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Register a new userInterplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. VI. The $gamma$ model and its phase diagram at $2 < gamma <3$
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In this paper, the sixth in series, we continue our analysis of the interplay between non-Fermi liquid and pairing in the effective low-energy model of fermions with singular dynamical interaction $V(Omega_m) = {bar g}^gamma/|Omega_m|^gamma$ (the $gamma$ model). The model describes low-energy physics of various quantum-critical metallic systems at the verge of an instability towards density or spin order, pairing of fermions at the half-filled Landau level, color superconductivity, and pairing in SYK-type models. In previous Papers I-V we analyzed the $gamma$ model for $gamma leq 2$ and argued that the ground state is an ordinary superconductor for $gamma <1$, a peculiar one for $1<gamma <2$, when the phase of the gap function winds up along real frequency axis due to emerging dynamical vortices in the upper half-plane of frequency, and that there is a quantum phase transition at $gamma =2$, when the number of dynamical vortices becomes infinite. In this paper we consider larger $2< gamma <3$ and address the issue what happens on the other side of this quantum transition. We argue that the system moves away from criticality in that the number of dynamical vortices becomes finite and decreases with increasing $gamma$. The ground state is again a superconductor, however a highly unconventional one with a non-integrable singularity in the density of states at the lower edge of the continuum. This implies that the spectrum of excited states now contains a level with a macroscopic degeneracy, proportional to the total number of states in the system. We argue that the phase diagram in variables $(T,gamma)$ contains two distinct superconducting phases for $gamma <2$ and $gamma >2$, and an intermediate pseudogap state of preformed pairs.
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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 models with an effective dynamical electron-electron interaction $V(Omega_m) propto 1/|Omega_m|^gamma$ (the $gamma$-model). We analyze both the original model and its extension, in which we introduce an extra parameter $N$ to account for non-equal interactions in the particle-hole and particle-particle channel. In two previous papers(arXiv:2004.13220 and arXiv:2006.02968), we considered the case $0 < gamma <1$ and argued that (i) at $T=0$, there exists an infinite discrete set of topologically different gap functions, $Delta_n (omega_m)$, all with the same spatial symmetry, and (ii) each $Delta_n$ evolves with temperature and terminates at a particular $T_{p,n}$. In this paper, we analyze how the system behavior changes between $gamma <1$ and $gamma >1$, both at $T=0$ and a finite $T$. The limit $gamma to 1$ is singular due to infra-red divergence of $int d omega_m V(Omega_m)$, and the system behavior is highly sensitive to how this limit is taken. We show that for $N =1$, the divergencies in the gap equation cancel out, and $Delta_n (omega_m)$ gradually evolve through $gamma=1$ both at $T=0$ and a finite $T$. For $N eq 1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $gamma >1$. Namely, the form of $Delta_n (omega_m)$ changes qualitatively, and the spectrum of condensation energies, $E_{c,n}$ becomes continuous at $T=0$. We introduce different extension of the model, which is free from singularities for $gamma >1$.
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.
This paper is a continuation and a partial summary of our analysis of the pairing at a quantum-critical point (QCP) in a metal for a set of quantum-critical systems, whose low-energy physics is described by an effective model with dynamical electron-electron interaction $V(Omega_m) = ({bar g}/|Omega_m|)^gamma$ (the $gamma$-model). Examples include pairing at the onset of various spin and charge density-wave and nematic orders and pairing in SYK-type models. In previous papers, we analyzed the physics for $gamma <2$. We have shown that the onset temperature for the pairing $T_p$ is finite, of order ${bar g}$, yet the gap equation at $T=0$ has an infinite set of solutions within the same spatial symmetry. As the consequence, the condensation energy $E_c$ has an infinite number of minima. The spectrum of $E_c$ is discrete, but becomes more dense as $gamma$ increases. Here we consider the case $gamma =2$. The $gamma=2$ model attracted special interest in the past as it describes the pairing by an Einstein phonon in the limit when the dressed phonon mass $omega_D$ vanishes. We show that for $gamma =2$, the spectrum of $E_c$ becomes continuous. We argue that the associated gapless longitudinal fluctuations destroy superconducting phase coherence at a finite $T$, such that at $0<T< T_p$ the system displays pseudogap behavior. We show that for each gap function from the continuum spectrum, there is an infinite array of dynamical vortices in the upper half-plane of frequency. For the electron-phonon case, our results show that $T_p =0.1827 {bar g}$, obtained in earlier studies, marks the onset of the pseudogap behavior of preformed pairs, while the actual superconducting $T_c$vanishes at $omega_D to 0$.
Recent experiments on electron- or hole-doped SrTiO$_{3}$ have revealed a hitherto unknown form of superconductivity, where the Fermi energy of the paired electrons is much lower than the energies of the bosonic excitations thought to be responsible for the attractive interaction. We show that this situation requires a fresh look at the problem calling for (i) a systematic modeling of the dynamical screening of the Coulomb interaction by ionic and electronic charges, (ii) a transverse optical phonon mediated pair interaction and (iii) a determination of the energy range over which the pairing takes place. We argue that the latter is essentially given by the limiting energy beyond which quasiparticles cease to be well defined. The model allows to find the transition temperature as a function of both, the doping concentration and the dielectric properties of the host system, in good agreement with experimental data. The additional interaction mediated by the transverse optical soft phonon is shown to be essential in explaining the observed anomalous isotope effect. The model allows to capture the effect of the incipient (or real) ferroelectric phase in pure, or oxygen isotope substituted SrTiO$_{3}$ .
We construct a two-dimensional lattice model of fermions coupled to Ising ferromagnetic critical fluctuations. Using extensive sign-problem-free quantum Monte Carlo simulations, we show that the model realizes a continuous itinerant quantum phase transition. In comparison with other similar itinerant quantum critical points (QCPs), our QCP shows much weaker superconductivity tendency with no superconducting state down to the lowest temperature investigated, hence making the system a good platform for the exploration of quantum critical fluctuations. Remarkably, clear signatures of non-Fermi-liquid behavior in the fermion propagators are observed at the QCP. The critical fluctuations at the QCP partially resemble Hertz-Millis-Moriya behavior. However, careful scaling analysis reveals that the QCP belongs to a different universality class, deviating from both (2+1)d Ising and Hertz-Millis-Moriya predictions.
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