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arXiv:0901.3530 · Physical Review B

Signatures of the nematic ordering transitions in the thermal conductivity of d -wave superconductors

Lars Fritz, Subir Sachdev A

Published Oct 5, 2009• 23 citations
arXivHTMLPDF

Abstract

We study experimental signatures of the Ising nematic quantum phase transition in $d$-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 ${v}_{F}$ 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.

DOI
10.1103/physrevb.80.144503

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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\ensuremathΔ{v}_{\ensuremath{\Delta}}v\ensuremathΔ​ flows to a maximally anisotropic fixed point; i.e., the renormalization-group fixed point is situated at (v\ensuremathΔ/vF)\ensuremath∗=0{({v}_{\ensuremath{\Delta}}/{v}_{F})}^{\ensuremath{\ast}}=0(v\ensuremathΔ​/vF​)\ensuremath∗=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.
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.
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.
We describe a search for renormalization group fixed points which control a second-order quantum phase transition between a dx2-y2-superconductor and some other superconducting ground state. Only a few candidate fixed points are found. In the finite temperature (T) quantum-critical region of some of these fixed points, the fermion quasiparticle lifetime is very short and the spectral function has an energy width of order kBT near the Fermi points. Under the same conditions, the thermal conductivity is infinite in the scaling limit. We thus provide simple, explicit, examples of quantum theories in two dimensions for which a purely fermionic quasiparticle description of transport is badly violated.
We examine the quantum theory of the spontaneous breaking of lattice rotation symmetry in ddd-wave superconductors on the square lattice. This is described by a field theory of an Ising nematic order parameter coupled to the gapless fermionic quasiparticles. We determine the structure of the renormalization group to all orders in a 1/Nf1/{N}_{f}1/Nf​ expansion, where Nf{N}_{f}Nf​ is the number of fermion spin components. Asymptotically exact results are obtained for the quantum critical theory in which, as in the large Nf{N}_{f}Nf​ theory, the nematic order has a large anomalous dimension, and the fermion spectral functions are highly anisotropic.
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.
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.