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. <>/XObject<>/Font<>/Pattern<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 2592 1728] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> We report results of low temperature (65 mK to 770 mK) magneto-transport measurements of the quantum Hall plateau in an n-type GaAsAlxGa1−x As heterostructure. We shall see that the hierarchical state can be considered as an integer quantum Hall state of these composite fermions. The so-called composite fermions are explained in terms of the homotopy cyclotron braids. We shall see that the fractional quantum Hall state can be considered as a Bose-condensed state of bosonized electrons. 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 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. endobj The approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. are added to render the monographic treatment up-to-date. This article attempts to convey the qualitative essence of this still unfolding phenomenon, known as the fractional quantum Hall effect. 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. Finally, a discussion of the order parameter and the long-range order is given. Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan . Great efforts are currently devoted to the engineering of topological Bloch bands in ultracold atomic gases. The numerical results of the spin models on honeycomb and simple cubic lattices show that the ground-state properties including quantum phase transitions and the critical behaviors are accurately captured by only O(10) physical and bath sites. All rights reserved. fractional quantum Hall effect to three- or four-dimensional systems [9–11]. The fractional quantum Hall effect (FQHE), i.e. The electron localization is realized by the long-range potential fluctuations, which are a unique and inherent feature of quantum Hall systems. 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. M uch is understood about the frac-tiona l quantum H all effect. We propose a numeric approach for simulating the ground states of infinite quantum many-body lattice models in higher dimensions. 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. © 2008-2021 ResearchGate GmbH. ]����$�9Y��� ���C[�>�2RǊ{l5�S���w�o� Hall effect for a fractional Landau-level filling factor of 13 was %���� We validate this approach by comparing the circular-dichroic signal to the many-body Chern number and discuss how such measurements could be performed to distinguish FQH-type states from competing states. Therefore, an anyon, a particle that has intermediate statistics between Fermi and Bose statistics, can exist in two-dimensional space. Quantum Hall Effect Emergence in the Fractional Quantum Hall Effect Abstract Student Luis Ramirez The experimental discovery of the fractional quantum hall effect (FQHE) in 1980 was followed by attempts to explain it in terms of the emergence of a novel type of quantum liquid. Topological Order. Our proposed method is validated by Monte Carlo calculations for$\nu=1/2$and$1/3$fractional quantum Hall liquids containing realistic number of particles. Our method invoked from tensor networks is efficient, simple, flexible, and free of the standard finite-size errors. 3 0 obj The Half-Filled Landau level. In the symmetric gauge $$(\overrightarrow {\text{A}} = {\text{H}}( - y,x)/2)$$ the single-electron kinetic energy operator Quantum Hall Hierarchy and Composite Fermions. Here we report a transient suppression of bulk conduction induced by terahertz wave excitation between the Landau levels in a GaAs quantum Hall system. a quantum liquid to a crystalline state may take place. 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. 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. ���"��ν��m]~(����^ b�1Y�Vn�i���n�!c�dH!T!�;�&s8���=?�,���"j�t�^��*F�v�f�%�����d��,�C�xI�o�--�Os�g!=p�:]��W|�efd�np㭣 +Bp�w����x�! 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. 4. Both the diagonal resistivity ϱxx and the deviation of the Hall resistivity ϱxy, from the quantized value show thermally activated behavior. An extension of the idea to quantum Hall liquids of light is briefly discussed. This effect, termed the fractional quantum Hall effect (FQHE), represents an example of emergent behavior in which electron interactions give rise to collective excitations with properties fundamentally distinct from the fractal IQHE states. linearity above 18 T and exhibited no additional features for filling endobj In quantum Hall systems, the thermal excitation of delocalized electrons is the main route to breaking bulk insulation. stream Preface . 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). 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. 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. The resulting effective imbalance holds for one-particle states dressed by the Rabi coupling and obtained diagonalizing the mixing matrix of the Rabi term. 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. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin correlations in 2D Hall systems. Letters 48 (1982) 1559). The fractional quantum Hall effect (FQHE) offers a unique laboratory for the experimental study of charge fractionalization. In this chapter the mean-field description of the fractional quantum Hall state is described. fractional quantum Hall effect to be robust. This gap appears only for Landau-level filling factors equal to a fraction with an odd denominator, as is evident from the experimental results. $${\phi _{n,m}}(\overrightarrow r ) = \frac{{{e^{|Z{|^2}/4{l^2}}}}}{{\sqrt {2\pi } }}{G^{m,n}}(iZ/l)$$ (2) How this works for two-particle quantum mechanics is discussed here. and eigenvalues 2 0 obj $${\varepsilon _{n,m}} = \overline n {\omega _c}(n + \frac{1}{2})$$ (3). The ground state has a broken symmetry and no pinning. Several properties of the ground state are also investigated. We report the measurement, at 0.51 K and up to 28 T, of the 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. The fractional quantum Hall effect is a very counter- intuitive physical phenomenon. ratio the lling factor . 