Work extraction and Landauer's principle in a quantum spin Hall device

Landauer's principle states that erasure of each bit of information in a system requires at least a unit of energy $k_{B}Tln2$ to be dissipated. In return, the blank bit may possibly be utilized to extract usable work of the amount $k_{B}Tln2$, in keeping with the second law of thermodynamics. While in principle any collection of spins can be utilized as information storage, work extraction by utilizing this resource in principle requires specialized engines that are capable of using this resource. In this work, we focus on heat and charge transport in a quantum spin Hall device in the presence of a spin bath. We show how a properly initialized nuclear spin subsystem can be used as a memory resource for a Maxwell's demon to harvest available heat energy from the reservoirs to induce charge current that can power an external electrical load. We also show how to initialize the nuclear spin subsystem using applied bias currents which necessarily dissipate energy, hence demonstrating Landauer's principle. This provides an alternative method of “energy storage” in an all-electrical device. We finally propose a realistic setup to experimentally observe a Landauer erasure/work extraction cycle.

Figure 1

QSHI with nuclear spins and electron-nuclear spin flip interaction. (a) The band structure of a typical QSHI system (using the BHZ model with tight-binding approximation). Red lines represent the edge states. (b) The band structure of the simplified Hamiltonian $h_{\mathrm{eff}}$ projected to a single edge (dashed blue lines). (c) Schematic description of the QSHI system with the edge currents interacting with the nuclear spins in the system, with the diamonds representing nuclear spins. (d) The spin flip interaction with the nuclear spins that form the MD.

Figure 2

(a) QSHI based quantum information engine, providing power to loads 1 and 2. Schematic description of (b) the charging and (c) the discharging phase of the QIE. In the charging phase, an applied bias current increases the number of right movers (solid lines) at the edges relative to left movers (dashed lines). This excess in turn creates a net nuclear spin polarization with opposite values in each edge. In the discharging phase, even without external bias, the net polarization of the nuclear spins increases the number of right movers, driving a net discharging current to the left.

Figure 3

Power, calculated using Eq. (16), as a function of the applied voltage and (a) as a function of mean polarization $m$ with $\zeta =1.0$ and (b) as a function of $\zeta$ with full polarization $m=0.5$. The lines divide the charging ($P>0$) and discharging ($P<0$) phases. Power can have negative values for $V<0(V>0)$ for a given mean polarization $m>0(m<0)$ (here, $e>0$), as an indication of the work extraction phase.

Figure 4

$P+k_{B}T{\dot{S}}_{\text{nuc}}$ as a function of $\tilde{V}$ and $m$ for $\zeta =1$. Note that $P+k_{B}T{\dot{S}}_{\text{nuc}}\ge 0$, in agreement with the second law.