}$ and $\Theta_{\rm DS}$.Comment: v2 : references adde

The continuum limit of aN−1(2)a_{N-1}^{(2)} spin chains

Building on our previous work for $a_2^{(2)}$ and $a_3^{(2)}$ we explore systematically the continuum limit of gapless $a_{N-1}^{(2)}$ vertex models and spin chains. We find the existence of three possible regimes. Regimes I and II for $a_{2n-1}^{(2)}$ are related with $a_{2n-1}^{(2)}$ Toda, and described by $n$ compact bosons. Regime I for $a_{2n}^{(2)}$ is related with $a_{2n}^{(2)}$ Toda and involves $n$ compact bosons, while regime II is related instead with $B^{(1)}(0,n)$ super Toda, and involves in addition a single Majorana fermion. The most interesting is regime III, where {\sl non-compact} degrees of freedom appear, generalising the emergence of the Euclidean black hole CFT in the $a_{2}^{(2)}$ case. For $a_{2n}^{(2)}$ we find a continuum limit made of $n$ compact and $n$ non-compact bosons, while for $a_{2n-1}^{(2)}$ we find $n$ compact and $n-1$ non-compact bosons. We also find deep relations between $a_{N-1}^{(2)}$ in regime III and the gauged WZW models $SO(N)/SO(N-1)$.Comment: 43 pages, 4 figure

Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model

The geometrical critical behaviour of the two-dimensional Q-state Potts model is usually studied in terms of the Fortuin-Kasteleyn (FK) clusters, or their surrounding loops. In this paper we study a quite different geometrical object: the spin clusters, defined as connected domains where the spin takes a constant value. Unlike the usual loops, the domain walls separating different spin clusters can cross and branch. Moreover, they come in two versions, "thin" or "thick", depending on whether they separate spin clusters of different or identical colours. For these reasons their critical behaviour is different from, and richer than, those of FK clusters. We develop a transfer matrix technique enabling the formulation and numerical study of spin clusters even when Q is not an integer. We further identify geometrically the crossing events which give rise to conformal correlation functions. We study the critical behaviour both in the bulk, and at a boundary with free, fixed, or mixed boundary conditions. This leads to infinite series of fundamental critical exponents, h_{l_1-l_2,2 l_1} in the bulk and h_{1+2(l_1-l_2),1+4 l_1} at the boundary, valid for 0 <= Q <= 4, that describe the insertion of l_1 thin and l_2 thick domain walls. We argue that these exponents imply that the domain walls are `thin' and `thick' also in the continuum limit. A special case of the bulk exponents is derived analytically from a massless scattering approach.Comment: 18 pages, 5 figures, 2 tables. Work based on the invited talk given by Jesper L. Jacobsen at STATPHYS-24 in Cairns, Australia (July 2010). Extended version of arXiv:1008.121