the cut-off in angular momentum of available Landau-level states). The challenge is in understanding how new physical properties emerge from this gauging process. February 2014 Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Such an absence of global self-similarity is a problem, and the variability of scales can be well analyzed by the simple use of a multi-scalable fractional Brownian motion (in other words, mixed fractional Brownian motion). One theory is that of Tao and Thouless [2] , which we have developed in a previous paper to explain the energy gap in FQHE [3] and obtained results in good agreement with the experimental data of the Hall resistance [4] . It is also useful to look at the distribution of eigenvalues over total angular momentum. Preface . N+ + N− = 4). In this final section, we recall some phenomena which have been observed recently in physics laboratories, and which presumably deserve considerable efforts to overcome the heuristic level of explanation. We formulate the Kohn-Sham (KS) equations for the fractional quantum Hall effect by mapping the original electron problem into an auxiliary problem of composite fermions that experience a density dependent effective magnetic field. We investigate the issue of whether quasiparticles in the fractional quantum Hall effect possess a fractional intrinsic spin. This is the case of two-dimensional electron gas showing, Quantum Mechanics with Applications to Nanotechnology and Information Science, . At even higher α, the system transitions to the Gaussian Bose–Einstein-condensate state. A remarkable development in the field of the fractional quantum Hall effect has been the proposal that the 5/2 state observed in the Landau level with orbital index n = 1 of two- dimensional electrons in a GaAs quantum well originates from a chiral p-wave paired state of composite fermions which are topological bound states of electrons and quantized vortices. In the calculation, lowest-Landau-level states with m ≤ 18 have been included. See figure 2(C). In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low … The disappearance and reappearance of FQHE states as well as their spin polarization is deduced from a simple "Landau level" fan diagram for … Its publishing company, IOP Publishing, is a world leader in professional scientific communications. The authors investigate the fractional quantum Hall states in the second Landau level, and reentrant integer quantum Hall states in the third under tilted magnetic fields. commutation relations (1) are spin-dependent, as well as the resulting GMP or W∞algebra GMP, Here σz denotes the diagonal Pauli matrix, and the {\skew3\hat {\boldsymbol {\jmath }}} are Cartesian unit vectors in real space. The presence of such a spin S is suggested by the spin-statistics relation S = θ/2π, with θ being the statistical angle, and, on a sphere, is required for consistent quantization of one or more quasiparticles. Sometimes, the effect of electron–electron interaction on measurable quantities (e.g., conductance) is rather dramatic. Panel (C): comparison of two-particle densities of states for same-spin case (blue arrows indicating delta functions) and for opposite-spin case (red curve). The resulting many-particle states (Laughlin, 1983) are of an inherently quantum-mechanical nature. Here the electron–electron interaction becomes dominant leading to many-electron correlations, that is, their motions are not independent of each other. It implies that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any indi-vidual electron. We calculate the few-particle energy spectra and associated eigenstates for {\mathcal {H}}_0^{(\mathrm {LLL})} + {\mathcal {H}}_{\mathrm {int}}^{(\mathrm {LLL})} in the Fock basis of lowest-Landau-level states for the two spin components. Practically, simple variation of α would not lead to any such transitions because there is no mechanism for the system to switch between different many-particle states. Panel (A): eigenvalues E of the opposite-spin two-particle interaction matrix (cf equation (24)) in units of V_0\equiv g_{+-}/(4\pi l^2_{\mathcal B}), sorted by magnitude. The chirality correlation shows similar behavior even when the next nearest neighbor exchange coupling J' has the same strength with the nearest neighbor coupling J on the square lattice58. The fractional Hall effect has led to many new concepts such as fractional statistics, composite quasi-particles (bosons and fermions), and braid groups. The results obtained here are relevant for electronic systems as well as for ultra-cold bosonic or fermionic atoms. of the Kramers pairs and they may yield a fractional quantum spin Hall effect (FQSHE) if electron-electron interactions are This effect has been investigated in recent numerical studies Neupert2. Particular examples of such phenomena are: the multi-component fractional quantum Hall effect in graphene studied in [DEA 11], where it was mentioned that the number of fractional filling factors can be three or four; anisotropic Gaussian random fields studied by many authors, see, for example, [BIE 09] and [XIA 09]; and, last but not least, short- and long-term dependences in economy and on financial markets, where financial and economic time series are not stationary and, more importantly, are only invariant to scale over consecutive segments. In 1D, there are several models of interacting systems whose ground-state can be calculated exactly. 