The spectrum seen there has to be compared with that given in panel (B) where only particles with the same spin interact. Fractionally charged skyrmions, which … The variational argument has shown that the antiferromagnetic exchange coupling J in the t – J model favors the appearance of the flux state. For more information, see, for example, [DOM 11] and the references therein. Higher mr is observed to correlate with lower energy, but there are many states even lower in energy than the trial state with largest mr that is compatible with the finite systems size. This site uses cookies. Peter Fulde, ... Gertrud Zwicknagl, in Solid State Physics, 2006, L. Triolo, in Encyclopedia of Mathematical Physics, 2006. 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. February 2014 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. Following this line of thought, some previous discussions of a putative fractional QSH physics [38, 42] have been based on an ad hoc adaptation of trial wave functions first proposed in [22]. It was realized early on that the small electronic g-factor in the GaAs/AlGaAs system further complicated the problem because the small Zeeman energy favors spin-unpolarized (or spin-reversed) fractional states at filling factors of v < 1 for which full polarization is otherwise expected (Halperin, 1983). To reveal the associated degeneracies of the spectrum shown in figure 2(B), we obtained the energy eigenvalues in the presence of a parabolic confinement. The flux in the unit square is similarly defined by, The flux state is defined from the long range order as < p123 > ≠ 0 or < P1234 > ≠ 0. 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. It works to advance physics research, application and education; and engages with policy makers and the public to develop awareness and understanding of physics. 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). Furthermore, with the aim of predicting the sequence of magic proton and neutron numbers accurately, physicists have constructed a higher-dimensional representation of a fractional rotation group with mixed derivative types. σ' = σ, we obtain, In contrast, for the matrix element involving opposite-spin particles (σ = −σ'), we find. Fractional quantum Hall effect Last updated January 14, 2020. The fractional quantum Hall effect is also understood as an integer quantum Hall effect, although not of electrons but of charge-flux composites known as composite fermions. Note the dependence of the eigenvalues on the systems size (i.e. (2)Department of Physics and Astronomy, … in panel (A). Cold-atom systems are usually studied while trapped by an external potential of tunable strength. Operators for the guiding-center locations can then be defined in the usual manner [34], {\bf{ R}}^{(\sigma )}= {\bf{ r}}- \sigma \,l_{\mathcal B}^2\, [{\hat {\bf{ z}}}\times {\boldsymbol {\pi }}^{(\sigma )}]/\hbar, and their components satisfy the commutation relations, Moreover, we find [R(σ)α,π(σ')α'] = 0. J.K. Jain, in Encyclopedia of Mathematical Physics, 2006, At small Zeeman energies, partially spin-polarized or spin-unpolarized FQHE states become possible. 1. Rigorous examination of the interacting two-particle system in the opposite-spin configuration (see below) shows that energy eigenstates are not eigenstates of COM angular momentum or relative angular momentum and, furthermore, have an unusual distribution. The fractional quantum Hall effect is the result of the highly correlated motion of many electrons in 2D ex-posed to a magnetic field. Any systematic difference between the results given in figures 1(A) and (B) is probably at least in part due to the fact that the representation using the COM and relative angular-momentum basis assumes an infinite number of single-particle angular-momentum modes to be available to the particles. 18.15.3 linked to the book web page), (4) the Kondo model (see Sec. The new densities are ρp = (N-1)/Ωc ρi = 1/Ωc. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles , and excitations have a fractional elementary charge and possibly also fractional statistics. Thus any feasible route toward realizing the fractional QSH effect using a spin-dependent uniform magnetic field [29, 32] should strive to eliminate interactions between the opposite-spin components. Finally, electron–electron interaction in zero-dimensional systems underlies the Coulomb blockade, spin blockade, and the Kondo effect in quantum dots. in [39–41], the total angular momenta for states from different components have opposite sign. Stronger interactions strengths between the spin components significantly change the character of the few-particle state at small α (panel (D)). The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. 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. A finite trapping potential lifts the energy degeneracies seen at α = 0 and singles out a unique lowest-energy state. It has been observed recently in some ceramic materials well above 100 K, and a clear model which takes into account the formation of pairs and the peculiar isotropy–anisotropy aspects of the normal conductivity and superconductivity is still lacking (Mattis 2003). Quantum Spin Hall Effect. Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. In the conceptually simplest realization of the QSH effect [22], particles exhibit an integer QH effect due to a spin-dependent perpendicular magnetic field that points in opposite directions for the two opposite-spin components. 9.5.8) in which the Hall conductance is quantized as σH=νe2∕h where the filling factor ν are rational numbers. Part of the motivation for our present theoretical work arises from these rapid developments of experimental capabilities. Low-lying energy levels for a system with N+ = N− = 3 in the sector of total angular momentum L = 0. The corrections to leading order in ρi to h0pP are hence contained in Δhpp evaluated using zeroth order quantities. 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. 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. These include: (1) the Heisenberg spin 1/2 chain, (2) the 1D Bose gas with delta-function interaction, (3) the 1D Hubbard model (see Sec. This is markedly different from the case of same-spin particles. However, we do not have sufficient data to draw a conclusion on this problem at the moment. Panel (A) corresponds to the case with g+− = 0. In the case where g+− = 0, the system reduces to two independent two-dimensional (electron or atom) gases that are each subject to a perpendicular magnetic field. The flux order parameter is defined from, for the elementary triangle with corners (1, 2, 3) in the lattice. However, gii(r) of the inhomogeneous plasma is really a three-particle problem, viz, g(r→1,r→2|0) since the ion-ion correlations are needed in the presence of the impurity (usually the “radiator” in plasma spectroscopy) held at the origin. Copyright © 2021 Elsevier B.V. or its licensors or contributors. However, there are several challenges in making this state an experimental reality: if one imagines the state in semiclassical terms, then spin-up and spin-down electrons are circling in opposite directions, and the most logical effect of Coulomb interactions is to form a Wigner crystal (an incompressible quantum solid rather than an incompressible quantum liquid). Fractional Quantum Hall Effect in a Relativistic Field Theory We construct a class of 2+1 dimensional relativistic quantum field theories which exhibit the fractional quantum Hall effect in the infrared, both in the continuum and on the lattice. In the calculation, lowest-Landau-level states with m ≤ 18 have been included. The idea of retaining the product form with a modified g(1,2) has also been examined21 in the context of triplet correlations in homogeneous plasmas but the present problem is in a sense simpler. 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. When interactions among same-spin and between opposite-spin particles have equal magnitude, the one-particle momentum distribution of the ground state differs markedly from that associated with a fractional-QH state. The resulting many-particle states (Laughlin, 1983) are of an inherently quantum-mechanical nature. The origin of the density of states is the interactions between electrons, the so-called many-body effects, for which quantitative theory is both complicated and computationally extremely time consuming. The fractional quantum Hall effect (FQHE) has been the subject of a number of theoretical treatments , . The correlation of chirality has been calculated in various choices of lattices in the quantum spin systems defined by the Hamiltonian. This case is illustrated in figure 2(B). A fractional phase in three dimensions must necessarily be a more complex state. We consider a gas of particles (e.g. We now consider single-particle states associated with spin component σ. The Fractional Quantum Hall Effect: PDF Laughlin Wavefunctions, Plasma Analogy, Toy Hamiltonians. Nevertheless, when the energy eigenvalues obtained for the finite system are plotted alongside the results for the analytical model (see magenta data points in figure 1(A)), both are seen to exhibit the same exponential behavior. There are in general several states with different spin polarizations possible at any given fraction. We focus here on the case of bosonic particles to be directly applicable to currently studied ultra-cold atom systems, but our general conclusions apply to systems of fermionic particles as well. 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. Considerable theoretical effort is currently going into lattice models that might realize the fractional two-dimensional phase. Since ρp = ρ0p- ρi we have, from Eq.. (5.3), We have used r0 instead of r3 in the last term in square brackets. 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]. The other is a kinematical effect and has opposite signs for the quasihole and quasielectron. The total spin thus agrees with a generalized spin-statistics theorem $(S_{qh} + S_{qe})/2 = \theta/2\pi$. 