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arXiv:1207.2314 · Physical Review B

Renormalization group flow for fermions into antiferromagnetically ordered phases: Method and mean-field models

Stefan Alexander Maier, Carsten Honerkamp A

Published Oct 9, 2012• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.86.134404

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

Gersch, R., Reiss, J., Carsten Honerkamp A

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.

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arXiv:cond-mat/0409725•Progress of Theoretical Physics•Dec 1, 2004•106 citations
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.
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.
This thesis is devoted to purely fermionic functional renormalization-group (fRG) studies of multiband effects and of antiferromagnetically ordered phases. The formal framework of the one-particle irreducible fRG scheme is recapitulated and extended for charge-conserving theories with a collinear spin symmetry. For multiband models, point-group symmetries and their interplay with the phases of Bloch basis states are discussed. Further, an improved truncation of the fRG flow equations is proposed that keeps track of three-particle interactions generated by virtual excitations of bands away from the Fermi surface. The feedback of this three-particle term on the two-particle interaction is first investigated in a g-ology approach to a simple two-band model for cuprate superconductors at weak coupling. The results suggest a potentially strong impact of the three-particle term. Subsequently, the three-band Emery model is analyzed within a channel-decomposition, again at weak coupling. In contrast to earlier work, the approach pursued here does not rely on a form-factor expansion, which allows for the observation of a deformed d-wave pairing gap for increasingly incommensurate ordering tendencies in the magnetic channel. In this more refined study, the three-particle feedback plays a minor role. Despite considerable qualitative agreement with results for the one-band Hubbard model with effective parameters, multi-orbital effects are found to play an important role on a quantitative level. Phases of antiferromagnetic ordering are first studied for a reduced mean-field model. In order to go beyond the mean-field picture, a hierarchy of approximations is devised, which starts from the corresponding channel-decomposed one-loop flow equations. Interaction terms breaking discrete symmetries are thereby successively neglected. In the course of these approximations, also an exchange parametrization is employed. For the spin-density wave phase of a two-pocket model initially proposed by Chubukov et al. in Phys. Rev. B 78:134512, the flow equations are then integrated numerically at the most approximate level of the hierarchy. This yields a gap that is significantly reduced compared to mean-field theory. The resulting violation of the SU(2) Ward identity appears acceptable on a qualitative level, suggesting that the approximations causing this violation are physically meaningful.
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.
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 describe a method to continue the fermionic renormalization group flow into phases with broken global symmetry. This method does not require a Hubbard-Stratonovich decoupling of the interaction. Instead an infinitesimally small symmetry-breaking component is inserted into the initial action, as an initial condition for the flow of the self-energy. Its flow is driven by the interaction, and at low scales it saturates at a nonzero value if there is a tendency for spontaneous symmetry breaking in the corresponding channel. For the reduced BCS model, we show how a small initial gap amplitude flows to the value given by the exact solution of the model. We also discuss the emergence of the Goldstone boson in this approach.