Fermion-parity duality and energy relaxation in interacting open systems

Abstract

We study the transient heat current out of a confined electron system into a weakly coupled electrode in response to a voltage switch. We show that the decay of the Coulomb interaction energy for this repulsive system exhibits signatures of electron-electron attraction, and is governed by an interaction-independent rate. This can only be understood from a general duality that relates the non-unitary evolution of a quantum system to that of a dual model with inverted energies. Deriving from the fermion-parity superselection postulate, this duality applies to a large class of open systems.PACS numbers: 73.63.Kv, Energy decay due to heat currents is of key importance in the continued downscaling of electronic devices [1]. The quantum [2][3][4][5][6] and interaction effects [7][8][9] that arise on the nanoscale give rise to new possibilities [10][11][12][13] and motivate both fundamental [14] and application oriented [3, 4, 8, 13, 15] studies on quantum heatengines, possibly realized in, e.g., cold atoms, trapped ions, or quantum dots. The successful control and exploitation of heat in nanodevices requires both a fundamental understanding and the practical ability to detect and manipulate few-electron heat currents. Under stationary conditions, progress has been achieved using various approaches [16][17][18], including heat transfer through molecular-scale devices [19] with electrostatic gating [20]. However, any device is eventually adjusted by some external agent that provokes a time-dependent response. In the context of electronic heat currents, this raises a very basic question that, despite recent promising theoretical [21][22][23][24][25][26][27] and experimental [28][29][30] studies, has not been answered so far: how does a small electron system, typically governed by a strong level-quantization and Coulomb interaction, dissipate in time its stored energy into a coupled electronic bath?The essence of time-dependent transport in such systems is already captured by the simple model sketched in Figs. 1(a,b). Here, an instant energy shift of a single electronic orbital in a quantum dot leads to a timedependent charge current I N (t) [31,32] and heat current I Q (t) into a tunnel-coupled electrode. In the weak coupling regime, expressions for these currents can be calculated straightforwardly, and in the case of the transient charge current I N (t) also allow for an intuitive physical understanding [33]. This is, however, not the case for the heat current I Q (t) = a c e −γct +a p e −γpt . Compared to the charge current I N (t) ∝ e −γct , the heat current contains a second decay mode. The mere presence of this mode can be expected: it originates from the dissipation of the Coulomb energy. However, what is quite remarkable is that its rate γ p turns out to be completely independent of the interaction strength U [34-37] -despite entering the heat current only as a consequence of the interaction. Even more surprisingly, as indicated by the blue dashed line in Fig. 1(e 2 dealing with a system governed by repulsive i...

Relaxation of quantum dots in a magnetic field at finite bias – Charge, spin, and heat currents

“…The surprising result of Ref. is that such a physical connection between modes and amplitude covectors indeed turns out to exist for a large class of fermionic systems. Namely, this applies to systems that (i) evolve in time according to a Hamiltonian of the form –, (ii) contain noninteracting electronic leads r that are initially uncorrelated with the subsystem, and each in equilibrium (temperatures , electrochemical potentials ), and (iii) have tunneling frequencies that are energy‐independent (wideband limit).…”

“…To formulate the duality clearly, we need some more suitable notation. Realizing that the set of operators acting on the Hilbert space of the open system forms a vector space –the so‐called Liouville space –we denote an operator x as a “ket” vector . Via the Hilbert–Schmidt scalar product, we can let a vector act on another vector to yield a scalar number: with denoting an operator argument, we define a “bra” or covector .…”

Fundamental aspects of steady-state conversion of heat to work at the nanoscale

“…[370] treats interacting systems with adiabatic driving. One extremely recent work used similar methods for non-equilibrium propagators in time (rather than energy) on the Keldysh contour [357], it claims that one can use a method known as real-time transport theory [371][372][373][374][375][376][377][378] to prove the second law of thermodynamics, and the fluctuation theorems in sections 8.10.2-8.10.5, for an arbitrary interacting quantum system with or without time-dependent driving. All these works raise a number questions, and we feel it is much too soon to write a definitive review of these methods.…”

Gauge freedom in observables and Landsberg's nonadiabatic geometric phase: Pumping spectroscopy of interacting open quantum systems

“…The magnetic field B is initially added for our comparison with the FCS approach in Appendix I. We later focus on zero magnetic field results [68,69,75,98], and make use of the supplemental information to [156] where more details can be found.…”

“…Therefore, driving of L /¯ and R /¯ effectively amounts to single parameter driving. using notation borrowed from [156] the pumping response covector (84) is obtained as N is the fermion-parity operator which plays a special role [156].…”