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Register a new userLattice study of sphaleron transitions in a 2D O(3) sigma model
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A lattice approach is developed to measure the sphaleron free energy. Its feasibility is demonstrated through a Monte Carlo study of the two-dimensional O(3) sigma model.
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We present our progress on a study of the $O(3)$ model in two-dimensions using the Tensor Renormalization Group method. We first construct the theory in terms of tensors, and show how to construct $n$-point correlation functions. We then give results for thermodynamic quantities at finite and infinite volume, as well as 2-point correlation function data. We discuss some of the advantages and challenges of tensor renormalization and future directions in which to work.
The 2d O(3) model is widely used as a toy model for ferromagnetism and for Quantum Chromodynamics. With the latter it shares --- among other basic aspects --- the property that the continuum functional integral splits into topological sectors. Topology can also be defined in its lattice regularised version, but semi-classical arguments suggest that the topological susceptibility $chi_{rm t}$ does not scale towards a finite continuum limit. Previous numerical studies confirmed that the quantity $chi_{rm t}, xi^{2}$ diverges at large correlation length $xi$. Here we investigate the question whether or not this divergence persists when the configurations are smoothened by the Gradient Flow (GF). The GF destroys part of the topological windings; on fine lattices this strongly reduces $chi_{rm t}$. However, even when the flow time is so long that the GF impact range --- or smoothing radius --- attains $xi/2$, we do still not observe evidence of continuum scaling.
Lattice artifacts in the 2d O(n) non-linear sigma-model are expected to be of the form O(a^2), and hence it was (when first observed) disturbing that some quantities in the O(3) model with various actions show parametrically stronger cutoff dependence, apparently O(a), up to very large correlation lengths. In a previous letter we described the solution to this puzzle. Based on the conventional framework of Symanziks effective action, we showed that there are logarithmic corrections to the O(a^2) artifacts which are especially large, (ln(a))^3, for n=3 and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3) and O(4) are also presented.
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 $ell=L_t/L_stoinfty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.
We study the O(3) sigma model in $D=2$ on the lattice with a Boltzmann weight linearized in $beta$ on each link. While the spin formulation now suffers from a sign-problem the equivalent loop model remains positive and becomes particularly simple. By studying the transfer matrix and by performing Monte Carlo simulations in the loop form we study the mass gap coupling in a step scaling analysis. The question addressed is, whether or not such a simplified action still has the right universal continuum limit. If the answer is affirmative this would be helpful in widening the applicability of worm algorithm methods.
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