Small-polaron transformation

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

When the system-bath coupling is linear in the bath coordinates, as in the spin-boson Hamiltonian, the physical interpretation is that the minimum position of each bath oscillator is shifted proportionately to the value of the system variable to which it is coupled. The small-polaron transformation redefines the Hamiltonian in terms of oscillators shifted adiabatically as a function of the system coordinate here the system coordinate is tr, so that the oscillators will be implicitly displaced equally but in opposite directions for each quantum state. Note that in the limit that the TLS coupling J vanishes, this transformation completely separates the system and bath. This makes it an effective transformation for cases of small coupling, and it has in fact been long and widely used in many types of physical problems, although typically in a nonvariational form [102]. Harris and Silbey showed that while simple enough to handle analytically, a variational small-polaron transformation contained the flexibility to treat the spin-boson problem effectively in most parameter regimes (see below) [45-47]. 

It is clear that the polaron shifts, fj, directly reduce the linear TLS-bath couplings, so that F would vanish entirely for fj = gf, this is the traditional (nonvariational) small-polaron transformation used in solid-state physics [102]. This choice is not generally optimal, however, because it leads to larger F+ and F couplings. In calculations presented below, the fj are always variationally optimized. 

Results. We can evaluate the effectiveness of the small-polaron transformation by considering a system of just two modes, for which exact free energies can also be obtained using traditional basis-set methods. The results are shown in Fig. 5 as a function of the bath frequency and the TLS coupling, J. Here the fj) in Eqs. (83)-(87) were varied to minimize the free energy Ag of the zero-order Hamiltonian, after the... 

Fourth, Jaime and Salamon (1999) have pointed out that a(T) increases more sharply than exponentially on cooling to rmax in fig. 30 and that the additional entropy transported increases on crossing the () -() phase boundary at tc. This observation is consistent with a progressive transformation from Zener to small polarons in the hole-poor phase as the hole concentration x = 0.30 in this phase is diluted by the trapping of Zener polarons in the hole-rich phase. Such a transformation would double the number of sites available to a polaron and would therefore increase the a of eq. (26) by reducing c = (1 — r)2x toward c = (1 — r)x, where r is the ratio of trapped to free polarons. In the O phase, most of the polarons appear to be small polarons at 7 N. 

Using the small polaron model, we can easily diagonalize the Hamiltonian (521) with respect to the phonon variables by canonical transformation [compare with expression (255)]... 

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