My study focuses on a two-site model. One work in which the role of PHTs was elaborated on is Ref. [262], where Sznajd and Becker showed that the generalization of PHT (iii) in Table A.3 is a PHS of the usual non-superconducting Hubbard-model on a bipartite lattice.
A.9 Additional data on finite bias transport simula-tion
In Sec. 5.4.4 I discussed that in general the non-local coupling terms hybridize the Andreev states on the separate quantum dots, leading to the appearance of anticrossings in the transport spectrum. Here I will show the similarities and the differences of the anticrossings originating form different coupling terms. Generally two dominating anti-crossings open, one on diagonal, i.e. on the εR = εL line and one on the skew-diagonal, εR =−U−εL, their relative size and their evolution asεα is tuned differ for each coupling mechanisms.
Fig. A.9 shows the differential conductance of the two side, GL andGR, in the absence of non-local couplings on panel (a) (same as Fig. 5.19a) and with only one coupling being finite on panels (b-d) (ΓCAR = 0.1U on panel (b), γEC = 0.15 on panel (b), tLR = 0.1U on panel (d)) as the function of εL and µN for fixed εR = −1.2U. Fig. A.10 shows the conductance for the three latter case along the diagonal and the skew-diagonal, to illustrate the evolution of the size of the anticrossings.
In case of CAR coupling (ΓCAR = 0.1U) the more pronounced anticrossing is located on the skew-diagonal at εL = 0.2U, constituting of 3 conductance lines, appearing along the dashed line on Fig. A.9b. Another smaller anticrossing is present on the diagonal (εR=εL) at εL=−1.2U, marked with gray dotted line.
When εR is tuned, the size of the anticrossing positioned on the skew-diagonal re-mains the same, while for the one moving on the diagonal is maximal in the particle-hole symmetric point (εL =εR =−U/2) and decrease as the on-site energies are tuned away from this point. These two different evolution are shown on panel d) and a) of Fig. A.10, respectively. The gray dashed and dotted lines also mark the correspondence of Fig. A.9 and Fig. A.10. Note that due to the symmetries for the special cuts, presented on Fig. A.10 the conductance of the two sides are equal, GL =GR.
Explanation of the anticrossings: At εL = 0.2U, εR = −1.2U the ground state of the QD–SC–QD system is a Signlet, dominantly |gsi ∝ |0,↑↓i +... and the first two excited states are Doublets, |es1/2i ∝ |0, σi ± |σ,↑↓i+..., which are the bonding and antibonding combination of |0, σi and |σ,↑↓i states, hybridized by the CAR. The lower and upper excitation lines of the three in the anticrossing corresponds to the |gsi → |es1i and |gsi → |es2i transitions.
The third, middle line running parallel with the upper one corresponds to the excitation to the third, singlet excited state, from the first one, |es1i → |es3i = |S(1,1)i+.... The excitation energy for the transition is lower than the |gsi → |es2i transition’s. Note that this third excited state, |es3i cannot be accessed from the ground state since it has the same parity.
At εL = εR = −1.2U the ground state is dominantly |gsi = | ↑↓,↑↓i+... and the excited states are|es1i=|σ,↑↓i+...and|es2i=| ↑↓, σi+.... These two components cannot
A.9. Additional data on finite bias transport simulation
-0.2 0 0.2 0.4
-0.05 0 0.05 0.10
-0.2 0 0.2 0.4
-0.1 0 0.1 0.2
-0.2 0 0.2 0.4
-0.1 0 0.1 0.2
-0.2 -0.1 0 0.1 0.2 0.3
-0.10 -0.05 0 0.05 0.10 0.15 ΓCAR = 0U, γEC = 0, tLR = 0U, εR = -1.2U
GL (G0) GR (G0)
GL (G0)GL (G0)GL (G0) GR (G0)GR (G0)GR (G0)
ΓCAR = 0.1U, γEC = 0, tLR = 0U, εR = -1.2U
ΓCAR = 0U, γEC = 0.15, tLR = 0U, εR = -1.2U
ΓCAR = 0U, γEC = 0, tLR = 0.1U, εR = -1.2U a)
b)
c)
d)
+
εL/U εL/U
εL/U εL/U
εL/U εL/U
εL/U εL/U
μN/U
μN/UμN/U μN/UμN/U
μN/UμN/U μN/U
Figure A.9:Differential conductance of the QD–SC–QD system to the N leads as a function of εLandµN for fixedεR=−1.2U, with ΓLAR,L= ΓLAR,R = 0.25U a) without non-local couplings the usual single-QD ABS is formed on the dots, same as Fig. 5.19a in the main text., b) with ΓCAR= 0.1U, c) withγEC= 0.15, withtLR = 0.1U. The presence of the non-local couplings lead to the hybridization of the ABSs on the QDs, indicated by the appearance of anticrossings. The gray dashed and dotted lines mark the correspondence with Fig. A.10. The white circles mark the triplet blockade related NDC lines and the white + sign on panel b) marks a not triplet related NDC.
