Generalized self-consistent reservoir model for normal and anomalous heat transport in quantum harmonic chains

Fictitious stochastic reservoirs incorporate scattering and dephasing mechanisms into the system in contact with these reservoirs. The reservoir-system coupling is described by the related self-energy in terms of the nonequilibrium Green's function formalism or equivalently the quantum Langevin equation formalism. In this study, we investigate thermal transport in a finite segment of an infinitely extended quantum harmonic chain with an equal self-energy at each site by using the self-consistent reservoir approach. In this setup, the entire system is lattice translation invariant so that mismatched boundaries are excluded from the model. Solving the Landauer-Büttiker equations under the self-consistent adiabatic condition, we quantitatively elucidate a thermally induced crossover of ballistic-to-diffusive transport and its scaling relation prescribed by a temperature-dependent mean free path. It is also shown that normal transport emerges in the diffusive limit for a linear self-energy, while nonlinear higher-order ones generically lead to anomalous transport. Physical implications of these observations are discussed in terms of the persistence of a massless Goldstone mode as well as the conservation of total linear momentum.

Figure 1

An infinitely extended quantum harmonic chain with an equal self-energy $\mathrm{\Sigma }$ at each site. A central portion consisting of $N$ lattice sites is regarded as the system, while two semi-infinite outer regions connecting to the system are viewed as leads, which are denoted as $L$ and $R$. In the system, the self-energy is ascribed to an inner reservoir serving as a temperature probe, which is symbolized by a gray bar in this figure.

Figure 2

Normalized conductance (a) $P_{jL}$ and (b) $P_{j{j}^{\prime }}$ as a function of intersite distance for $n=1$. In the calculation, the coupling strength is taken as $\gamma /s=0.1$, while the mean temperature is varied as $k_B\theta /s={10}^{-4}, {10}^{-3}, {10}^{-2}, \cdots , {10}^2$.

Figure 3

Probed temperatures for $n=1$. The upper panels show ${\mathrm{\Theta }}_j$ for (a) $N=10$ and (b) $N=1000$. In the calculation, the coupling strength is taken as $\gamma /s=0.1$, while the mean temperature is varied as $k_B\theta /s={10}^{-4}, {10}^{-3}, {10}^{-2}, \cdots , {10}^2$. The gray line indicates the linear profile without boundary jumps as a reference. The lower panels display ${\mathrm{\Theta }}_{1N}$ as a function of $k_B\theta$ for (c) $N=10$ and (d) $N=1000$. In the calculation, $\gamma$ is varied as $\gamma /s={10}^{-2}, {10}^{-1}, {10}^0$, and ${10}^1$.

Figure 4

Internal temperature difference ${\mathrm{\Theta }}_{1N}$ scaled with (a) ${\xi }_N$, (b) ${\eta }_N$, (c) ${\mathrm{\Delta }}_N$ and (d) ${\xi }_{\infty }/(N+1)$ for $n=1$. In the calculation, the parameters are varied in the range $10\le N\le 1000, {10}^{-2}\le \gamma /s\le 10$, and ${10}^{-4}\le k_B\theta /s\le {10}^2$.

Figure 5

Probed temperatures for $n=3$. The upper panels show ${\mathrm{\Theta }}_j$ for (a) $N=10$ and (b) $N=1000$. In the calculation, the coupling strength is taken as $\gamma s=0.1$, while the mean temperature is varied as $k_B\theta /s={10}^{-4}, {10}^{-3}, {10}^{-2}, \cdots , {10}^2$. The gray line indicates the linear profile without boundary jumps as a reference. The lower panels display ${\mathrm{\Theta }}_{1N}$ as a function of $k_B\theta$ for (c) $N=10$ and (d) $N=1000$. In the calculation, $\gamma$ is varied as $\gamma s={10}^{-2}, {10}^{-1}, {10}^0$, and ${10}^1$.

Figure 6

Internal temperature difference ${\mathrm{\Theta }}_{1N}$ scaled with (a) ${\xi }_N$, (b) ${\eta }_N$, (c) ${\mathrm{\Delta }}_N$ and (d) ${\xi }_{\infty }/(N+1)$ for $n=3$. In the calculation, the parameters are varied in the range $10\le N\le 1000, {10}^{-2}\le \gamma s\le 10$ and ${10}^{-4}\le k_B\theta /s\le {10}^2$. In (d), the data in the range $N\ge 100$ are chosen to show the asymptotic behavior.

Figure 7

Effective length scale ${\xi }_{\infty }$ as a function of temperature $k_B\theta$ for (a) $n=1$ and (b) $n=3$. In the calculation, $\gamma$ is varies as ${10}^{-4}, {10}^{-3}, {10}^{-2}, \cdots , {10}^2$ in units where $s=1$.

Figure 8

Thermal conductivities [(a) and (b)] ${\kappa }_N$ and [(c) and (d)] ${\kappa }_{1N}$ as a function of system size $N$. The numerical results for $n=1$ are shown in (a) and (c), while those for $n=3$ in (b) and (d). In the calculation, $\gamma$ is taken as 0.1 in units where $s=1$, while $k_B\theta /s$ is varied as ${10}^{-4}, {10}^{-3}, {10}^{-2}, \cdots , {10}^2$. The thin horizontal line in (a) and (c) indicates ${\kappa }_{\infty }$ for $n=1$ as a reference.

Figure 9

Density of states $D\left(\varepsilon \right)$ for (a) $n=1$ and (b) $n=3$. In the calculation, $\gamma$ is varied as ${10}^{-4}, {10}^{-3}, {10}^{-2}, \cdots , {10}^2$ in units where $s=1$. The gray line shows $D\left(\varepsilon \right)$ for $\gamma =0$ as a reference.