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Additional is-wave Cooper pairing driven by nematic order in d-wave superconductors

Jing-Rong Wang, Guo-Zhu Liu

Published Jul 29, 2012• 0 citations
OpenAlex

Abstract

We make a non-perturbative analysis of the strong interaction between gapless nodal fermions and nematic order parameter in two-dimensional $d_{x^2 - y^2}$ superconductors. We predict that the critical nematic fluctuation can generate a dynamical nodal gap if the fermion flavor $N$ is smaller than a threshold $N_c$. Such gap generation leads to an additional $is$-wave Cooper pairing instability, and induces a fully gapped $d_{x^2 - y^2} + is$ superconducting dome in the vicinity of nematic quantum critical point. The opening of dynamical gap has important consequences, including saturation of fermion velocity renormalization, weak confinement of fermions, and suppression of observable quantities.

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We make a non-perturbative analysis of the strong interaction between gapless nodal fermions and nematic order parameter in two-dimensional dx2−y2d_{x^2 - y^2}dx2−y2​ superconductors. We predict that the critical nematic fluctuation can generate a dynamical nodal gap if the fermion flavor NNN is smaller than a threshold NcN_cNc​. Such gap generation leads to an additional isisis-wave Cooper pairing instability, and induces a fully gapped dx2−y2+isd_{x^2 - y^2} + isdx2−y2​+is superconducting dome in the vicinity of nematic quantum critical point. The opening of dynamical gap has important consequences, including saturation of fermion velocity renormalization, weak confinement of fermions, and suppression of observable quantities.
We perform a non-perturbative analysis of the strong interaction between gapless nodal fermions and the nematic order parameter in two-dimensional d x 2 − y 2 superconductors. We predict that the critical nematic fluctuation can generate a dynamical nodal gap if the fermion flavor N is smaller than a threshold N c . Such gap generation leads to an additional is-wave Cooper pairing instability, which induces a fully gapped d x 2 − y 2 + is superconducting dome in the vicinity of the nematic quantum critical point. The opening of a dynamical gap has important consequences, including the saturation of fermion velocity renormalization, a weak confinement of fermions and the suppression of observable quantities.
The ddd-wave high-temperature cuprate superconductor may exhibit a nematic ordering transition at zero temperature. Near the quantum critical point, the coupling between gapless nodal quasiparticles and nematic order parameter fluctuation can result in unusual behaviors, such as extreme anisotropy of fermion velocities. We study the disorder effects on the nematic quantum critical behavior and especially on the flow of fermion velocities. The disorders that couple to nodal quasiparticles are divided into three types: random mass, random gauge field, and random chemical potential. A renormalization-group analysis shows that random mass and random gauge field are both irrelevant and, thus, do not change the fixed point of extreme velocity anisotropy. However, the marginal interaction due to random chemical potential destroys this fixed point and makes the nematic phase transition unstable.
Abstract The proximity of many strongly correlated superconductors to density-wave or nematic order has led to an extensive search for fingerprints of pairing mediated by dynamical quantum-critical (QC) fluctuations of the corresponding order parameter. Here we study anisotropic s-wave superconductivity induced by anisotropic QC dynamical nematic fluctuations. We solve the non-linear gap equation for the pairing gap Δ(θ,ωm)\Delta (\theta ,{\omega }_{m})Δ(θ,ωm​) Δ ( θ , ω m ) and show that its angular dependence strongly varies below Tc{T}_{{\rm{c}}}Tc​ T c . We show that this variation is a signature of QC pairing and comes about because there are multiple s-wave pairing instabilities with closely spaced transition temperatures Tc,n{T}_{{\rm{c}},n}Tc,n​ T c , n . Taken alone, each instability would produce a gap Δ(θ,ωm)\Delta (\theta ,{\omega }_{m})Δ(θ,ωm​) Δ ( θ , ω m ) that changes sign 8n8n8n 8 n times along the Fermi surface. We show that the equilibrium gap Δ(θ,ωm)\Delta (\theta ,{\omega }_{m})Δ(θ,ωm​) Δ ( θ , ω m ) is a superposition of multiple components that are nonlinearly induced below the actual Tc=Tc,0{T}_{{\rm{c}}}={T}_{{\rm{c}},0}Tc​=Tc,0​ T c = T c , 0 , and get resonantly enhanced at T=Tc,n < TcT={T}_{{\rm{c}},n}\ <\ {T}_{{\rm{c}}}T=Tc,n​ < Tc​ T = T c , n < T c . This gives rise to strong temperature variation of the angular dependence of Δ(θ,ωm)\Delta (\theta ,{\omega }_{m})Δ(θ,ωm​) Δ ( θ , ω m ) . This variation progressively disappears away from a QC point.
At the nematic quantum critical point that exists in the -wave superconducting dome of cuprates, the massless nodal fermions interact strongly with the quantum critical fluctuation of nematic order. We study this problem by means of the renormalization group approach and show that, the fermion damping rate vanishes more rapidly than the energy ω and the quasiparticle residue in the limit . The nodal fermions thus constitute an unconventional non-Fermi liquid that represents an even weaker violation of Fermi liquid theory than a marginal Fermi liquid. We also investigate the interplay of quantum nematic critical fluctuation and gauge-potential-like disorder, and find that the effective disorder strength flows to the strong coupling regime at low energies. Therefore, even an arbitrarily weak disorder can drive the system to become a disorder controlled diffusive state. Based on these theoretical results, we are able to understand a number of interesting experimental facts observed in curpate superconductors.
We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions (m)(m)(m) and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in ddd spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension d=dc(m)d={d}_{c}(m)d=dc​(m). The fixed points are identified from the RG flow equations, as functions of ddd and mmm. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of (d=3,m=2)(d=3,\phantom{\rule{0.16em}{0ex}}m=2)(d=3,m=2) and (d=2,m=1)(d=2,\phantom{\rule{0.16em}{0ex}}m=1)(d=2,m=1). In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.
Gapless nodal quasiparticles emerge at a low-energy regime of high-Tc{T}_{c}Tc​ cuprate superconductors due to the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ gap symmetry. We study the unusual renormalizations of the Fermi velocity vF{v}_{F}vF​ and gap velocity ${v}_{\ensuremath{\Delta}}$ of these quasiparticles close to various quantum critical points in a superconducting dome. Special attention is paid to the behavior of the velocity ratio, ${v}_{\ensuremath{\Delta}}/{v}_{F}$, since it determines a number of observable quantities. We perform a renormalization-group analysis and show that the velocity ratio may vanish, approach unity, or diverge at different quantum critical points. The corresponding superfluid densities and critical temperatures are suppressed, slightly increased, or significantly enhanced. The effects of three types of static disorders, namely, random mass, random gauge potential, and random chemical potential, on the stability of the system are also addressed. An analogous analysis reveals that both random mass and random gauge potential are irrelevant. This implies that these fixed points of the velocity ratio are stable, and hence observable effects ignited by them are unchanged. However, the random chemical potential is marginal. As a result, these fixed points are broken, and thus, the instabilities of quantum phase transitions are triggered.