Lattice Realization of Complex Conformal Field Theories: Two-Dimensional Potts Model with Q>4 States.

The two-dimensional Q-state Potts model with real couplings has a first-order transition for Q>4. We study a loop-model realization in which Q is a continuous parameter. This model allows for the collision of a critical and a tricritical fixed point at Q=4, which then emerge as complex conformally invariant theories at Q>4, or even complex Q, for suitable complex coupling constants.

Heisenberg spin chain with random-sign couplings.

We study the 1D quantum Heisenberg chain with randomly ferromagnetic or antiferromagnetic couplings [a model previously studied by approximate strong-disorder renormalization group (RG)]. We find that, at least for sufficiently large spin , the ground state has "spin glass" order. The spin waves on top of this state have the dynamical exponent [Formula: see text], intermediate between the values = 1 of the antiferromagnet and = 2 of the ferromagnet.

Observation of nonscalar and logarithmic correlations in two- and three-dimensional percolation.

In bulk percolation, we exhibit operators that insert N clusters with any given symmetry under the symmetric group S_{N}. At the critical threshold, this leads to predictions that certain combinations of two-point correlation functions depend logarithmically on distance, without the usual power law. The behavior under rotations of certain amplitudes of correlators is also determined exactly.

Duality and the universality class of the three-state Potts antiferromagnet on plane quadrangulations.

We provide a criterion based on graph duality to predict whether the three-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge c=1. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the three-state Potts ferromagnet.

Entanglement in Nonunitary Quantum Critical Spin Chains.

Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to "nonunitary quantum mechanics," which has seen growing interest from areas as diverse as open quantum systems, noninteracting electronic disordered systems, or nonunitary conformal field theory (CFT). We propose and investigate such an extension here, by focusing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains.

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≥25 to c≤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 Dorn-Otto-Zamolodchikov-Zamolodchikov 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.

Recursive percolation.

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and find compelling numerical evidence that it can be repeated recursively any number n of generations. In two dimensions, we determine the percolation thresholds up to n=5. The corresponding critical clusters become more and more compact as n increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any n.

Exact overlaps in the Kondo problem.

It is well known that the ground states of a Fermi liquid with and without a single Kondo impurity have an overlap that decays as a power law of the system size, expressing the Anderson orthogonality catastrophe. Ground states with two different values of the Kondo couplings have, however, a finite overlap in the thermodynamic limit. This overlap, which plays an important role in quantum quenches for impurity systems, is a universal function of the ratio of the corresponding Kondo temperatures, which is not accessible using perturbation theory or the Bethe ansatz.

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 TB, 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.

Two-dimensional Potts antiferromagnets with a phase transition at arbitrarily large q.

We exhibit infinite families of two-dimensional lattices (some of which are triangulations or quadrangulations of the plane) on which the q-state Potts antiferromagnet has a finite-temperature phase transition at arbitrarily large values of q. This unexpected result is proven rigorously by using a Peierls argument to measure the entropic advantage of sublattice long-range order. Additional numerical data are obtained using transfer matrices, Monte Carlo simulation, and a high-precision graph-theoretic method.

Puzzle of bulk conformal field theories at central charge c = 0.

Nontrivial critical models in 2D with a central charge c=0 are described by logarithmic conformal field theories (LCFTs), and exhibit, in particular, mixing of the stress-energy tensor with a "logarithmic" partner under a conformal transformation. This mixing is quantified by a parameter (usually denoted b), introduced in Gurarie [Nucl. Phys.

Integrable spin chain for the SL(2,R)/U(1) black hole sigma model.

We introduce a spin chain based on finite-dimensional spin-1/2 SU(2) representations but with a non-Hermitian "Hamiltonian" and show, using mostly analytical techniques, that it is described at low energies by the SL(2,R)/U(1) Euclidian black hole conformal field theory. This identification goes beyond the appearance of a noncompact spectrum; we are also able to determine the density of states, and show that it agrees with the formulas in [J. Maldacena, H.

