# P16: Geometric phases and gauge-structure in adiabatically driven open quantum systems

## Project Description

Geometric phases appearing in driven, closed quantum systems are relatively well-understood in relation to the phase gauge-transformations in standard quantum mechanics. A general framework of such type of physical theories is offered by the mathematical approach of fiber-bundles in differential geometry and topology. This allows, for instance, for a useful classification of various theories of physics (electromagnetism, particle-physics, gravitation) enabling a fruitful exchange of insights between them. Most nanoscale devices, however, are inherently open quantum systems by their coupling to (unwanted) environments or (wanted) electrodes for driving charge, spin, and energy currents. In various standard approaches to such open quantum systems, geometric contributions were identified ^{(1)-(4)} and also here the fiber-bundle framework is applicable and advantageous ^{(5),(6)} . So far, these were limited by physical assumptions, e.g., mean-field interaction (scattering matrix approach) ^{(3),(4)} or Markovian-Lindblad

time-evolutions for the reduced density matrix ^{(1),(2)}, or lowest order tunneling processes ^{(7),(8)}. A more general starting point is the real-time approach to the time-evolution of a strongly interacting subsystem formulated as a quantum-field theory (QFT) in Liouville space ^{(9)} (which also provides the starting point for RTRG used in P11 and P14). The key advantage of this approach is that the *derivation* of the memory-kernel for the subsystem is entirely transparent in all orders of the perturbation theory by explicit diagrammatic rules, including non-Markovian and time-dependent driving effects (see also P8 and P14). The crucial properties of the quantum field *super*operators that occur in this approach and generate the many-body Liouville space of density operators were analyzed in detail in a recent study by one of us ^{(10)}. This seems to provide an interesting new starting point for the analysis of local gauge-transformations of open quantum systems.

The following objectives aim to combine the real-time Liouville space approach with the mathematical fiber-bundle framework.

(i) We will work out the general structure of subsystem dynamics, both in the traditional language of gauge transformations as well as in the framework of general fiber-bundles. By identifying the gauge-group and the connection imposed by the open system dynamics the geometric contributions to the *dissipative* time-evolutions

can then be constructed from the so-called holonomy group.

(ii) Next, we will identify how geometric contributions appear in *measurable* quantities in *adiabatically* driven, strongly interacting nanostructures^{(11)} , such as the adiabatically pumped charge, spin or energy but also the current-current fluctuations studied in P8. Here we also envisage applications in project P14 which focuses on* non*-adiabatic dynamics instead.

(iii) Just as the Aharonov-Bohm effect can be related to the geometric phase, the above fundamental analysis of geometric phases in the Liouville space QFT may subsequently be related to electromagnetic gauge potentials/connections. We may consider the important task of deriving Ward identities for correlation functions

of Liouville space *super*fields. These are of importance for perturbative as well as non-perturbative RG studies, as

checks on their conservation of gauge-structure of various approximation schemes, and may perhaps simplify them. This part of the project directly connects to project P11.

Above it was the adiabatic limit of some physical time-evolution equation that dictated the gauge-potential. However, different gauge-potentials ("fiber-bundle connections'') for density operators were studied already early on (12),(13) in the context of mixed-state purification (rather than time-evolution). The proposed research also aims to benefit from this branch of quantum-information theory, and may in the end contribute to it.

(1) M. S. Sarandy and D. A. Lidar, Phys. Rev. A **71**, 012331 (2005)

(2) M. S. Sarandy and D. A. Lidar, Phys. Rev. A** 73**, 062101 (2006)

(3) B. L. Altshuler and L. I. Glazman, Science **283**, 1864 (1999)

(4) H.-Q. Zhou, S. Y. Cho and R. H. McKenzie, Phys. Rev. Lett. **91**, 186803 (2003)

(5) H. Goto and K. Ichimura, Phys. Rev. A **76**, 012120 (2007)

(6) J. E. Avron, M. Fraas and G. M. Graf, J. Stat. Phys. **148** 800 (2012)

(7) R. S. Whitney, Y. Makhlin, A. Shnirman and Y. Gefen, Phys. Rev. Lett. **94**, 070407 (2005)

(8) M. Campisi, S. Denisov and P. Hänggi, Phys. Rev. A **86**, 032114 (2012)

(9) H. Schoeller, Eur. Phys. J. Special Topics **168**, 179 (2009)

(10) R. B. Saptsov and M. R. Wegewijs, Phys. Rev. B **86**, 235432 (2012)

(11) J. Splettstoesser, M. Governale, J. König and R. Fazio, Phys. Rev. B **74**, 085305 (2006)

(12) A. Uhlmann, Rep. Math. Phys. **24**, 229 (1986)

(13) A. Uhlmann, Ann. Phys. **7**, 63 (1989)