Duality for open fermion systems: Energy-dependent weak coupling and quantum master equations

Abstract

Open fermion systems with energy-independent bilinear coupling to a fermionic environment have been shown to obey a general duality relation [Phys. Rev. B 93, 81411 (2016)] which allows for a drastic simplification of time-evolution calculations. In the weak-coupling limit, such a system can be associated with a unique dual physical system in which all energies are inverted, in particular the internal interaction. This paper generalizes this fermionic duality in two ways: we allow for weak coupling with arbitrary energy dependence and describe both occupations and coherences coupled by a quantum master equation for the density operator. We also show that whenever generalized detailed balance holds (Kolmogorov criterion), the stationary probabilities for the dual system can be expressed explicitly in terms of the stationary recurrence times of the original system, even at large bias.We illustrate the generalized duality by a detailed analysis of the rate equation for a quantum dot with strong onsite Coulomb repulsion, going beyond the commonly assumed wideband limit. We present predictions for (i) the decay rates for transient charge and heat currents after a gatevoltage quench and (ii) the thermoelectric linear response coefficients in the stationary limit. We show that even for pronouncedly energy-dependent coupling, all nontrivial parameter dependence in these problems is entirely captured by just two well-understood stationary variables, the average charge of the system and of the dual system. Remarkably, it is the latter that often dictates the most striking features of the measurable quantities (e.g., positions of resonances), underscoring the importance of the dual system for understanding the actual one. PACS numbers: 85.75.-d, 73.63.Kv, 85.35.-p 1 Previously referred to as "fermion-parity duality" by us [16-18].arXiv:1808.10223v2 [cond-mat.mes-hall]

Fermionic duality: General symmetry of open systems with strong dissipation and memory

“…In the following we will denote this direct consideration of the exact propagator Π(t) as approach (i) to duality. Applied to weakly coupled but locally interacting open systems, the fermionic duality (3) has already provided several interesting insights and predictions [10,[15][16][17]. For example, the time-dependent response of a "kicked" quantum dot with repulsive Coulomb interaction was shown to exhibit effects of electron-attraction.…”

“…The same formulas are very difficult to understand directly in terms of the real repulsive interaction, but are easily rationalized by electron-pairing induced by the attraction in the fictitious dual system defined by the duality mapping. More generally, the thermoelectric response of a quantum dot-although studied long ago-entails several features that turned out to have a very simple explanation in terms of an effective attractive model that is dual to the repulsive system of interest [15,16]. These conclusions hold even beyond linear response to electro-thermal biases where, e.g., Onsager relations no longer apply, and the effects can be understood by extending the weak-coupling fermionic duality beyond the wide-band limit [16].…”

Thermovoltage in quantum dots with attractive interaction

“…Finally, the thermoelectric response of repulsive quantum dots 16,19 has successfully been analyzed using a mapping based on a fermionic duality relation, 25 providing simple analytical formulas. Here, this relation enables us to explain the thermoelectric response of a quantum dot with attractive interaction in terms of the well understood physics of a repulsive dot.…”

“…We analyze the full thermoelectric response of the quantum dot using expressions obtained from a recently established, general fermionic duality. 16,19,25,28 This purely dissipative symmetry, applied to a weak-coupling master-equation description, maps the transport dynamics of fermionic open nonequilibrium systems to those of dual systems with sign-inverted local energies, chemical potentials, and energy dependencies of the tunnel couplings, replacing ½ẽ; ÀjUj; l a ; C a ðEÞ ! ½Àẽ; jUj; Àl a ; C a ðÀEÞ.]…”

“…This is typical for insights offered by fermionic duality. 16,19,25 The mapping to a repulsive system via n d enables a further prediction for experiments. The peaks in @S=@e are shifted by approximately 6T ln(2) away from the zero-temperature resonances e ¼ 0; jUj of a repulsive dot.…”