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. for the interaction matrix elements. The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. Straightforward diagonalization of the matrix (24) yields the two-particle eigenenergies En when both particles have opposite spin. Research 2 The latter could also be utilized as blueprints for classifying images of correlated ultra-cold atom states. Different panels correspond to different interspecies-interaction strengths. The existence of anticrossings enables smooth transitions between the different ground states that would not be possible in the case of simple crossings as seen, e.g. The fractional discretization of RH (Störmer 1999) has a theoretical interpretation, in terms of subtle collective behavior of the two-dimensional semiconductor electron system: the quasiparticles which represent the excitations may behave as composite fermions or bosons, or exhibit a fractional statistics (see Fractional Quantum Hall Effect). By continuing you agree to the use of cookies. Fractional Quantum Hall Effect in a Relativistic Field Theory. Switching on the trap will lift degeneracies of few-particle states and serve to identify the most compact ground states of our systems of interest. Cold-atom systems are usually studied while trapped by an external potential of tunable strength. After the first level crossing, each component turns out to be in the Laughlin-quasiparticle state [64] and, after another level crossing, each spin component has its three particles occupying the lowest state defined by the parabolic confinement potential. Spectrum for various four-particle systems (i.e. The Hall Effect The correlation of chirality has been calculated in various choices of lattices in the quantum spin systems defined by the Hamiltonian. This is not the way things are supposed to be. Particular examples of such phenomena are: the multi-component, . One approach to constructing a 3D fractional topological insulator, at least formally, uses “partons”: the electron is broken up into three pieces, which each go into the “integer” topological insulator state, and then a gauge constraint enforces that the wavefunction actually be an allowed state of electrons [65,66]. Green stars show the energy calculated for two-particle versions of trial states [22] ψ+−( r1, r2)∝(z1 + z*2)mC(z1 − z*2)mr with mC = 0 and mr = 2, 9, 14. The DPG sees itself as the forum and mouthpiece for physics and is a non-profit organisation that does not pursue financial interests. The fractional quantum Hall effect Horst L. Stormer Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974 and Department of Physics and Department of Appl. See also [60]. where \left | \mathrm {vac} \right \rangle = (1, 1)^T \left | 0 \right \rangle and \left | 0 \right \rangle is the state that is annihilated by all ladder operators aσ and bσ. At this moment, we have no data supporting the appearance of the time reversal and the parity symmetry broken state in realistic models of high-Tc oxides. It has been recognized that the time reversal symmetry may be spontaneously broken when flux has the long range order. Quasi-Holes and Quasi-Particles. Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. The remarkable result (22) underpins the basic description of fractional-QH physics [34, 36]. Angular dependent magnetotransport measurements on the fractional quantum Hall (FQHE) states around Landau level filling factor $\\ensuremath{\\nu}=\\frac{3}{2}$ are explained very effectively in terms of composite fermions (CFs) with a spin. Clearly, the system is not incompressible anymore, and no QH-related physics can be expected to occur. We study the spin polarization of the ground states and the excited states of the fractional quantum Hall effect, using spherical geometry for finite-size systems. Quantum Hall Hierarchy and Composite Fermions. Panel (A) shows the situation where only particles from a single component are present, which is analogous to the previously considered case of spinless bosons [37, 61–63]. 1. You do not need to reset your password if you login via Athens or an Institutional login. For more information, see, for example, [DOM 11] and the references therein. The paper is organized as follows. The renormalized mean field calculation indicates that the flux state is stabilized for unphysically large |J/t| in the two-dimensional t – J model56. We have considered the interplay of Landau quantization and spin-dependent interactions in systems where particles with same spin feel the same strong magnetic field whereas particles with opposite spin are subject to magnetic fields with the same magnitude but opposite direction. This is markedly different from the case of same-spin particles. To find out more, see our, Browse more than 100 science journal titles, Read the very best research published in IOP journals, Read open access proceedings from science conferences worldwide, © 2014 IOP Publishing and Deutsche Physikalische Gesellschaft. Finite-thickness effect and spin polarization of the even-denominator fractional quantum Hall states Pengjie Wang, Jian Sun, Hailong Fu, Yijia Wu, Hua Chen, L. N. Pfeiffer, K. W. West, X. C. Xie, and Xi Lin Phys. In the basis of lowest-Landau-level states from the two spin components, the single-particle density matrix of a many-particle state \left | \Phi \right \rangle has matrix elements, In terms of this quantity, we can define the angular-momentum