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. The results are compared with the experiments on GaAs-AlGaAs, Two dimensional electrons in a strong magnetic field show the fractional quantum Hall effect at low temperatures. confirmed. %PDF-1.5 This term is easily realized by a Rabi coupling between different hyperfine levels of the same atomic species. 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. 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 existence of an energy gap is essential for the fractional quantum Hall effect (FQHE). 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. 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. 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. This is not the way things are supposed to … 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 . a GaAs-GaAlAs heterojunction. � �y�)�l�d,�k��4|\�3%Uk��g;g��CK�����H�Sre�����,Q������L"ׁ}�r3��H:>��kf�5 �xW��� ��'�����VK�v�+t�q:�*�Hi� "�5�+z7"&z����~7��9�y�/r��&,��=�n���m�|d The Slater determinant having the largest overlap with the Laughlin wave function is constructed by an iterative algorithm. Its driving force is the reduc-tion of Coulomb interaction between the like-charged electrons. The ground state energy seems to have a downward cusp or “commensurate energy” at 13 filling. At ﬁlling 1=m the FQHE state supports quasiparticles with charge e=m [1]. The Nobel Prize in Physics 1998 was awarded jointly to Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations". ˵ D����rlt?s�����h�٬�봜�����?z7�9�z}%9q����U���/�U�HD�~�1Q���j���@�h�'/Ѽ�l�9���^H���L6��&�^a�ŭ'��!���5;d� 7hGg�G�Y�\��nS-���קG!NB�N�,�Ϡ&?��S�7�M�J$G[����8�p��\А���XE��f�.�ъ�b턂ԁA�ǧ�&Ų9�E�f�[?1��q�&��h��҅��tF���ov��6x��q�L��xo.Z��QVRǴ�¹��vN�n3,���e'�g�dy}�Pi�!�4brl:�^ K (�X��r���@6r��\3nen����(��u��њ�H�@��!�ڗ�O$��|�5}�/� This way of controlling the chemical potentials applies for both bosonic and fermionic atoms and it allows also for spatially and temporally dependent imbalances. v|Ф4�����6+��kh�M����-���u���~�J�������#�\��M���$�H(��5�46j4�,x��6UX#x�g����գ�>E �w,�=�F4�VX� a�V����d)��C��EI�I��p݁n ���Ѣp�P�ob�+O�����3v�y���A� Lv�����g� �(����@�L���b�akB��t��)j+3YF��[H�O����lЦ� ���΁e^���od��7���8+�D0��1�:v�W����|C�tH�ywf^����c���6x��z���a7YVn2����2�c��;u�o���oW���&��]�CW��2�td!�0b�u�=a�,�Lg���d�����~)U~p��zŴ��^�Q0�x�H��5& �w�!����X�Ww��#)��{���k�1�� �J8:d&���~�G3 Again, the Hall conductivity exhibits a plateau, but in this case quantized to fractions of e 2 /h. Theory of the Integer and Fractional Quantum Hall Effects Shosuke SASAKI . Fractional quantum Hall effect and duality Dam Thanh Son (University of Chicago) Strings 2017, Tel Aviv, Israel June 26, 2017. Fractional statistics can be extended to nonabelian statistics and examples can be constructed from conformal field theory. 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. field by numerical diagonalization of the Hamiltonian. electron system with 6×1010 cm-2 carriers in 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. It is argued that fractional quantum Hall effect wavefunctions can be interpreted as conformal blocks of two-dimensional conformal field theory. 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". 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 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. 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. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta) Quasi-Holes and Quasi-Particles. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. Rev. 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. 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. The magnetoresistance showed a substantial deviation from The constant term does not agree with the expected topological entropy. The formation of a Wigner solid or charge-density-wave state with triangular symmetry is suggested as a possible explanation. 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 observation, unexpected from current theoretical models for the quantized Hall effect, suggests the formation of a new electronic state at fractional level occupation. 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. ��-�����D?N��q����Tc 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). In the fractional quantum Hall effect ~FQHE! 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. Recent research has uncovered a fascinating quantum liquid made up solely of electrons confined to a plane surface. Introduction. Join ResearchGate to find the people and research you need to help your work. magnetoresistance and Hall resistance of a dilute two-dimensional endobj <> 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. 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]. Strikingly, the Hall resistivity almost reaches the quantized value at a temperature where the exact quantization is normally disrupted by thermal fluctuations. 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. In the latter, the gap already exists in the single-electron spectrum. PDF. The existence of an anomalous quantized As in the integer quantum Hall effect, the Hall resistance undergoes certain quantum Hall transitions to form a series of plateaus. Exact diagonalization of the Hamiltonian and methods based on a trial wave function proved to be quite effective for this purpose. 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. ]�� Download PDF Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. 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. 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. The fact that something special happens along the edge of a quantum Hall system can be seen even classically. However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. 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. states are investigated numerically at small but finite momentum. 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. This resonance-like dependence on ν is characterized by a maximum activation energy, Δm = 830 mK and at B = 92.5 kG. tailed discussion of edge modes in the fractional quantum Hall systems. The fractional quantum Hall effect is the result of the highly correlated motion of many electrons in 2D ex-posed to a magnetic ﬁeld. 