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Register a new userAsymptotic scattering and duality in the one-dimensional three-state quantum Potts model on a lattice
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We determine numerically the single-particle and the two-particle spectrum of the three-state quantum Potts model on a lattice by using the density matrix renormalization group method, and extract information on the asymptotic (small momentum) S-matrix of the quasiparticles. The low energy part of the finite size spectrum can be understood in terms of a simple effective model introduced in a previous work, and is consistent with an asymptotic S-matrix of an exchange form below a momentum scale $p^*$. This scale appears to vanish faster than the Compton scale, $mc$, as one approaches the critical point, suggesting that a dangerously irrelevant operator may be responsible for the behavior observed on the lattice.
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A phenomenological approach to the ferromagnetic two dimensional Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent {alpha} allow to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.
The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is SLE with kappa=3. We hypothesise that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries which are also SLE in the scaling limit, but with kappa=10/3. To test this, we generate samples using the Wolff algorithm and test them against predictions of SLE: we examine the statistics of the Loewner driving function, estimate the fractal dimension and test against Schramms formula. The results are in support of our hypothesis.
We perform a detailed investigation of the scaling Potts field theory using the truncated conformal space approach.
The surface and bulk properties of the two-dimensional Q > 4 state Potts model in the vicinity of the first order bulk transition point have been studied by exact calculations and by density matrix renormalization group techniques. For the surface transition the complete analytical solution of the problem is presented in the $Q to infty$ limit, including the critical and tricritical exponents, magnetization profiles and scaling functions. According to the accurate numerical results the universality class of the surface transition is independent of the value of Q > 4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and in the disordered phases are identical at the transition point.
The scaling limit as T->0 of the antiferromagnetic three-state Potts model on the square lattice is described by the sine-Gordon quantum field theory at a specific value of the coupling. We show that the correspondence follows unambigously from an analysis of the sine-Gordon operator space based on locality, and that the scalar operators carrying solitonic charge play an essential role in the description of the lattice model. We then evaluate the correlation functions within the form factor approach and give a number of universal predictions that can be checked in numerical simulations.
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