A.9. Additional data on finite bias transport simulation
-0.1 0 0.1 0.2 0.3 0.4 0.5
-0.1 0 0.1 0.2 0.3 0.4 0.5
a) ΓCAR = 0.1U b) γEC = 0.15 c) tLR = 0.1U
GL = GR (G0)GL = GR (G0)
d) e) f)
μN/U μN/U μN/UμN/U
μN/U μN/U
εL/U
εL/U εL/U
εL/U εL/U
εL/U
Figure A.10: Differential conductance of the QD–SC–QD system along the diagonal, εR = εL (panel a),b) and c)) and the skew-diagonal εR = −U −εL lines (panel d) e) and f)) for different non-local parameters: a),d) with ΓCAR = 0.1U, b),e) with γEC = 0.15 and c),f) with tLR = 0.1U. ΓLAR,L = ΓLAR,R = 0.25U as previously. The gray dashed and dotted lines mark the correspondence with Fig. S1. The size of anticrossing a), decrease b) increase, c),d) stays constant and e) decrease as εR is tuned away from the particle-hole symmetric point. f) the first two excitation are degenerate, there is no anticrossing present. Panel d-f) are the same as Fig. 5.20 in the main text.
A.9. Additional data on finite bias transport simulation directly by coupled by CAR, since they have the same number of electrons. However they can be coupled in second order, using the fact that LAR mix these state with |σ,0i and
|0, σirespectively. The CAR matrix element is finite between the|σ,0i(|0, σi) and| ↑↓, σi (|σ,↑↓i) states. The absence of the direct coupling between the excited states results in the suppression of the size of the anticrossing compared to the one on the skew-diagonal.
The splitting of further conductance lines is comparable to the line width, which is further broadened in an experiment due to the coupling to the normal leads – which is neglected in the present model –, therefore I neglect the detailed analysis of the fine structure here.
In case of EC the anticrossings are positioned similarly as for CAR (see Fig. A.9c withγEC = 0.15), a larger one constituting of 3 conductance line positioned on the skew-diagonal (dashed line) and a smaller one on the skew-diagonal (dotted line). The main difference between EC and CAR lies in behavior of the size of the anticrossing asεR is tuned. While the pronounced one have the same size for CAR (Fig. A.10d), for EC is maximal in the particle-hole symmetric point and it decreases (see Fig. A.10e) as εR is tuned away from this point. For the anticrossing positioned on the diagonal the behavior is strictly different from the case of CAR: On one hand the higher energy anticrossing line has a negative differential conductance (see the black lines on Fig. A.10b) for εL>0 (negative bias) and εL < −U (positive bias). Furthermore the two excitation lines move apart as εL increased from 0 (or decreased below −U) for EC, while the distance between the two lines decreases for CAR (Fig. A.10a). Note that second excitation is not visible for opposite bias directions as discussed previously on Fig. A.10b.
The third conductance line in the anticrossing positioned on the skew-diagonal is originating from the excitation to the third excited state similarly to the case of CAR, but positioned at higher energy than the first two excitations for EC.
Interestingly for the anticrossing positioned on the diagonal the splitting disappears close toεL=εR≈ −U (see Fig. A.10b). At these parameters the hybridized excited states are dominantly formed from by the |σ,↑↓i and | ↑↓, σi states, which are coupled by the HeffEC
(σ,↑↓)−(↑↓,σ) matrix element, which vanishes at εL=εR=−U (see Eq. A.16).
Note that due to the different definition of ΓCAR andγEC one cannot directly compare the magnitude of the splittings (see Eq. (A.15)).
Finally Fig. A.9d shows the effect of the IT, withtLR = 0.1U. On contrary to the CAR and EC, in this case only one anticrossing can be found – along the diagonal (dotted line) – and there is no anticrossing on the skew-diagonal (dashed line). The size of the anticrossing does not depend on the value of εL, except the εL ≈ −U/2 region (see Fig. A.10c). On the skew diagonal the first two excitations are exactly degenerate, independently of εR (see Fig. A.10f).
As I have shown above the main, common signature of the non-local couplings is the appearance of anticrossings in the excitation spectrum of the QD–SC–QD system, and the main difference of the three couplings lie in the behavior of these anticrossings. For CAR and EC they are positioned similarly, but their size change differently when εR is changed, while for IT only one anticrossing is present. This difference in the anticrossings allows one to determine the dominant coupling term in an experiment.
A not triplet related NDC line is shown on Fig. A.9b at εL ≈ −0.2U, µN ≈ −0.3U (marked with a white + sign), where the blocking occurs when the|es1i → |es4itransition becomes available, where|es1i is a singlet, and|es4i is a doublet state.
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