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Bulletin of the American Physical Society

Thermoelectric transport near the pair breaking quantum phase transition out of a d-wave superconductor

Daniel K. Podolsky, Ashvin Vishwanath, Joel E. Moore, Subir Sachdev A

Published Mar 15, 2006• 0 citations
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Abstract

Department of Physics, Harvard University, Cambridge, MA 02138(Dated: Printed February 2, 2008)We study electric, thermal, and thermoelectric conductivities in the vicinity of a z = 2superconductor-diffusive metal transition in two dimensions, both in the high and low frequencylimits. We find violation of the Wiedemann-Franz law and a dc thermoelectric conductivity α thatdoes not vanish at low temperatures, in contrast to Fermi liquids. We introduce a Langevin equationformalism to study critical dynamics over a broad region surrounding the quantum critical point.INTRODUCTION

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Department of Physics, Harvard University, Cambridge, MA 02138(Dated: Printed February 2, 2008)We study electric, thermal, and thermoelectric conductivities in the vicinity of a z = 2superconductor-diffusive metal transition in two dimensions, both in the high and low frequencylimits. We find violation of the Wiedemann-Franz law and a dc thermoelectric conductivity α thatdoes not vanish at low temperatures, in contrast to Fermi liquids. We introduce a Langevin equationformalism to study critical dynamics over a broad region surrounding the quantum critical point.INTRODUCTION
We study electric, thermal, and thermoelectric conductivities in the vicinity of a z=2z=2z=2 superconductor-diffusive metal transition in two dimensions, both in the high- and low-frequency limits. We find violation of the Wiedemann-Franz law and a dc thermoelectric conductivity $\ensuremath{\alpha}$ that does not vanish at low temperatures, in contrast to Fermi liquids. We introduce a Langevin equation formalism to study critical dynamics over a broad region surrounding the quantum critical point.
We define a ``hyperconductor'' to be a material whose electrical and thermal dc conductivities are infinite at zero temperature and finite at any nonzero temperature. The low-temperature behavior of a hyperconductor is controlled by a quantum critical phase of interacting electrons that is stable to all potentially gap-generating interactions and potentially localizing disorder. In this paper, we compute the low-temperature dc and ac electrical and thermal conductivities in a one-dimensional hyperconductor, studied previously by the present authors, in the presence of both disorder and umklapp scattering. We identify the conditions under which the transport coefficients are finite, which allows us to exhibit examples of violations of the Wiedemann-Franz law. The temperature dependence of the electrical conductivity, which is characterized by the parameter ${\mathrm{\ensuremath{\Delta}}}_{X}$, is a power law, $\ensuremath{\sigma}\ensuremath{\propto}1/{T}^{1\ensuremath{-}2(2\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{X})}$ when ${\mathrm{\ensuremath{\Delta}}}_{X}\ensuremath{\ge}2$, down to zero temperature when the Fermi surface is commensurate with the lattice. There is a surface in parameter space along which ${\mathrm{\ensuremath{\Delta}}}_{X}=2$ and ${\mathrm{\ensuremath{\Delta}}}_{X}\ensuremath{\approx}2$ for small deviations from this surface. In the generic (incommensurate) case with weak disorder, such scaling is seen at high temperatures, followed by an exponential increase of the conductivity $ln\ensuremath{\sigma}\ensuremath{\sim}1/T$ at intermediate temperatures and, finally, $\ensuremath{\sigma}\ensuremath{\propto}1/{T}^{2\ensuremath{-}2(2\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{X})}$ at the lowest temperatures. In both cases, the thermal conductivity diverges at low temperatures.
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.
Quantum dot devices allow one to access the critical point of the two-channel Kondo model. The effective critical theory involves a free Majorana fermion quasiparticle localized on the dot. As a consequence, this critical point shows both the phenomenon of non-Fermi-liquid physics and fractionalization. Although a violation of the Wiedemann-Franz law is often considered to be a sign of non-Fermi-liquid systems, we show by exact calculations that it holds at the critical point, thereby providing a counterexample to this lore. Furthermore, we show that the fractionalized Majorana character of the critical point can be unambiguously detected from the heat conductance, opening the door to a direct experimental measurement of the elusive Majorana central charge c=12c=\frac{1}{2}c=21​.
Transport measurements on the cuprates suggest the presence of a quantum critical point (QCP) hiding underneath the superconducting dome near optimal hole doping. We provide numerical evidence in support of this scenario via a dynamical cluster quantum Monte Carlo study of the extended two-dimensional Hubbard model. Single-particle quantities, such as the spectral function, the quasi-particle weight and the entropy, display a crossover between two distinct ground states: a Fermi liquid at low filling and a non-Fermi liquid with a pseudo-gap at high filling. Both states are found to cross over to a marginal Fermi-liquid state at higher temperatures. For finite next-nearest-neighbour hopping t', we find a classical critical point at temperature Tc. This classical critical point is found to be associated with a phase-separation transition between a compressible Mott gas and an incompressible Mott liquid corresponding to the Fermi liquid and the pseudo-gap state, respectively. Since the critical temperature Tc extrapolates to zero as t' vanishes, we conclude that a QCP connects the Fermi liquid to the pseudo-gap region, and that the marginal Fermi-liquid behaviour in its vicinity is the analogue of the supercritical region in the liquid-gas transition. © 2011 The Royal Society.
We study the properties of a quantum critical point which develops in a BCS superconductor when pair breaking suppresses the transition temperature to zero. The pair fluctuations are characterized by a dynamical critical exponent z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2. Except for very low temperatures, anomalous contribution to the conductivity is proportional to T\sqrt{T}T​ in three dimensions, but to 1/T1/T1/T in two dimensions. At lowest temperatures, the conductivity correction varies as T1/4{T}^{1/4}T1/4 in three dimensions, and as ln(1/T)\mathrm{ln}(1/T)ln(1/T) in two.