The fractional Quantum Hall effect (FQHE) has been predicted in a cloud of bosons rotating at a frequency equal to that of the confining harmonic potential. Achieving this regime experimentally, by directly spinning the cloud, is a challenge. In this thesis, we probe an alternate approach to realizing the FQHE physics in cold gases by examining cold bosons confined to a rotating optical lattice.
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We derive modified Bose-Hubbard Hamiltonians to study the system in the tightbinding limit and develop analogies to Bloch electrons in the presence of a magnetic field. The primary effect of rotation is to introduce non-abelian character in the problem. This is because the system picks up a phase not always equal to a multiple of 2π when a particle goes around a plaquette.
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We numerically study hardcore bosons in the tight-binding limit by diagonalizing a Hamiltonian constructed using a truncated Fock basis. We show that vorticity can exist in the tight-binding limit though angular momentum is no longer quantized. Using the discrete rotational symmetry of the system, we identify quasi-angular momentum as a good quantum number. Quantum phase transitions are expected when the discrete rotational symmetry of the ground state changes. Using a perturbative analysis, we show that the effect of a weak lattice is to spatially modulate the Hall resistance. A linear-response analysis of the system\textquoterights response to an AC perturbation shows that the system does indeed display FQHE characteristics. We attempt to extend the analysis to larger systems using Monte Carlo methods. We show that the sign problem is inherent in the World Line Monte Carlo method as applied to non-abelian systems.
The fractional Quantum Hall effect (FQHE) has been predicted in a cloud of bosons rotating at a frequency equal to that of the confining harmonic potential. Achieving this regime experimentally, by directly spinning the cloud, is a challenge. In this thesis, we probe an alternate approach to realizing the FQHE physics in cold gases by examining cold bosons confined to a rotating optical lattice.
\
We derive modified Bose-Hubbard Hamiltonians to study the system in the tightbinding limit and develop analogies to Bloch electrons in the presence of a magnetic field. The primary effect of rotation is to introduce non-abelian character in the problem. This is because the system picks up a phase not always equal to a multiple of 2π when a particle goes around a plaquette.
\
We numerically study hardcore bosons in the tight-binding limit by diagonalizing a Hamiltonian constructed using a truncated Fock basis. We show that vorticity can exist in the tight-binding limit though angular momentum is no longer quantized. Using the discrete rotational symmetry of the system, we identify quasi-angular momentum as a good quantum number. Quantum phase transitions are expected when the discrete rotational symmetry of the ground state changes. Using a perturbative analysis, we show that the effect of a weak lattice is to spatially modulate the Hall resistance. A linear-response analysis of the system\textquoterights response to an AC perturbation shows that the system does indeed display FQHE characteristics. We attempt to extend the analysis to larger systems using Monte Carlo methods. We show that the sign problem is inherent in the World Line Monte Carlo method as applied to non-abelian systems.