Violation of Onsager's theorem in approximate master equation approaches

The consistency with Onsager's theorem is examined for commonly used perturbative approaches, such as the Redfield and second-order von Neumann master equations, for thermoelectric transport through nanostructures. We study a double quantum dot, which requires coherences between states for a correct description, and we find that these perturbative approaches violate Onsager's theorem. We show that the deviations from the theorem scale with the lead-coupling strength in an order beyond the one considered systematically in the respective approach.

Figure 1

Spin-polarized double-dot structure. The energy of the dot states is shifted by a gate voltage $V_g$. Both dots are coupled to each other ($\mathrm{\Omega }$) and to one lead each ($\mathrm{\Gamma }$). The two leads are described as electron reservoirs at distinct temperatures $T_{\ell }$ and chemical potentials ${\mu }_{\ell }$, where $\ell \in \{L,R\}$. A difference in either of the two parameters between the two leads can result in the flow of particle and energy currents, $I_{\ell }$ and ${\dot{E}}_{\ell }$, respectively. An interaction energy $U$ can be present for the double-occupied state (not shown).

Figure 2

Off-diagonal Onsager coefficient $L_1$ (top row) and the deviation from Onsager's theorem, $\mathrm{\Delta }=L_1^{\prime }-L_1$ (bottom row), as functions of the charging energy $U$ and the gate voltage ${\tilde{V}}_g=V_g+U/2$, calculated using the indicated approaches. The maximum deviation ${\mathrm{\Delta }}_{\max }$ uncovers significant violations of the Onsager relation (4) for the first-order Redfield and 1vN approaches. Neglecting the principal-value integrals (“No $\mathcal{P}$”) or including second-order contributions (“2vN”) significantly reduces the violation. In all calculations we set $\mathrm{\Gamma }=2\mathrm{\Omega }=T/2$.

Figure 3

Dependence of the peak values $L_{1,\mathrm{peak}},L_{1,\mathrm{peak}}^{\prime }$ of the Onsager coefficients and their difference ${\mathrm{\Delta }}_{\mathrm{peak}}$ on $U$ (left column) and $\mathrm{\Omega }$ (right column) upon varying the gate voltage $V_g$ for given parameters $T, \mathrm{\Omega }, U$. For increasing $U$, the deviation ${\mathrm{\Delta }}_{\mathrm{peak}}$ saturates in all approaches. Increasing $\mathrm{\Omega }$ leads to saturated ${\mathrm{\Delta }}_{\mathrm{peak}}$ for Redfield and 1vN approaches and decaying ${\mathrm{\Delta }}_{\mathrm{peak}}$ for the 2vN and “No $\mathcal{P}$” approaches.

Figure 4

$\mathrm{\Gamma }$ dependence of the peak values $L_{1,\mathrm{peak}}, L_{1,\mathrm{peak}}^{\prime }$, and ${\mathrm{\Delta }}_{\mathrm{peak}}$ for $U=0$ (left column) and $U=5T$ (right column) for $2\mathrm{\Omega }=T/2$. The Pauli ME (not shown) gives $L_1=L_1^{\prime }\propto \mathrm{\Gamma }$ with $L_1$ and $L_1^{\prime }$ having the same slope as in all other approaches at $\mathrm{\Gamma }=0$. For the Redfield and 1vN approaches the scaling of ${\mathrm{\Delta }}_{\mathrm{peak}}/L_{1,\mathrm{peak}}$ is of order $\mathrm{\Gamma }$ and for “No $\mathcal{P}$” and 2vN approaches the scaling is of order ${\mathrm{\Gamma }}^2$. However, for $U=0$ Onsager's theorem is satisfied in the 2vN approach as the conductances agree with the exact transmission formalism, Eq. (7).

Figure 5

Dependence of ${\mathrm{\Delta }}_{\mathrm{peak}}/L_{1,\mathrm{peak}}$ on $\mathrm{\Gamma }$ shown in a double logarithmic plot. The figure corresponds to Fig. 4 zoomed into the region of small $\mathrm{\Gamma }/T$. The values of parameters are $U=5T$ and $2\mathrm{\Omega }=T/2$.