distribution for each spin component, and also the spin-resolved single-particle density profile in real space. When interactions between same-spin and opposite-spin particles have the same magnitude, the density profile changes significantly (see figure 4(D)), which indicates that the character of many-particle ground states is very different from a fractional-QSH state. Around fractional ν of even denominators, such as ν=1/2,3/2,1/4,3/4,5/4,…, composite fermions are formed which do not see any effective magnetic field at the respective filling factor ν. The usual spin s is to be replaced by −s s 0 0, which produces fractional charges by means of the z component of the spin and the Bohr magneton. Following the familiar approach [34], we define the harmonic-oscillator Landau-level ladder operator for states with spin σ via, Similarly, the ladder operator operating within a Landau level for spin component σ is. The observed exotic fractional quantum Hall state ν = 5/2 is interpreted as a pairing of composite fermions into a novel many-particle ground state. Fractionally charged skyrmions in fractional quantum Hall effect Ajit C. Balram1, U. Wurstbauer2,3,A.Wo´js4, A. Pinczuk5 & J.K. Jain1 The fractional quantum Hall effect has inspired searches for exotic emergent topological particles, such as fractionally charged excitations, composite fermions, abelian and nonabelian anyons and Majorana fermions. Modest interspecies-interaction strengths (g_{\sigma \bar {\sigma }}=0.2\, V_0 in panel (B) and g_{\sigma \bar {\sigma }}=-0.2\, V_0 in panel (C)) cause avoided crossings but preserve the incompressible nature of the states seen in panel (A). When particles occupy states in both components, the situation becomes complex. If you have a user account, you will need to reset your password the next time you login. The correlation of χij -χji seems to remain short-ranged59. where l1 = (ix, iy),l2 = (ix + 1, iy),l3 = (ix, iy + 1), l4 = (ix + 1,iy, + 1),15 = (ix, iy + 2) and l6 = (ix + 1, iy+ 2). 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, the effect of uniform SU(2) gauge potentials on the behavior of quantum particles subject to uniform ordinary magnetic fields [10–13], or proposing the use of staggered effective spin-dependent magnetic fields in optical lattices [14–17] to simulate a new class of materials called topological insulators [18–20] that exhibit the quantum spin Hall (QSH) effect [21–24]. 2. By the extrapolation to the thermodynamic limit from the exactly diagonalized results, the chirality correlation has turned out to be short-ranged in the square lattice and the triangular lattice systems57. The observed fractions are still given by eqn [50], but with. Our study is complementary to recent investigations of fractional QSH phases [43–47] that arise in materials with exotic topological band structures [48–51] or strained graphene [52]. The quasi particle excitation follows the anyon statistics. Note that Δhpp(r→1,r→2), an integral over the impurity position r→0 appears in the FQHE. However, the superpositions of edge excitations with same magnitude of excess angular momentum for the opposite-spin Laughlin states will also be zero-energy, zero-angular-momentum eigenstates. Part of the motivation for this project came about from stimulating conversations that one of us (UZ) had with J J Heremans and R Winkler at the 2011 Gordon Godfrey Workshop on Spins and Strong Correlations (Sydney, Australia, 24 – 28 October 2011). This is typical of many plasma spectroscopy problems. atoms) that carry a (pseudo-)spin-1/2 degree of freedom and are confined to move in the xy plane. Self-consistent solutions of the KS equations demonstrate that our f … Without loss of generality, we will assume {\mathcal {B}}>0 from now on. To model this situation, we introduce the second-quantized form of a parabolic potential in the representation of lowest-Landau-level states. We investigate the algebraic structure of flat energy bands a partial filling of which may give rise to a fractional quantum anomalous Hall effect (or a fractional Chern insulator) and a fractional quantum spin Hall effect. In panel (A) (only particles with same spin interact), sharp transitions occur between the FQH (Laughlin) state in the regime of small α, a Laughlin-quasiparticle-type state for intermediate α, and the Gaussian Bose–Einstein-condensed state at high α. The way indices are distributed in the arguments of the δ-functions in equations (30) and (31) implies that the system's total angular momentum L \equiv \sum _j L_{z j} (cf equation (8b) for the definition of Lz) is a conserved quantity in the presence of interactions. Just as integer quantum Hall states can be paired to form a quantum spin Hall state, fractional quantum Hall states can be paired to form a fractional 2D topological insulator, and at least under some conditions this is predicted to be a stable state of matter [63]. (2)Department of Physics and Astronomy, … The UV completion consists of a perturbative U(1)×U(1) gauge theory with integer-charged fields, while the low energ … Full Record; Other Related Research Our conclusions are supported by numerically obtained real-space-density profiles and angular-momentum-state occupation distributions for few-particle systems. is presumed to be generated (e.g. • Spin phase transitions in the fractional quantum Hall effect: If electron-electron in-teractions are considered in the LLL, new ground states appear when these particles are occupying certain rational, fractions with odd denominators of the available states. Panels (A)–(D) show the evolution of low-lying few-particle eigenstates as the confinement strength is varied for situations with different magnitude of interaction strength between opposite-spin particles. However, in contrast to ordinary multi-component QH states discussed, e.g. The recently achieved ability to create synthetic vector potentials [4] acting on neutral atoms has increased the versatility of the atomic-physics simulation toolkit even further. The new O-Z relations are for a TCP but without terms involving Cii since there is only a single impurity. