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arXiv:0911.5047

Functional renormalization group with counterterms for competing order parameters

Matthias Ossadnik, Carsten Honerkamp A

Published Nov 26, 2009• 0 citations
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Abstract

We explore the possibilities of using the fermionic functional renormalization group to compute the phase diagram of systems with competing instabilities. In order to overcome the ubiquituous divergences encountered in RG flows, we propose to use symmetry breaking counterterms for each instability, and employ a self-consistency condition for fixing the counterterms. As a validity check, results are compared to known exact results for the case of one-dimensional systems. We find that whilst one-dimensional peculiarities, in particular algebraically decaying correlation functions, can not be reproduced, the phase boundaries are reproduced accurately, encouraging further explorations for higher-dimensional systems.

DOI
10.48550/arxiv.0911.5047

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We explore the possibilities of using the fermionic functional renormalization group to compute the phase diagram of systems with competing instabilities. In order to overcome the ubiquituous divergences encountered in RG flows, we propose to use symmetry breaking counterterms for each instability, and employ a self-consistency condition for fixing the counterterms. As a validity check, results are compared to known exact results for the case of one-dimensional systems. We find that whilst one-dimensional peculiarities, in particular algebraically decaying correlation functions, can not be reproduced, the phase boundaries are reproduced accurately, encouraging further explorations for higher-dimensional systems.
First-order phase transitions in many-fermion systems are not detected in the susceptibility analysis of common renormalization-group (RG) approaches. Here, we introduce a counterterm technique within the functional renormalization-group (fRG) formalism which allows access to all stable and metastable configurations. It becomes possible to study symmetry-broken states which occur through first-order transitions as well as hysteresis phenomena. For continuous transitions, the standard results are reproduced. As an example, we study discrete-symmetry breaking in a mean-field model for a commensurate charge-density wave. An additional benefit of the approach is that away from the critical temperature for the breaking of discrete symmetries large interactions can be avoided at all RG scales.
First-order phase transitions in many-fermion systems are not detected\n in the susceptibility analysis of common renormalization-group (RG)\n approaches. Here, we introduce a counterterm technique within the\n functional renormalization-group (fRG) formalism which allows access to\n all stable and metastable configurations. It becomes possible to study\n symmetry-broken states which occur through first-order transitions as\n well as hysteresis phenomena. For continuous transitions, the standard\n results are reproduced. As an example, we study discrete-symmetry\n breaking in a mean-field model for a commensurate charge-density wave.\n An additional benefit of the approach is that away from the critical\n temperature for the breaking of discrete symmetries large interactions\n can be avoided at all RG scales.
We derive an efficient and unbiased method for computing order parameters in correlated electron systems with competing instabilities. Charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are taken into account by a functional renormalization group flow, while the formation of order below that scale is treated in mean-field theory. The method captures fluctuation-driven instabilities such as ddd-wave superconductivity. As a first application we study the competition between antiferromagnetism and superconductivity in the ground state of the two-dimensional Hubbard model.
In this thesis, the functional renormalization group (FRG) method is applied to quantum spin chains. FRG is a well-known method in the treatment of correlated fermion systems, but the method has been applied to spin systems only recently[1]. The method has been used, however, only as a qualitative tool to find phase boundaries for frustrated spin systems with multiple competing interactions. The main goal of this work is to critically assess the general accuracy of the method by studying toy models, for which quantitative comparison with exact results is possible. After a full derivation of the flow equations for spin systems, the numerical solution of the resulting non-linear system of differential equations is presented. It is shown that in the antiferromagnetic two-site Heisenberg model, FRG can reproduce some known properties of the singlet state but the ground state energy, for example, disagrees with the exact result by 36 %. In addition, the quasiparticle self-energy and dynamic spin-spin correlation functions are of an incorrect form. Unphysical fluctuations in the particle number are shown to be large. In the antiferromagnetic Heisenberg chain, it is shown that whereas the use of properly truncated flow equations regularizes the unphysical antiferromagnetic instability, the resulting ground state is disordered and not critical as expected. [1] J. Reuther and P. Wölfle, Phys. Rev. B 81, 144410 (2010).