What is the difference between lattice models and tight-binding simulations? In condensed-matter physics, people use different methods to solve the many-particle Schrödinger equation. I was wondering about two of those methods, the lattice model and tight-binding simulation. Can someone explain what is the difference between the two?
Another method to solve the many-particle Schrödinger equation is density functional theory (DFT). As I understand, DFT is a method for calculating properties (for example bandstructures) of weakly correlated systems. However it is computational very expensive method. Is that why one uses tight-binding simulations for larger systems? What is the system size (number of particles) that tight-binding simulations, DFT and lattice models can solve?
 A: I believe the lattice model that you refer to is that the solid is modeled as having the constituent atoms/molecules all sit at the points of a regular repeating lattice.  The tight binding calculation is a related concept but not an alternative - it refers to use of atomic (or molecular) orbitals as the starting point for the calculation - as the basis set for building the full orbitals.  The tight bonding calculation can be done on a lattice, or it could be done in a disordered solid.  There are other calculations besides tight binding that can be done on the lattice (i.e. plane wave basis).
My experience aligns with yours re: tight-binding vs. DFT.  I could code up, do a quick-and-dirty tight-binding calculation on my laptop, in Matlab.  DFT required substantial dedicated software and/or hardware to run.  I don't know a rough number of atoms where you would be forced to switch from DFT to tight binding, for me it was more a matter of if you have the DFT infrastructure in place already (or access to it) you would just use that; if you don't have DFT you make do with what you have (i.e. tight-binding).  Another advantage of tight-binding is it can be easier to get intuitive answers and understanding about the system because you can look at the resulting wavefunctions as they are related to more familiar/simpler atomic/molecular orbitals.  For example doing a tight-binding calculation it might be obvious that one of the resulting bands is due to p-orbitals from the constituent atoms, whereas another is due to s-orbitals, and that provides insight into how they might be expected to behave.
Also important to note, that for solids the number of particles is generally the number of particles in the unit cell of the lattice.  So you could be modeling an arbitrarily large number of atoms using a lattice model, but each point on the lattice only has 1 particle, so DFT is extremely feasible.  Conversely, if you have many 10,000's of particles in each unit cell of the lattice you might not be able to do DFT.
