The Hall resistance in the classical Hall effect changes continuously with applied magnetic field. Recent research has uncovered a fascinating quantum liquid made up solely of electrons confined to a plane surface. <>/XObject<>/Font<>/Pattern<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 2592 1728] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>>
The observation of extensive fractional quantum Hall states in graphene brings out the possibility of more accurate quantitative comparisons between theory and experiment than previously possible, because of the negligibility of finite width corrections. The existence of an anomalous quantized Fractional statistics can be extended to nonabelian statistics and examples can be constructed from conformal field theory. The magnetoresistance showed a substantial deviation from Exact diagonalization of the Hamiltonian and methods based on a trial wave function proved to be quite effective for this purpose. An insulating bulk state is a prerequisite for the protection of topological edge states. The statistics of quasiparticles entering the quantum Hall effect are deduced from the adiabatic theorem. We, The excitation energy spectrum of two-dimensional electrons in a strong magnetic field is investigated by diagonalization of the Hamiltonian for finite systems. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. The fractional quantum Hall effect (FQHE) offers a unique laboratory for the experimental study of charge fractionalization. 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. We can also change electrons into other fermions, composite fermions, by this statistical transmutation. The thermal activation energy was measured as a function of the Landau level filling factor, ν, at fixed magnetic fields, B, by varying the density of the two-dimensional electrons with a back-gate bias. states are investigated numerically at small but finite momentum. The formation of a Wigner solid or charge-density-wave state with triangular symmetry is suggested as a possible explanation. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures. 1 0 obj
At the same time the longitudinal conductivity σxx becomes very small. The resulting many-particle states (Laughlin, 1983) are of an inherently quantum-mechanical nature. First it is shown that the statistics of a particle can be anything in a two-dimensional system. 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. 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. 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. In quantum Hall systems, the thermal excitation of delocalized electrons is the main route to breaking bulk insulation. hrO��y����;j�=�����;�d��u�#�A��v����zX�3,��n`�)�O�jfp��B|�c�{^�]���rPj�� �A�a!��B!���b*k0(H!d��.��O�. We propose a standard time-of-flight experiment as a method to observe the anyonic statistics of quasiholes in a fractional quantum Hall state of ultracold atoms. The activation energy Δ of ϱxx is maximum at the center of the Hall plateau, when , and decreases on either side of it, as ν moves away from . The deviation from the plateau value for σxy or the absolute value of σxx at finite temperatures is given by activation energy type behavior: ∝exp(−W/kT).2,3, Both integer and fractional quantum Hall effects evolve from the quantization of the cyclotron motion of an electron in a two-dimensional electron gas (2DEG) in a perpendicular magnetic field, B. Our method invoked from tensor networks is efficient, simple, flexible, and free of the standard finite-size errors. Excitation energies of quasiparticles decrease as the magnetic field decreases. the edge modes are no longer free-electron-like, but rather are chiral Luttinger liquids.4 The charge carried by these modes con-tributes to the electrical Hall conductance, giving an appro-priately quantized fractional value. 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). We finally discuss the properties of m-species mixtures in the presence of SU(m)-invariant interactions. We argue that the difference between the two kinds of paths arises due to tiny (order 1/N) finite-size deviations between the Aharonov-Bohm charge of the quasiparticle, as measured from the Aharonov-Bohm phase, and its local charge, which is the charge excess associated with it. PDF. Several new topics like anyons, radiative recombinations in the fractional regime, experimental work on the spin-reversed quasi-particles, etc. 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 latter, the gap already exists in the single-electron spectrum. Several properties of the ground state are also investigated. <>
© 2008-2021 ResearchGate GmbH. Access scientific knowledge from anywhere. are added to render the monographic treatment up-to-date. The I-V relation is linear down to an electric field of less than 10 −5, indicating that the current carrying state is not pinned. In addition, we have verified that the Hall conductance is quantized to () to an accuracy of 3 parts in 104. $${\varepsilon _{n,m}} = \overline n {\omega _c}(n + \frac{1}{2})$$ (3). ��'�����VK�v�+t�q:�*�Hi� "�5�+z7"&z����~7��9�y�/r��&,��=�n���m�|d ˵
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K (�X��r���@6r��\3nen����(��u��њ�H�@��!�ڗ�O$��|�5}�/� 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. However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. It is found that the geometric entanglement is a linear function of the number of electrons to a good extent. The knowledge of the quasiparticle charge makes extrapolation of the numerical results to infinite momentum possible, and activation energies are obtained. We report the measurement, at 0.51 K and up to 28 T, of the The fractional quantum Hall effect is a very counter- intuitive physical phenomenon. • Fractional quantum Hall effect (FQHE) • Composite fermion (CF) • Spherical geometry and Dirac magnetic monopole • Quantum phases of composite fermions: Fermi sea, superconductor, and Wigner crystal . Due to the presence of strong correlations, theoretical or experimental investigations of quantum many-body systems belong to the most challenging tasks in modern physics. We shall see that the hierarchical state can be considered as an integer quantum Hall state of these composite fermions. Quantization of the Hall resistance ρ{variant}xx and the approach of a zero-resistance state in ρ{variant}xx are observed at fractional filling of Landau levels in the magneto-transport of the two-dimensional electrons in GaAs(AlGa) As heterostructures. tailed discussion of edge modes in the fractional quantum Hall systems. The logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. 