טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentRan Budnik
SubjectEffective Hamiltonians for the Kagone and Triangular
Lattices
DepartmentDepartment of Physics
Supervisors Professor Emeritus Fishman Shmuel (Deceased)
Full Professor Auerbach Assa


Abstract

Geometric frustration of classical antiferromagnets is characterized by a macroscopic degeneracy of the ground state, resulting in extensive entropy. In the quantum limit of these systems the quantum fluctuations may play a role of perturbations, choosing a specific type of ground state. The Kagome lattice is an extreme example of such a system, and has been the subject of many different researches trying to illuminate its low temperature behavior.
In this work we study the low energy properties of the Heisenberg Antiferromagnet (spin half) on the Kagome lattice using the Contractor Renormalization (CORE) method. We find different effective Hamiltonians and analyze them. Due to the specific method we use we get the effective Hamiltonians set on a new triangular lattice (the super lattice) while each super site has two quantum degrees of freedom: A spin and a pseudo spin, both replacing the three original spin degrees of freedom. Our main result shows that the preferred ground state of the effective Hamiltonians tend to be a columnar dimer (local singlet) covering of this triangular lattice. This can be treated as a "quantum clock model" of six steps, consisting on the six different columnar dimer configurations. We also study the low excitations of this state, showing that the excitations within the dimer coverings of this triangular lattice are extremely close to O(2) symmetry, a property which offers an exponentially small excitations spectrum with no inherent energy scale within a spin gap, as expected from previous works. We also offer a way to connect this effective model to a Quantum Dimer Model, giving for the first time a rigorous way from the Heisenberg Hamiltonian into a Quantum Dimer Model. For completeness, we repeat the contractor renormalization procedure for the triangular lattice Heisenberg Antiferromagnet as well, showing that the results are in agreement with the expected ground state of Nèel order.