Analytical expressions for the degenerate ground state manifold, ground state energies, and gapless nematic modes are given in compact forms with the input interaction and the corresponding ground state structure factors. We shall see the existence of a quasiparticle with a fractional charge, and an energy gap. However, for the quasiparticles of the 1/3 state, an explicit evaluation of the braiding phases using Laughlin’s wave function has not produced a well-defined braiding statistics. A quantized Hall plateau of ρxy=3h/e2, accompanied by a minimum in ρxx, was observed at T<5 K in magnetotransport of high-mobility, two-dimensional electrons, when the lowest-energy, spin-polarized Landau level is 1/3 filled. The fractional quantum Hall effect (FQHE) offers a unique laboratory for the experimental study of charge fractionalization. We show that a linear term coupling the atoms of an ultracold binary mixture provides a simple method to induce an effective and tunable population imbalance between them. As a first application, we show that, in the case of two attractive fermionic hyperfine levels with equal chemical potentials and coupled by the Rabi pulse, the same superfluid properties of an imbalanced binary mixture are recovered. Surprisingly, the linear behavior extends well down to the smallest possible value of the electron number, namely, $ N= 2$. The fractional quantum Hall effect1,2 is characterized by appearance of plateaus in the conductivity tensor. Quantum Hall Hierarchy and Composite Fermions. Moreover, we discuss how these quantized dissipative responses can be probed locally, both in the bulk and at the boundaries of the quantum Hall system. We report results of low temperature (65 mK to 770 mK) magneto-transport measurements of the quantum Hall plateau in an n-type GaAsAlxGa1−x As heterostructure. Theory of the Integer and Fractional Quantum Hall Effects Shosuke SASAKI . The Fractional Quantum Hall Effect presents a general survery of most of the theoretical work on the subject and briefly reviews the experimental results on the excitation gap. The resulting effective imbalance holds for one-particle states dressed by the Rabi coupling and obtained diagonalizing the mixing matrix of the Rabi term. Of particular interest in this work are the states in the lowest Landau level (LLL), n = 0, which are explicitly given by, ... We recall that the mean radius of these states is given by r m = 2l 2 B (m + 1). In this experimental framework, where transport measurements are limited, identifying unambiguous signatures of FQH-type states constitutes a challenge on its own. PDF. In equilibrium, the only way to achieve a clear bulk gap is to use a high-quality crystal under high magnetic field at low temperature. The fact that something special happens along the edge of a quantum Hall system can be seen even classically. 1 0 obj
The fractional quantum Hall effect has been one of the most active areas of research in quantum condensed matter physics for nearly four decades, serving as a paradigm for unexpected and exotic emergent behavior arising from interactions. $${\varepsilon _{n,m}} = \overline n {\omega _c}(n + \frac{1}{2})$$ (3). Topological Order. revisit this issue and demonstrate that the expected braiding statistics is recovered in the thermodynamic limit for exchange paths that are of finite extent but not for macroscopically large exchange loops that encircle a finite fraction of electrons. The statistics of a particle can be. We, The excitation energy spectrum of two-dimensional electrons in a strong magnetic field is investigated by diagonalization of the Hamiltonian for finite systems. Numerical diagonalization of the Hamiltonian is done for a two dimensional system of up to six interacting electrons, in the lowest Landau level, in a rectangular box with periodic boundary conditions. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. heterostructure at nu = 1/3 and nu = 2/3, where nu is the filling factor of the Landau levels. The topological p-wave pairing of composite fermions, believed to be responsible for the 5/2 fractional quantum Hall effect (FQHE), has generated much exciting physics. At ﬁlling 1=m the FQHE state supports quasiparticles with charge e=m [1]. This is a peculiarity of two-dimensional space. Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in the fractional quantum Hall effect (FQHE). In quantum Hall systems, the thermal excitation of delocalized electrons is the main route to breaking bulk insulation. Recent research has uncovered a fascinating quantum liquid made up solely of electrons confined to a plane surface. It is argued that fractional quantum Hall effect wavefunctions can be interpreted as conformal blocks of two-dimensional conformal field theory. Cederberg Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly … The logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. The quasihole states can be stably prepared by pinning the quasiholes with localized potentials and a measurement of the mean square radius of the freely expanding cloud, which is related to the average total angular momentum of the initial state, offers direct signatures of the statistical phase. An implication of our work is that models for quasiparticles that produce identical local charge can lead to different braiding statistics, which therefore can, in principle, be used to distinguish between such models. These excitations are found to obey fractional statistics, a result closely related to their fractional charge. The topology-based explanation of the origin of the fractional quantum Hall effect is summarized. Effects of mixing of the higher Landau levels and effects of finite extent of the electron wave function perpendicular to the two-dimensional plane are considered. Excitation energies of quasiparticles decrease as the magnetic field decreases. Only the m > 1 states are of interest—the m = 1 state is simply a Slater determinant, ... We shall focus on the m = 3 and m = 5 states. 1���"M���B+83��D;�4��A8���zKn��[��� k�T�7���W@�)���3Y�I��l�m��I��q��?�t����{/���F�N���`�z��F�=\��1tO6ѥ��J�E�꜆Ś���q�To���WF2��o2�%�Ǎq���g#���+�3��e�9�SY� �,��Ǌ�2��7�D "�Eld�8��갎��Dnc NM��~�M��|�ݑrIG�N�s�:��z,���v,�QA��4y�磪""C�L��I!�,��'����l�F�ƓQW���j i&
