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Condensed Matter Physics

Continuous unitary transformation approach to pairing interactions in statistical physics

T. Domański

Published Jun 1, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.5488/cmp.11.2.195

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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.
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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 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 address the linear response of the electron system with the pairing instabilities using nonperturbative framework of the continuous unitary transformation technique. For the case of the Bose-Einstein condensed pairs we have analytically derived the BCS result, which in the static and long wavelength limit accounts for the Meissner effect. We next apply our treatment to the mixture of the non-condensed (preformed) pairs interacting with the mobile electrons through the charge-exchange Andreev scattering. We determine the leading contributions to the current-current response function, expressing the vertices by the set of the corresponding flow equations. Our study indicates that the residual diamagnetic behavior detected by the torque magnetometry well above the transition temperature [L. Li et al, Phys. Rev. B 81, 054510 (2010)] can originate from the noncondensed preformed pairs.
On the basis of a scenario where electron pairing is induced by resonant two-particle scattering (the boson-fermion model), we show how precursors of the superconducting state-in the form of overdamped Bogoliubov modes-emerge in the normal state upon approaching the transition temperature from above. This result is obtained by a renormalization technique based on continuous unitary transformations (the flow equations), projecting out the coherent contributions in the electron spectral function from an incoherent background.
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.