light heavy+small small light+ small heavy H L 0 2 4 6 8 10 ω$_{0}$ 0 2 4 6 8 10 λ 0 . 25 1 M = 2 M = 1 0 R = 0 . 5 R= 3 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 M=2 M=5 M=10 M=20 R=1 R=2 R=3 4 5 6 7 0 2 4 6 8 10 0 2 4 6 8 10 M=2 M=5 M=10 M=20 R=0.1 R=0.2 R=0.4 R=0.6 R=1 5 FIG. 7: (Color online) Overview of different polaron regimes in 1D. Only for the notation in this figure, we call a polaron light (heavy) if M < 2 ( M > 10), and large (small) if R > 3 ( R < 0 . 5). With this terminology, heavy large polarons are found in the region “ HL ”. The gray dashed rectangle gives the parameter region shown in the left panel in Fig. 6. For better visibility, the region of smaller ω 0 , λ values is magnified through a continuous variation of the scaling along each axis. FIG. 6: (Color online) Curves of constant mass, and curves of constant radius in 1D. The left panel is a magnification of the adiabatic regime ω 0 ≤ 0 . 25, which corresponds to the small orange rectangle in the right panel. VI. OPTICAL RESPONSE IN 1D V. SURVEY OF THE 1D POLARON The physics of adiabatic, or large and heavy, polarons is essentially different from that of small polarons in the antiadiabatic strong coupling regime. The difference ex- tends beyond the differences in mass or size of the po- laron discussed so far, but affects its very structure, and thus relates to the question in which sense a polaron can be understood as an itinerant quasiparticle $^{22,44,45}$. For a characterization of the quasiparticle properties we can study its response to electric fields. The (regular part of the) optical conductivity σ$_{reg}$ ( ω ) is defined through We have seen in the previous two sections that in the adiabatic regime in 3D light polarons evolve into very heavy and very small polarons through a rapid crossover. Only in 1D a continuous renormalization of the polaron mass and radius occurs in an extended re- gion of parameter space. A map of that region is given in Fig. 6 where we plot curves of constant mass and of constant radius. If we assign names to certain regimes bordered by the M = 2 , 10 and R = 0 . 5 , 3 curves, we get Fig. 7, which indicates the tendencies in the evolu- tion from a light to a heavy, or a large to a small po- laron with changing model parameters. Independent of the precise choice of wording, large and heavy polarons exist for sufficiently small ω$_{0}$ . The existence of such po- larons is specific for 1D, and a similar regime does not appear in 3D, where heavy polarons are always small as we discussed in the previous section. Even in 1D, it re- quires very small phonon frequencies. In combination with the comparison of the electron-phonon correlation function in Fig. 5 we can therefore refine the conven- tional adiabaticity condition ω$_{0}$ < t to a much stricter statement, namely ω$_{0}$/t ≪ 0 . 1 or ω$_{0}$/t ≲ 0 . 01. The spe- cific 1D physics of large heavy polarons thus occurs only in a tiny region of parameter space. Turning to larger phonon frequencies, the polaron is always small. For ω$_{0}$/t ≫ 1, the polaron is almost point-like. Rearrang- ing Eq. (3) gives χ (0) / ∑ $_{n}$χ ( n ) ≥ 1 − R $^{2}$, so that e.g. for R = 0 . 5 at least 75% of the total lattice displace- ment are located at the electron. In this limit, the only significant polaron parameter is the average number of phonons g 2 = ϵ$_{p}$/ω$_{0}$ – instead of the two parameters ω$_{0}$ and λ –, the mass is given by the asymptotic Lang-Firsov expression M = exp( g $^{2}$), and the differences between di- mensions vanish. The reader should be aware that in a numerical cal- culation of σ$_{reg}$ ( ω ) it is impossible to capture the exact position and weight of all excited states. For our results, the resulting error in integrated quantities, e.g. the first few moments of σ$_{reg}$ ( ω ), is negligible. For σ$_{reg}$ ( ω ) itself a compromise between finite spectral resolution and the finite level spacing in the numerical Hilbert space has to be found. Consequently, the shape, envelope and peak positions in the pictures for σ$_{reg}$ ( ω ) (see Fig. 9 below) with the current operator ˆ  = i t ∑ n c † n $_{+1}$c$_{n}$ − c † $_{n}$c$_{n}$$_{+1}$ and the eigenstates | ψ$_{l}$ 〉 with excitation energy ω$_{l}$ . Further information is contained in the f-sum rule σ$_{reg}$ ( ω ) = π ∑ l> 0 〈 ψ$_{l}$ | ˆ  | ψ$_{0}$ 〉| 2 ω$_{l}$ [ δ ( ω − ω$_{l}$ ) + δ ( ω + ω$_{l}$ )] , (5) 2 π D + 2 ∫ ∞ 0 σ$_{reg}$ ( ω ) dω = − π E$_{kin}$ , (6)