Coupled minimal models with and without disorder

We analyze in this article the critical behavior of M q₁-state Potts models coupled to N q₂-state Potts models (q₁, q₂ ε [2, …, 4]) with and without disorder. The techniques we use are based on perturbed conformal theories. Calculations have been performed at two loops. We already find some interesting situations in the pure case for some peculiar values of M and N with new tricritical points. When adding weak disorder, the results we obtain tend to show that disorder makes the models decouple. Therefore, no relations emerges, at a perturbation level, between for example the disordered q₁ × q₂-state Potts model and the two disordered q₁, q₂-state Potts models (q₁ ≠ q₂), despite the fact that their central charges are similar according to recent numerical investigations.     

Boundary critical behaviour of two-dimensional random Potts models

Using extensive Monte Carlo simulations, transfer matrix techniques and conformal invariance, ferromagnetic random q-state Potts models for are studied in the vicinity of the critical temperature. In particular the surface and bulk magnetization exponents and are found monotonically increasing with q. At the critical temperature, different moments (n) of the magnetization profiles are calculated which are all found to accurately follow predictions of conformal invariance. The critical correlation functions show multifractal behaviour, the decay exponents of the different moments both in the volume and at the surface, are n-dependent. Received 4 June 1999     

Moduli integrals in liouville gravity

We consider moduli integrals appearing in four-point correlation functions of the (p, q) minimal models coupled to Liouville gravity on a sphere, which is sometimes called 2D minimal gravity or minimal string theory on a sphere. Liouville gravity on a sphere is the quantized metric of the spherical topology in the conformal gauge. Reviewing the previous results on such four-point functions (Y. Ishimoto and Sh. Yamaguchi: Phys. Lett. B607 (2005) 172), we show logarithmic correlation functions of ‘tachyons’ in the Liouville sector, and its moduli integrals of the full correlation functions, in particular in the Majorana fermion model coupled to 2D gravity. Further discussions and related results are given in the final section and in Y. Ishimoto and Al. Zamolodchikov: Theor. Math. Phys.147 (2006) 755.     

Phase transitions and critical phenomena in a three-dimensional site-diluted Potts model

The phase transitions and critical phenomena in the three-dimensional (3D) site-diluted q-state Potts models on a simple cubic lattice are explored. We systematically study the phase transitions of the models for q=3 and q=4 on the basis of Wolff high-effective algorithm by the Monte–Carlo (MC) method. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.00–0.65. It is shown that introducing of weak disorder (p∼0.95) into the system is sufficient to change the first order phase transition into a second order one for the 3D 3-state Potts model, while for the 3D 4-state Potts model, such a phase transformation occurs when introducing strong disorder (p∼0.65). Results for 3D pure 3-state and 4-state Potts models (p=1.00) agree with conclusions of mean field theory. The static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the exponent of the correlation radius ν are calculated for the samples on the basis of finite-size scaling theory.     

Conformal invariance and surface critical behavior

Conformal invariance constrains the form of correlation functions near a free surface. In two dimensions, for a wide class of models, it completely determines the correlation functions at the critical point, and yields the exact values of the surface critical exponents. They are related to the bulk exponents in a non-trivial way. For the Q-state Potts model (0 Q 4) we find η_<|; = 2/(3v − 1), and for the O(N) model (−2 N 2), η_<|; = (2v − 1)/(4v − 1).     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

A study of the critical behavior of the O(N) linear and nonlinear σ models

A study of the large N behavior of both the O(N) linear and nonlinear σ models is presented. The purpose is to investigate the relationship between the disordered (ordered) phase of the linear and nonlinear sigma models. Utilizing operator product expansions and stability analyses, it is shown that for 2 ≤ d < 4, it is the dimensionless renormalized quartic coupling and $\text{λ}{}^1$ is the IR fixed point) limit of the linear σ model which yields the nonlinear σ model. It is also shown that stable large N linear σ models with λ < 0 (σ is the bare quartic coupling) can exist (at least in the context of no tachyonic states being present). A criteria valid for all dimensionalities d, less than four, is derived which determines when λ < 0 models are tachyonic free. Arguments are given showing that the d = 4 large N linear (for λ > 0) and nonlinear models are trivial. This result (i.e., triviality) is well known but only for one and two component models. Interestingly enough, the λ < 0, d = 4 linear σ model remains nontrivial and tachyonic free.     

Phase structure and critical behavior of multi-Higgs U(1) lattice gauge theory in three dimensions

We study the three-dimensional (3D) compact U(1) lattice gauge theory coupled with N-flavor Higgs fields by means of the Monte Carlo simulations. This model is relevant to multi-component superconductors, antiferromagnetic spin systems in easy plane, inflational cosmology, etc. It is known that there is no phase transition in the N = 1 model. For N = 2, we found that the system has a second-order phase transition line in the c₂ (gauge coupling)-c₁ (Higgs coupling) plane, which separates the confinement phase and the Higgs phase. Numerical results suggest that the phase transition belongs to the universality class of the 3D XY model as the previous works by Babaev et al. and Smiseth et al. suggested. For N = 3, we found that there exists a critical line similar to that in the N = 2 model, but the critical line is separated into two parts; one for c₂<c_2tc=2.4±0.1 with first-order transitions, and the other for c_2tc<c₂ with second-order transitions, indicating the existence of a tricritical point. We verified that similar phase diagram appears for the N = 4 and N = 5 systems. We also studied the case of anistropic Higgs coupling in the N = 3 model and found that there appear two second-order phase transitions or a single second-order transition and a crossover depending on the values of the anisotropic Higgs couplings. This result indicates that an “enhancement” of phase transition occurs when multiple phase transitions coincide at a certain point in the parameter space.     