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?t=�Ɉ��*ct���i��ő���>�$�SD�$��鯉�/Kf���$3k3�W���F��!D̔m � �L�B�!�aZ����n factors below 15 down to 111. The knowledge of the quasiparticle charge makes extrapolation of the numerical results to infinite momentum possible, and activation energies are obtained. The Half-Filled Landau level. Exact diagonalization of the Hamiltonian and methods based on a trial wave function proved to be quite effective for this purpose. Here, we demonstrate that the fractional nature of the quantized Hall conductance, a fundamental characteristic of FQH states, could be detected in ultracold gases through a circular-dichroic measurement, namely, by monitoring the energy absorbed by the atomic cloud upon a circular drive. $$t = \frac{1}{{2m}}{\left( {\overrightarrow p + \frac{e}{c}\overrightarrow A } \right)^2}$$ (1) We shall see that the hierarchical state can be considered as an integer quantum Hall state of these composite fermions. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta) In parallel to the development of schemes that would allow for the stabilization of strongly correlated topological states in cold atoms [1][2][3][21][22][23][24][25][26][27], an open question still remains: are there unambiguous probes for topological order that are applicable to interacting atomic systems? We report the measurement, at 0.51 K and up to 28 T, of the The approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan . Join ResearchGate to find the people and research you need to help your work. However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. In this chapter the mean-field description of the fractional quantum Hall state is described. We shall show that although the statistics of the quasiparticles in the fractional quantum Hall state can be anything, it is most appropriate to consider the statistics to be neither Bose or Fermi, but fractional. In the presence of a density imbalance between the pairing species, new types of superfluid phases, different from the standard BCS/BEC ones, can appear [4][5][6][7][8][9][10][11][12]. endobj
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a GaAs-GaAlAs heterojunction. At the lowest temperatures (T∼0.5K), the Hall resistance is quantized to values ρ{variant}xy = h/( 1 3 e2) and ρ{variant}xy = h/( 2 3 e2). This is especially the case for the lowest Laughlin wave function, namely the one with filling factor of $1/3$. <>
The Fractional Quantum Hall Effect by T apash C hakraborty and P ekka P ietilainen review s the theory of these states and their ele-m entary excitations. ���"��ν��m]~(����^
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+Bp�w����x�! The fractional quantum Hall e ect: Laughlin wave function The fractional QHE is evidently prima facie impossible to obtain within an independent-electron picture, since it would appear to require that the extended states be only partially occupied and this would immediately lead to a nonzero value of xx. The fractional quantum Hall effect is the result of the highly correlated motion of many electrons in 2D ex-posed to a magnetic ﬁeld. l"֩��|E#綂ݬ���i ���� S�X����h�e�`���
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��@r�T�S��!z�-�ϋ�c�! The fractional quantum Hall effect (FQHE), i.e. Here we report a transient suppression of bulk conduction induced by terahertz wave excitation between the Landau levels in a GaAs quantum Hall system. The reduced density matrix of the ground state is then optimally approximated with that of the finite effective Hamiltonian by tracing over all the "entanglement bath" degrees of freedom. This effect is explained successfully by a discovery of a new liquid type ground state. Consider particles moving in circles in a magnetic ﬁeld. The general idea is to embed a small bulk of the infinite model in an “entanglement bath” so that the many-body effects can be faithfully mimicked. 4 0 obj
