Lang-Firsov Approaches to Polaron Physics: From Variational Methods to Unbiased Quantum Monte Carlo Simulations

In this chapter, we shall mainly review different versions of a recently developed quantum Monte Carlo (QMC) method applicable to Holstein-type models with one, two or many electrons. The appealing advantages of QMC over other numerical methods include the accessibility of rather large systems, the exact treatment of bosonic degrees of freedom (i.e., no truncation is necessary), and the possibility to consider finite temperatures to study phase transitions. The important new aspect here is the use of canonically transformed Hamiltonians, which permits the introduction of exact sampling for the phonon degrees of freedom, enabling us to carry out accurate simulations in practically all interesting parameter regimes.

Additionally, based on a generalization of the Lang-Firsov transformation, we shall present a simple variational approach to the polaron and the bipolaron problem which yields surprisingly accurate results.

This review is organized as follows. In Sect. 2, we present the general model Hamiltonian. Section 3 is devoted to a discussion of the Lang-Firsov transformation, and Sect. 4 contains the derivation of the variational approach. The QMC method is introduced in Sect. 5. Section 6 gives a selection of results for the cases of one, two and many electrons. Finally, we summarize in Sect.7.