#### Date of Award

Fall 2012

#### Degree Type

Thesis

#### Degree Name

Master of Science (MSc)

#### Department

Physics and Astronomy

#### Supervisor

Sung-Sik Lee

#### Language

English

#### Committee Member

Erik Sorensen, Duncan O'Dell

#### Abstract

In Part I, the relationship between the topology of the Fermi surface and the entanglement entropy S is examined. Spinless fermionic systems on one and two dimensional lattices at fixed chemical potential are considered. The lattice is partitioned into sub-system of length L and environment, and the entanglement of the subsystem with the environment is calculated via the correlation matrix. S is plotted as a function of the next-nearest or next-next nearest neighbor hopping parameter, t. In 1 dimension, the entanglement entropy jumps at lifshitz transitions where the number of Fermi points changes. In 2 dimensions, a neck-collapsing transition is accompanied by a cusp in S, while the formation of electron or hole-like pockets coincides with a kink in the S as a function of the hopping parameter. The entanglement entropy as a function of subsystem length L is also examined. The leading order coefficient of the LlnL term in 2 dimensions was seen to agree well with the Widom conjecture. Of interest is the difference this coefficient and the coefficient of the term linear in L near the neck-collapsing point. The leading order term changes like |t-t_{c}|^{1/2} whereas the first sub-leading term varies like |t-t_{c}|^{1/3}, where t_{c} is the critical value of the hopping parameter at the transition.

In Part II, we study the statistics of fractionalized excitations in a bosonic model which describes strongly interacting excitons in a N-band insulator. The elementary excitations of this system are strings, in a large N limit. A string is made of a series of bosons whose flavors are correlated such that the end points of a string carries a fractionalized flavor quantum number. When the tension of a string vanishes, the end points are deconfined. We determine the statistics of the fractionalized particles described by the end points of strings. We show that either bosons or Fermions can arise depending on the microscopic coupling constants. In the presence of the cubic interaction in the Hamiltonian as the only higher order interaction term, it was shown that bosons are emergent. In the presence of the quartic interaction with a positive coupling constant, it was revealed that the elementary excitations of the system possess Fermion statistics.

#### Recommended Citation

Rodney, Marlon A., "THE ENTANGLEMENT ENTROPY NEAR LIFSHITZ QUANTUM PHASE TRANSITIONS & THE EMERGENT STATISTICS OF FRACTIONALIZED EXCITATIONS" (2012). *Open Access Dissertations and Theses.* Paper 7487.

http://digitalcommons.mcmaster.ca/opendissertations/7487

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