Polarons and slow quantum phonons Andreas Alvermann and Holger Fehske Institut f¨ ur Physik, Ernst-Moritz-Arndt-Universit¨ at Greifswald, 17489 Greifswald, Germany Stuart A. Trugman Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA We describe the formation and properties of Holstein polarons in the entire parameter regime. Our presentation focuses on the polaron mass and radius, which we obtain with an improved numerical technique. It is based on the combination of variational exact diagonalization with an improved construction of phonon states, providing results even for the strong coupling adiabatic regime. In particular we can describe the formation of large and heavy adiabatic polarons. A comparison of the polaron mass for the one and three dimensional situation explains how the different properties in the static oscillator limit determine the behavior in the adiabatic regime. The transport properties of large and small polarons are characterized by the f-sum rule and the optical conductivity. Our calculations are approximation-free and have negligible numerical error. This allows us to give a conclusive and impartial description of polaron formation. We finally discuss the implications of our results for situations beyond the Holstein model. PACS numbers: 71.38.-k,71.38.Cn,71.38.Ht In many materials it is the strong coupling of elec- tronic and lattice degrees of freedom that determines the charge transport or the optical properties $^{1–6}$. A basic ef- fect is the renormalization of the charge carrier mass $^{7}$. The mass renormalization is especially large in materials with a short-ranged interaction of electrons with optical phonons, as in molecular crystals or the colossal mag- netoresistive manganites $^{8}$. In such materials it happens that charge carriers, which otherwise resemble free elec- trons apart from a certain mass renormalization, give way to a kind of new quasiparticle with entirely different properties, the Holstein polaron $^{9}$. The mass of Holstein polarons can exceed that of a Bloch electron by several orders of magnitude, and may thus explain a low conduc- tivity or thermally activated transport $^{10}$. On the other hand, polarons have specific optical signatures $^{11}$. From early work we understand the fundamental mech- anism of polaron formation very well in two opposite cases: In the antiadiabatic regime of large phonon fre- quencies, where strong coupling perturbation theory pro- vides the picture of a Lang-Firsov polaron $^{12}$, and in the strict adiabatic limit of static oscillators, where the elec- tron and phonon wavefunctions factor $^{13,14}$. Much analytical and numerical work has been per- formed to extend our understanding to finite phonon frequencies and the intermediate regime, including ex- tended variational approaches $^{15–17}$, exact diagonaliza- tion (ED) $^{18–24}$, density matrix renormalization group$^{25}$, quantum Monte Carlo $^{26–29}$, and different perturbative or approximate schemes $^{30–35}$. Numerical calculations pro- vide reliable data for a wide range of phonon frequencies and coupling strengths, against which any discussion of Holstein polarons must be checked (for a recent review, see e.g. Refs. 36,37). It is however very difficult to per- form comparable calculations for small phonon frequen- The paper is organized as follows. In Sec. II we explain our numerical technique in application to the Holstein model. The differences between the 1D and 3D case are discussed through the comparison of the polaron mass in Sec. III, and the polaron radius in Sec. IV. The study of In the present contribution we use a new numerical technique to extend previous studies deep into the adia- batic regime. The key idea is an improved construction of phonon states that allows for the simultaneous rep- resentation of small and large oscillator shifts with few elements in a Hilbert space. The construction overcomes the problem that the number of oscillator energy eigen- states needed grows fast at small frequency. We com- bine this construction with the phonon state selection in variational exact diagonalization (VED) $^{19}$. Based on this method we present here results for phonon frequen- cies down to a hundredth of the electron transfer integral and still at large coupling, extending the frequency range accessed in previous studies by more than one order of magnitude. We explore the Holstein polaron in parame- ter regimes that, so far, were only accessible to approxi- mate treatments. The validity of common concepts, and thus our very picture of polaron formation, can now be assessed. This includes the question which properties of the static limit remain at small but finite phonon fre- quencies, and how the limit itself is approached. To our knowledge it is for the first time that the formation of large and heavy 1D polarons is studied with unbiased numerics. cies, since the increasing number of phonon excitations poses a severe restriction on any, even large-scale, nu- merics. So far this prevented a thorough study of the experimentally most relevant adiabatic regime, in partic- ular of large and heavy polarons as predicted by adia- batic approximations for one-dimensional (1D) systems. Also, numerical data for optical or transport properties are scarce $^{38,39}$. arXiv:1001.2482v2 [cond-mat.str-el] 26 Apr 2010 I. INTRODUCTION