Abstract In superconductors with relatively low superfluid density, such as cuprate high-
Tc 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.
We consider the scaling properties of the linear ac and nonlinear dc conductivities of a superconductor near the transition temperature
Tc. We first review a scaling theory of the conductivity near a critical point recently proposed by Fisher, Fisher, and Huse. By combining these scaling assumptions with causality arguments, we show that the phase ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}$ of the linear ac conductivity takes on a universal value near the critical point, ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}$=\ensuremath{\pi}(2-d+z)/2z, where d is the spatial dimension and z is the dynamic critical exponent. Hence, a measurement of the phase lag between the current and the voltage near the critical point provides a measurement of the exponent z. Then, using relaxational dynamics, we calculate these conductivities for 2d4, using (a) a Gaussian approximation, in which the quartic term in the Ginzburg-Landau Hamiltonian is neglected, and (b) an O(2n) generalization of the Ginzburg-Landau Hamiltonian, which for n\ensuremath{\rightarrow}\ensuremath{\infty} produces a Hartree approximation for the quartic term. These results agree with the scaling theory, and provide explicit expressions for the universal scaling functions. In addition, it is possible to study the crossover behavior from Gaussian to critical fluctuations as the critical temperature is approached. The prospects for going beyond the mean-field level by using renormalization-group methods are discussed, along with possible applications to the high-
Tc superconductors.
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.
We calculate the relaxational dynamical critical behavior of systems of O(n_{ parallel}) plus sign in circleO(n_{ perpendicular}) symmetry including conservation of magnetization by renormalization group theory within the minimal subtraction scheme in two-loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point, strong dynamical scaling holds, with the asymptotic dynamical critical exponent z=2varphinu-1 , where varphi is the crossover exponent and nu the exponent of the correlation length. The critical dynamics at n_{ parallel}=1 and n_{ perpendicular}=2 is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen, e.g., in the temperature dependence of the magnetic transport coefficients.
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=4/3 and an anomalous scaling dimension $\ensuremath{\Phi}=\ensuremath{-}2/3$ for the charge density operator.
Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where
F is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient
b of the fourth-order term in
F, however, shows that
b is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for
T>0 and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at
Tc, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in
F. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.