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npj Quantum Materials

Multiple intertwined pairing states and temperature-sensitive gap anisotropy for superconductivity at a nematic quantum-critical point

Avraham Klein, Yiming Wu, Andrey V. Chubukov

Published Nov 6, 2019• 24 citations
Publisher

Abstract

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 $$\Delta (\theta ,{\omega }_{m})$$ Δ ( θ , ω m ) and show that its angular dependence strongly varies below $${T}_{{\rm{c}}}$$ 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 $${T}_{{\rm{c}},n}$$ T c , n . Taken alone, each instability would produce a gap $$\Delta (\theta ,{\omega }_{m})$$ Δ ( θ , ω m ) that changes sign $$8n$$ 8 n times along the Fermi surface. We show that the equilibrium gap $$\Delta (\theta ,{\omega }_{m})$$ Δ ( θ , ω m ) is a superposition of multiple components that are nonlinearly induced below the actual $${T}_{{\rm{c}}}={T}_{{\rm{c}},0}$$ T c = T c , 0 , and get resonantly enhanced at $$T={T}_{{\rm{c}},n}\ <\ {T}_{{\rm{c}}}$$ T = T c , n < T c . This gives rise to strong temperature variation of the angular dependence of $$\Delta (\theta ,{\omega }_{m})$$ Δ ( θ , ω m ) . This variation progressively disappears away from a QC point.

DOI
10.1038/s41535-019-0192-x

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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.
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.
We investigate the interplay between superconductivity and nematic order in a two-dimensional electron gas with competing pairing and quadrupolar forward-scattering interactions. The model includes both sss-wave and ddd-wave superconducting channels. We compute the mean-field free energy density and determine the phase diagrams as functions of interaction strengths and temperature by solving a set of coupled self-consistent equations. At zero-temperature, we find that the nematic order competes strongly with ddd-wave superconductivity, leading to a direct first-order phase transition, while its interplay with sss-wave pairing allows for a coexistence phase characterized by an anisotropic Fermi surface with a uniform superconducting gap. At finite-temperatures, quadrupolar interactions promote the emergence of additional superconducting components, giving rise to regimes where sss-wave, ddd-wave, and nematic orders coexist. Our results highlight the role of symmetry and interaction strength in shaping the phase structure and provide a minimal framework to describe intertwined nematic and superconducting phases in correlated electron systems.
In fermionic systems with different types of quasi-particles, attractive interactions can give rise to exotic superconducting states, such as pair density wave (PDW) superconductivity and breached pairing. In recent years the search for these new types of ground states in cold atoms and in metallic systems has been intense. In the case of metals the different quasi-particles may be the up and down spin bands in an external magnetic field or bands arising from distinct atomic orbitals that coexist at a common Fermi surface. These systems present a complex phase diagram as a function of the difference between the Fermi wavevectors of the different bands. This can be controlled by external means, varying the density in the two-component cold atom system or, in a metal, by applying an external magnetic field or pressure. Here we study the zero temperature instability of the normal system as the Fermi wavevector mismatch of the quasi-particles (bands) is reduced and find a second order quantum phase transition to a PDW superconducting state. From the nature of the quantum critical fluctuations close to the superconducting quantum critical point (SQCP), we obtain its dynamic critical exponent. It turns out to be z = 2 and this allows us to fully characterize the SQCP for dimensions d ≥ 2.
We study experimental signatures of the Ising nematic quantum phase transition in ddd-wave superconductors, associated with the change of lattice symmetry from tetragonal to orthorhombic in the superconducting state. The characteristic feature of this transition is that the ratio between the Fermi velocity vF{v}_{F}vF​ and gap velocity ${v}_{\ensuremath{\Delta}}$ flows to a maximally anisotropic fixed point; i.e., the renormalization-group fixed point is situated at ${({v}_{\ensuremath{\Delta}}/{v}_{F})}^{\ensuremath{\ast}}=0$. Our main point is that the logarithmic approach to this fixed point has visible signatures in the thermal transport. The analysis of the quasiparticle contribution to the thermal transport is carried out in the framework of a kinetic approach, which shows that the thermal conductivity is enhanced near the nematic critical point. Another aspect of our study is the interplay of dilute disorder and electronic interactions in the measured thermal transport coefficients.
We investigate the superfluid properties of a one-component Fermi gas with a uniaxially anisotropic ppp-wave pairing interaction, Ux>Uy=Uz{U}_{x}>{U}_{y}={U}_{z}Ux​>Uy​=Uz​ [where Ui(i=x,y,z){U}_{i}\phantom{\rule{0.28em}{0ex}}(i=x,y,z)Ui​(i=x,y,z) is a pi{p}_{i}pi​-wave pairing interaction]. This type of interaction is considered to be realized in a 40K^{40}\mathrm{K}40K Fermi gas. Including pairing fluctuations within a strong-coupling TTT-matrix theory, we determine the px{p}_{x}px​-wave superfluid phase transition temperature Tcpx{T}_{\mathrm{c}}^{{p}_{x}}Tcpx​​, as well as the other phase transition temperature Tcpx+ipy(<Tcpx){T}_{\mathrm{c}}^{{p}_{x}+i{p}_{y}}\phantom{\rule{0.28em}{0ex}}(<{T}_{\mathrm{c}}^{{p}_{x}})Tcpx​+ipy​​(<Tcpx​​), below which the superfluid order parameter has the px+ipy{p}_{x}+i{p}_{y}px​+ipy​-wave symmetry. In the normal state near Tcpx,px{T}_{\mathrm{c}}^{{p}_{x}}, {p}_{x}Tcpx​​,px​-wave pairing fluctuations are shown to induce an anisotropic pseudogap phenomenon, where a dip structure in the angle-resolved density of states around $\ensuremath{\omega}=0$ is the most remarkable in the px{p}_{x}px​ direction. In the px{p}_{x}px​-wave superfluid phase $({T}_{\mathrm{c}}^{{p}_{x}+i{p}_{y}}<T\ensuremath{\le}{T}_{\mathrm{c}}^{{p}_{x}})$, while the pseudogap in the px{p}_{x}px​ direction continuously changes to the superfluid gap, the pseudogap in the perpendicular direction to the px{p}_{x}px​ axis is found to continue developing because of enhanced py{p}_{y}py​-wave and pz{p}_{z}pz​-wave pairing fluctuations around the node of the px{p}_{x}px​-wave superfluid order parameter. Since pairing fluctuations are always suppressed in the isotropic sss-wave superfluid state, this phenomenon is peculiar to an unconventional Fermi superfluid with a nodal superfluid order parameter. Since the ppp-wave Fermi superfluid is the most promising non-sss-wave pairing state in an ultracold Fermi gas, our results would contribute to understanding how the anisotropic pairing fluctuations, as well as the existence of plural superfluid phases, affect many-body properties of this unconventional Fermi superfluid.