# Hubbard model dynamical correlation function via Lanczos Techniques

### given by Mario Collura

**Term: **0

**Start:** 11-01-2021
**End:** 26-02-2021
**Room:** 131

**Credits: **3

**Schedule:** TBA

## Program

This hands-on course has the aim of guiding the students to apply numerical iterative techniques

to face with a standard problem which can be encounter in typical condensed matter setup: Compute dynamical correlation functions.

The approach will be the following:

- Consider the Hubbard Hamiltonian with PBC and implement symmetries in

constructing the Hilbert space of the system. - Implement an efficient procedure to compute the action of the Hamiltonian to a generic state in the

previously constructed symmetric Hilbert space. - Write an iterative algorithm (e.g. Davidson) to find out the ground state |GS> of the model.

Working at HALF-FILLING, find out the |GS> -
Consider the operator

c(q)_s = Sum_{j = 1}^{N} \exp(-i q j) c(j)_s

where “s” is the spin index; and compute the dynamical correlation function (or its Fourier transform)

< GS | c(0,t)_s c^{\dag}(0,t’)_s | GS >

with t > t’, where c(0,t)_s = exp(i H t) c(0)_s exp(-i H t)via the Green function

< GS | c(0)_s (z - H)^{-1} c^{\dag}(0)_s | GS >

using the continued fraction methods. Notice that the number of particle is left unchanged;so the dynamical procedure works in the half-filling sector.

- Study the previous points as function of the Hubbard interaction U.
- [Optional] Study the previous points as function of the momenta “q”.

Reference for deep details: Rev.Mod.Phys.66.763; Rev.Mod.Phys.68.13;