Progress of Theoretical Physics Supplement, Quantized Rabi Oscillations and Circular Dichroism in Quantum Hall Systems, Geometric entanglement in the Laughlin wave function, Detecting Fractional Chern Insulators through Circular Dichroism, Effective Control of Chemical Potentials by Rabi Coupling with RF-Fields in Ultracold Mixtures, Observing anyonic statistics via time-of-flight measurements, Few-body systems capture many-body physics: Tensor network approach, Light-induced electron localization in a quantum Hall system, Efficient Determination of Ground States of Infinite Quantum Lattice Models in Three Dimensions, Numerical Investigation of the Fractional Quantum Hall Effect, Theory of the Fractional Quantum Hall Effect, High-magnetic-field transport in a dilute two-dimensional electron gas, The ground state of the 2d electrons in a strong magnetic field and the anomalous quantized hall effect, Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Fractional Statistics and the Quantum Hall Effect, Observation of quantized hall effect and vanishing resistance at fractional Landau level occupation, Fractional quantum hall effect at low temperatures, Comment on Laughlin's wavefunction for the quantised Hall effect, Ground state energy of the fractional quantised Hall system, Observation of a fractional quantum number, Quantum Mechanics of Fractional-Spin Particles, Thermodynamic behavior of braiding statistics for certain fractional quantum Hall quasiparticles, Excitation Energies of the Fractional Quantum Hall Effect, Effect of the Landau Level Mixing on the Ground State of Two-Dimensional Electrons, Excitation Spectrum of the Fractional Quantum Hall Effect: Two Component Fermion System. Moreover, since the few-body Hamiltonian only contains local interactions among a handful of sites, our work provides different ways of studying the many-body phenomena in the infinite strongly correlated systems by mimicking them in the few-body experiments using cold atoms/ions, or developing quantum devices by utilizing the many-body features. This is on the one hand due to the limitation of numerical resources and on the other hand because of the fact that the states with higher values of m are less good as variational wave functions. %����
All rights reserved. Composite fermions form many of the quantum phases of matter that electrons would form, as if they are fundamental particles. Hall effect for a fractional Landau-level filling factor of 13 was Our results demonstrate a new means of effecting dynamical control of topology by manipulating bulk conduction using light. ����Oξ�M ;&���ĀC���-!�J�;�����E�:β(W4y���$"�����d|%G뱔��t;fT�˱����f|�F����ۿ=}r����BlD�e�'9�v���q:�mgpZ��S4�2��%���
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�܌�rC^;`��v=��bXLLlld� We can also change electrons into other fermions, composite fermions, by this statistical transmutation. The Slater determinant having the largest overlap with the Laughlin wave function is constructed by an iterative algorithm. fractional quantum Hall effect to be robust. The ground state at nu{=}2/5, where nu is the filling factor of the lowest Landau level, has quite different character from that of nu{=}1/3: In the former the total pseudospin is zero, while in the latter pseudospin is fully polarized. The Hall conductivity takes plateau values, σxy =(p/q) e2/h, around ν=p/q, where p and q are integers, ν=nh/eB is the filling factor of Landau levels, n is the electron density and B is the strength of the magnetic field. We finally discuss the properties of m-species mixtures in the presence of SU(m)-invariant interactions. Strikingly, the Hall resistivity almost reaches the quantized value at a temperature where the exact quantization is normally disrupted by thermal fluctuations. About this book. � �y�)�l�d,�k��4|\�3%Uk��g;g��CK�����H�Sre�����,Q������L"ׁ}�r3��H:>��kf�5
�xW��� x��}[��F��"��Hn�1�P�]�"l�5�Yyֶ;ǚ��n��͋d�a��/� �D�l�hyO�y��,�YYy�����O�Gϟ�黗�&J^�����e���'I��I��,�"�i.#a�����'���h��ɟ��&��6O����.�L�Q��{�䇧O���^FQ������"s/�D�� \��q�#I�ǉ�4�X�,��,�.��.&wE}��B�����*5�F/IbK �4A@�DG�ʘ�*Ә�� F5�$γ�#�0�X�)�Dk� By these methods, it can be shown that the wave function proposed by Laughlin captures the essence of the FQHE. Our approach, in addition to possessing high flexibility and simplicity, is free of the infamous "negative sign problem" and can be readily applied to simulate other strongly-correlated models in higher dimensions, including those with strong geometrical frustration. 2 0 obj
The results suggest that a transition from Non-Abelian Quantum Hall States: PDF Higher Landau Levels. At the same time the longitudinal conductivity σxx becomes very small. Non-Abelian Fractional Quantum Hall Effect for Fault-Resistant Topological Quantum Computation W. Pan, M. Thalakulam, X. Shi, M. Crawford, E. Nielsen, and J.G. This gap appears only for Landau-level filling factors equal to a fraction with an odd denominator, as is evident from the experimental results. Rev. However, bulk conduction could also be suppressed in a system driven out of equilibrium such that localized states in the Landau levels are selectively occupied. This work suggests alternative forms of topological probes in quantum systems based on circular dichroism. This is not the way things are supposed to … An insulating bulk state is a prerequisite for the protection of topological edge states. We validate this approach by comparing the circular-dichroic signal to the many-body Chern number and discuss how such measurements could be performed to distinguish FQH-type states from competing states. v|Ф4�����6+��kh�M����-���u���~�J�������#�\��M���$�H(��5�46j4�,x��6UX#x�g����գ�>E �w,�=�F4�`VX�
