Non-zero temperature transport near quantum critical points

TL;DR

The paper addresses universal finite-temperature transport near a two-dimensional quantum critical point, showing that inelastic scattering among thermally excited carriers drives a hydrodynamic, incoherent regime with a universal scaling function $\Sigma(\hbar\omega/k_B T)$. Using a quantum Boltzmann equation and an $\varepsilon=3-d$ expansion in a disorder-free boson model of the superfluid-insulator transition, it computes the DC conductivity as a universal multiple of $e^2/h$ and maps out a crossover function linking the hydrodynamic and collisionless regimes via $\Sigma(\bar{\omega})$. The work highlights a nontrivial $\,\omega/k_B T$ dependence at criticality, clarifies misinterpretations that treated DC and AC limits as identical, and discusses implications for spin transport, Luttinger liquids, and potential self-duality, with concrete experimental tests proposed for AC conductivity across quantum critical points. Overall, it provides the first controlled calculation of universal finite-$T$ transport at a 2D quantum critical point and establishes a framework for interpreting finite-temperature quantum critical dynamics in related systems.

Abstract

We describe the nature of charge transport at non-zero temperatures ($T$) above the two-dimensional ($d$) superfluid-insulator quantum critical point. We argue that the transport is characterized by inelastic collisions among thermally excited carriers at a rate of order $k_B T/\hbar$. This implies that the transport at frequencies $ω\ll k_B T/\hbar$ is in the hydrodynamic, collision-dominated (or `incoherent') regime, while $ω\gg k_B T/\hbar$ is the collisionless (or `phase-coherent') regime. The conductivity is argued to be $e^2 / h$ times a non-trivial universal scaling function of $\hbar ω/ k_B T$, and not independent of $\hbar ω/k_B T$, as has been previously claimed, or implicitly assumed. The experimentally measured d.c. conductivity is the hydrodynamic $\hbar ω/k_B T \to 0$ limit of this function, and is a universal number times $e^2 / h$, even though the transport is incoherent. Previous work determined the conductivity by incorrectly assuming it was also equal to the collisionless $\hbar ω/k_B T \to \infty$ limit of the scaling function, which actually describes phase-coherent transport with a conductivity given by a different universal number times $e^2 / h$. We provide the first computation of the universal d.c. conductivity in a disorder-free boson model, along with explicit crossover functions, using a quantum Boltzmann equation and an expansion in $ε=3-d$. The case of spin transport near quantum critical points in antiferromagnets is also discussed. Similar ideas should apply to the transitions in quantum Hall systems and to metal-insulator transitions. We suggest experimental tests of our picture and speculate on a new route to self-duality at two-dimensional quantum critical points.

Figures (6)

• Figure 1: A sketch of the expected form of the real part, $\Sigma'$, of the universal scaling function $\Sigma$ appearing in the scaling form (\ref{['scaling']}) for the conductivity, as a function of $\overline{\omega} = \hbar \omega / k_B T$. There is a Drude-like peak from the inelastic scatterings between thermally excited carriers at $\overline{\omega}$ of order unity. At larger $\overline{\omega}$, there is a crossover to the collisionless regime where $\Sigma' \sim \overline{\omega}^{(d-2)/z}$ as $\overline{\omega} \rightarrow \infty$.
• Figure 2: Universal form of the conductivity $\sigma ( \omega , T \rightarrow 0 )$ in $d=2$; the vertical scale is measured in units of $\hbar/Q^2$. Only the $\omega =0$ value is given by the universal number $\Sigma ( 0 )$. For all $\omega > 0$, $(\hbar/Q^2) \sigma = \Sigma ( \infty )$. $Q$ is the 'charge' of the order parameter: for the superfluid-insulator transition $Q=2e$, while for quantum antiferromagnets $Q = g \mu_B$.
• Figure 3: Structure of the real part, $\Sigma'$, of the universal scaling function $\Sigma$ in (\ref{['scaling']}) for the conductivity at the quantum critical coupling of the model ${\cal S}$ defined in (\ref{['action']}). The spatial dimensionality $d=3-\epsilon$, and $\epsilon$ is assumed to be small. As before $\overline{\omega} = \hbar\omega/ k_B T$. The Drude peak at small $\overline{\omega}$ has a width of order $\epsilon^2$ and a height of order $1/\epsilon^2$: this feature of the conductivity is denoted later in the paper by $\sigma_I$. The collisionless contribution (denoted $\sigma_{II}$ later) begins at $\overline{\omega}$ of order $\epsilon^{1/2}$; as $\overline{\omega} \rightarrow \infty$, this contribution is a number of order unity times $\overline{\omega}^{1-\epsilon}$
• Figure 4: The real part, $\Sigma'$, of the universal scaling function $\Sigma$ (see (\ref{['scaling']})) for the conductivity at the quantum critical coupling of the model ${\cal S}$, correct to first order in $\epsilon=3-d$. The numerical values are obtained from (\ref{['k1a']}) and (\ref{['k5a']}) with $d=2$ ($\epsilon = 1$). There is a delta function precisely at $\omega/T = 0$ represented by the heavy arrow: the weight of this delta function is given in (\ref{['k4']}) and (\ref{['k6']}). The delta function contributes to $\sigma_I$, and the higher frequency continuum to $\sigma_{II}$
• Figure 5: Real part of the universal function $\overline{k}^3 \Psi ( \overline{k}, \widetilde{\omega})$ as a function of $\overline{k}$ for a few values of $\widetilde{\omega}$. The function $\Psi$ is defined in (\ref{['scalepsi']}) and (\ref{['cless2']}), and was obtained by numerical solution of the linearized quantum Boltzmann equation (\ref{['inteq1']}). At $\widetilde{\omega} = 0$, $\Psi$ is real, but is complex for general $\widetilde{\omega}$. Here $\overline{k} = k/T$, and $\widetilde{\omega} = \overline{\omega}/\epsilon^2 = \omega/\epsilon^2 T$ (in physical units $\overline{k} = \hbar c k / k_B T$, $\widetilde{\omega} = \hbar \omega/ \epsilon^2 k_B T$).