Nonlocal current algebra and N = 2 superconformal field theory in two dimensions

We present a new construction of the unitary highest weight representations of the N = 2 superconformal algebras in two dimensions. This construction is based on the non-local current in the Z_N conformal theory and a free scalar field. It provides a physical realization of all the unitary N = 2 superconformal field theories by critical systems. The correlation functions of the theories can be calculated through this construction.     

Conformal field theories and modular invariant partition functions for parafermion field theories

Parafermion conformal field theories with D_2N discrete symmetry are examined in detail. The structure of field space of parafermion field theories is studied with the help of a projection operator G. Characters of the representations of the twist sector of parafermion algebra and projected characters are given. A new class of modular invariant partition functions, therefore conformal field theories, for parafermion theories are found. We argue that the principal theories correspond to the generic critical SOS models of Andrew, Baxter and Forrest.     

Scale and conformal covariance in quantum electrodynamics

The paper consists of two independent parts. First, we review the situationin scale invariant massless QED from an axiomatic standpoint. Assuming that the τ-functions (or, equivalently, the Euclidean Schwinger functions) transform covariantly under dilatations, we deduce that the current j_τ(x) has zero n-point Wightman functions, but nonvanishing τ-functions. Assuming in addition conformal invariance of the current-field vertex function, we write down a bootstrap equation similar to the one derived in [7] from the point of view of pertubation theory. Next we consider a non-Lagrangian, conformal invariant model of interacting antisymmetric tensor field F_μv (of scale dimension d) and Dirac field ψ (of dimension d'. The model involves two conserved currents (an “electric” and a “magnetic” one) and two effective coupling constants. We demonstrate that it is free of ultravioletdivergences in the range of dimensions $\text{2 < d < 3,}\text{3}\text{2}\text{< d′ <}\text{5}\text{2}$.     

The W N Minimal Model Classification

We first rigourously establish, for any N ≥ 2, that the toroidal modular invariant partition functions for the (not necessarily unitary) W _N (p, q) minimal models biject onto a well-defined subset of those of the SU(N) × SU(N) Wess-Zumino-Witten theories at level (p − N, q − N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W ₃(p, q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W ₃(p, p + 1), were classified. For all N our correspondence yields for free an extensive list of W _N (p, q) modular invariants. The W ₃ modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3) × SU(3) modular invariants, and find there a new infinite series of exceptionals.     

Large energy cumulants in the 2D Potts model and their effects in finite size analysis

We develop an ansatz for expressing the free energy of the two-dimensional q-states Potts model for q > 4 near its first order phase transition point. We notice that for the moderate values of q < 15, the energy profile at the phase transition is not expressible as a sum of gaussians. We discuss how this affects the traditional finite size analysis of this phase transition. In particular, the dominant length scale governing the finite size corrections turns out to be much (∼ 6 times) larger than the largest correlation length in the problem.     

Boundary correlation functions of the six and nineteen vertex models with domain wall boundary conditions

Boundary correlation functions of the six and nineteen vertex models on an N×N lattice with domain wall boundary conditions are studied. The general expression of the boundary correlation functions is obtained for the six vertex model by the use of the quantum inverse scattering method. For the nineteen vertex model, the boundary correlation functions are shown to be expressed in terms of those for the six vertex model.     

Critical behaviour of random-bond Potts models: a transfer matrix study

We study the two-dimensional Potts model on the square lattice in the presence of quenched random-bond impurities. For q > 4 the first-order transitions of the pure model are softened due to the impurities, and we determine the resulting universality classes by combining transfer matrix data with conformal invariance. The magnetic exponent β/v varies continuously with q, assuming non-Ising values for q > 4, whereas the correlation length exponent ν is numerically consistent with unity. We present evidence for the correctness of a formerly proposed phase diagram, unifying pure, percolative and non-trivial random behaviour.     

Asymptotic correlation functions and FFLO signature for the one-dimensional attractive spin-1/2 Fermi gas

We investigate the long distance asymptotics of various correlation functions for the one-dimensional spin-1/2 Fermi gas with attractive interactions using the dressed charge formalism. In the spin polarized phase, these correlation functions exhibit spatial oscillations with a power-law decay whereby their critical exponents are found through conformal field theory. We show that spatial oscillations of the leading terms in the pair correlation function and the spin correlation function solely depend on ΔkF and 2ΔkF, respectively. Here ΔkF=π(n_↑−n_↓) denotes the mismatch between the Fermi surfaces of spin-up and spin-down fermions. Such spatial modulations are characteristics of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. Our key observation is that backscattering among the Fermi points of bound pairs and unpaired fermions results in a one-dimensional analog of the FFLO state and displays a microscopic origin of the FFLO nature. Furthermore, we show that the pair correlation function in momentum space has a peak at the point of mismatch between both Fermi surfaces k=ΔkF, which has recently been observed in numerous numerical studies.     

Monte Carlo study of the phase transitions in 2D ferro- and antiferromagnetic Potts models on a triangular lattice

The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.     

First-order phase transitions and integrable field theory. The dilute q-state Potts model

We consider the two-dimensional dilute q-state Potts model on its first-order phase transition surface for 0 < q ⩽ 4. After determining the exact scattering theory which describes the scaling limit, we compute the two-kink form factors of the dilution, thermal and spin operators. They provide an approximation for the correlation functions whose accuracy is illustrated by evaluating the central charge and the scaling dimensions along the tricritical line.