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arXiv:physics/9910031

Renormalization Group Studies of Vertex Models

Mitra, Saibal

Published Oct 21, 1999• 0 citations
arXivHTMLPDF

Abstract

In this thesis we use renormalization group methods to study the critical behaviour of the staggered F-model. This model is a six vertex model that can be used to calculate the thermodynamic properties of a facet of a BCC crystal in the (100) direction.
DOI
10.48550/arxiv.physics/9910031

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The F-model is one of the six vertex models, a range of two-dimensional models designed to describe the positions of hydrogen atoms in ice. It exhibits a Kosterlitz-Thouless phase transition, and can easily be mapped onto many other models including the roughening transition in the body centered solid-on-solid model. In this thesis the critical behaviour in the F-model is examined through a computational approach, combined with finite-size scaling and real space renormalization. The simulations were run using two Monte Carlo algorithms; the so-called short loop algorithm, and the full lattice cluster algorithm. The ergodicity of the latter is revealed to be problematic, but this is overcome by combining both algorithms. Also, finite-size scaling is applied, returning a critical inverse temperature of \\epsilon \\beta = 0.79 plus or minus 0.02 compared to the known analytical value of log[2]. Furthermore, although the ice rules prevent local renormalization such as by Kadanoff block spins, a functioning coarse graining prescription based on the height function is proposed. This turns out to renormalise the high temperature states into states with temperatures below the critical point, thus preventing the determination of critical exponents from the renormalization flow.
The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
We present a new method for the numerical calculation of the equilibrium crystal shape around a facet. Our method is based on the transfer matrix method with the product-wave-function renormalization group (PWFRG) algorithm for the asymmetric density matrix, which is an extension of the density matrix renormalization group (DMRG) algorithm. By applying this method to the staggered body-centered-cubic solid-on-solid model, which is known to exhibit the inverse roughening phenomena, we obtain the facet shape, shape exponents, and step tension.