Analytical calculation of the Green’s function and Drude weight for a correlated fermion-boson system

In classical Drude theory the conductivity is determined by the mass of the propagating particles and the mean free path between two scattering events. For a quantum particle this simple picture of diffusive transport loses relevance if strong correlations dominate the particle motion. We study a situation where the propagation of a fermionic particle is possible only through creation and annihilation of local bosonic excitations. This correlated quantum transport process is outside the Drude picture, since one cannot distinguish between free propagation and intermittent scattering. The characterization of transport is possible using the Drude weight obtained from the f-sum rule, although its interpretation in terms of free mass and mean free path breaks down. For the situation studied we calculate the Green’s function and Drude weight using a Green’s functions expansion technique, and discuss their physical meaning. 1. Motivation The motion of a fermionic particle, such as an electron or hole, that interacts strongly with some background medium is a constantly recurring theme in condensed matter physics. The present contribution is motivated partly by the study of the motion of a hole in an antiferromagnetic spin background, as in a doped Mott insulator. Mott insulators occur for electrons on a lattice subject to strong Coulomb repulsion [1], and are commonly studied in the context of the Hubbard model. Since putting two electrons at the same lattice site costs a large energy if Coulomb repulsion is strong, the number of doubly occupied lattice sites is negligibly small. At half-filling we can therefore assume that every lattice site is occupied by a single electron. The system is an insulator because electron motion, which creates doubly occupied sites, is energetically forbidden. If electron motion is suppressed, the electron spin is the only relevant degree of freedom. Perturbation theory in the kinetic energy shows that spins couple antiferromagnetically in a Mott insulator. Starting from the Hubbard model, spin dynamics is described by the antiferromagnetic Heisenberg model with complex properties arising from the interplay of strong correlations and quantum spin fluctuations. In a very simple picture, we may alternatively consider the Néel state of a classical antiferromagnet (see Fig. 1, upper panel). If an electron is removed from a two-dimensional (2D) Mott insulator in a Néel state, the hole can move without creating doubly occupied sites. But the moving hole distorts the antiferromagnetic spin background, creating misaligned spins with substantial energy (see Fig. 1, lower panel). The ‘string’ of misaligned spins that forms along the path of the hole strongly Figure 1. Upper panel: Hole in a Néel antiferromagnet on a square lattice. Lower panel: When the hole moves by nearestneighbour hopping, it creates misaligned spins that do not match the Néel order. These distortions of the spin background lead to parallel spins on neighbouring sites – indicated with a dashed line – with a substantial increase in energy. To ‘unwind’ the string of background distortions, the hole in principle has to retrace the path it moved, and is therefore bound to its origin. The existence of (quantum) spin fluctuations, and spinorder restoring processes (see Fig. 2) weakens, or completely destroys, this ‘string effect’. restricts propagation, and tends to bind the hole to its origin. Therefore the hole does not move as a free particle. In a quantum antiferromagnet, which does not possess the strict order of the Néel antiferromagnet, the spin distortions can ‘heal out’ or relax by quantum spin fluctuations. It is then the relaxation rate of spin distortions that determines how fast the hole can move: The hole is continuously creating a string of distortions but can move on ‘freely’ at a speed which gives the distortions time to decay. In the absence of quantum spin fluctuations free-particle like motion of a hole is completely suppressed. Suprisingly enough, a hole can propagate through so-called ‘Trugman paths’ [2], which realize a certain combinatorial rearrangement of spins compatible with the spin order. The simplest of these processes, where the hole moves by two sites and restores the antiferromagnetic spin alignment, is shown in Fig. 2. In contrast to the situation for an almost free particle propagation through such processes depends entirely on the successive rearrangement of spins in precise coordination with the hole motion, just like executing a complicated dance pattern. With respect to the correlations between spin and hole motion we call this a correlated transport process. The focus of the present contribution is on similar correlated transport processes in a simpler fermion-boson model. This model is introduced in the next section (Sec. 2). In Sec. 3 we define the Drude weight as the appropriate measure of the efficiency of a correlated transport process. In Sec. 4 we derive the one-fermion Green’s function, which is discussed together with the Drude weight in Sec. 5, before we conclude in Sec. 6. 2. The model The picture sketched here — how the motion of a particle depends on the creation of background distortions and their decay — is very general, and applies e.g. to the polaron problem of electron Figure 2. The simplest ‘Trugman path’ that allows for propagation of a hole in a Néel antiferromagnet [2]. After six steps, the hole has moved by two sites to a next-nearest neighbour and the Néel spin order is restored.