Asymptotic freedom with discrete spin variables?

We study the critical behaviour of the 2d dodecahedron spin model and investigate the conjecture that the discrete model describes the same continuum theory as the O(3) non-linear sigma model. In particular, we found that the anisotropy of the magnetization A(z) measured in a fixed physical volume decreases with increasing correlation length, at least up to \xi \approx 1000.Comment: 5 pages, 4 figure

Isospin susceptibility in the O(nn) sigma-model in the delta-regime

We compute the isospin susceptibility in an effective O($n$) scalar field theory (in $d=4$ dimensions), to third order in chiral perturbation theory ($\chi$PT) in the delta--regime using the quantum mechanical rotator picture. This is done in the presence of an additional coupling, involving a parameter $\eta$, describing the effect of a small explicit symmetry breaking term (quark mass). For the chiral limit $\eta=0$ we demonstrate consistency with our previous $\chi$PT computations of the finite-volume mass gap and isospin susceptibility. For the massive case by computing the leading mass effect in the susceptibility using $\chi$PT with dimensional regularization, we determine the $\chi$PT expansion for $\eta$ to third order. The behavior of the shape coefficients for long tube geometry obtained here might be of broader interest. The susceptibility calculated from the rotator approximation differs from the $\chi$PT result in terms vanishing like $1/\ell$ for $\ell=L_t/L_s\to\infty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.Comment: 34 page

New fixed point action for SU(3) lattice gauge theory

We present a new fixed point action for SU(3) lattice gauge theory, which has --- compared to earlier published fixed point actions --- shorter interaction range and smaller violations of rotational symmetry in the static q\bar{q}-potential even at shortest distances

Non-trivial \theta-Vacuum Effects in the 2-d O(3) Model

We study \theta-vacua in the 2-d lattice O(3) model using the standard action and an optimized constraint action with very small cut-off effects, combined with the geometric topological charge. Remarkably, dislocation lattice artifacts do not spoil the non-trivial continuum limit at \theta\ non-zero, and there are different continuum theories for each value of \theta. A very precise Monte Carlo study of the step scaling function indirectly confirms the exact S-matrix of the 2-d O(3) model at \theta = \pi.Comment: 4 pages, 3 figure

Casimir squared correction to the standard rotator Hamiltonian for the O(nn) sigma-model in the delta-regime

In a previous paper we found that the isospin susceptibility of the O($n$) sigma-model calculated in the standard rotator approximation differs from the next-to-next to leading order chiral perturbation theory result in terms vanishing like $1/\ell\,,$ for $\ell=L_t/L\to\infty$ and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Heged\"us for $n=3,4$ and by Gromov, Kazakov and Vieira for $n=4$. We also consider the case of 3 dimensions.Comment: 25 pages, 1 figur

Study of theta-Vacua in the 2-d O(3) Model

We investigate the continuum limit of the step scaling function in the 2-d O(3) model with different theta-vacua. Since we find a different continuum value of the step scaling function for each value of theta, we can conclude that theta indeed is a relevant parameter of the theory and does not get renormalized non-perturbatively. Furthermore, we confirm the result of the conjectured exact S-matrix theory, which predicts the continuum value at theta = pi. To obtain high precision data, we use a modified Hasenbusch improved estimator and an action with an optimized constraint, which has very small cut-off effects. The optimized constraint action combines the standard action of the 2-d O(3) model with a topological action. The topological action constrains the angle between neighboring spins and is therefore invariant against small deformations of the field.Comment: 7 pages, 4 figures, The 30 International Symposium on Lattice Field Theory - Lattice 2012, June 24-29, 2012, Cairns, Australi