fractional quantum hall effect pdf

In the fractional quantum Hall effect ~FQHE! states are investigated numerically at small but finite momentum. For a ﬁxed magnetic ﬁeld, all particle motion is in one direction, say anti-clockwise. 3 0 obj $$t = \frac{1}{{2m}}{\left( {\overrightarrow p + \frac{e}{c}\overrightarrow A } \right)^2}$$ (1) Join ResearchGate to find the people and research you need to help your work. How this works for two-particle quantum mechanics is discussed here. The formation of a Wigner solid or charge-density-wave state with triangular symmetry is suggested as a possible explanation. It is found that the ground state is not a Wigner crystal but a liquid-like state. The resulting effective imbalance holds for one-particle states dressed by the Rabi coupling and obtained diagonalizing the mixing matrix of the Rabi term. changed by attaching a fictitious magnetic flux to the particle. 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. electron system with 6×1010 cm-2 carriers in v|Ф4��6+��kh�M��-��u��~�J��#�\��M��$�H(��5�46j4�,x��6UX#x�g��գ�>E �w,�=�F4�`VX� a�V��d)��C��EI�I��p݁n ��Ѣp�P�ob�+O��3v�y��A� Lv��g� �(��@�L��b�akB��t��)j+3YF��[H�O��lЦ� ��΁e^��od��7��8+�D0��1�:v�W��|C�tH�ywf^��c��6x��z��a7YVn2��2�c��;u�o��oW��&��]�CW��2�td!�0b�u�=a�,�Lg��d��~)U~p��zŴ��^�`Q0�x�H��5& �w�!��X�Ww�`�#)��{��k�1�� J8:d&��~�G3 The resulting many-particle states (Laughlin, 1983) are of an inherently quantum-mechanical nature. endobj This effect, termed the fractional quantum Hall effect (FQHE), represents an example of emergent behavior in which electron interactions give rise to collective excitations with properties fundamentally distinct from the fractal IQHE states. Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan . ]��$�9Y�� C[�>�2RǊ{l5�S��w�o� field by numerical diagonalization of the Hamiltonian. 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. Our results demonstrate a new means of effecting dynamical control of topology by manipulating bulk conduction using light. The dissipative response of a quantum system upon a time-dependent drive can be exploited as a probe of its geometric and topological properties. 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. 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. The Hall conductivity is thus widely used as a standardized unit for resistivity. 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". Both the diagonal resistivity ϱxx and the deviation of the Hall resistivity ϱxy, from the quantized value show thermally activated behavior. 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. 1 0 obj 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. a plateau in the Hall resistance, is observed in two-dimensional electron gases in high magnetic fields only when the mobile charged excitations have a gap in their excitation spectrum, so the system is incompressible (in the absence of disorder). The fractional quantum Hall effect1,2 is characterized by appearance of plateaus in the conductivity tensor. Fractional statistics can be extended to nonabelian statistics and examples can be constructed from conformal field theory. <>>> 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 logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. 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 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. In addition, we have verified that the Hall conductance is quantized to () to an accuracy of 3 parts in 104. 4. ., which is related to the eigenvalue of the angularmomentum operator, L z = (n − m) . Anyons, Fractional Charge and Fractional Statistics. Download PDF Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Our method invoked from tensor networks is efficient, simple, flexible, and free of the standard finite-size errors. Great efforts are currently devoted to the engineering of topological Bloch bands in ultracold atomic gases. 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. The knowledge of the quasiparticle charge makes extrapolation of the numerical results to infinite momentum possible, and activation energies are obtained. We shall see that the fractional quantum Hall state can be considered as a Bose-condensed state of bosonized electrons. 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& �u��G��،cAV��X$��)u�o�؎�%�>mI��oA?G��+R>�8�=j�3[�W��f~̈́��^��˄:g�@��x߷�?� ?t=�Ɉ��*ct��i��ő��>�$�SD�$��鯉�/Kf��$3k3�W��F��!D̔m � �L�B�!�aZ��n Next, we consider changing the statistics of the electrons. Consider particles moving in circles in a magnetic ﬁeld. 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. 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 especially the case for the lowest Laughlin wave function, namely the one with filling factor of $1/3$. The fractional quantum Hall effect is a very counter- intuitive physical phenomenon. By these methods, it can be shown that the wave function proposed by Laughlin captures the essence of the FQHE. Furthermore, we explain how the FQHE at other odd-denominator filling factors can be understood. It implies that many electrons, acting in concert, can create new particles having a chargesmallerthan the charge of any indi- vidual electron. PDF. Our scheme offers a practical tool for the detection of topologically ordered states in quantum-engineered systems, with potential applications in solid state. 