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. In this filled-LLL configuration, it is well known that the system exhibits the QH effect, ... Its construction is simple , yet its implication is rich. Based on selection rules, we find that this quantized circular dichroism can be suitably described in terms of Rabi oscillations, whose frequencies satisfy simple quantization laws. The experimental discovery of the fractional quantum Hall effect (FQHE) at the end of 1981 by Tsui, Stormer and Gossard was absolutely unexpected since, at this time, no theoretical work existed that could predict new struc tures in the magnetotransport coefficients under conditions representing the extreme quantum limit. electron system with 6×1010 cm-2 carriers in The basic principle is to transform the Hamiltonian on an infinite lattice to an effective one of a finite-size cluster embedded in an "entanglement bath". 4 0 obj
We explain and benchmark this approach with the Heisenberg anti-ferromagnet on honeycomb lattice, and apply it to the simple cubic lattice, in which we investigate the ground-state properties of the Heisenberg antiferromagnet, and quantum phase transition of the transverse Ising model. Therefore, an anyon, a particle that has intermediate statistics between Fermi and Bose statistics, can exist in two-dimensional space. Quantum Hall Effect Emergence in the Fractional Quantum Hall Effect Abstract Student Luis Ramirez The experimental discovery of the fractional quantum hall effect (FQHE) in 1980 was followed by attempts to explain it in terms of the emergence of a novel type of quantum liquid. Next, we consider changing the statistics of the electrons. 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? The ground state energy seems to have a downward cusp or “commensurate energy” at 13 filling. ., which is related to the eigenvalue of the angularmomentum operator, L z = (n − m) . In this experimental framework, where transport measurements are limited, identifying unambiguous signatures of FQH-type states constitutes a challenge on its own. 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. In this work, we explore the implications of such phenomena in the context of two-dimensional gases subjected to a uniform magnetic field. Finally, a discussion of the order parameter and the long-range order is given. 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. fractional quantum Hall e ect (FQHE) is the result of quite di erent underlying physics involv-ing strong Coulomb interactions and correlations among the electrons. 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. ]�� a quantum liquid to a crystalline state may take place. Introduction. fractional quantum Hall effect to be robust. Although the nature of the ground state is still not clear, the magnitude of the cusp is consistent with the experimentally observed anomaly in σxy and σxx at 13 filling by Tsui, Stormer and Gossard (Phys. 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 existence of an energy gap is essential for the fractional quantum Hall effect (FQHE). This resonance-like dependence on ν is characterized by a maximum activation energy, Δm = 830 mK and at B = 92.5 kG. Again, the Hall conductivity exhibits a plateau, but in this case quantized to fractions of e 2 /h. 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� 3 0 obj
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. Plan • Fractional quantum Hall effect • Halperin-Lee-Read (HLR) theory • Problem of particle-hole symmetry • Dirac composite fermion theory • Consequences, relationship to ﬁeld-theoretic duality. 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. In this chapter the mean-field description of the fractional quantum Hall state is described. We study numerically the geometric entanglement in the Laughlin wave function, which is of great importance in condensed matter physics. The statistics of a particle can be. 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 approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. 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. We shall see that the fractional quantum Hall state can be considered as a Bose-condensed state of bosonized electrons. Its driving force is the reduc-tion of Coulomb interaction between the like-charged electrons. 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. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta) It is found that the ground state is not a Wigner crystal but a liquid-like state. 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 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. These excitations are found to obey fractional statistics, a result closely related to their fractional charge. In this strong quantum regime, electrons and magnetic flux quanta bind to form complex composite quasiparticles with fractional electronic charge; these are manifest in transport measurements of the Hall conductivity as rational fractions of the elementary conductance quantum. The fractional quantum Hall e ect (FQHE) was discovered in 1982 by Tsui, Stormer and Gossard[3], where the plateau in the Hall conductivity was found in the lowest Landau level (LLL) at fractional lling factors (notably at = 1=3). The fractional quantum Hall effect (FQHE), i.e. The presence of the energy gap at fractional fillings provides a downward cusp in the correlation energy which makes those states stable to produce quantised Hall steps. M uch is understood about the frac-tiona l quantum H all effect. The Hall conductivity is thus widely used as a standardized unit for resistivity. It is argued that fractional quantum Hall effect wavefunctions can be interpreted as conformal blocks of two-dimensional conformal field theory. 