Tunneling amplitude

In the tunneling argument from Section IIIA, we have suggested a WKB-type expression for the tunneling amplitude ... 

As is clear from the previous discussion, the long-time power law behavior of the heat capacity is determined by the slow two-level systems corresponding to the higher barrier end of the tunneling amplitude distribution, argued to be of the form shown in Eq. (24). If one assumes that this distribution is valid for the... 

Here e is the new value of the energy splitting, the co, are the ripplon frequencies, and the A,- are tunneling amplitudes of transitions that excite the corresponding vibrational mode of the domain wall. Those amplitudes will be discussed in due time for now, we repeat, the expression above will be correct in the limit Ai/Ha>i —> 0. Finally, the renormalized value e was used in the denominator. While, according to Feenberg s expansion [118], including e in the resolvent is actually more accurate, we do it here mostly for convenience. 

Our renormalization procedure is internally consistent in that the physical value of the tunneling amplitude depends on the scaling variable—the bare coupling Aq—only logarithmically. This bare coupling must scale with the only quantum scale in the problem—the Debye frequency, as pointed out in the first section. 

Let us now turn to low temperatures T [Pg.144]

More realistic choice of random Gaussian correlated tunneling amplitudes tk1, k2 leads to the same results after averaging over tk1>k2-... 

Since the probability amplitude for tunneling is exponentially sensitive to the position of the dot, the maximum of the tunnel exchange interaction between an electron on the dot and an electron in one lead occurs when the tunneling coupling to the other lead is negligible. This is why we will assume the following property of tunneling amplitude ALjR t) to be fulfilled ... 

The total tunneling amplitude along the incoming trajectory at energy E includes contributions from all tunneling paths initiated in the reactant valley... 

The tunneling amplitude for each straight-line path is approximated using a primitive semiclassical expression... 

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