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Coherent states have been used in variational 15–17 and numerical approaches to the Holstein model $^{20,42}$. The latter two references describe a construction similar to ours. It appears that our construction is simpler, in that it avoids variational optimization of coherent state pa- rameters, while the additional virtues of the VED con- struction allow us to perform calculations for 1D and 3D systems in the far adiabatic regime. 

The fundamental quantity characterizing polaron for- mation is the polaron mass m defined by 

where E ( k ), the polaron dispersion, is the groundstate energy with momentum k . For a free electron with ϵ$_{p}$ = 0, the mass is m$_{0}$ = 1 / (2 t ). With increasing cou- pling, the polaron mass m grows, and the mass renor- malization M = m/m$_{0}$ becomes larger than unity. Our numerical method works on the infinite lattice with con- tinuous momentum k , which allows for a direct evaluation of the k -derivative in Eq. (2). 

In the antiadiabatic strong coupling limit ( ω$_{0}$ → ∞ with g 2 fixed) the polaron mass is given by the asymp- totic Lang-Firsov formula M ≃ exp( g $^{2}$), independent of dimension. In contrast, the behavior in the adiabatic regime ( ω$_{0}$ ≪ t ) depends on dimension, since it is gov- erned by the static limit ω$_{0}$ = 0. In the static limit the polaron in 1D has infinite mass for all λ > 0, while a phase transition – the self-trapping transition – from the free electron ( M = 1) to a polaron ( M = ∞ ) occurs at a finite coupling strength λ$_{c}$ ≈ 0 . 9 in 3D. Since this differ- ence is the most significant aspect of polaron formation 

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## III. POLARON MASS 

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in different dimensions, we leave out the 2D case here and concentrate on the 1D and 3D situation. 

In Fig. 1 we show how the polaron mass at finite ω$_{0}$ interpolates between the two limits. Both for 1D and 3D the polaron mass shows the asymptotic Lang-Firsov be- havior for ω$_{0}$ ≫ t . In 1D, the curves get steeper for smaller ω$_{0}$ , while keeping their functional form. The curves for ω$_{0}$ ≪ t and ω$_{0}$ ≫ t differ only quantitatively. In particular, a singular value of λ (or ω$_{0}$ ) associated with a qualitative change does not exist. In this sense the 1D behavior is scale-free. 

Quite different in 3D: The curves develop a sharp bend close to λ = 1 if ω ≲ 0 . 5, whose position converges to λ$_{c}$ for even smaller ω$_{0}$ . In this case a small change of λ , in the vicinity of λ$_{c}$ , results in a dramatic change of m . Although a true phase transition does not occur 43 for any ω$_{0}$ > 0, already at ω$_{0}$ = 0 . 5 the behavior of the polaron mass is, for all practical purposes, indistinguishable from that for a phase transition. In the adiabatic 3D situation one deals either with a quasi-free electron or a very heavy polaron. 

Plotting the polaron mass as a function of phonon fre- quency (Fig. 2) demonstrates the behavior discussed here in a very clear way. In 1D the mass diverges for all λ > 0 if ω$_{0}$ goes to zero. In 3D the mass diverges only if λ > λ$_{c}$ , while it approaches m$_{0}$ if λ < λ$_{c}$ . Just as before, now a small change in ω$_{0}$ results in a large change of m for λ close to λ$_{c}$ . In both situations the mass at finite fre- quency converges correctly to that in the static limit. 

