Çekici Etkileşimli, Hızla Dönen bir Bose-Bose Karışımının Tam Köşegenleştirme Yöntemiyle İncelenmesi

Turhan, Umut Can

Published Aug 27, 2025• 0 citations

Abstract

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.

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Abd Ali, Maher Mahmood

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Toward a New Theory of the Fractional Quantum Hall Effect

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Quantum simulation of conductivity plateaux and fractional quantum Hall effect using ultracold atoms

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We analyze the role of impurities in the fractional quantum Hall effect using a highly controllable system of ultracold atoms. We investigate the mechanism responsible for the formation of plateaux in the resistivity/conductivity as a function of the applied magnetic field in the lowest Landau level regime. To this aim, we consider an impurity immersed in a small cloud of an ultracold quantum Bose gas subjected to an artificial magnetic field. We consider scenarios corresponding to experimentally realistic systems with gauge fields induced by rotation of the trapping parabolic potential. Systems of this kind are adequate to simulate quantum Hall effects in ultracold atom setups. We use exact diagonalization for few atoms and to emulate transport equations, we analyze the time evolution of the system under a periodic perturbation. We provide a theoretical proposal to detect the up-to-now elusive presence of strongly correlated states related to fractional filling factors in the context of ultracold atoms. We analyze the conditions under which these strongly correlated states are associated with the presence of the resistivity/conductivity plateaux. Our main result is the presence of a plateau in a region, where the transfer between localized and non-localized particles takes place, as a necessary condition to maintain a constant value of the resistivity/conductivity as the magnetic field increases.

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Quantum simulation of conductivity plateaux and fractional quantum Hall effect using ultracold atoms

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Exactly solving one-dimensional models for the fractional quantum Hall effect

Joseph R. Cruise

Nearly fifty years after its experimental debut, the quantum hall effect has produced perhaps more unique descriptions than any other subject of condensed matter physics. From topological band theory to non-commutative geometry, topological and conformal field theories, random matrix models and more, each approach captures a subset of the essential features, and compliments the others to produce a beautiful mosaic description. In this thesis, we revisit the fractional quantum hall (FQH) effect from its oldest vantage point, that of the trial wavefunctions and the parent Hamiltonians which stabilize them. Laughlin's observation of the importance of the theory of holomorphic functions to describing incompressible states in a partially filled Landau level has produced a coherent paradigm for realizing FQH parent Hamiltonians as frustration-free exactly-solvable one-dimensional models through dimensional reduction. Trial wavefunctions are then classified in terms of a small number of product states which generate all other states in the orbital-occupation expansion of the full state by a sequence of angular-momentum-preserving squeezing operations. These unentangled states are called the DNA of the FQH state, and they can be described in terms of simple local rules called generalized Pauli principles (GPP) of the form "at most r electrons can occupy k consecutive orbitals". It was only recently shown that an analysis of this form can be carried out in second-quantization for states with Landau level mixing, such as the Jain composite-fermion and parton series. In this case, the GPPs are generalized once further to entangled Pauli principles (EPP) for which the DNA may be entangled with respect to the extra Landau level index. In the first part of this thesis, we prove a theorem that allows for a rigorous computation of EPPs given a frustration free k-body parent Hamiltonian by examining only k-particle zero-modes, and give an example of its utility by demonstrating the exact-solvability of a correlated pair-hopping model that can be derived as a special limit of a FQH parent Hamiltonian. In the second part, taking advantage of the second quantized EPP analysis from the continuum, we derive a class of bosonic lattice models for states in multiple Landau levels which, in the presence of a contact interaction, are shown to stabilize non-Abelian ground states. Furthermore, we show how the EPP can be used to compute the degeneracies of quasihole excitations, and therefore the topological order through the bulk-boundary correspondence, which we verify with simple numerics. To the best of our knowledge, this work represents the first exponentially-local exactly-solvable fractional Chern insulator model for non-Abelian states with only two-body interactions.

