Numerically exact full counting statistics of the energy current in the Kondo regime

We use the inchworm quantum Monte Carlo method to investigate the full counting statistics of particle and energy currents in a strongly correlated quantum dot. Our method is used to extract the heat fluctuations and entropy production of a quantum thermoelectric device, as well as cumulants of the particle and energy currents. The energy-particle current cross correlations reveal information on the preparation of the system and the interplay of thermal and electric currents. We furthermore demonstrate the signature of a crossover from Coulomb blockade to Kondo physics in the energy current fluctuations, and show how the conventional master equation approach to full counting statistics systematically fails to capture this crossover.

Figure 1

Illustration of a quantum junction comprising leads $L$ and $R$ coupled by a central quantum dot region $D$, with a chemical potential bias at no temperature bias (left) and an opposite temperature bias (right). Depending on the choice of parameters, the particle and heat currents, $I^p$ and $I^h$, may be expected to flow in either the same direction (left) or opposite directions (right).

Figure 2

Real and imaginary parts of the moment generating function $Z\left(t;\lambda =0.5,\chi =0.1\right)$ in lead $L$ are shown for calculations using the QMC (thick black) QME (dashed blue), and PINEGF (dotted red) methods. We choose the voltage $V=1\mathrm{\Gamma }$, interaction strength $U=0\mathrm{\Gamma }$, and the empty initial condition. The two leads are held at equal inverse temperature ${\beta }_L={\beta }_R$. The left panels show the high-temperature case ${\beta }_L=0.4/\mathrm{\Gamma }$. The right panels show the low-temperature case ${\beta }_L=50/\mathrm{\Gamma }$.

Figure 3

Real (upper panels) and imaginary (lower panels) parts of the moment generating function $Z\left(t;\lambda ,\chi \right)$ in lead $L$ for the same parameters as in the upper panels of Fig. 2, but with applied thermal bias with ${\beta }_L{\beta }_R$ and left-to-right lead temperature ratio $T_L/T_R=0.1$. The counting fields are set to $\lambda =0.5$ and $\chi =0.1$ (left panels), and applied temperature bias and $\lambda =0.5$ and $\chi =0$ (right panels).

Figure 4

First cumulant of particle number change in lead $L$ (left panels) and $R$ (right panels) for different initial conditions, at voltage $V=1\mathrm{\Gamma }$, on-site interaction strength $U=8\mathrm{\Gamma }$, and with no temperature gradient, ${\beta }_L={\beta }_R$. The upper panels show the high-temperature case of $\beta =0.4/\mathrm{\Gamma }$. The lower panels show the low-temperature case of $\beta =50.0/\mathrm{\Gamma }$. Solid lines denote iQMC results, and dashed lines denote QME data.

Figure 5

First cumulant of energy change in lead $L$ (left panels) and $R$ (right panels) for different initial conditions, at the same parameters as Fig. 4. Solid lines denote iQMC results, and dashed lines QME.

Figure 6

Cross correlations between particle and energy change in lead $L$ (left panels) and $R$ (right panels) for different initial conditions, at the same parameters as Fig. 4. Solid lines denote iQMC results, and dashed lines QME.

Figure 7

First cumulant of heat transfer into lead $L$ for different initial conditions, at voltage $V=1\mathrm{\Gamma }$ and interaction strength $U=8\mathrm{\Gamma }$, and for the iQMC (thick lines) and QME (dashed lines) methods. We show the high-temperature regime of ${\beta }_L=0.4/\mathrm{\Gamma }$, with the temperature ratios set to $T_L/T_R=1.0$ (upper panel) and $T_L/T_R=0.1$ (lower panel). Solid lines denote iQMC results, and dashed lines QME.

Figure 8

Second cumulants of heat transfer for the same parameters as in Fig. 7. Solid lines denote iQMC results, and dashed lines QME.

Figure 9

Steady state particle current (upper panels) and current noise (lower panels) as a function of temperature ratio $T_L/T_R$. QME data are shown at $U=8\mathrm{\Gamma }$ (dashed black) and $U=4\mathrm{\Gamma }$ (dashed green). Black crosses indicate iQMC data at $U=8\mathrm{\Gamma }$, and green crosses denote the iQMC results for $U=4\mathrm{\Gamma }$. The left lead is held at constant high (left panels) or low (right panels) temperature, and the bias voltage is fixed at $V=1\mathrm{\Gamma }$.

Figure 10

Steady state heat current (upper panels), heat current noise (middle panels), and natural logarithm of the entropy production rate normalized by the left lead temperature (lower panels), plotted as functions of the temperature ratio $T_L/T_R$. All parameters and labeling conventions are as in Fig. 9.