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arXiv:cond-mat/0412179 · Physical Review Letters

Breakdown of One-Parameter Scaling in Quantum Critical Scenarios for High-Temperature Copper-Oxide Superconductors

Philip Phillips, Claudio Chamon

Published Aug 31, 2005• 62 citations
arXivHTMLPDF

Abstract

We show that if the excitations which become gapless at a quantum critical point also carry the electrical current, then a resistivity linear in temperature, as is observed in the copper-oxide high-temperature superconductors, obtains only if the dynamical exponent z satisfies the unphysical constraint, z < 0. At fault here is the universal scaling hypothesis that, at a continuous phase transition, the only relevant length scale is the correlation length. Consequently, either the electrical current in the normal state of the cuprates is carried by degrees of freedom which do not undergo a quantum phase transition, or quantum critical scenarios must forgo this basic scaling hypothesis and demand that more than a single-correlation length scale is necessary to model transport in the cuprates.

DOI
10.1103/physrevlett.95.107002

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We show that if the excitations which become gapless at a quantum critical point also carry the electrical current, then a resistivity linear in temperature, as is observed in the copper-oxide high-temperature superconductors, obtains only if the dynamical exponent z satisfies the unphysical constraint, z < 0. At fault here is the universal scaling hypothesis that, at a continuous phase transition, the only relevant length scale is the correlation length. Consequently, either the electrical current in the normal state of the cuprates is carried by degrees of freedom which do not undergo a quantum phase transition, or quantum critical scenarios must forgo this basic scaling hypothesis and demand that more than a single-correlation length scale is necessary to model transport in the cuprates.
Abstract In superconductors with relatively low superfluid density, such as cuprate high-TcT_cTc​ superconductors, the phase fluctuations of the superconducting order parameter are remarkable, presumably playing a nonnegligible role in shaping many distinctive physical properties. Despite existing phenomenological arguments for many years, a comprehensive quantitative analysis of the electrical transport properties resulting from thermal superconducting phase fluctuations in two-dimensional superconductors has been lacking. This study employs Monte Carlo simulations to explore the numerically exact transport properties of a microscopic model of superconductivity, where the thermal phase fluctuations of the superconducting order parameter are described by the classical XY model. For both s-wave and d x 2 − y 2 -wave pairings, the electrical resistivity exhibits a universal scaling behavior at temperatures above the critical temperature T c . Our numerical results demonstrate that the scaling behavior of the quasiparticle lifetime is closely linked to the correlation length of the superconducting order parameter, yielding the universal scaling behavior of electrical resistivity determined by the Berezinskii-Kosterlitz-Thouless critical scaling of the correlation length. While this behavior aligns formally with the phenomenological framework of Halperin and Nelson in the critical region, our findings suggest that the resistivity scales with the inverse of the correlation length with a power less than one. Additionally, we examine the dependence of the electrical resistivity coefficient on the pairing amplitude and potential implications for recent transport experiments.
Using the scaling theory of quantum critical phenomena we explore the occurrence of universal critical behavior at the insulator-to-superconductor and superconductor-to-normal state transitions at zero temperature.Experimentally, these phase transitions are driven by doping and correspond to critical end points of the phase transition łine in the temperature-hole concentration plane.Provided that the order parameter is a complex scalar in two dimensions, and that the London relation between superfluid number density and magnetic penetration depth holds, the scaling theory predicts universal behavior close to the insulator-to-superconductor transition.In particular, transition temperature and zero temperature penetration depth are universally related and the sheet resistance adopts a universal value.These predictions agree remarkably well with available experimental data and provide useful constraints for a microscopic theory.
Central to the normal state of cuprate high-temperature superconductors is the collapse of the pseudo-gap, briefly reviewed here, at a critical point and the subsequent onset of the strange metal characterized by a resistivity that scales linearly with temperature. A possible clue to the resolution of this problem is the inter-relation between two facts: (i) a robust theory of T-linear resistivity resulting from quantum criticality requires an additional length scale outside the standard one-parameter scaling scenario and (ii) breaking the Landau correspondence between the Fermi gas and an interacting system with short-range repulsions requires non-fermionic degrees. We show that a low-energy theory of the Hubbard model that correctly incorporates dynamical spectral weight transfer has the extra degrees of freedom needed to describe this physics. The degrees of freedom that mix into the lower band as a result of dynamical spectral weight transfer are shown to either decouple beyond a critical doping, thereby signalling Mottness collapse, or unbind above a critical temperature, yielding strange metal behaviour characterized by T-linear resistivity.
We consider the scaling of thermal transport in the presence of electric and magnetic fields near the finite temperature transition from the metallic normal state to the superconductor. We do so with fully relaxational, model A, dynamics for the order parameter and particle-hole symmetry. This enables us to determine the exact scaling dimension of the heat current operator and hence critical exponents for all transport coefficients in terms of the (approximately) known values of the correlation length exponent and the dynamic scaling exponent. In particular, we determine the critical behavior for the Nernst coefficient.
We show that the anomalous temperature scaling of five distinct transport quantities in the strange metal regime of the cuprate superconductors can be reproduced with only two nontrivial critical exponents. The quantities are (i) the electrical resistivity, (ii) the Hall angle, (iii) the Hall-Lorenz ratio, (iv) the magnetoresistance, and (v) the thermopower. The exponents are the dynamical critical exponent z=4/3z=4/3z=4/3 and an anomalous scaling dimension $\ensuremath{\Phi}=\ensuremath{-}2/3$ for the charge density operator.
It is claimed by a recent quantum Monte Carlo simulation that the linear-in-temperature DC resistivity observed in the high-TcT_{c}Tc​ cuprate superconductors can be reproduced in the pure two dimensional Hubbard model\cite{Huang}. Here we show perturbatively that such a translational invariant electronic model can not support a steady state current in the presence of a uniform electric field at any finite temperature. Instead, the Hubbard model is perfectly conducting in the linear response regime and will undergo Bloch oscillation at finite electric field for any finite temperature. Nevertheless, the quantum Monte Carlo simulation can provide us the key information on the temperature dependence of the Drude weight, a quantity of central importance in the holographic description of the transport properties of the strange metal phase.