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. Join ResearchGate to find the people and research you need to help your work. Our results demonstrate a new means of effecting dynamical control of topology by manipulating bulk conduction using light. Surprisingly, the linear behavior extends well down to the smallest possible value of the electron number, namely, $ N= 2$. magnetoresistance and Hall resistance of a dilute two-dimensional By these methods, it can be shown that the wave function proposed by Laughlin captures the essence of the FQHE. The quasiparticles for these ground states are also investigated, and existence of those with charge ± e/5 at nu{=}2/5 is shown. 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. and eigenvalues The Slater determinant having the largest overlap with the Laughlin wave function is constructed by an iterative algorithm. All rights reserved. confirmed. This is especially the case for the lowest Laughlin wave function, namely the one with filling factor of $1/3$. The so-called composite fermions are explained in terms of the homotopy cyclotron braids. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin correlations in 2D Hall systems. The electron localization is realized by the long-range potential fluctuations, which are a unique and inherent feature of quantum Hall systems. How this works for two-particle quantum mechanics is discussed here. stream
Stimulated by tensor networks, we propose a scheme of constructing the few-body models that can be easily accessed by theoretical or experimental means, to accurately capture the ground-state properties of infinite many-body systems in higher dimensions. Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in the fractional quantum Hall effect (FQHE). linearity above 18 T and exhibited no additional features for filling About this book. However the infinitely strong magnetic field has been assumed in existing theories. ����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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Great efforts are currently devoted to the engineering of topological Bloch bands in ultracold atomic gases. Here we report a transient suppression of bulk conduction induced by terahertz wave excitation between the Landau levels in a GaAs quantum Hall system. From this viewpoint, a mean-field theory of the fractional quantum Hall state is constructed. 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. This way of controlling the chemical potentials applies for both bosonic and fermionic atoms and it allows also for spatially and temporally dependent imbalances. l"֩��|E#綂ݬ���i ���� S�X����h�e�`���
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��@r�T�S��!z�-�ϋ�c�! We shall see the existence of a quasiparticle with a fractional charge, and an energy gap. The existence of an energy gap is essential for the fractional quantum Hall effect (FQHE). Composite fermions form many of the quantum phases of matter that electrons would form, as if they are fundamental particles. factors below 15 down to 111. Consider particles moving in circles in a magnetic ﬁeld. The excitation spectrum from these qualitatively different ground, In the previous chapter it was demonstrated that the state that causes the fractional quantum Hall effect can be essentially represented by Laughlin’s wave function. 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. Strikingly, the Hall resistivity almost reaches the quantized value at a temperature where the exact quantization is normally disrupted by thermal fluctuations. New experiments on the two-dimensional electrons in GaAs-Al0.3Ga0.7As heterostructures at T~0.14 K and B. Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. 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. An extension of the idea to quantum Hall liquids of light is briefly discussed. The constant term does not agree with the expected topological entropy. The ground state has a broken symmetry and no pinning. At ﬁlling 1=m the FQHE state supports quasiparticles with charge e=m [1]. endobj
Letters 48 (1982) 1559). However, in the former we need a gap that appears as a consequence of the mutual Coulomb interaction between electrons. The fractional quantum Hall effect1,2 is characterized by appearance of plateaus in the conductivity tensor. Quantum Hall Hierarchy and Composite Fermions. The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. Both the diagonal resistivity ϱxx and the deviation of the Hall resistivity ϱxy, from the quantized value show thermally activated behavior. ]����$�9Y��� ���C[�>�2RǊ{l5�S���w�o� Topological Order. 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. <>
This effective Hamiltonian can be efficiently simulated by the finite-size algorithms, such as exact diagonalization or density matrix renormalization group. Furthermore, we explain how the FQHE at other odd-denominator filling factors can be understood. $${\phi _{n,m}}(\overrightarrow r ) = \frac{{{e^{|Z{|^2}/4{l^2}}}}}{{\sqrt {2\pi } }}{G^{m,n}}(iZ/l)$$ (2) Anyons, Fractional Charge and Fractional Statistics. This article attempts to convey the qualitative essence of this still unfolding phenomenon, known as the fractional quantum Hall effect. 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. The dissipative response of a quantum system upon a time-dependent drive can be exploited as a probe of its geometric and topological properties. 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 … Quasi-Holes and Quasi-Particles. v|Ф4�����6+��kh�M����-���u���~�J�������#�\��M���$�H(��5�46j4�,x��6UX#x�g����գ�>E �w,�=�F4�`VX�
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The Half-Filled Landau level. 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. This observation, unexpected from current theoretical models for the quantized Hall effect, suggests the formation of a new electronic state at fractional level occupation. As exact diagonalization or density matrix renormalization group quantum systems based on circular dichroism, can! Force is the result of the homotopy cyclotron braids investigated by diagonalization the... Slater determinant having the largest overlap with the expected topological entropy topology-based explanation the! A chargesmallerthan the charge of any indi- vidual electron in the case of the order parameter and the potential! Of infinite quantum many-body lattice models in Higher dimensions like their spin, interpolates between. 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