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arXiv:cond-mat/0202010

Renormalization group approach to the interacting boson fermion systems

T. Domański, J. Ranninger

Published Feb 1, 2002• 0 citations
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Abstract

We study a pseudogap region of the mixed boson fermion system using a recent formulation of the renormalization group technique through the set of infinitesimal unitary transformations. Renormalization of fermion energies gives rise to a depletion of the low energy states (pseudogap) for temperatures $T^{*}>T>T_{c}$ which foreshadows appearance of the pairwise correlations with their long range phase coherence being missed. With a help of the flow equations for boson and fermion operators we analyze spectral weights and finite life times of these quasiparticles caused by interactions.

DOI
10.48550/arxiv.cond-mat/0202010

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We study a pseudogap region of the mixed boson fermion system using a recent formulation of the renormalization group technique through the set of infinitesimal unitary transformations. Renormalization of fermion energies gives rise to a depletion of the low energy states (pseudogap) for temperatures T∗>T>TcT^{*}>T>T_{c}T∗>T>Tc​ which foreshadows appearance of the pairwise correlations with their long range phase coherence being missed. With a help of the flow equations for boson and fermion operators we analyze spectral weights and finite life times of these quasiparticles caused by interactions.
We study the interplay between the single-particle and fermion-pair features in the boson fermion model, both above and below the transition temperature Tc{T}_{c}Tc​, using the flow equation method. Upon lowering the temperature the single-particle fermionic spectral function (a) gradually develops a depletion of the low-energy states (pseudogap) for T∗>T>TcT*>T>{T}_{c}T∗>T>Tc​ and a true superconducting gap for T<TcT<{T}_{c}T<Tc​ and (b) exhibits a considerable transfer of spectral weight between the incoherent background and the narrow coherent peak(s) signifying long-lived quasiparticle features. The Cooperon spectral function consists of a $\ensuremath{\delta}$-function peak, centered at the renormalized boson energy $\ensuremath{\omega}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{E}}_{\mathbf{q}}$ and a surrounding incoherent background which is spread over a wide energy range. When the temperature approaches Tc{T}_{c}Tc​ from above, this peak for q=0\mathbf{q}=\mathbf{0}q=0 moves to $\ensuremath{\omega}=0$, so that the static pair susceptibility diverges (Thouless criterion for the broken symmetry phase transition). Upon decreasing the temperature below Tc{T}_{c}Tc​ the Cooperon peak becomes the collective (Goldstone) mode ${E}_{\mathbf{q}}\ensuremath{\propto}\ensuremath{\mid}\mathbf{q}\ensuremath{\mid}$ in the small-momentum region and simultaneously splits off from the incoherent background states which are expelled to the high-energy sector $\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}\ensuremath{\geqslant}2{\ensuremath{\Delta}}_{sc}(T)$. We discuss the smooth evolution of these features upon approaching Tc{T}_{c}Tc​ from above and consider its feedback on the single-particle spectrum where a gradual formation of damped Bogoliubov modes (above Tc{T}_{c}Tc​) is observed.
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.
By means of the continuous unitary transformation similar to a general scheme of the Renormalization Group (RG) procedure we study the issue of symmetry breaking and pairing instability in the system of interacting fermions. Constructing a generalized version of the Bogoliubov transformation we show that formation of the fermion pairs and their superconductivity/superfluidity can appear at different temperatures. It is shown that strong quantum fluctuations can destroy the long-range order without breaking the fermion pairs which may still exist as incoherent and/or damped entities. Such unusual phase is characterized by a partial suppression of the density of states near the Fermi energy and by residual collective features like the sound-wave mode in the fermion pair spectrum.
We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter.For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently.We further generalize this procedure to the case of fermions interacting with the discrete boson mode.Andreev-type interactions are responsible for developing a gap in the excitation spectrum.However, the long-range coherence is destroyed due to strong quantum fluctuations.
A renormalization-group analysis combined with bosonization is performed for fermions interacting with repulsive long-range interaction of the form V(R)\ensuremath{\simeq}1/${\mathit{R}}^{(1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\epsilon}}\mathrm{^})}$, 0^\ensuremath{\le}1; we find that in any dimension the system is a non-Fermi liquid with a quasiparticle weight which vanishes like ${\mathit{w}}^{\mathrm{\ensuremath{\epsilon}}\mathrm{^}/4}$ when w\ensuremath{\rightarrow}0. For \ensuremath{\epsilon}^\ensuremath{\rightarrow}0 a marginal Fermi liquid is obtained. A nonperturbative solution is obtained using the bosonization method in d\ensuremath{\ge}1 dimensions. Within the same formalism we investigate also the electron-phonon interaction in dimensions d=3-\ensuremath{\epsilon}. When the coupling constant is imaginary, one obtains a time-dependent potential similar to what is obtained for fermions coupled to transversed U(1) gauge bosons. A non-Fermi-liquid behavior is found for d\ensuremath{\le}3.