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Non-Abelian gauge fields

Fabrice Gerbier, Nathan Goldman, Maciej Lewenstein +1 more

Building a universal quantum computer is a central goal of emerging quantum technologies, which has the potential to revolutionize science and technology. Unfortunately, this future does not seem to be very close at hand. However, quantum computers built for a special purpose, i.e. quantum simulators , are currently developed in many leading laboratories. Many schemes for quantum simulation have been proposed and realized using, e.g., ultracold atoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions in arrays of traps, quantum dots, photonic networks, or superconducting circuits. The progress in experimental implementations is more than spectacular. Particularly interesting are those systems that simulate quantum matter evolving in the presence of gauge fields. In the quantum simulation framework, the generated (synthetic) gauge fields may be Abelian, in which case they are the direct analogues of the vector potentials commonly associated with magnetic fields. In condensed matter physics, strong magnetic fields lead to a plethora of fascinating phenomena, among which the most paradigmatic is perhaps the quantum Hall effect. The standard Hall effect consists in the appearance of a transverse current, when a longitudinal voltage difference is applied to a conducting sample. For quasi-two-dimensional semiconductors at low temperatures placed in very strong magnetic fields, the transverse conductivity, the ratio between the transverse current and the applied voltage, exhibits perfect and robust quantization, independent for instance of the material or of its geometry. Such an integer quantum Hall effect, is now understood as a deep consequence of underlying topological order. Although such a system is an insulator in the bulk, it supports topologically robust edge excitations which carry the Hall current. The robustness of these chiral excitations against backscattering explains the universality of the quantum Hall effect. Another interesting and related effect, which arises from the interplay between strong magnetic field and lattice potentials, is the famous Hofstadter butterfly: the energy spectrum of a single particle moving on a lattice and subjected to a strong magnetic field displays a beautiful fractal structure as a function of the magnetic flux penetrating each elementary plaquette of the lattice. When the effects of interparticle interactions become dominant, two-dimensional gases of electrons exhibit even more exotic behaviour leading to the fractional quantum Hall effect. In certain conditions such a strongly interacting electron gas may form a highly correlated state of matter, the prototypical example being the celebrated Laughlin quantum liquid. Even more fascinating is the behaviour of bulk excitations (quasi-hole and quasi-particles): they are neither fermionic nor bosonic, but rather behave as anyons with fractional statistics intermediate between the two. Moreover, for some specific filling factors (ratio between the electronic density and the flux density), these anyons are proven to have an internal structure (several components) and non-Abelian braiding properties. Many of the above statements concern theoretical predictions—they have never been observed in condensed matter systems. For instance, the fractional values of the Hall conductance is seen as a direct consequence of the fractional statistics, but to date direct observation of anyons has not been possible in two-dimensional semiconductors. Realizing these predictions in experiments with atoms, ions, photons etc, which potentially allow the experimentalist to perform measurements complementary to those made in condensed matter systems, is thus highly desirable! Non-Abelian gauge fields couple the motional states of the particles to their internal degrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons, etc). In this sense external non-Abelian fields extend the concept of spin–orbit coupling (Rashba and Dresselhaus couplings), familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating phenomena such as the quantum spin Hall effect, three-dimensional topological insulators, topological superconductors and superfluids of various kinds. One also expects here the appearance of excitations in a form of topological edge states that can support robust transport, or entangled Majorana fermions in the case of topological superconductors or superfluids. Again, while many kinds of topological insulators have been realized in condensed matter systems, a controlled way of creating them in AMO systems and studying quantum phase transitions between various kinds of them is obviously very appealing and challenging. The various systems listed so far correspond to static gauge fields, which are externally imposed by the experimentalists. Even more fascinating is the possibility of generating synthetically dynamical gauge fields, i.e. gauge fields that evolve in time according to an interacting gauge theory, e.g., a full lattice gauge theory (LGT). These dynamical gauge fields can also couple to matter fields, allowing the quantum simulation of such complex systems (notoriously hard to simulate using 'traditional' computers), which are particularly relevant for modern