0 0.5 1 1.5 2 1 10 100 m / m 0 ω 0 =0.05 ω $_{0}$=1 ω 0 =0.01 $_{0}$=5 $_{$_{ω}$0}$=0.5 ω 0 =0.1 0 0.5 1 1.5 2 1 10 100 m / m 0 0.5 0 =0.25 0.75 ω 0 =1 $_{$_{ω}$0}$=2.5 $_{0}$=5 3 0.01 0.1 1 5 0 1 10 100 m / m 0 λ =0.25 λ =0.5 λ =0.75 λ =1 λ =1.5 λ =2 0.1 1 5 0 1 10 100 m / m 0 =0.5 =0.85 λ =1.5 λ =1 λ =1.25 FIG. 2: (Color online) Polaron mass as a function of phonon frequency for different coupling, in 1D (left panel) and 3D (right panel), for λ as indicated. FIG. 1: (Color online) Polaron mass as a function of cou- pling for different phonon frequency, in 1D (left panel) and 3D (right panel), for ω 0 as indicated. N gen N E (SOS) E (Fock) 11 5063 -2.53737656 -2.4463726 12 10391 -2.53776224 -2.4588902 13 21247 -2.53792335 -2.4702848 14 43310 -2.53798172 -2.4807877 15 88052 -2.53800006 -2.4904380 16 178617 -2.53800511 -2.4991678 17 361644 -2.53800635 -2.5068954 18 731027 -2.53800663 -2.5135778 19 1475635 -2.53800668 -2.5192232 20 2975103 -2.53800669 -2.5238825 TABLE I: Variation of numerical groundstate energy E for ω 0 = 0 . 1, λ = 1 in 1D, with the number of iterations N gen in the VED state selection, and corresponding number N of phonon states per electron site in the Hilbert space. With Fock states, convergence is not achieved. With SOS, 6 digits of the energy are converged already for N = 10 5 states. The groundstate energy on the infinite lattice is obtained with a relative error of less than 10 − $^{8}$. 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. 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. The fundamental quantity characterizing polaron for- mation is the polaron mass m defined by III. POLARON MASS m − 1 = d 2 d | k | $_{2}$E ( k ) ∣ k =0 , (2) 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. 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 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.