Quantum critical thermal transport in the unitary Fermi gas

Strongly correlated systems are often associated with an underlying quantum critical point which governs their behavior in the finite-temperature phase diagram. Their thermodynamical and transport properties arise from critical fluctuations and follow universal scaling laws. Here, we develop a microscopic theory of thermal transport in the quantum critical regime expressed in terms of a thermal sum rule and an effective scattering time. We explicitly compute the characteristic scaling functions in a quantum critical model system, the unitary Fermi gas. Moreover, we derive an exact thermal sum rule for heat and energy currents and evaluate it numerically using the nonperturbative Luttinger-Ward approach. For the thermal scattering times we find a simple quantum critical scaling form. Together, the sum rule and the scattering time determine the heat conductivity, thermal diffusivity, Prandtl number, and sound diffusivity from high temperatures down into the quantum critical regime. The results provide a quantitative description of recent sound attenuation measurements in ultracold Fermi gases.

Figure 1

Thermal diffusivity $D_T$ (red) and sound diffusivity $D_{\text{sound}}$ (blue) vs temperature $T/T_F$ in the quantum critical regime of the unitary Fermi gas. Theoretical results from Luttinger-Ward calculations are shown in comparison with sound diffusion measurements [23].

Figure 2

Phase diagram of the spin-balanced, unitary Fermi gas at finite temperatures [36]. The QCP at $T=0$, $\mu =0$ is the starting point for the phase boundary of the homogeneous superfluid at $T_c=0.4\mu$ (solid line). The dashed lines mark the crossover to the quantum critical regime above the QCP.

Figure 3

Diagrammatic representation of the current correlation function (Kubo formula) and the dressed, amputated current vertex ${\tilde{\mathcal{T}}}^q$. The total response is given by the sum of the fermionic and pair contributions.

Figure 4

Thermal sum rule scaling function $f_{{\chi }_{qq}^T}$ vs $\beta \mu$ for the unitary Fermi gas: the interacting Luttinger-Ward result (25) is substantially larger than the free-fermion result (32) in the quantum degenerate regime.

Figure 5

Thermal and viscous scattering times ${\tau }_{\kappa ,\eta }T/\hslash$ vs $\beta \mu$ in the quantum critical region of the unitary Fermi gas. In this regime, both Boltzmann and large-$N$ calculations give nearly identical results. Furthermore, the large-$N$ viscous scattering time (blue) agrees well even with the strong-coupling Luttinger-Ward computation [16] (red).

Figure 6

Thermal conductivity $\kappa$ vs temperature $T/T_F$ for the unitary Fermi gas from Luttinger-Ward calculations (red line) and from experiment [23]. $\kappa$ saturates in the quantum degenerate regime and exhibits a shallow minimum of $\kappa /n\approx 8.7\hslash /m$ at $T/T_F\approx 0.6$.

Figure 7

Landau-Placzek ratio $\text{LP}=c_p/c_V-1$ vs temperature $T/T_F$ from Luttinger-Ward calculations (green below $T_c$ [32], red above $T_c$ [34]) and from experiment [37]. The dashed line indicates the high-temperature limit $\text{LP}=\frac{2}{3}$.

Figure 8

Prandtl number $\text{Pr}=D_{\eta }/D_T$ vs temperature $T/T_F$ from Luttinger-Ward calculations (red line) and sound attenuation measurements [23]; the dashed line marks the high-temperature limit $\text{Pr}=\frac{2}{3}$.

Figure 9

Spin scattering times ${\tau }_s$ from Boltzmann (dashed) and large-$N$ calculations (blue) coincide and agree well with Luttinger-Ward results (red) in the quantum critical regime.

Figure 10

Schmidt number $\text{Sc}=D_{\eta }/D_s$ in the quantum critical regime, combining quantum critical scattering times from our large-$N$ calculation with the Luttinger-Ward equation of state. The dashed line denotes the Boltzmann limit $\text{Sc}=\frac{5}{2}$.