# P15: Quantum transport of many-particle correlations

## Project Description

Recently it has become clear that for the physical understanding of spintronic systems, one-particle currents of the charge and the spin-dipole moment are not sufficient^{(1)-(3)}. This was noted first for a strongly interacting spin-valve structure including a quantum dot that locally supports a spin \(S>1/2\). Here, not only charge and spin-dipole moment are flowing, but there are additional currents of spatial spin-spin correlations^{(1)}. These are quantified by* currents* of the spin-quadrupole moment, which satisfy a corresponding continuity equation ^{(1),(2)}. These flows of correlations are physical in the sense that they ``drive'' the dynamics of more directly measurable charge- and spin-accumulations and when neglected unphysical results are obtained. These complex spintronic systems ^{(4)} were analyzed using the real-time diagrammatic approach ^{(5)} which can account for their strong local Coulomb and spin-spin interactions. However, also in effectively non-interacting systems quadrupole currents can arise and studying such systems allows for a better understanding of this new, rather complicated (tensorial) quantity. Applying the same real-time approach to a much simpler, non-interacting model of a spin-valve ^{(2)}, it was found that when applying a thermal bias a non-zero quadrupole current may flow due to the Pauli-principle, even for *zero spin-current*.

Quadrupole currents in spintronic devices are thus related to the \emph{changes} in the various spin-spin correlations that can arise from interactions as well as from quantum-statistics. The goal of this project is to find measurable quantities and setups that probe most clearly and directly this transport of many-spin correlations. By reformulating such problems in several approaches, we expect a fruitful comparison with studies of cross-correlations in, e.g, mesoscopic ``2-particle electron colliders'' ^{(6)} and Hanbury-Brown-Twiss setups ^{(7),(8)}. Another interesting question

is to what extent quadrupole currents are related to generation of quantum-entanglement ^{(9)} or even its transport, since the spin quadrupole operators in a double quantum dot are closely related to Bell-state density operators. Finally, we note that spin-quadrupole currents seem to be related to biquadratic four-spin interactions, an interesting connection with P4 which could be explored.

We plan to proceed in two main steps.

(i) To better understand the physical concept, we first address simpler non-interacting but Pauli-correlated spin-valve systems using two approaches that are complementary to the real-time approach. We will extend the Landauer-Büttiker scattering formalism ^{(10)} by formulating the calculation of the two-particle spin-quadrupole current as defined in Ref. (2). This will simplify the task of identifying measurable quantities and setups. For the same systems, we will also work out the Keldysh Green's function formulation, going beyond one-particle correlators. This is more closely related to important spintronic theories ^{(11)} and is more convenient for considering perturbative corrections due to interactions. Using both approaches, the calculation of electric, thermal, and magnetic transport are of interest. The outcome of this part may provide a starting point for extending those existing transport theories of spintronics that are based on effective one-particle/mean-field approximations. For instance, a quantity lacking in many such formulations is the magnetic anisotropy of the transported spins, a two-spin quantity. These approaches may also provide a convenient starting point for considering spin-quadrupole currents in exotic materials (e.g. quantum-Hall edge-states, topological insulators) studied in P5 and P6.

(ii) In the second step we reconsider the measurement quantities identified in step (i) and extend the analysis to setups combining interacting quantum dots with spin-valves using the real-time approach. This will allow the analysis of the transport of spin-spin correlations that arise from the Pauli-principle as well as those generated by strong interactions (spin-spin, Coulomb). This is promising, as recent work by one of us ^{(3)} has demonstrated: the magnetic anisotropy, crucial for the manipulation of nanomagnets, may be entirely *generated* by the quantum coherent transfer of spin-spin correlations from external ferromagnets. This quadrupolar analog of the spin-torque effect thus ``spintronically'' produces nanomagnets that are electrically tunable.

(1) M. M. E. Baumgärtel, M. Hell, S. Das and M. R. Wegewijs, Phys. Rev. Lett.** 107**, 087202 (2011)

(2) M. Hell, S. Das and M. R. Wegewijs, Phys. Rev. B **88**, 115435 (2013)

(3) M. Misiorny, M. Hell and M. R. Wegewijs, Nature Physics (DOI: 10.1038/nphys2766) (2013)

(4) B. Sothmann and J. König, Phys. Rev. B **82**, 245319 (2010)

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

(6) S. Ol'khovskaya, J. Splettstoesser, M. Moskalets and M. Büttiker, Phys. Rev. Lett.** 101**, 166802 (2008)

(7) P. Samuelsson, E.V. Sukhorukov and M. Büttiker, Phys. Rev. Lett. **92**, 026805 (2004)

(8) I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu and V. Umansky, Nature **448**, 333 (2007)

(9) G. Burkard, D. Loss and E.V. Sukhorukov, Phys. Rev. B **61**, 16303 (2000)

(10) Ya. M. Blanter and M. Büttiker, Phys. Rep. **336** 1 (2000)

(11) A. Brataas, G. E. W. Bauer and P.J. Kelly, Phys. Rep.** 427**, 157 (2006)