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تسجيل مستخدم جديدInterplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. IV: The $gamma$ model and its phase diagram at $1<gamma <2$
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In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction $V(Omega_m) propto 1/|Omega_m|^gamma$ (the $gamma$-model). In previous papers we studied the cases $0<gamma <1$ and $gamma approx 1$. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists an infinite number of topologically distinct solutions for the gap function $Delta_n (omega_m)$ at $T=0$ ($n=0,1,2...$), each with its own condensation energy $E_{c,n}$. Here we extend the analysis to larger $1< gamma <2$. We argue that the discrete set of solutions survives, and the spectrum of $E_{c,n}$ gets progressively denser as $gamma$ approaches $2$ and eventually becomes continuous at $gamma to 2$. This increases the strength of longitudinal gap fluctuations, which tend to reduce the actual superconducting $T_c$ and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis for $gamma >1$ which become critical at $gammato 2$. First, the density of states evolves towards a set of discrete $delta-$functions. Second, an array of dynamical vortices emerges in the upper frequency half-plane. These two features come about because on a real axis, the real part of the interaction, $V (Omega) propto cos(pi gamma/2)/|Omega|^gamma$, becomes repulsive for $gamma >1$, and the imaginary $V^{} (Omega) propto sin(pi gamma/2)/|Omega|^gamma$, gets progressively smaller at $gamma to 2$. The features on the real axis are consistent with the development of a continuum spectrum of $E_{c,n}$ obtained using $Delta_n (omega_m)$ on the Matsubara axis. We consider the case $gamma =2$ separately in the next paper.
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