.00 , Figure 3 . 9 : Temperature profile for a 2-site chain in the stationary state, as a function of the temperature TL of the left bath, for T R = 1 respectively T R = 3 . Parameters are Jxy = 0 . 2 , bz = 2 . A l l three measurements of the local temperature T, agree, and find TL ~ T i and T 2 « T R . model. However, the heat transport must be investigated in much more detail before a definite picture can emerge. We now investigate in more detail the temperature profiles in non-equilibrium steady state situations. We continue with a model with Jxy = 0 . 2 , bx = —2, T R either 1 or 3 , and different values of TL- Results for 2 , 3 and 4 spin-chains are shown in Figs. 3 . 9 and 3 . 1 0 . The local temperatures T, were calculated from using the three methods discussed in the previous section, namely from matching (i) trs(aiT\ > T2 > T 3 > T 4 > TR. As alluded to before, this situation may change if there is a very large temperature differential across the chain. • Finally, in 3.13 we show temperature profiles for three and four-site chains with Jxy = 2 and bz = 0.1, as a function of TL when TR is fixed at 3. The magnetic field is very small, and this combined with the tendency of the system towards antiferromagnetic ordering leads to even smaller (o~iiZ)T values. As dis-cussed, in such cases some of the (o~iiZ)T expectation values are non-monotonic functions of T, leading to multiple solutions for Tj. In this case, we plot the value of Tj closest to the temperature of the bath near which the spin is located. Even so, fits from this are very peculiar for the three-site chain, as they suggest that T2 and T3 cool below TR, and that for large TL, the middle site is coldest than T i . This must be wrong, since it is impossible to have the chain colder than the baths it is connected to. For the four site chain, the fits to ( a i , z ) r Chapter 3. Spin chains coupled to thermal baths 47 • • m • o • CT • I A 1 • i i A 1 « * CT T CT • • • * • M * • : " i i * t l • • t • • • A f • ... • • .. m • Figure 3.12: Steady-state temperature profile for 3-site (upper panel) and 4-site (lower panel) chains as a function of TL, for TR = 3, JXY = 4 and bz = 2. Estimates from the tanh fit (green symbols) are completely wrong. Estimates from fits of (/I ,)T (blue symbols) and (cri T2 > T 4 > T 3 , but at least they are now all in between Tr, and TR. However, the values obtained from ( / IJ ) T are still reasonable and in agreement with previous data, and also with the results for the four-site chains. Clearly more investigations are needed before a coherent picture of the heat transport in spin chains emerges. First, we still need to understand how reason-able are the definitions of the local temperatures in cases where there is signif-icant difference between their predictions. It is to be hoped that using a small "thermometer" to measure local temperatures wil l help solve this issue. The dependence on chain length and on the various parameters Jz, Jxy, bz, TR, TL must be investigated thoroughly. For Jz = 0 this task may be aided by map-ping to non-interacting fermions and the possibility to use known results from there. Better modeling of the coupling to the leads (so that r is not the same irrespective of the energy of the transition) is also of interest. Finally, different types of baths may be analyzed. We already have results for "fermionic" baths, for the same type of quantities described here. These results are obtained by simply replacing n(E) —> (e / 3 ( £ - ' J ) + l ) - 1 , in the occupation numbers. Such baths could be thought of as collections of independent two-level systems (for instance, modeling coupling to nuclear spins) that are biased by their own local field, fixed by p. The results we have (not shown) reveal some similarities but also clear differences from the results for the phonon baths. These shall also be further investigated in future work. 50 Chapter 4 Conclusion In this thesis we have made some initial steps in the effort of developing a formalism for studying transport properties in systems biased by coupling to several baths. While the approach we propose if quite general, we have chosen short spin-chains as specific models in this work. Our approach is to generalize the projection operator technique to cases with multiple baths. The key approximation made is that the coupling to the baths is weak, so that second-order perturbation is sufficient to describe it. We have also made a Markov approximation to simplify the calculation. This approximation should be reasonable for the steady-state case that we focus on, but its effects on short term evolution remain to be analyzed. This requires numerical integration of some differential equations and it therefore seems feasible, at least for not too large systems. If only the steady-state is of interest, the calculation simplifies even more. In this case there is no need to solve the entire time evolution, instead the eigenstate corresponding to the zero eigenvalue of a certain matrix must be found. Using Maple, we could investigate systems with up to 4 spins, however we estimate that up to 14 or 15 spin chains could be studied quite straightforwardly if we write our own programs. We have shown that in a non-equilibrium steady-state, the reduced density matrix is non-diagonal, as one would expect since otherwise there would never be any currents flowing through the system. This observation raises some doubts about the relevance of studying Quantum Master Equations to learn something about non-equilibrium behavior of quantum systems, for cases where the steady-state is not an equilibrium state. One particular problem in the investigation of heat properties, which are our focus here, is the definition of the local temperature when the system is in a non-equilibrium steady state. For reasons explained in the text, we disagree with the simple definition used in the literature. We have introduced two other methods to calculate the local temperatures. In some cases, the predictions of the two methods agree with one another, and one can be reasonably certain that they are correct. When the two definitions disagree (as seen for small magnetic fields), the results obtained from the fit of the local energy to its equilibrium value are still reasonable, however it would be re-assuring to have a third method to measure the local temperatures that would give similar results. Such work is in progress, based on the idea of using a small "thermometer" coupled to individual sites to measure their temperatures. Of course, all this assumes that the notion of "local temperature" is meaningful. This also remains to be tested, but one expects it to be so at least in cases where the bias across the system is Chapter 4. Conclusion 51 not too large. System-size and available time constraints have severely limited the number of results we could show in this thesis. The results we show suggest that inter-esting physics may be uncovered for spin chains, but clearly significantly more effort is needed before a coherent picture emerges. For xy chains, in particular, one may also use mapping to non-interacting spinless fermions to try to make more progress, although the meaning of the coupling to the baths after such a transformation may be rather peculiar. In general, for all the results we obtain, we plan to also attempt the calculation of similar quantities using other appro-priate techniques, such as linear response for small biases, or non-equilibrium Keldysh formalism for large biases. Such comparisons will also be undertaken, soon. 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