2D quantum spin models and Jordan-Wigner fermions

We apply the 2D Jordan-Wigner fermionization to examine the ground state and thermodynamic properties of the square-lattice s=1/2 anisotropic XY model. We compare our findings with the results of different analytical and numerical approaches.     

Standard-like models from intersecting D4-branes

We construct a one-parameter set of intersecting D4-brane models, with six stacks, that yield the (non-supersymmetric) standard model plus extra vector-like matter. Twisted tadpoles and gauge anomalies are cancelled, and the model contains all of the Yukawa couplings to the tachyonic Higgs doublets that are needed to generate mass terms for the fermions. A string scale in the range 1–10 TeV and a Higgs mass not much greater than the current bound is obtained for certain values of the parameters, consistently with the observed values of the gauge coupling constants.     

Classifying bases for 6D F-theory models

We classify six-dimensional F-theory compactifications in terms of simple features of the divisor structure of the base surface of the elliptic fibration. This structure controls the minimal spectrum of the theory. We determine all irreducible configurations of divisors (??clusters??) that are required to carry nonabelian gauge group factors based on the intersections of the divisors with one another and with the canonical class of the base. All 6D F-theory models are built from combinations of these irreducible configurations. Physically, this geometric structure characterizes the gauge algebra and matter that can remain in a 6D theory after maximal Higgsing. These results suggest that all 6D supergravity theories realized in F-theory have a maximally Higgsed phase in which the gauge algebra is built out of summands of the types su(3), so(8), f₄, e₆, e₈, e₈, (g₂ ?? su(2)); and su(2) ?? so(7) ?? su(2), with minimal matter content charged only under the last three types of summands, corresponding to the non-Higgsable cluster types identified through F-theory geometry. Although we have identified all such geometric clusters, we have not proven that there cannot be an obstruction to Higgsing to the minimal gauge and matter configuration for any possible F-theory model. We also identify bounds on the number of tensor fields allowed in a theory with any fixed gauge algebra; we use this to bound the size of the gauge group (or algebra) in a simple class of F-theory bases.     

Aging in 1D discrete spin models and equivalent systems

We derive exact expressions for a number of aging functions that are scaling limits of nonequilibrium correlations, R(t(w),t(w)+t) as t(w)-->infinity, t/t(w)-->theta, in the 1D homogenous q-state Potts model for all q with T = 0 dynamics following a quench from T = infinity. One such quantity is (0)(t(w));sigma-->(n)(t(w)+t)> when n/square root of ([t(w))-->z. Exact, closed-form expressions are also obtained when an interlude of T = infinity dynamics occurs. Our derivations express the scaling limit via coalescing Brownian paths and a "Brownian space-time spanning tree," which also yields other aging functions, such as the persistence probability of no spin flip at 0 between t(w) and t(w)+t.     

On the large-N limit of 3D and 4D hermitian matrix models

The large-N limit of the hermitian matrix model in three and four euclidean space-time dimensions is studied with the help of the approximate Renormalization Group recursion formula. The planar graphs contributing to wave-function, mass and coupling-constant renormalization are identified and summed in this approximation. In four dimensions the model fails to have an interacting continuum limit, but in three dimensions there is a non-trivial fixed point for the approximate RG relations. The critical exponents of the three-dimensional model at this fixed point are ν = 0.67 and η = 0.20. The existence (or non-existence) of the fixed point and the critical exponents display a fairly high degree of universality since they do not seem to depend on the specific (non-universal) assumptions made in the approximation.     

Exactly solvable 1D frustrated quantum spin models

A class of the frustrated quantum s = ½ models with nearest and next nearest neighbor couplings is investigated. An exact wave function of the singlet ground state at the transition point from the ferromagnetic to the singlet state is presented. The recurrence technics of expectation value calculations is developed and the simple expressions for spin-correlation function at N → ∞ are obtained. A long range double-spiral ordering is demonstrated. We show that in one particular case the model reduces to the effective spin-1 model and the exact singlet ground state wave function is presented for this model. The behavior of the system in the vicinity of the transition point is investigated.     

