The dilute Temperley-Lieb O(n=1n=1) loop model on a semi infinite strip: the ground state

We consider the integrable dilute Temperley-Lieb (dTL) O($n=1$) loop model on a semi-infinite strip of finite width $L$. In the analogy with the Temperley-Lieb (TL) O($n=1$) loop model the ground state eigenvector of the transfer matrix is studied by means of a set of $q$-difference equations, sometimes called the $q$KZ equations. We compute some ground state components of the transfer matrix of the dTL model, and show that all ground state components can be recovered for arbitrary $L$ using the $q$KZ equation and certain recurrence relation. The computations are done for generic open boundary conditions.Comment: 25 pages, 30 figures, Updated versio

Exact characterization of O(n) tricriticality in two dimensions

We propose exact expressions for the conformal anomaly and for three critical exponents of the tricritical O(n) loop model as a function of n in the range $-2 \leq n \leq 3/2$. These findings are based on an analogy with known relations between Potts and O(n) models, and on an exact solution of a 'tri-tricritical' Potts model described in the literature. We verify the exact expressions for the tricritical O(n) model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.Comment: submitted to Phys. Rev. Let

End to end distance on contour loops of random gaussian surfaces

A self consistent field theory that describes a part of a contour loop of a random Gaussian surface as a trajectory interacting with itself is constructed. The exponent \nu characterizing the end to end distance is obtained by a Flory argument. The result is compared with different previuos derivations and is found to agree with that of Kondev and Henley over most of the range of the roughening exponent of the random surface.Comment: 7 page

Lattice Ising model in a field: E8_8 scattering theory

Zamolodchikov found an integrable field theory related to the Lie algebra E$_8$, which describes the scaling limit of the Ising model in a magnetic field. He conjectured that there also exist solvable lattice models based on E$_8$ in the universality class of the Ising model in a field. The dilute A$_3$ model is a solvable lattice model with a critical point in the Ising universality class. The parameter by which the model can be taken away from the critical point acts like a magnetic field by breaking the \Integer_2 symmetry between the states. The expected direct relation of the model with E$_8$ has not been found hitherto. In this letter we study the thermodynamics of the dilute A$_3$ model and show that in the scaling limit it exhibits an appropriate E$_8$ structure, which naturally leads to the E$_8$ scattering theory for massive excitations over the ground state.Comment: 11 pages, LaTe

Long range order in the classical kagome antiferromagnet: effective Hamiltonian approach

Following Huse and Rutenberg [Phys. Rev. B 45, 7536 (1992)], I argue the classical Heisenberg antiferromagnet on the kagom\'e lattice has long-range spin order of the $\sqrt{3}\times\sqrt{3}$ type (modulo gradual orientation fluctuations of the spins' plane). I start from the effective quartic Hamiltonian for the soft (out of plane) spin fluctuation modes, and treat as a perturbation those terms which depend on the discrete coplanar state. Soft mode correlations, which become the coefficients of a discrete effective Hamiltonian, are estimated analytically.Comment: 4pp, no figures. Converted to PRB format, extensive revisions/some reorderings to improve clarity; some cut

Conducting-angle-based percolation in the XY model

We define a percolation problem on the basis of spin configurations of the two dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.Comment: 16 pages, 14 figure