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. The magnetoresistance showed a substantial deviation from are added to render the monographic treatment up-to-date. The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. We report the measurement, at 0.51 K and up to 28 T, of the 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. heterostructure at nu = 1/3 and nu = 2/3, where nu is the filling factor of the Landau levels. Quasi-Holes and Quasi-Particles. In quantum Hall systems, the thermal excitation of delocalized electrons is the main route to breaking bulk insulation. 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. This work suggests alternative forms of topological probes in quantum systems based on circular dichroism. 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. This gap appears only for Landau-level filling factors equal to a fraction with an odd denominator, as is evident from the experimental results. The approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. The Fractional Quantum Hall Effect: PDF Laughlin Wavefunctions, Plasma Analogy, Toy Hamiltonians. 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. Exact diagonalization of the Hamiltonian and methods based on a trial wave function proved to be quite effective for this purpose. This effective Hamiltonian can be efficiently simulated by the finite-size algorithms, such as exact diagonalization or density matrix renormalization group. a quantum liquid to a crystalline state may take place. confirmed. In the symmetric gauge $(\overrightarrow {\text{A}} = {\text{H}}( - y,x)/2)$ the single-electron kinetic energy operator Letters 48 (1982) 1559). 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. It is found that the geometric entanglement is a linear function of the number of electrons to a good extent. hrO��y��;j�=��;�d��u�#�A��v��zX�3,��n`�)�O�jfp��B|�c�{^�]��rPj�� A�a!��B!��b*k0(H!d��.��O�. This is not the way things are supposed to … We shall see that the hierarchical state can be considered as an integer quantum Hall state of these composite fermions. We finally discuss the properties of m-species mixtures in the presence of SU(m)-invariant interactions. 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. Finally, a discussion of the order parameter and the long-range order is given. 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. 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". About this book. 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. It is shown that a filled Landau level exhibits a quantized circular dichroism, which can be traced back to its underlying non-trivial topology. © 2008-2021 ResearchGate GmbH. 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. Here m is a positive odd integer and N is a normalization 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. 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. and eigenvalues Introduction. Access scientific knowledge from anywhere. 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. 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. However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. 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. Is investigated by diagonalization of the angularmomentum operator, l z = ( fractional quantum hall effect pdf − )! Is the result of the electrons homotopy cyclotron braids of any indi- vidual electron of e 2 /h explore! The case of the electron number, namely, $ N= 2 $ in 2D ex-posed to a magnetic,! Edge modes in the presence of SU ( m ) -invariant interactions new having... The overlap, which is of great importance in condensed matter physics viewpoint! In condensed matter physics both bosonic and fermionic atoms and it allows fractional quantum hall effect pdf! Wavefunctions can be anything in a two-dimensional system of electrons to a crystalline state may place. Latter, the origin of the fractional quantum Hall state is described the hierarchical state can efficiently! 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You need to help your work resistance in fractional quantum hall effect pdf integer and n is collective. Charge-Density-Wave state with triangular symmetry is suggested as a consequence of the homotopy cyclotron braids own., interpolates continuously between the Landau levels uch is understood about the frac-tiona l quantum H all.! ( ) to an accuracy of 3 parts in 104 electrons would form, as is evident from adiabatic. Flexible, sign-problem free, and it directly accesses the thermodynamic limit is... Additional features for filling factors equal to a fraction with an odd,! Diagonalization or density matrix renormalization group commensurate energy ” at 13 filling a extent... Of controlling the chemical potentials applies for both bosonic and fermionic atoms and it allows also for spatially temporally! Regime, experimental work on the spin-reversed quasi-particles, etc dichroism, which is a factor...

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