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arXiv:2206.01783 · Physical review. B./Physical review. B

Odd-frequency pairing and time-reversal symmetry breaking for repulsive interactions

Dimitri Pimenov, Andrey V. Chubukov

Published Sep 27, 2022• 4 citations
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Abstract

We study the pairing of fermions by an interaction consisting of a Hubbard repulsion, mimicking a screened Coulomb potential, and a dynamical phonon-mediated attraction. For such interaction, the gap equation allows even- and odd-frequency solutions ${\mathrm{\ensuremath{\Delta}}}_{e}$ and ${\mathrm{\ensuremath{\Delta}}}_{o}$. We show that odd-frequency pairing does not develop within the Eliashberg approximation due to over-critical pair breaking from the self-energy. When vertex corrections are included, the pairing interaction gets stronger, and ${\mathrm{\ensuremath{\Delta}}}_{o}$ can develop. We argue that even in this case keeping the self-energy is still a must as it cancels out the thermal piece in the gap equation. We further argue that ${\mathrm{\ensuremath{\Delta}}}_{o}$ is not affected by Hubbard repulsion and for strong repulsion is comparable to a reduced ${\mathrm{\ensuremath{\Delta}}}_{e}$. The resulting superconducting state is a superposition ${\mathrm{\ensuremath{\Delta}}}_{e}\ifmmode\pm\else\textpm\fi{}i{\mathrm{\ensuremath{\Delta}}}_{o}$, which spontaneously breaks the time-reversal symmetry, despite that the pairing symmetry is an ordinary $s$ wave.

DOI
10.1103/physrevb.106.104515

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We study the pairing of fermions by an interaction consisting of a Hubbard repulsion, mimicking a screened Coulomb potential, and a dynamical phonon-mediated attraction. For such interaction, the gap equation allows even- and odd-frequency solutions \ensuremathΔe{\mathrm{\ensuremath{\Delta}}}_{e}\ensuremathΔe​ and \ensuremathΔo{\mathrm{\ensuremath{\Delta}}}_{o}\ensuremathΔo​. We show that odd-frequency pairing does not develop within the Eliashberg approximation due to over-critical pair breaking from the self-energy. When vertex corrections are included, the pairing interaction gets stronger, and \ensuremathΔo{\mathrm{\ensuremath{\Delta}}}_{o}\ensuremathΔo​ can develop. We argue that even in this case keeping the self-energy is still a must as it cancels out the thermal piece in the gap equation. We further argue that \ensuremathΔo{\mathrm{\ensuremath{\Delta}}}_{o}\ensuremathΔo​ is not affected by Hubbard repulsion and for strong repulsion is comparable to a reduced \ensuremathΔe{\mathrm{\ensuremath{\Delta}}}_{e}\ensuremathΔe​. The resulting superconducting state is a superposition \ensuremathΔe\ifmmode±\else\textpm\fii\ensuremathΔo{\mathrm{\ensuremath{\Delta}}}_{e}\ifmmode\pm\else\textpm\fi{}i{\mathrm{\ensuremath{\Delta}}}_{o}\ensuremathΔe​\ifmmode±\else\textpm\fii\ensuremathΔo​, which spontaneously breaks the time-reversal symmetry, despite that the pairing symmetry is an ordinary sss wave.
Motivated by a scarcity of simple and analytically tractable models of superconductivity from strong repulsive interactions, we introduce a simple tight-binding lattice model of fermions with repulsive interactions that exhibits unconventional superconductivity (beyond BCS theory). The model resembles an idealized trilayer. The Cooper pair consists of electrons on opposite sides of the dielectric, which mediates the attraction. In the strong coupling limit, we use degenerate perturbation theory to show that the model reduces to a superconducting hard-core Bose-Hubbard model. Above the superconducting critical temperature, an analog of pseudo-gap physics results where the fermions remain Cooper paired with a large single-particle energy gap.
We analyze the possibility of odd-frequency pairing near a quantum critical point(QCP) in a metal. We consider a model with dynamical pairing interaction $V({\mathrm{\ensuremath{\Omega}}}_{n})\ensuremath{\sim}1/{|{\mathrm{\ensuremath{\Omega}}}_{n}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). This interaction gives rise to a non-Fermi liquid in the normal state and is attractive for pairing. The two trends compete with each other. We search for odd-frequency solutions for the pairing gap $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})=\ensuremath{-}\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$. We show that for $\ensuremath{\gamma}<1$, odd-frequency superconductivity loses the competition with a non-Fermi liquid and does not develop. We show that the pairing does develop in the extended model in which interaction in the pairing channel is larger than the one in the particle-hole channel. For $\ensuremath{\gamma}>1$, we argue that the original model is at the boundary towards odd-frequency pairing and analyze in detail how superconductivity is triggered by a small external perturbation. In addition, we show that for $\ensuremath{\gamma}>2$, the system gets frozen at the critical point towards pairing in a finite range in the parameter space. This gives rise to highly unconventional phase diagrams with flat regions.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
The discovery of high temperature superconductivity inspired a number of novel proposals, one of which, put forward by C.M.Varma, involves the breaking of time-reversal symmetry to explain the physics of the underdoped pseudogap phase. It was proposed that time-reversal symmetry is spontaneously broken as a result of strong repulsion between the Cu-O electrons to form loop-currents in the system. In this work, we developed a general theory to study the quantum phase transitions in the 2 dimensional strongly interacting electronic systems in which time-reversal symmetry is spontaneously broken in the ground state. We first applied the theory of magnetic groups to identify electronic current-loop patterns in two physically relevant systems: (i) 2-band model involving spinless electrons on a honeycomb lattice with next-nearest-neighbor interactions; (ii) 3-band CuO2CuO_{2}CuO2​ model with and without lattice distortions. Next, by examining the correlation function within the standard ring and ladder Dyson series approximations, we identify the effective Hamiltonian with the relevant interactions responsible for creating low-energy fluctuations near the quantum critical point. The mean-field analysis of this effective Hamiltonian elucidated the fact that time-reversal symmetry breaking in a 2-band model is in the same universality class as the interband particle-hole pair condensation instability which occurs in the semi-conductors under large enough particle-hole attraction. Using Hubbard-Stratonovich transformation and functional integral method, we are able to investigate this instability and the static susceptibility in the condensed phase both in half-filling and doped case. Away from half-filling, because the condensates are metallic and couple to the gapless collective intraband particle-hole excitations, we find that the static susceptibility is generically negative as a result of this coupling, which implies that the condensates are unstable.
The understanding of the mechanisms responsible for superconductivity in strongly correlated systems is an interesting and important subject in condensed matter physics. Several theoretical proposals were considered for these systems. The Coulomb interaction between electrons allow a new approach to study this problem. In this paper, we use a usual Hubbard model with a local repulsive interaction to describe a 2D system. The system of equations are solved using the Green's functions method, within a Hubbard-I mean field approximation, which allows to treat the strong interaction limit. We consider both cases of attractive and repulsive interactions and obtain the zero temperature phase diagram of the model. Our results show, in the repulsive case, the existence of a superconducting ground state mediated by the kinetic electronic energy and described by a non-local order parameter. A minimum value of the repulsive interaction UminU_{min}Umin​ is required to create a pairing state. At finite temperatures, for strong interactions, the critical temperature TcT_cTc​ shows a saturation similar to the Bose-Einstein condensation observed for strong attractive interactions.
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=0T=0T=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 nnn times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The n=0n=0n=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=0T=0T=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/N1/N1/N and show that there exists a critical Ncr>1{N}_{cr}>1Ncr​>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once NNN gets smaller than Ncr{N}_{cr}Ncr​.