Finite-temperature phase transition in a class of four-state Potts antiferromagnets.

We argue that the four-state Potts antiferromagnet has a finite-temperature phase transition on any Eulerian plane triangulation in which one sublattice consists of vertices of degree 4. We furthermore predict the universality class of this transition. We then present transfer-matrix and Monte Carlo data confirming these predictions for the cases of the Union Jack and bisected hexagonal lattices.

Demixing of compact polymers chains in three dimensions.

We present a Monte Carlo algorithm that provides efficient and unbiased sampling of polymer melts consisting of two chains of equal length that jointly visit all the sites of a cubic lattice with rod geometry L × L × rL and nonperiodic (hard wall) boundary conditions. Using this algorithm for chains of length up to 40,000 monomers and aspect ratios 1 ≤ r ≤ 10 , we show that in the limit of a large lattice the two chains phase separate. This demixing phenomenon is present already for r=1 and becomes more pronounced, albeit not perfect, as r is increased.

Exact solution of the anisotropic special transition in the O(n) model in two dimensions.

The effect of surface exchange anisotropies is known to play an important role in magnetic critical and multicritical behavior at surfaces. We give an exact analysis of this problem in d=2 for the O(n) model using the Coulomb gas, conformal invariance, and integrability techniques. We obtain the full set of critical exponents at the anisotropic special transition-where the symmetry on the boundary is broken down to O(n1)xO(n-n1)--as a function of n1.

Semiflexible fully packed loop model and interacting rhombus tilings.

Motivated by a recent adsorption experiment [M. O. Blunt, Science 322, 1077 (2008)10.

Critical interfaces in the random-bond Potts model.

We study geometrical properties of interfaces in the random-temperature q-states Potts model as an example of a conformal field theory weakly perturbed by quenched disorder. Using conformal perturbation theory in q-2 we compute the fractal dimension of Fortuin-Kasteleyn (FK) domain walls. We also compute it numerically both via the Wolff cluster algorithm for q=3 and via transfer-matrix evaluations.

Unbiased sampling of globular lattice proteins in three dimensions.

We present a Monte Carlo method that allows efficient and unbiased sampling of Hamiltonian walks on a cubic lattice. Such walks are self-avoiding and visit each lattice site exactly once. They are often used as simple models of globular proteins, upon adding suitable local interactions.

Role of conformational entropy in force-induced biopolymer unfolding.

A statistical mechanical description of flexible and semiflexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. We predict the existence of many intermediate states at low temperatures stabilized by the force. A unified response to pulling and compressing forces has been obtained in the constant distance ensemble.

Random trimer tilings.

We study tilings of the square lattice by linear trimers. For a cylinder of circumference m , we construct a conserved functional of the base of the tilings, and use this to block diagonalize the transfer matrix. The number of blocks increases exponentially with m .

Secondary structures in long compact polymers.

Compact polymers are self-avoiding random walks that visit every site on a lattice. This polymer model is used widely for studying statistical problems inspired by protein folding. One difficulty with using compact polymers to perform numerical calculations is generating a sufficiently large number of randomly sampled configurations.

Classical dimers with aligning interactions on the square lattice.

We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers. The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase. With large-scale Monte Carlo and transfer matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature.

Unconventional continuous phase transition in a three-dimensional dimer model.

Phase transitions occupy a central role in physics, due both to their experimental ubiquity and their fundamental conceptual importance. The explanation of universality at phase transitions was the great success of the theory formulated by Ginzburg and Landau, and extended through the renormalization group by Wilson. However, recent theoretical suggestions have challenged this point of view in certain situations.

Interacting classical dimers on the square lattice.

We study a model of close-packed dimers on the square lattice with a nearest neighbor interaction between parallel dimers. This model corresponds to the classical limit of quantum dimer models [D. S.

Fermionic field theory for trees and forests.

We prove a generalization of Kirchhoff's 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).