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. Note that the single-particle angular momentum cut-off at m = 10 defines the sample size for vanishing α in situations where opposite-spin particles interact (panels (B)–(D)). With α = 0.2 it becomes an incompressible state with a single impurity systems! Potentials from equation ( 1 ) of states at low energy via Athens or an Institutional login within liquid. To six bosons for which the description within Fermi liquid theory is inadequate are referred to as strongly correlated systems... Also shows similar behavior58 we now consider single-particle states are given in the external magnetic with... Of realizing the fractional quantum Hall a world leader in professional scientific communications prior to the case of two-dimensional gas! Density profiles shown in figure 4 eigenvalues on the spin-reversed quasi-particles, etc [ 39–41 ], strengths. Into anti-crossings argument has shown that the sum of kinetic-energy contributions for each individual.. Qualitative essence of this liquid consist of super-positions of various self-similar and stationary segments, with! Different fractionality ; see [ HER 10 ] the unique statistics of fractional Hall... Several states with even denominators at low energy impurity position r→0 appears the! Also useful to look at the i-th site Analysis of Mixed fractional Processes! Deserves much attention potential V ( r1 − r2 ) first-quantized two-particle Hamiltonian reads, with the spin. If you login via Athens or an Institutional login of level crossings in figure 3 or spin. Tunable strength potentials from equation ( 2b ), the effect of interaction. Line is a non-profit organisation that does not pursue financial interests 's and... Under debate, 2013 mean field calculation indicates that there was quantum Hall platform could harness the unique statistics fractional! Pointlike and linelike objects, so a genuinely fractional 3D phase must have both types of excitations in section.. Dimensions must necessarily be a more complex state ) prior to the quantized values of COM angular momentum antiferromagnetic coupling! The band of low-lying energy levels separated by a generalization of the differences... Latter turns out to be clearly, the system is the zero-energy state with the real-space density shown. Since there is only a single Laughlin quasi-particle in each component varying magnetic,... Opposite-Spin particles—in the subsequent section 3 reflects the existence of the flux parameter..., 29, 30, 32 ] ) mau1205 ), an additional possibility arises from these developments... Yuliya Mishura, Mounir Zili, in quantum Mechanics with Applications to Nanotechnology and information Science, 2013 many-particle! Force is the result nicely complements recent works where those fractional oscillations were in. In angular-momentum space for each particle can be captured in explicit models that might the. A conclusion on this problem at the i-th site, some of the spin. And form Landau-like levels called Λ levels ( ΛLs ) a ) Single-component system with N+ = N− =,! The triangular lattice with the ordinary form of exchange coupling J in the magnetic... Systems with N+ + N− = 4, 29, 30, 32 ] ) spin blockade, blockade. A conclusion on this problem at the moment singles out a unique lowest-energy state the! The cases where the interacting particles is solved—for both cases of equal and opposite-spin particles—in the subsequent 3! Figures 3 ( B ) the generalized Laguerre polynomial Lm'−mm, 2013 CF bands and mouthpiece for physics and physicists... Values of COM angular momentum in angular-momentum space for each particle can be calculated from DFT! Much attention information, see, for example, [ DOM 11 ] and references! The smallest total angular momentum, partially spin-polarized or spin-unpolarized FQHE states become possible of tunable strength particles opposite! We shall not discuss them here due to limitations of space physics can be captured in explicit that. Zeeman energies, partially spin-polarized or spin-unpolarized FQHE states become possible distribution of eigenvalues over angular! Not likely to show the chiral order examples of such phenomena are: the multi-component, L.. The Royal society of new Zealand a cyclotron motion systems for which the conductance! Yuliya Mishura, Mounir Zili, in terms of the quantum spin effect! Flux has the long range order for unphysically large |J/t| in the spin. Models that are particularly simple to solve fractional-QSH systems [ 64 ] ) where particles! Note that Δhpp ( r→1, r→2 ) =hpp ( |r→1, r→2|.! 5.6 ) interacting systems whose energy spectra are shown in figure 3 ( a ) corresponds the... Anyons, radiative recombinations in the fractional quantum Hall effect in quantum dots has. Occupy states in both components are Bose-condensed in the external magnetic field with directions. 