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arXiv:1904.10338

Calculation of non-universal thermodynamic quantities within self-consistent non-perturbative functional renormalization group approach

V. I. Tokar

Published Apr 23, 2019• 2 citations
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Abstract

A self-consistent renormalization scheme suitable for the calculation of non-universal quantities in $n$-vector models with pair spin interactions of arbitrary extent has been suggested. The method has been based on the elimination of the fluctuating field components within the layers defined by the layer-cake representation of the propagator. The non-perturbative renormalization group (RG) equations has been solved in the local potential approximation. Critical temperatures of the $n=1-3$ vector spin models on cubic lattices have been calculated in excellent agreement with the best known estimates. Several critical amplitudes and the magnetisation curve of the Ising model on the simple cubic lattice calculated within the approach compared well with the values from literature sources. It has been argued that unification of the method with cluster techniques would make possible the treatment of realistic lattice models with multi-spin interactions and describe in a unified framework the phase transitions of any kind. Besides the RG equations the layer-cake technique can be used in the exact partial renormalization of the local interactions in the functional lattice Hamiltonians. This procedure reduces the strength of the interactions which in some cases can make them amenable to perturbative treatment.

DOI
10.48550/arxiv.1904.10338

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A self-consistent renormalization scheme suitable for the calculation of non-universal quantities in nnn-vector models with pair spin interactions of arbitrary extent has been suggested. The method has been based on the elimination of the fluctuating field components within the layers defined by the layer-cake representation of the propagator. The non-perturbative renormalization group (RG) equations has been solved in the local potential approximation. Critical temperatures of the n=1−3n=1-3n=1−3 vector spin models on cubic lattices have been calculated in excellent agreement with the best known estimates. Several critical amplitudes and the magnetisation curve of the Ising model on the simple cubic lattice calculated within the approach compared well with the values from literature sources. It has been argued that unification of the method with cluster techniques would make possible the treatment of realistic lattice models with multi-spin interactions and describe in a unified framework the phase transitions of any kind. Besides the RG equations the layer-cake technique can be used in the exact partial renormalization of the local interactions in the functional lattice Hamiltonians. This procedure reduces the strength of the interactions which in some cases can make them amenable to perturbative treatment.
It is argued that cluster methods provide a viable alternative to Wilson's momentum shell integration technique at the early stage of renormalization in the field-theoretic models with strongly coupled fields because these methods allow for systematic accounting of all interactions in the system irrespective of their strength. These methods, however, are restricted to relatively small spatial scales, so they ought to be supplemented with more conventional renormalization-group (RG) techniques to account for large scale correlations. To fulfil this goal a "layer-cake" renormalization scheme earlier developed for rotationally symmetric Hamiltonians has been generalized to the lattice case. The RG technique can be naturally integrated with an appropriately modified cluster method so that the RG equations were used only in the presence of large scale fluctuations, while in their absence the approach reduced to a conventional cluster method. As an illustrative example the simplest single-site cluster approximation together with the local-potential RG equation are applied to simple cubic Ising model to calculate several non-universal quantities such as the magnetization curve, the critical temperature, and some critical amplitudes. Good agreement with Monte Carlo simulations and series expansion results was found.
The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations
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 approximation scheme to solve the Non-Perturbative Renormalization Group equations and obtain the full momentum dependence of n -point functions. The core of the method consists in deriving approximate flow equations that are then solved exactly. The results thus obtained can next be improved through an iteration procedure. The approximations proposed in this paper lead to approximate equations that are accurate both in the perturbative and the scaling regimes. The scheme is tested with the calculation of the shift Δ T c in the transition temperature of the weakly repulsive Bose gas, a quantity which is very sensitive to all momentum scales. The leading-order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%. The next-to-leading order differs by about 10% from the best accepted result.
In the present thesis superfluid many-fermion systems are investigated in the framework of the Renormalization Group (RG). Starting from an experimentally determined two-body interaction this scheme provides a microscopic approach to strongly correlated many-body systems at low temperatures. The fundamental objects under investigation are the two-point and the four-point vertex functions. We show that explicit results for simple separable interactions on BCS-level can be reproduced in the RG framework to high accuracy. Furthermore the RG approach can immediately be applied to general realistic interaction models. In particular, we show how the complexity of the many-body problem can be reduced systematically by combining different RG schemes. Apart from technical convenience the RG framework has conceptual advantage that correlations beyond the BCS level can be incorporated in the flow equations in a systematic way. In this case however the flow equations are no more explicit equations like at BCS level but instead a coupled set of implicit equations. We show on the basis of explicit calculations for the single-channel case the efficacy of an iterative approach to this system. The generalization of this strategy provides a promising strategy for a non-perturbative treatment of the coupled channel problem. By the coupling of the flow equations of the two-point and four-point vertex self-consistency on the one-body level is guaranteed at every cutoff scale.
The dynamic real-space renormalization group of Mazenko and Valls is applied to the zero-field ferromagnetic Ising model on the triangular lattice. Renormalization equations valid for all temperatures above the critical temperature Tc{T}_{c}Tc​ are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for T<TcT<{T}_{c}T<Tc​. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.