By Jonas Fransson

The purpose of this e-book is to offer a formula of the non-equilibrium physics in nanoscale structures when it comes to many-body states and operators and, furthermore, talk about a diagrammatic method of eco-friendly features expressed through many-body states. The purpose isn't to offer an account of strongly correlated structures as such. therefore, the focal point of this publication ensues from the common questions that come up while addressing nanoscale structures from a realistic perspective, e.g. current-voltage asymmetries, destructive differential conductance, spin-dependent tunneling. the point of interest is on nanoscale platforms produced from complexes of subsystems interacting with each other, lower than non-equilibrium stipulations, during which the neighborhood houses of the subsystems are ideally being defined by way of its (many-body) eigenstates.

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**Additional info for Non-Equilibrium Nano-Physics: A Many-Body Approach (Lecture Notes in Physics)**

**Example text**

Gαα (t, t ) = (−i) Tdα (t)dα† (t ) (dα )pq (dα† )q p (−i) TX pq (t)X q p (t ) . 3) pp qq As we know from the earlier sections, the transition matrix elements (dα )pq take care of the amplitude/importance of each of the propagators (−i) TX pq (t)X q p (t ) . From time to time it will be inconvenient to use the four index notation. We therefore define the Latin multi-indices a, b, c, . . , such that X a = Xpq , and the ¯ c, conjugate indices a, ¯ b, ¯ . . , such that X a¯ = Xqp = (X pq )† . Then, we can write ¯ the Hubbard operator Green functions as Ga b¯ (t, t ) = (−i) TXa (t)X b (t ) , such that the above identity becomes ¯ (dα )a (dα† )b Ga b¯ (t, t ).

G. (−i) T([h0 + hσ ]ckσ )(t)Xa¯ (t ) U . We may deal with those according to the decoupling procedure applied in Sect. 2, or by the Wick’s theorem following the approach described in detail in [2]. The latter of those procedures, however, is inappropriate to use in non-equilibrium situations. Therefore, we will follow another approach that provides a diagrammatic expansion of the propagators. g. 29) where the transitions a, b are arbitrary. Varying this Green function with respect to the sources Uξ (t) in the action S = exp{−i C ξ Uξ (t)Z ξ dt }, we obtain ¯ δGa b¯ (t, t ) = −i C ξ TSZ ξ (t )X a (t)X b (t ) δUξ (t ) (−i) TS ¯ − (−i) TSXa (t)Xb (t ) TSZ ξ (t ) TS TS dt .

The above analysis of the Pauli spin blockade achieved in the serial weakly coupled quantum dot can be interpreted in terms of transition matrix elements. Namely, when the quantum dot has a one electron it can host a second electron in several References 21 different configurations, and in the considered bias regime it is most likely that the second electron enters the quantum dot from the left lead. The quantum dot may make a transition into any of the triplet or singlet states, all transitions occur with a rather high probability.