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arXiv:1302.6137 · Physical Review B

Velocity renormalization of nodal quasiparticles in d -wave superconductors

Jing Wang

Published Feb 25, 2013• 19 citations
arXivHTMLPDF

Abstract

Gapless nodal quasiparticles emerge at a low-energy regime of high-${T}_{c}$ cuprate superconductors due to the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ gap symmetry. We study the unusual renormalizations of the Fermi velocity ${v}_{F}$ 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.

DOI
10.1103/physrevb.87.054511

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Gapless nodal quasiparticles emerge at a low-energy regime of high-Tc{T}_{c}Tc​ cuprate superconductors due to the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ gap symmetry. We study the unusual renormalizations of the Fermi velocity vF{v}_{F}vF​ and gap velocity v\ensuremathΔ{v}_{\ensuremath{\Delta}}v\ensuremathΔ​ 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\ensuremathΔ/vF{v}_{\ensuremath{\Delta}}/{v}_{F}v\ensuremathΔ​/vF​, 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.
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.
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.
Fixed points and critical temperature near quantum critical points in ddd-wave cuprate superconductors
We study the critical behavior driven by potential quantum critical points (QCPs) termed as τ0,x,y,zτ_{0,x,y,z}τ0,x,y,z​-Type QCPs in the ddd-wave cuprate superconductors. Within the framework of the renormalization group approach, we construct the coupled flow equations of all interaction parameters and obtain several unique properties. On one hand, the interaction parameters flow toward several fixed points (FPs) at certain critical energy scales. We identify two different types of FPs designated at the clean limit. FP-I is characterized by the divergence of the quadratic parameter and exhibits robustness against variations in interaction parameters. In contrast, FP-II is dominated by the cubic and quartic interaction parameters, and it is sensitive to initial conditions, leading to five subclasses. Besides, we find that disorder scattering can influence fermion velocities and critical energy scales, and even destabilize certain FPs around the τx,zτ_{x,z}τx,z​-QCPs, driving the system toward a preempted disorder-induced FP. On the other hand, we find that quantum fluctuations play a critical role in shaping the critical temperature (TcT_cTc​) as the system approaches these QCPs. Near the τxτ_xτx​-QCP, TcT_cTc​ is considerably suppressed for both FP-I and FP-II. In contrast, near the τ0τ_0τ0​-QCP, TcT_cTc​ undergoes a substantial decrease for FP-I but only a slight decrease for FP-II. Conversely, TcT_cTc​ exhibits an increasing trend near the τyτ_yτy​-QCP, with a pronounced peak at vΔ0/vF0∼0.25v_{Δ0}/v_{F0} \sim 0.25vΔ0​/vF0​∼0.25. However, numerical analysis suggests that the τzτ_zτz​-QCP is unlikely to be physically realizable. Additionally, we realize that TcT_cTc​ can also be modified by the emergence of disorder-induced FPs in the vicinity of the τxτ_xτx​-QCP. These findings would provide valuable insights into the critical low-energy properties of ddd-wave cuprate superconductors and related materials.
Quantum critical behaviors induced by a putative quantum phase transition are vigilantly investigated, which separates a d-wave superconducting state and d-wave superconducting+X state below the superconducting dome of the d−wave superconductors with tuning the non-thermal doping variable. Within the framework of renormalization group approach, we start with a phenomenological effective theory originated from the Landau-Ginzburg-Wilson theory and practice one-loop calculations to construct a set of coupled flows of all interaction parameters. After extracting related physical information from these coupled evolutions, we address that both fermion velocities and critical temperatures exhibit critical behaviors, which are robust enough against the initial conditions due to strong quantum fluctuations. At first, the evolution of Yukawa coupling between X-state order parameter and nodal fermions in tandem with quantum fluctuations heavily renormalize fermion velocities and generally drive them into certain finite anisotropic fixed point at the lowest-energy limit, whose concrete value relies upon the very quantum phase transition. In addition, these unique properties of fermion velocities largely reshape the fate of superfluid density, giving rise to either an enhancement or a dip of critical temperature. Moreover, we find that fermion-fermion interactions bring non-ignorable quantitative corrections to quantum critical behaviors despite they are subordinate to quantum fluctuations of order parameters.
Motivated by the strong, low temperature damping of nodal quasiparticles observed in some cuprate superconductors, we study quantum phase transitions in d(x(2)-y(2)) superconductors with a spin-singlet, zero momentum, fermion bilinear order parameter. We present a complete, group-theoretic classification of such transitions into seven distinct cases (including cases with nematic order) and analyze fluctuations by the renormalization group. We find that only two, the transitions to d(x(2)-y(2))+is and d(x(2)-y(2))+id(xy) pairing, possess stable fixed points with universal damping of nodal quasiparticles; the latter leaves the gapped quasiparticles along (1,0), (0,1) essentially undamped.
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.