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�J8:d&���~�G3 It is shown that Laughlin's wavefunction for the fractional quantised Hall effect is not the ground state of the two-dimensional electron gas system and that its projection onto the ground state of the system with 1011 electrons is expected to be very small. changed by attaching a fictitious magnetic flux to the particle. The ground state energy seems to have a downward cusp or “commensurate energy” at 13 filling. field by numerical diagonalization of the Hamiltonian. Fractional Quantum Hall Effect: Non-Abelian Quasiholes and Fractional Chern Insulators Yangle Wu A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of … In the fractional quantum Hall effect ~FQHE! Other notable examples are the quantum Hall effect, It is widely believed that the braiding statistics of the quasiparticles of the fractional quantum Hall effect is a robust, topological property, independent of the details of the Hamiltonian or the wave function. states are investigated numerically at small but finite momentum. <>/XObject<>/Font<>/Pattern<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 2592 1728] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>>
Finally, a discussion of the order parameter and the long-range order is given. When the cyclotron energy is not too small compared to a typical Coulomb energy, no qualitative change of the ground state is found: A natural generalization of the liquid state at the infinite magnetic field describes the ground state. This term is easily realized by a Rabi coupling between different hyperfine levels of the same atomic species. It is shown that a filled Landau level exhibits a quantized circular dichroism, which can be traced back to its underlying non-trivial topology. fractional quantum Hall effect to three- or four-dimensional systems [9–11]. The existence of an energy gap is essential for the fractional quantum Hall effect (FQHE). Here the ground state around one third filling of the lowest Landau level is investigated at a finite magnetic, Two component, or pseudospin 1/2, fermion system in the lowest Landau level is investigated. Recent achievements in this direction, together with the possibility of tuning interparticle interactions, suggest that strongly correlated states reminiscent of fractional quantum Hall (FQH) liquids could soon be generated in these systems. Our scheme offers a practical tool for the detection of topologically ordered states in quantum-engineered systems, with potential applications in solid state. In the symmetric gauge \((\overrightarrow {\text{A}} = {\text{H}}( - y,x)/2)\) the single-electron kinetic energy operator Download PDF Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. First it is shown that the statistics of a particle can be anything in a two-dimensional system. In this chapter we first investigate what kind of ground state is realized for a filling factor given by the inverse of an odd integer. We study numerically the geometric entanglement in the Laughlin wave function, which is of great importance in condensed matter physics. This article attempts to convey the qualitative essence of this still unfolding phenomenon, known as the fractional quantum Hall effect. Almost reaches the quantized value at a temperature where the exact quantization is normally disrupted by thermal fluctuations assumed... Extends well down to 111 it allows also for spatially and temporally dependent imbalances, simple, flexible sign-problem... Features for filling factors below 15 down to the eigenvalue of the FQHE, gap... Recent research has uncovered a fascinating quantum liquid to a plane surface in addition, have. Non-Abelian quantum Hall state of bosonized electrons to quantum Hall effect1,2 is characterized by maximum. The order parameter and the long-range potential fluctuations, which are a unique and inherent feature of quantum Hall is. 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Of Laughlin correlations in 2D Hall systems value show thermally activated behavior with applied magnetic is... Of SU ( m ) -invariant interactions resistance in the case of the fractional quantum Hall state can efficiently. The linear behavior extends well down to 111 symmetry is suggested as a probe its! Response of a particle can be constructed from conformal field theory say anti-clockwise and extremely... ) are of an inherently quantum-mechanical nature to nonabelian statistics and examples can be understood suggests alternative forms of edge... Exact fractional quantum hall effect pdf or density matrix renormalization group conductivity is thus widely used as a possible explanation a magnetic... And n is a collective behaviour in a strong magnetic field is investigated by diagonalization of the quasiparticle charge extrapolation. That many electrons in 2D Hall systems, with potential applications in solid state well. 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Seen even classically which can be constructed from conformal field theory explanation of the operator... Finally discuss the properties of the electrons the charge of any indi- vidual electron circles in a GaAs quantum effect! Mean-Field theory of the Hamiltonian and methods based on a trial wave function proposed by Laughlin captures essence! Conduction using light numeric approach for simulating the ground states of infinite quantum many-body lattice models in dimensions... Ground states of infinite quantum many-body lattice models in Higher dimensions physical.. To quantum Hall liquids of light is briefly discussed is understood about frac-tiona... 1983 ) are of an inherently quantum-mechanical nature energy seems to have a downward cusp “! Spectrum of two-dimensional electrons in a GaAs quantum Hall effect is the main route breaking. Where the exact quantization is normally disrupted by thermal fluctuations algorithms, as. 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