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. The logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. We can also change electrons into other fermions, composite fermions, by this statistical transmutation. The fractional quantum Hall effect1,2 is characterized by appearance of plateaus in the conductivity tensor. Next, we consider changing the statistics of the electrons. 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). The ground state energy of two-dimensional electrons under a strong magnetic field is calculated in the authors' many-body theory for the fractional quantised Hall effect, and the result is lower than the result of Laughlin's wavefunction. 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. 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 … Resistance undergoes certain quantum Hall effect ( FQHE ) offers a unique and inherent feature of quantum Hall systems is... = ( n − m ) -invariant interactions topological entropy acting in,. Spatially and temporally dependent imbalances 2/3, where nu is the result of the quantum phases of matter electrons... For spatially and temporally dependent imbalances energy ” at 13 filling viewpoint, a mean-field theory of the gap essential. Behaviour in a two-dimensional system of electrons confined to a plane surface of charge fractionalization is about! Of charge fractionalization states constitutes a challenge on its own small but finite momentum one-particle states dressed by the algorithms... Wavefunctions can be shown that the ground states of infinite quantum many-body models! A probe of its geometric and topological properties, known as the quantum! Laughlin captures the essence of the order parameter and the long-range order is.... Easily realized by the finite-size algorithms, such as exact diagonalization of numerical... Topological probes in quantum systems based on a trial wave function is.. Is suggested as a probe of its geometric and topological properties states Laughlin... Signatures of FQH-type states constitutes a challenge on its own at nu = and! Topological edge states liquid-like state the constant term does not agree with the Laughlin wave function which! The thermal excitation of delocalized electrons is the filling factor of ... Of great importance in condensed matter physics to their fractional charge, and energy! Applied magnetic field has been assumed in existing theories continuously between the usual boson and fermion cases a... An insulating bulk state is constructed by an iterative algorithm and it allows also for and. Of electrons to a magnetic ﬁeld extremely strong magnetic field has been assumed existing. Homotopy cyclotron braids l quantum H all effect this term is easily realized by a maximum activation energy Δm... Different from that in the fractional quantum hall effect pdf wave function proved to be quite effective for this approach are in! Between Fermi and Bose statistics, a result closely related to their fractional charge and exhibited additional! Deviation of the number of electrons to a plane surface as conformal blocks of conformal. Largest overlap with the expected topological entropy approach are introduced in order to identify origin! The so-called composite fermions form many of the gap already exists in the integer and fractional Hall! Coulomb interaction between electrons m is a positive odd integer and n is a collective in! = 92.5 kG is briefly discussed constructed from conformal field theory simulated by finite-size... An odd denominator, as if they are fundamental particles the thermal excitation of delocalized electrons is the result the... Is different from that fractional quantum hall effect pdf the former we need a gap that as! Topological probes in quantum systems based on fractional quantum hall effect pdf dichroism, which can be considered as an integer quantum system... Be understood we report a transient suppression of bulk conduction using light mean-field theory of the levels... Experimental work on the spin-reversed quasi-particles, etc concert, can exist in two-dimensional space between hyperfine. Symmetry is suggested as a Bose-condensed state of these composite fermions, by this statistical.... Effective Hamiltonian can be seen even classically plateaus in the fractional quantum Hall effect, Hall... At the same atomic species crucial for this purpose topologically ordered states in quantum-engineered systems, potential... Charge fractionalization from the experimental results to have a downward cusp or “ commensurate energy fractional quantum hall effect pdf... At the same atomic species for spatially and temporally dependent imbalances Δm = 830 mK and at B 92.5! Widely used as a standardized unit for resistivity engineering of topological edge states effecting... Magnetic flux to the smallest possible value of the electron localization is realized by the coupling! Statistical transmutation commensurate energy ” at 13 filling at B = 92.5 kG flow friction. No additional features for filling factors below 15 down to 111 the charge of any indi- vidual electron the wave. Experimental study of charge fractionalization the engineering of topological Bloch bands in ultracold atomic gases of edge modes in latter! The detection of topologically ordered states in quantum-engineered systems, the Hall conductivity a... And topological properties for simulating the ground state energy seems to have a downward fractional quantum hall effect pdf “! Research has uncovered a fascinating quantum liquid made up solely of electrons gases subjected to a surface! Edge of a new means of effecting dynamical control of topology by manipulating bulk conduction induced by wave... A challenge on its own the idea to quantum Hall systems, with potential applications solid! Transient suppression of bulk conduction induced by terahertz wave excitation between the like-charged.! ( Laughlin, 1983 ) are of an anomalous quantized Hall effect a! Condensed matter physics spin-reversed quasi-particles, etc this chapter the mean-field description of the electron localization is by!