Nonstationary 2D models of the electron-wave interaction

A 2D nonstationary model of induced emission of ribbon-shaped electron beams and clusters in free space is developed in the quasi-optical approximation. On the basis of this model, the problem of enhancement of a short electromagnetic pulse propagating along a quasi-stationary ribbon electron flow, the theory of a BWT-type oscillator with radiation channeling by an electron beam, and the process of collective acceleration of a short electron cluster in the field of an intense cocurrent wave are considered.     

Phase transition properties of 3D Potts models

Using multicanonical Metropolis simulations we estimate phase transition properties of 3D Potts models for q=4 to 10: The transition temperatures, latent heats, entropy gaps, normalized entropies at the disordered and ordered endpoints, interfacial tensions, and spinodal endpoints.     

Criticality of D=2 and D=3 Ising models: cluster structure versus populations

The energy and the specific heat of two- and three-dimensional Ising systems are analyzed in terms of cluster properties. The energy and the specific heat are decomposed into two components, which are defined by quantities pertaining to cluster populations and cluster structure expressed in terms of average cluster perimeters. It is shown that the structural component of the energy as well as of the specific heat represents the dominant contribution. Indications are presented that the critical exponent of structural and populational components of specific heat matches the exponent of the entire specific heat.     

Randomized Wilson loops,reduced models and the large D expansion

Reduced models are matrix integrals believed to be related to the large N limit of gauge theories. These integrals are known to simplify further when the number of matrices D (corresponding to the number of space–time dimensions in the gauge theory) becomes large. Even though this limit appears to be of little use for computing the standard rectangular Wilson loop (which always singles out two directions out of D), a meaningful large D limit can be defined for a randomized Wilson loop (in which all D directions contribute equally). In this article, a proof-of-concept demonstration of this approach is given for the simplest reduced model (the original Eguchi–Kawai model) and the simplest randomization of the Wilson loop (Brownian sum over random walks). The resulting averaged Wilson loop displays a scale behavior strongly reminiscent of the area law.     

Conformal algebra and multipoint correlation functions in 2D statistical models

Based on the conformal algebra approach, a general technique is given for the calculation of multipoint correlation functions in 2D statistical models at the critical point. Particular conformal operator algebras are found for operators of the 2D q-component Potts model (1 < q < 4), and the O(N) model (0 < N < 2) at the critical point. A number of four-point correlation functions are calculated for these models.     

Efficient algorithm for random-bond ising models in 2D

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.     

SUSY flavor structure of generic 5D supergravity models

We perform a comprehensive and systematic analysis of the SUSY flavor structure of generic 5D supergravity models on S ¹/Z ₂ with multiple Z ₂-odd vector multiplets that generate multiple moduli. The SUSY flavor problem can be avoided due to contact terms in the 4D effective K?hler potential peculiar to the multi-moduli case. A?detailed phenomenological analysis is provided based on an illustrative model.     

The quantum group structure of 2D gravity and minimal models I

On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) _q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) _q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of “q-deformed” factorials and binomial coefficients. Laboratoire Propre du Centre National de la Recherche Scientifique, associé à l'école Normale Supérieure et à l'Université de Paris-Sud     

The quantum group structure of 2D gravity and minimal models I

On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) _q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) _q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of q-deformed factorials and binomial coefficients.     

Inequivalence of first- and second-order formulations in D=2 gravity models

The usual equivalence between the Palalini and metric (or affinity and vielbein) formulations of Einstein theory fails in two spacetime dimensions for its Kaluza-Klein reduced (as well as for its standard) version. Among the differences is the necessary vanishing of the cosmological constant in the first-order forms. The purely affine Eddington formulation of Einstein theory also fails here.The present results were reported in the Proceedings of the Markov Memorial Quantum Gravity Seminar.This work was supported bt the NSF under grant #PHY-9315811.