Bethe Equations "on the Wrong Side of Equator"

We analyse the famous Baxter's $T-Q$ equations for $XXX$ ($XXZ$) spin chain and show that apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe-Ansatz equations, there exists also the second solution which should corresponds to Bethe-Ansatz beyond $N/2$. This second solution of Baxter's equation plays essential role and together with the first one gives rise to all fusion relations.Comment: 13 pages, original paper was spoiled during transmissio

Sexual negotiation in the AIDS era: negotiated safety revisited

Objective: To test the safety of the 'negotiated safety' strategy - the strategy of dispensing with condoms within HIV-seronegative concordant regular sexual relationships under certain conditions. Method: Data from a recently recruited cohort of homosexually active men (Sydney Men and Sexual Health cohort, n = 1037) are used to revisit negotiated safety. The men were surveyed using a structured questionnaire and questions addressing their sexual relationships and practice, their own and their regular partner's serostatus, agreements entered into by the men concerning sexual practice within and outside their regular relationship, and contextual and demographic variables. Results: The findings indicate that a significant number of men used negotiated safety as an HIV prevention strategy. In the 6 months prior to interview, of the 181 men in seroconcordant HIV-negative regular relationships, 62% had engaged in unprotected anal intercourse within their relationship, and 91% (165 men) had not engaged in unprotected anal intercourse outside their relationship. Of these 165 men, 82% had negotiated agreements about sex outside their relationship. The safety of negotiation was dependent not only on seroconcordance but also on the presence of an agreement; 82% of the men who had not engaged in unprotected anal intercourse outside their regular relationship had entered into an agreement with their partner, whereas only 56% of those who had engaged in unprotected anal intercourse had an agreement. The safety of negotiation was also related to the nature of the safety agreement reached between the men and on the acceptability of condoms. Agreements between HIV-negative seroconcordant regular partners prohibiting anal intercourse with casual partners or any form of sex with a casual partner were typically complied with, and men who had such negotiated agreements were at low risk of HIV infection. Conclusions: The adoption of the strategy of negotiated safety among men in HIV-seronegative regular relationships may help such men sustain the safety of their sexual practice

Bethe Ansatz Equations for the Broken ZNZ_{N}-Symmetric Model

We obtain the Bethe Ansatz equations for the broken ${\bf Z}_{N}$-symmetric model by constructing a functional relation of the transfer matrix of $L$-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.Comment: 43 pages, latex, 11 figure

General scalar products in the arbitrary six-vertex model

In this work we use the algebraic Bethe ansatz to derive the general scalar product in the six-vertex model for generic Boltzmann weights. We performed this calculation using only the unitarity property, the Yang-Baxter algebra and the Yang-Baxter equation. We have derived a recurrence relation for the scalar product. The solution of this relation was written in terms of the domain wall partition functions. By its turn, these partition functions were also obtained for generic Boltzmann weights, which provided us with an explicit expression for the general scalar product.Comment: 24 page

Diamagnetic susceptibility obtained from the six-vertex model and its implications for the high-temperature diamagnetic state of cuprate superconductors

We study the diamagnetism of the 6-vertex model with the arrows as directed bond currents. To our knowledge, this is the first study of the diamagnetism of this model. A special version of this model, called F model, describes the thermal disordering transition of an orbital antiferromagnet, known as d-density wave (DDW), a proposed state for the pseudogap phase of the high-Tc cuprates. We find that the F model is strongly diamagnetic and the susceptibility may diverge in the high temperature critical phase with power law arrow correlations. These results may explain the surprising recent observation of a diverging low-field diamagnetic susceptibility seen in some optimally doped cuprates within the DDW model of the pseudogap phase.Comment: 4.5 pages, 2 figures, revised version accepted in Phys. Rev. Let

A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded. Version 3 shows that the worst-case running time is sub-exponential in the number of vertice

Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the q-colouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ~ 3^m, where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q^{-79}. On the triangular lattice, we verify Baxter's analytical expression for the bulk free energy (to order q^{-40}), and we are able to conjecture exact product formulae for the surface and corner free energies.Comment: 17 pages. Version 2: added 4 further term to the serie

Scattering Rule in Soliton Cellular Automaton associated with Crystal Base of Uq(D4(3))U_q(D_4^{(3)})

In terms of the crystal base of a quantum affine algebra $U_q(\mathfrak{g})$, we study a soliton cellular automaton (SCA) associated with the exceptional affine Lie algebra $\mathfrak{g}=D_4^{(3)}$. The solitons therein are labeled by the crystals of quantum affine algebra $U_q(A_1^{(1)})$. The scatteing rule is identified with the combinatorial $R$ matrix for $U_q(A_1^{(1)})$-crystals. Remarkably, the phase shifts in our SCA are given by {\em 3-times} of those in the well-known box-ball system.Comment: 25 page

Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions

We address the general problem of hard objects on random lattices, and emphasize the crucial role played by the colorability of the lattices to ensure the existence of a crystallization transition. We first solve explicitly the naive (colorless) random-lattice version of the hard-square model and find that the only matter critical point is the non-unitary Lee-Yang edge singularity. We then show how to restore the crystallization transition of the hard-square model by considering the same model on bicolored random lattices. Solving this model exactly, we show moreover that the crystallization transition point lies in the universality class of the Ising model coupled to 2D quantum gravity. We finally extend our analysis to a new two-particle exclusion model, whose regular lattice version involves hard squares of two different sizes. The exact solution of this model on bicolorable random lattices displays a phase diagram with two (continuous and discontinuous) crystallization transition lines meeting at a higher order critical point, in the universality class of the tricritical Ising model coupled to 2D quantum gravity.Comment: 48 pages, 13 figures, tex, harvmac, eps

A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models

In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.Comment: 22 pages, Latex2e. This replacement is a revised version that includes a simple explicit expression for the Q matrix for the 6-vertex mode