high-energy physics. So far, most of the theoretical proposals concern the simulation of Abelian gauge theories, however, several groups have recently proposed extensions to the non-Abelian scenarios. The scope of the present focused issue of Journal of Physics B is to cover all of these developments, with particular emphasis on the non-Abelian gauge fields. The 14 papers in this issue include contributions from the leading theory groups working in this field; we believe that this collection will provide the reference set for quantum simulations of gauge fields. Although the special issue contains exclusively theoretical proposals and studies, it should be stressed that the progress in experimental studies of artificial Abelian and non-Abelian gauge fields in recent years has been simply spectacular. Multiple leading groups are working on this subject and have already obtained a lot of seminal results. The papers in the special issue are ordered according to the date of acceptance. The issue opens with a review article by Zhou et al [1] on unconventional states of bosons with synthetic spin–orbit coupling. Next, the paper by Maldonado-Mundo et al [2] studies ultracold Fermi gases with artificial Rashba spin–orbit coupling in a 2D gas. Anderson and Charles [3], in contrast, discuss a three-dimensional spin–orbit coupling in a trap. Orth et al [4] investigate correlated topological phases and exotic magnetism with ultracold fermions, again in the presence of artificial gauge fields. The paper of Nascimbène [5] does not address the synthetic gauge fields directly, but describes an experimental proposal for realizing one-dimensional topological superfluids with ultracold atomic gases; obviously, this problem is well situated in the general and growing field of topological superfluids, in particular those realized in the presence of non-Abelian gauge fields/spin–orbit coupling. Graß et al [6] consider in their paper fractional quantum Hall states of a Bose gas with spin–orbit coupling induced by a laser. Particular attention is drawn here to the possibility of realizing states with non-Abelian anyonic excitations. Zheng et al [7] study properties of Bose gases with Raman-induced spin–orbit coupling. Kiffner et al [8] in their paper touch on another kind of system, namely ultracold Rydberg atoms. In particular they study the generation of Abelian and non-Abelian gauge fields in dipole–dipole interacting Rydberg atoms. The behaviour of fermions in synthetic non-Abelian gauge potentials is discussed by Shenoy and Vyasanakere [9]. The paper starts with the study of Rashbon condensates (i.e. Bose condensates in the presence of Rashba coupling) and also introduces novel kinds of exotic Hamiltonians. Goldman et al [10] propose a concrete setup for realizing arbitrary non-Abelian gauge potentials in optical square lattices; they discuss how such synthetic gauge fields can be exploited to generate Chern insulators. Zygelman [11], similarly as Kiffner et al [8], discusses in his paper non-Abelian gauge fields in Rydberg systems. Marchukov et al [12] return to the subject of spin–orbit coupling, and investigate spectral gaps of spin–orbit coupled particles in the realistic situations of deformed traps. The last two papers, in contrast, are devoted to different subjects. Edmonds et al [13] consider a 'dynamical' density-dependent gauge potential, and study the Josephson effect in a Bose–Einstein condensate subject to such a potential. Last, but not least, Mazzucchi et al [14] study the properties of semimetal-superfluid quantum phase transitions in 3D lattices with Dirac points. References [1] Zhou X, Li Y, Cai Z and Wu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001 [2] Maldonado-Mundo D, Öhberg P and Valiente M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134002 [3] Anderson B M and Clark C W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003 [4] Orth P P, Cocks D, Rachel S, Buchhold M, Le Hur K and Hofstetter W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134004 [5] Nascimbène S 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134005 [6] Graß T, Juliá-Díaz B, Burrello M and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134006 [7] Zheng W, Yu Z-Q, Cui X and Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007 [8] Kiffner M, Li W and Jaksch D 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134008 [9] Shenoy V B and Vyasanakere J P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134009 [10] Goldman N, Gerbier F and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134010 [11] Zygelman B 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134011 [12] Marchukov O V, Volosniev A G, Fedorov D V, Jensen A S and Zinner N T 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134012 [13] Edmonds M J, Valiente M and Öhberg P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134013 [14] Mazzucchi G, Lepori L and Trombettoni A 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134014

Journal of Physics B Atomic Molecular and Optical PhysicsJun 24, 201311 citations

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^2ν), 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^2ν/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 Chapter2, 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 Chapter4, 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.