Magnetothermoelectric response near quantum critical points

Following on from our previous work [M. J. Bhaseen et al., Phys. Rev. Lett. 98, 166801 (2007)] we examine the finite temperature magnetothermoelectric response in the vicinity of a quantum critical point (QCP). We begin with general scaling considerations relevant to an arbitrary QCP, either with or without Lorentz invariance, and in arbitrary dimension. In view of the broad connections to high-temperature superconductivity and cold atomic gases, we focus on the quantum critical fluctuations of the relativistic Landau-Ginzburg theory. This paradigmatic model arises in many contexts and describes the (particle-hole symmetric) superfluid-Mott insulator quantum phase transition in the Bose-Hubbard model. The application of a magnetic field opens up a wide range of physical observables, and we present a detailed overview of the charge and thermal transport and thermodynamic response. We combine several different approaches including the epsilon expansion and associated quantum Boltzmann equation, entropy drift, and arguments based on Lorentz invariance. The results differ markedly from the zero-field case, and we include an extended discussion of the finite thermal conductivity which emerges in the presence of a magnetic field. We derive an integral equation that governs its response and explore the crossover upon changing the magnetic field. This equation may be interpreted as a projection equation in the low-field limit, and clearly highlights the important role of collision invariants (or zero modes) in the hydrodynamic regime. Using an epsilon expansion around three dimensions, our analytic and numerical results interpolate between our previously published value and the exact limit of two-dimensional relativistic magnetohydrodynamics.

Figure 1

Superfluid-Mott insulator quantum phase transition in $2+1$ dimensions, as tuned by a control parameter $g$, such as film thickness or doping. The diagram shows the finite temperature 2D Kosterlitz-Thouless transition, and the crossover between the Mott insulator and the normal state. We examine the universal finite temperature magnetothermoelectric response in the vicinity of the (particle-hole symmetric) $2+1$ $XY$ QPT, as shown by the vertical arrow.

Figure 2

Mean-field phase diagram of the Bose-Hubbard model. At the tips of the Mott insulating lobes (indicated by dots) the system is particle-hole symmetric and described by the relativistic quantum Landau-Ginzburg theory.

Figure 3

Motion of a single relativistic charged particle in crossed electric and magnetic fields. (a) In the regime $\left|\mathbf{E}\right|<c\left|\mathbf{B}\right|$ the particle executes cyclotron orbits and has a well defined transverse drift velocity $v_D$. (b) In the regime $\left|\mathbf{E}\right|>c\left|\mathbf{B}\right|$ the particle is continually accelerated by the electric field.

Figure 4

Longitudinal scaling function, $G\left(\overline{k}\right)$, required for computation of the thermal conductivity in the drift regime. This figure was obtained by truncating the upper limit of integration at ${\overline{k}}_1=100$.

Figure 5

Evolution of the dimensionless thermal conductivity prefactor, $g$, as the ratio, $r\propto B^2/{ϵ}^4$, varies. Within the accuracy of our three-dimensional Monte Carlo integrations (approximately 3%) the result interpolates between the value reported in our previous letter (Ref. 5), $g_{\infty }\approx 5.55$, and the magnetohydrodynamic value (Ref. 30), $g_0=8\pi {\left(2{\pi }^2/45\right)}^2/\left(1.037\right)\approx 4.66$. The latter corresponds to the exact relation, ${\overline{\kappa }}_{xx}\left(B\right)=T{\mathcal{S}}_{\text{ch}}^2/B^2{\sigma }_Q$, for $B⪡{ϵ}^2$.