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Register a new userChiral perturbation theory, finite size effects and the three-dimensional $XY$ model
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We study finite size effects of the d=3 $XY$ model in terms of the chiral perturbation theory. We calculate by Monte Carlo simulations physical quantities which are, to order of $(1/L)^2$, uniquely determined only by two low energy constants. They are the magnetization and the helicity modulus (or the Goldstone boson decay constant) in infinite volume. We also pay a special attention to the region of the validity of the two possible expansions in the theory.
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A lagrangian which describes interactions between a soliton and a background field is derived for sigma models whose target is a symmetric space. The background field modifies the usual moduli space approximation to soliton dynamics in two ways: by introducing a potential energy, and by inducing a Kaluza-Klein metric on the moduli space. In the particular case of the Skyrme model, this lagrangian is quantised and shown to agree with the leading pion-nucleon term in the chiral effective lagrangian, which is widely used in theoretical nuclear physics. Thus chiral perturbation theory could be considered a low energy limit of the Skyrme model.
Although taste violations significantly affect the results of staggered calculations of pseudoscalar and heavy-light mesonic quantities, those entering staggered calculations of baryonic quantities have not been quantified. Here I develop staggered chiral perturbation theory in the light-quark baryon sector by mapping the Symanzik action into heavy baryon chiral perturbation theory. For 2+1 dynamical quark flavors, the masses of flavor-symmetric nucleons are calculated to third order in partially quenched and fully dynamical staggered chiral perturbation theory. To this order the expansion includes the leading chiral logarithms, which come from loops with virtual decuplet-like states, as well as terms the order of the cubed pion mass, which come from loops with virtual octet-like states. Taste violations enter through the meson propagators in loops and tree-level terms the order of the squared lattice spacing. The pattern of taste symmetry breaking and the resulting degeneracies and mixings are discussed in detail. The resulting chiral forms are appropriate to lattice results obtained with operators already in use and could be used to study the restoration of taste symmetry in the continuum limit. I assume that the fourth root of the fermion determinant can be incorporated in staggered chiral perturbation theory using the replica method.
The dimensionless parameter $xi = M^2/(16 pi^2 F^2)$, where $F$ is the pion decay constant in the chiral limit and $M$ is the pion mass at leading order in the quark mass, is expected to control the convergence of chiral perturbation theory applicable to QCD. Here we demonstrate that a strongly coupled lattice gauge theory model with the same symmetries as two-flavor QCD but with a much lighter $sigma$-resonance is different. Our model allows us to study efficiently the convergence of chiral perturbation theory as a function of $xi$. We first confirm that the leading low energy constants appearing in the chiral Lagrangian are the same when calculated from the $epsilon$-regime and the $p$-regime. However, $xi lesssim 0.002$ is necessary before 1-loop chiral perturbation theory predicts the data within 1%. However, for $xi > 0.0035$ the data begin to deviate qualitatively from 1-loop chiral perturbation theory predictions. We argue that this qualitative change is due to the presence of a light $sigma$-resonance in our model. Our findings may be useful for lattice QCD studies.
We revisit two-color, two-flavor chiral perturbation theory at finite isospin and baryon density. We investigate the phase diagram obtained varying the isospin and the baryon chemical potentials, focusing on the phase transition occurring when the two chemical potentials are equal and exceed the pion mass (which is degenerate with the diquark mass). In this case, there is a change in the order parameter of the theory that does not lend itself to the standard picture of first order transitions. We explore this phase transition both within a Ginzburg-Landau framework valid in a limited parameter space and then by inspecting the full chiral Lagrangian in all the accessible parameter space. Across the phase transition between the two broken phases the order parameter becomes an $SU(2)$ doublet, with the ground state fixing the expectation value of the sum of the magnitude squared of the pion and the diquark fields. Furthermore, we find that the Lagrangian at equal chemical potentials is invariant under global $SU(2)$ transformations and construct the effective Lagrangian of the three Goldstone degrees of freedom by integrating out the radial fluctuations.
We present and discuss the results of a Monte-Carlo simulation of the phase transition in pure compact U(1) lattice gauge theory with Wilson action on a hypercubic lattice with periodic boundary conditions. The statistics are large enough to make a thorough analysis of the size dependence of the gap. In particular we find a non-zero latent heat in the infinite volume limit. We also find that the critical exponents $ u$ and $alpha$ are consistent with the hyperscaling relation but confirm that the critical behavior is different from a conventional first-order transition.
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