An Exact Diagonalization Study of a Rapidly Rotating Bose-Bose Mixture with Attractive Interactions

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2025-7-23

Author

Turhan, Umut Can

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The Hall experiment can be explained by classical transport theory for lower magnetic fields (B ≲ 1 T). In Chapter 1 of this thesis, we review the Drude model to obtain cyclotron motion and to derive the conductivity tensor σ as a function of the magnetic field B. The quantum version of the classical Hall effect initiated a remarkable era in the search for new topological phases of two-dimensional materials and strongly correlated phases, treated via Landau quantization, which generates highly degenerate energy levels known as Landau levels (LLs). Strong, uniform magnetic fields lead to the quantization of the Hall resistivity ρ_H = h/(e^2*ν), where ν is the filling fraction of electrons, taking integer or fractional values depending on the system. This quantization reflects the presence of fully and partially filled LLs. Chapter 2 addresses the role of uniform magnetic and electric fields in a 2D system by choosing an appropriate vector potential A gauge. Also, we investigate the quantum mechanical current operator that allows us to quantize the conductivity tensor element σ_xy = e^2*ν/h. Subsequent sections present the derivations of the wave functions in both Landau and symmetric gauges and the corresponding eigenvalues. In the final sections of Chapter~2, we discuss the percolation concept, which explains Hall plateaux, and examine the role of edge states and random impurities in Hall samples by introducing confinement and random electrostatic potentials, which lead to backscattering effects at the boundaries and localized states in the bulk. Many-body physics plays a crucial role in the partially filled LL, highlighting the strong interactions between electrons in the context of the Fractional Quantum Hall Effect (FQHE). Chapter 3 begins with an exact solution for the two-body problem at filling fraction v = 1. Laughlin’s ansatz offers an elegant solution for the many-body problem at filling fraction v = 1/m, where m is even for bosons and odd for fermions. We review Laughlin’s incompressible wave function, which is an essentially exact solution for short-range interactions, and discuss the plasma analogy, which describes the Laughlin state as a quantum droplet with uniform density. The last section of Chapter 3 addresses the Haldane pseudopotential, which characterizes strong interactions in the FQHE regime. Rotating and trapped bosonic condensates exhibit unique properties for strongly correlated states in the FQHE regime, where the ground state lies in the lowest LL. The effective magnetic field generated by the Coriolis force mimics that of Quantum Hall (QH) systems, enabling the emergence of incompressible states for neutral bosons. In Chapter~4, we also study a three-dimensional rotating BEC in a harmonic trap, highlighting the connection between the FQHE and rapidly rotating BECs. In the final chapter, we investigate an equal-number binary bosonic gas with a fixed total angular momentum L in the z-direction. The tunable coupling ratio g_12/g allows us to identify a critical threshold beyond which the gas becomes unstable in the strongly attractive inter-species interaction regime in disc geometry. The collapse mode refers to the instability in which the binary gas undergoes spatial contraction. Overlap calculations for the ground state show that the Laughlin square wave function, which describes two species via separate Laughlin states, vanishes near the critical coupling ratio g_12/g ∼ -1, as expected. We introduce a molecular Laughlin ansatz that describes center-of-mass pairing between inter-species particles. Our molecular ansatz yields higher overlaps with both the ground state and low-lying excited states compared to the Laughlin square wave function. We also confirm that the ground state of the gas in this regime corresponds to a collapse mode, where the system rotates around its common center of mass. The dependence of the energy spectrum on g_12/g and the corresponding real-space density profiles are also presented.

Subject Keywords

Strongly Correlated States, Quantum Hall Systems, Bose-Einstein Condensate, Rotated 2D Bosonic Gas, Two Component Bosonic Gas, Kuantum Hall Sistemleri, Güçlü İlintili Kuantum Durumları, Bose-Einstein Yoğuşması, 2 Boyutlu Dönen Gaz, İki Bileşenli Boson Gazı

URI

https://hdl.handle.net/11511/115534

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Graduate School of Natural and Applied Sciences, Thesis