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Symmetry breaking in interacting Fermi systems with the functional renormalization group

Roland Gersch

Published Jan 1, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.18419/opus-6655

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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 present a functional renormalization group analysis of superconductivity in the ground state of the attractive Hubbard model on a square lattice. Spontaneous symmetry breaking is treated in a purely fermionic setting via anomalous propagators and anomalous effective interactions. In addition to the anomalous interactions arising already in the reduced BCS model, effective interactions with three incoming legs and one outgoing leg (and vice versa) occur. We accomplish their integration into the usual diagrammatic formalism by introducing a Nambu matrix for the effective interactions. From a random-phase approximation generalized through use of this matrix we conclude that the impact of the 3+1 effective interactions is limited, especially considering the effective interactions which are important for the determination of the order parameter. The exact hierarchy of flow equations for one-particle irreducible vertex functions is truncated on the two-particle level, with higher-order self-energy corrections included in a scheme proposed by Katanin (2004 Phys. Rev. B 70 115109). Using a parametrization of effective interactions by patches in momentum space, the flow equations can be integrated numerically to the lowest scales without encountering divergences. Momentum-shell as well as interaction-flow cutoff functions are used, including a small external field or a large external field and a counterterm, respectively. Both approaches produce momentum-resolved order parameter values directly from the microscopic model. The size of the superconducting gap is in reasonable agreement with expectations from other studies.
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.
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 quantum phase 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. 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 non-analytic 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 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 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.
We present a consistent fusion of functional renormalization group and mean-field theory which explicitly introduces a bosonic field via a Hubbard-Stratonovich transformation at the critical scale, at which the order sets in. We show that a minimal truncation of the flow equations, that neglects order-parameter fluctuations, is integrable and fulfills fundamental constraints as the Goldstone theorem and the Ward identity connected with the broken global symmetry. To introduce the bosonic field, we present a technique to factorize the most singular part of the vertex, even when the full dependence on all its arguments is retained. We test our method on the two-dimensional attractive Hubbard model at half-filling and calculate the superfluid gap as well as the Yukawa couplings and residual two-fermion interactions in the ordered phase as functions of fermionic Matsubara frequencies. Furthermore, we analyze the gap and the condensate fraction for weak and moderate couplings and compare our results with previous functional renormalization group studies, and with quantum Monte Carlo data. Our formalism constitutes a convenient starting point for the inclusion of order-parameter fluctuations by keeping a full, nonsimplified, dependence on fermionic momenta and/or frequencies.