## - Quantum Fields on a Lattice

So far we have discussed particles moving in space that is a continuum. In solidstate physics it is more useful to employ what is known as a tight-binding picture where an electron is assumed to be tied to a particular atom, occasionally hopping to a neighboring atom. In this picture, the electron’s position is a discrete quantity-it is either on one atom or on its neighbor and not anywhere in between. This tightbinding picture may be derived using appropriate basis functions. For our purposes, we postulate that the kinetic motion of the electron is brought about by hopping. Therefore the kinetic energy is

c†iσc jσ, (9.1)

where < i j > signifies that sites i and j are nearest neighbors. The negative sign implies that hopping lowers the energy of the system by an amount t. But there is a price to be paid for hopping, in the form of a potential energy. The potential energy is assumed to be short-ranged, namely it is present only if a site has two electrons so that they repel with some energy U . But Pauli’s exclusion principle forbids two electrons with the same spin from residing at the same site. Thus while hopping lowers energy, hopping onto a site already occupied by an electron is either forbidden or has an energy cost U . The potential energy may be written as

V =U∑ i

ni↑ni↓, (9.2)

where niσ = c † iσciσ is the number of electrons in site i. These two put together, form

the famous Hubbard model of condensed matter physics. It is mathematically well defined given that short distances have a lower bound, namely the lattice spacing. The phases of this model depend on the ratio U/t and also the number of electrons per site (and temperature, if present). One can have variants of this model as well.

The extended Hubbard model includes interaction between the nearest neighbors. Then we have a term such as