Critical points in coupled Potts models and critical phases in coupled loop models

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.Comment: 23 pages, 10 figure

Massless Flows I: the sine-Gordon and O(n) models

The massless flow between successive minimal models of conformal field theory is related to a flow within the sine-Gordon model when the coefficient of the cosine potential is imaginary. This flow is studied, partly numerically, from three different points of view. First we work out the expansion close to the Kosterlitz-Thouless point, and obtain roaming behavior, with the central charge going up and down in between the UV and IR values of $c=1$. Next we analytically continue the Casimir energy of the massive flow (i.e. with real cosine term). Finally we consider the lattice regularization provided by the O(n) model in which massive and massless flows correspond to high- and low-temperature phases. A detailed discussion of the case $n=0$ is then given using the underlying N=2 supersymmetry, which is spontaneously broken in the low-temperature phase. The ``index'' \tr F(-1)^F follows from the Painleve III differential equation, and is shown to have simple poles in this phase. These poles are interpreted as occuring from level crossing (one-dimensional phase transitions for polymers). As an application, new exact results for the connectivity constants of polymer graphs on cylinders are obtained.Comment: 39 pages, 7 uuencoded figures, BUHEP-93-5, USC-93/003, LPM-93-0

Thermodynamics of the Complex su(3) Toda Theory

We present the first computation of the thermodynamic properties of the complex su(3) Toda theory. This is possible thanks to a new string hypothesis, which involves bound states that are non self-conjugate solutions of the Bethe equations. Our method provides equivalently the solution of the su(3) generalization of the XXZ chain. In the repulsive regime, we confirm that the scattering theory proposed over the past few years - made only of solitons with non diagonal S-matrices - is complete. But we show that unitarity does not follow, contrary to early claims, eigenvalues of the monodromy matrix not being pure phases. In the attractive regime, we find that the proposed minimal solution of the bootstrap equations is actually far from being complete. We discuss some simple values of the couplings, where, instead of the few conjectured breathers, a very complex structure (involving E_6, or two E_8) of bound states is necessary to close the bootstrap.Comment: 6 pages, 2 figures; some minor changes; accepted for publication in Phys. Lett.

Critical exponents of domain walls in the two-dimensional Potts model

We address the geometrical critical behavior of the two-dimensional Q-state Potts model in terms of the spin clusters (i.e., connected domains where the spin takes a constant value). These clusters are different from the usual Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross and branch. 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. This leads to an infinite series of fundamental critical exponents h_{l_1-l_2,2 l_1}, valid for 0 </- Q </- 4, that describe the insertion of l_1 thin and l_2 thick domain walls.Comment: 5 pages, 3 figures, 1 tabl

Loop models and their critical points

Loop models have been widely studied in physics and mathematics, in problems ranging from polymers to topological quantum computation to Schramm-Loewner evolution. I present new loop models which have critical points described by conformal field theories. Examples include both fully-packed and dilute loop models with critical points described by the superconformal minimal models and the SU(2)_2 WZW models. The dilute loop models are generalized to include SU(2)_k models as well.Comment: 20 pages, 15 figure

Self-duality in quantum impurity problems

We establish the existence of an exact non-perturbative self-duality in a variety of quantum impurity problems, including the Luttinger liquid or quantum wire with impurity. The former is realized in the fractional quantum Hall effect, where the duality interchanges electrons with Laughlin quasiparticles. We discuss the mathematical structure underlying this property, which bears an intriguing resemblance with the work of Seiberg and Witten on supersymmetric non-abelian gauge theory.Comment: 4 page

A unified framework for the Kondo problem and for an impurity in a Luttinger liquid

We develop a unified theoretical framework for the anisotropic Kondo model and the boundary sine-Gordon model. They are both boundary integrable quantum field theories with a quantum-group spin at the boundary which takes values, respectively, in standard or cyclic representations of the quantum group $SU(2)_q$. This unification is powerful, and allows us to find new results for both models. For the anisotropic Kondo problem, we find exact expressions (in the presence of a magnetic field) for all the coefficients in the ``Anderson-Yuval'' perturbative expansion. Our expressions hold initially in the very anisotropic regime, but we show how to continue them beyond the Toulouse point all the way to the isotropic point using an analog of dimensional regularization. For the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid, we find the non-equilibrium conductance for all values of the Luttinger coupling.Comment: 36 pages (22 in double-page format), 7 figures in uuencoded file, uses harvmac and epsf macro

Detailed analysis of the continuum limit of a supersymmetric lattice model in 1D

We present a full identification of lattice model properties with their field theoretical counter parts in the continuum limit for a supersymmetric model for itinerant spinless fermions on a one dimensional chain. The continuum limit of this model is described by an $\mathcal{N}=(2,2)$ superconformal field theory (SCFT) with central charge c=1. We identify states and operators in the lattice model with fields in the SCFT and we relate boundary conditions on the lattice to sectors in the field theory. We use the dictionary we develop in this paper, to give a pedagogical explanation of a powerful tool to study supersymmetric models based on spectral flow. Finally, we employ the developed machinery to explain numerically observed properties of the particle density on the open chain presented in Beccaria et al. PRL 94:100401 (2005).Comment: 28 pages, 7 figures, 3 tables, 1 appendix, this work is based on chapter 4 of the authors PhD Thesis: L. Huijse, A supersymmetric model for lattice fermions, University of Amsterdam (2010

Hyperelliptic curves for multi-channel quantum wires and the multi-channel Kondo problem

We study the current in a multi-channel quantum wire and the magnetization in the multi-channel Kondo problem. We show that at zero temperature they can be written simply in terms of contour integrals over a (two-dimensional) hyperelliptic curve. This allows one to easily demonstrate the existence of weak-coupling to strong-coupling dualities. In the Kondo problem, the curve is the same for under- and over-screened cases; the only change is in the contour.Comment: 7 pages, 1 figure, revte

Exact non-equilibrium DC shot noise in Luttinger liquids and fractional quantum Hall devices

A point contact in a Luttinger liquid couples the left- and right-moving channels, producing shot noise. We calculate exactly the DC shot noise at zero temperature in the out-of-equilibrium steady state where current is flowing. Integrability of the interaction ensures the existence of a quasiparticle basis where quasiparticles scatter ``one by one'' off the point contact. This enables us to apply a direct generalization of the Landauer approach to shot noise to this interacting model. We find a simple relation of the noise to the current and the differential conductance. Our results should be experimentally-testable in a fractional quantum Hall effect device, providing a clear signal of the fractional charge of the Laughlin quasiparticles.Comment: 4 pages in revtex two-column, one figure