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arXiv:cond-mat/0409725 · Progress of Theoretical Physics

Renormalization Group Flows into Phases with Broken Symmetry

Manfred Salmhofer B, Carsten Honerkamp A, W. Metzner, O. Lauscher

Published Dec 1, 2004• 106 citations
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Abstract

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.

DOI
10.1143/ptp.112.943

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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.
We describe a way to continue the fermionic renormalization group flow into phases with broken global symmetry. The method does not require a Hubbard-Stratonovich decoupling of the interaction. Instead an infinitesimally small symmetry-breaking component is inserted in the initial action, as an initial condition for the flow of the selfenergy. 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.
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.
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 generalize our recently developed super-field functional renormalization group (RG) method involving both Fermi and Bose fields (Schütz, Bartosch and Kopietz 2005 Phys. Rev. B 72 035105) to include the possibility that some bosonic components of the field have a finite vacuum expectation value. We derive an exact hierarchy of flow equations for the one-line irreducible vertices and the vacuum expectation value of the field. We apply our method to an interacting Fermi system where the interaction can be decoupled in the zero-sound channel and is then mediated by a collective bosonic field. The vacuum expectation value of the zero-frequency and zero-momentum component of the bosonic field is then closely related to the fermionic density. This can be exploited to calculate the compressibility of the interacting system. By using a cutoff in the bosonic propagator, the RG can be set up such that the self-consistent Hartree approximation is imposed as an initial condition for the RG flow and the corrections to this approximation are generated as the remaining degrees of freedom are successively eliminated by the RG procedure.
The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the occurrence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. In this thesis, we perform a comprehensive renormalization group analysis of two and three-dimensional Fermi systems at low and zero temperature. We examine systems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantumphase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which - for continuous symmetry-breaking - splits into two components corresponding to longitudinal and transversal Goldstone fluctuations. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become nonanalytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid.
We include spontaneous symmetry breaking in the functional renormalization group equations for the irreducible vertices of Ginzburg–Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each renormalization group (RG) step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Σ( k ) and interpolates between lowest-order perturbation theory at large momenta k and the critical scaling regime for small k . Close to the critical point, our method yields the self-energy in the scaling form Σ( k ) = k c 2 σ − (| k |ξ,| k |/ k c ), where ξ is the order parameter correlation length, k c is the Ginzburg scale, and σ − ( x , y ) is a dimensionless two-parameter scaling function for the broken symmetry phase which we calculate explicitly within our truncation.