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arXiv:cond-mat/0609520 · New Journal of Physics

Fermionic functional renormalization-group for first-order phase transitions: a mean-field model

R Gersch, J Reiss, Carsten Honerkamp A

Published Dec 14, 2006• 17 citations
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Abstract

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.

DOI
10.1088/1367-2630/8/12/320

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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 present a functional renormalization group (fRG) flow for many-fermion lattice models into phases with broken spin-rotational symmetry. The flow is expressed purely in terms of fermionic vertex functions. The symmetry breaking is seeded by a small initial anomalous self-energy which grows at the transition scale and which prevents a runaway flow at nonzero scales. Focusing on the case of commensurate antiferromagnetism, we discuss how the interaction vertex can be parametrized efficiently. For reduced models with long-range bare interactions, we show the results of standard mean-field theory are reproduced by the fRG and how anisotropies in the spin sector change the flows. We then describe a more efficient decomposition of the interaction vertex that should allow for the treatment of more general models.
We present a functional renormalization group (fRG) formalism for interacting fermions on lattices that captures the flow into states with commensurate spin-density-wave order. During the flow, the growth of the order parameter is fed back into the flow of the interactions and all modes can be integrated out. This extends previous fRG flows in the symmetric phase that run into a divergence at a nonzero RG scale, i.e., that have to be stopped at the ordering scale. We use the corresponding Ward identity to check the accuracy of the results. We apply our method to a model with two Fermi pockets that have perfect particle-hole nesting. The results obtained from the fRG are compared with those in random-phase approximation.
We extend the functional renormalization group technique in a modi cation of the one‐particle irreducible scheme to study discrete symmetry breaking at nite temperature. As an instructive example, we employ the technique to access both the symmetric and the symmetry‐broken phase of a charge‐density wave mean‐ eld model. We study the half‐ lled case, and thus the breaking of a discrete symmetry, at nite temperature. A small external symmetry‐breaking eld allows us to access the symmetry‐broken state without encountering any divergence in the o w. We show diagrammatically that our method is equivalent to an exact resummation treatment. We numerically study the dependence of the o w on the external eld and on temperature.
The fermionic functional renormalization group in the one-particle irreducible formulation is truncated following an idea by Katanin. In this truncation, mean-field models are solved exactly in the symmetric phase and also for broken symmetry if the flow is seeded with symmetry-broken initial conditions. The method is applied using the momentum-shell cutoff to a charge-density-wave mean-field model at zero and finite temperature in order to understand the flows it produces. A small external field is applied to provide the symmetry-broken starting point. Increasing it suppresses the maximum of the effective interaction in the flow while increasing the final result for the order parameter. The effect of temperature on the flow as well as on the impact of the external field is studied. By employing an interaction-flow cutoff instead of the momentum-shell cutoff, the introduction of an off-diagonal counterterm is made effective. Results at zero external field thus become attainable via the functional renormalization group despite the symmetry-broken starting point. The maximum of the effective interaction flow can be shifted to the end of the flow where the order parameter flow is saturated. Spontaneous first-order transitions and hysteresis effects, outside the scope of the momentum-shell procedure, can be studied. A resummation of perturbation theory is used to exactly solve a reduced model combining Cooper and forward scattering. Anomalous effective interactions, including ones with an odd number of incoming legs, and the Goldstone boson are discussed. The latter anomalous effective interactions are the focus of the following study of the attractive Hubbard model in two dimensions at zero temperature, where the resummation framework is used as an approximation. The anomalous effective interactions with an odd number of incoming legs are of small importance for qualitative aspects, but have an impact quantitatively. The latter is found to vary in strength. Both renormalization-group procedures are used to study the superconducting order in the attractive Hubbard model. Although neither one is exact in the Hubbard case, stable flows yielding approximations comparable to ones found in the recent literature are discovered in both cases. The momentum-shell flows of the order parameter as well as of the normal and anomalous effective interactions are analyzed. Anomalous effective interactions and collective modes are studied in particular. The interaction-flow approach is used to obtain results for the order parameter which are compared with results from previous studies in detail.
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.