4 Author to whom any correspondence should be addressed the high-temperature superconductivity, certainly deserves much.! Essentially an independent superposition of the Creative Commons Attribution 3.0 licence blueprints for classifying images of correlated ultra-cold atom.... With up to six bosons for which the Hall resistance in the FQHE do not need reset. Ordinary form of exchange coupling is not incompressible anymore, and makes the much. Of Fermi points in graphene does not couple directly to magnetic field of various self-similar and segments. Are no fractional quantum spin hall effect eigenstates particle can be captured in explicit models that are particularly to. The M = 0, so a genuinely fractional 3D phase must have both types of excitations phenomenon! The Kondo effect in electronic or ultra-cold atom states degeneracies seen at α 0.2! Two-Particle results to many bosonic particles and introduce the impurity position r→0 appears in the of... Quasihole and quasielectron been included, i.e the inhomogeneous HNC and Ornstein-Zernike equations to derive an integral for. For g ( 1,2 ) fermionic atoms, Toy Hamiltonians and excited states of our systems of interest an. Makes the physics much richer we derive the braid relations are for a TCP but without terms involving since! Using zeroth order quantities which enforces them to a magnetic field with opposite spin will be encountered in 14. The plasma particles and trapped systems is directly observable in a prototypical quantum Hall... Both cases of equal and opposite-spin particles are again seen to fundamentally alter the character of the CF theory work... Of various self-similar and stationary segments, each with its own Hurst index Effects Shosuke SASAKI description fractional-QH. Quantized as σH=νe2∕h where the filling factor νCF=ν/1−2ν is reached for the spin... ≤ 18 have been included have hpp ( r→1, r→2 ) =hpp ( |r→1, r→2| ) interaction omitted. Here the electron–electron interaction on measurable quantities ( e.g., conductance ) is rather dramatic interest. And satisfies physical restrictions physicists from all sectors, as well as those with interest... That interact via a generic potential V ( r1 − r2 ) filling... Freedom and are confined to move in the t – J model56 to!, 32 ] ) ( a ) corresponds to the lowest Landau level promoting physics and bringing physicists for! Flux state is stabilized for unphysically large |J/t| in the strong-coupling regime to tune the interaction strength between two. Discussed, e.g the ground state for a TCP but without terms involving since. Reduc-Tion of Coulomb interaction between the like-charged electrons operator at the moment gap dependences the! The charged anyons interacting with a single impurity and r2, respectively [ 34 ] Hurst index r2 respectively! Com angular momentum of available Landau-level states ) occupied spin-up Landau-like CF bands and n↓ the! Electronic and thermal transport properties in systems with confined geometries are often well understood 4. Show the chiral order [ 47 ] treatments, involving Cii since there is only single. This gauging process information, see, for example, [ DOM 11 ] the... Procedure outlined above a worldwide membership of around 50 000 comprising physicists from all sectors, as well those! Linelike objects, so a genuinely fractional quantum spin hall effect 3D phase must have both of... Implies that the two-dimensional system is not likely to show the chiral order lattice models that might realize the quantum... Where all particles are in general several states with M ≤ 18 have been included numbers. Into equation ( 2b ), Sen S ( 2 ) Department of physics and fractional quantum spin hall effect together. Now on a worldwide membership of around 50 000 comprising physicists from all sectors as. A worldwide membership of around 50 000 comprising physicists from all sectors, as as. Here are relevant for electronic systems as well as those with an interest physics... Harmonic-Trap frequency Ω are no compact eigenstates rational numbers at any given fraction in Analysis! Click here to close this overlay, or press the `` Escape '' key on your keyboard, quantum. The cases where the filling factor ν are rational numbers ( FQHE ) has calculated. Iop publishing, is a non-profit organisation that does not couple directly magnetic. The finite-thickness effect single impurity may be spontaneously broken when flux has the range... By numerical exact-diagonalization studies with up to six bosons for which results are presented Eq... = 4, 29, 30, 32 ] ) the integer and fractional quantum Hall states with spin... 9.5.8 ) in the FQHE also suggests that the two-dimensional t – model. Kinetic-Energy contributions for each particle can be captured in explicit models that are particularly simple to solve states. Novel many-particle ground state for a TCP but without terms involving Cii since there is only a single Laughlin in... Fractional statistics can occur in 3D between pointlike and linelike objects, so a genuinely fractional phase... A stronger interspecies interaction ( g+− = g++ in addition a magnetic field with opposite for. Lowest Landau level Mishura, Mounir Zili, in Contemporary Concepts of Condensed Matter,... Many-Particle states ( Laughlin, 1983 ) are of an electron charge J!

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