Exact Diagonalization of Heisenberg SU(N) models

par Pierre Nataf (EPF Lausanne)

vendredi 22 janv. 2016, 10:30 → 11:30 Europe/Paris

Auditorium (Annecy-le-Vieux)

Systems of multicolor fermions have recently raised considerable interest due to the pos- sibility to experimentally study those systems on optical lattices with ultracold atoms. To describe the Mott insulating phase of $N$-colors fermions, one can start with the $SU(N)$ Heisenberg Hamiltonian. In the case of one particule per site, the $SU(N)$ Heisenberg Hamiltonian takes the form of a Quantum permutation Hamiltonian $H = J \langle i,j \rangle P_{ij}$, where the transposition operator $P_{ij}$ exchanges two colors on neighboring sites. We have developped a method to implement the $SU(N)$ symmetry in an Exact Diagonalization algorithm. In particular, the method enables one to diagonalize the Hamiltonian directly in the irreducibe representations of $SU(N)$, thanks to the use of standard Young tableaux, which are shown to form a very convenient basis to diagonalize the problem. It allowed us to prove that the ground state of the Heisenberg $SU(5)$ model on the square lattice is long range color ordered and it provided evidence that the phase of the Heisenberg $SU(6)$ model on the Honeycomb lattice is a plaquette phase. We have also extended the method to the case where there are $m \geq 1$ particles per site in the fully antisymmetric and symmetric irreps of $SU(N)$ in order to study $SU(N)$ critical chains. In this seminar, i will not only present our results but also devote some time to describe other numerical methods (such as DMRG, Quantum Monte Carlo, Variationnal Monte Carlo, etc...) which are useful in the simulation of those systems and which illustrate the kind of numerical tools that theoretical physicists are currently developping in the field of strongly correlated systems.