Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. I. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> model and its phase diagram at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>: The case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>γ</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>
Ar. Abanov, Andrey V. Chubukov
Abstract
Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In this paper, the first in the series, we consider the case $T=0$ and $0<\ensuremath{\gamma}<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ ($n=0,1,2,...,\ensuremath{\infty}$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign $n$ times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The $n=0$ solution ${\mathrm{\ensuremath{\Delta}}}_{0}({\ensuremath{\omega}}_{m})$ is sign preserving and tends to a finite value at ${\ensuremath{\omega}}_{m}=0$, like in BCS theory. The $n=\ensuremath{\infty}$ solution corresponds to an infinitesimally small $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$, which oscillates down to the lowest frequencies as $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremath{\propto}|{\ensuremath{\omega}}_{m}{|}^{\ensuremath{\gamma}/2}cos[2\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}log(|{\ensuremath{\omega}}_{m}|/{\ensuremath{\omega}}_{0})]$, where $\ensuremath{\beta}=O(1)$ and ${\ensuremath{\omega}}_{0}$ is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at $T=0$ on the entire frequency axis for all $0<\ensuremath{\gamma}<1$, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor $1/N$ and show that there exists a critical ${N}_{cr}>1$, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once $N$ gets smaller than ${N}_{cr}$.