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. This situation of opposite-spin particles being subjected to oppositely directed magnetic fields corresponds directly to setups considered for a semiconductor heterostructure [22, 54] and in neutral-atom systems [27–29, 32]. The notation used in equations (3b)–(3d) can be related to that which is often adopted in the atom-gas literature [58, 59] by setting g0 ≡ c0, g2 ≡ c2, and g1 ≡ 0. It implies that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any indi-vidual electron. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. With varying magnetic field, these composite fermions survive and they now feel an effective magnetic field which enforces them to a cyclotron motion. See the following subsection for details.). MAU1205), administered by the Royal Society of New Zealand. 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. Spin transition of a two-dimensional hole system in the fractional quantum Hall effect K. Muraki and Y. Hirayama NTT Basic Research Laboratories, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan ~Received 8 June The one-particle density profiles in coordinate space and in angular-momentum space are useful quantities to enable greater understanding of the properties of specific many-body quantum states [65, 66]. The simplest approach22 to the present problem is to consider a two-component plasma (TCP) where one of the components (impurity) has a vanishingly small concentration. 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. If there are N particles in the correlation sphere of volume Ωc then quantities of the order of 1/N have to be retained since the impurity density is also of the order of 1/N. 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. In 1988, it was proposed that there was quantum Hall. The spin-1/2 antiferromagnetic system is the relevant model in the half-filled band. 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). The spin polarization of fractional states was measured experimentally by varying the Zeeman energy by rotating the magnetic field away from the normal (Clarke et al., 1989; Eisenstein et al., 1989) or by applying hydrostatic pressure (Morawicz et al., 1993). While the Landau quantization of single-particle energies is the origin of the integer QH effect, incompressibility at fractional filling factors is caused by the discrete spectrum of interaction energies for two particles occupying states from the same Landau level [35–37]. Yehuda B. The braid relations are used to calculate the quasiparticle's spin in the fractional quantum Hall states on Riemann surfaces. The various published calculations for the FQHE do not seem to have included all the terms presented in Eq.. (5.6). The starting point of such an analysis is the Fourier decomposition of a spin-dependent interaction potential given by, because its matrix elements can then be directly related to the corresponding matrix elements of the exponential in the integrand of (13). when n+ = n− = 0. The results obtained here are relevant for electronic systems as well as for ultra-cold bosonic or fermionic atoms. (C) Same situation as for (B) but with a finite trapping potential (α = 0.02) switched on in addition, revealing the energy degeneracies in (B). It reports on theoretical calculations making detailed quantitative predictions for two sets of phenomena, namely spin polarization transitions and the phase diagram of the crystal. We explore the ramifications of this fact by numerical exact-diagonalization studies with up to six bosons for which results are presented in section 4. In this article, we give the interpretation of the data on quantum Hall effect and describe some new spin properties which lead to fractional charge. Figure 1(A) shows a logarithmic plot of the En, ordered by decreasing magnitude, for different values mmax of the cut-off value for COM and relative angular momentum. Before presenting a formal analysis of the interacting two-particle system subject to a strong spin-dependent magnetic field in the following subsection, we provide a heuristic argument for how the cases where the two particles feel the same and opposite magnetic fields differ. AB - Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. The correlation of χij -χji seems to remain short-ranged59. Note the \mathcal {M}-dependence of the obtained values. (D) Same situation as for (B) but with finite interspecies interaction g+− = g++ in addition. But microfield calculations19 require Δhpp(r→1,r→2|r→0) prior to the r→0 integration. It is straightforward to show that the sum of kinetic-energy contributions for each particle can be re-arranged in terms of the linear combinations. in terms of the Euler Gamma function Γ(x). 4. 3. Fractional quantum Hall effect: Experimental progress and quantum computing applications ( Nanowerk News ) The Hall effect, discovered in 1879, is observable when a Hall voltage perpendicular to the current is produced across a conductor under a magnetic field. 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 uniform flux P+ and the staggered flux P– defined from, have relationship to the chirality order C± in the half-filled band as, On the square lattice, the uniform and staggered flux of the plaquette is defined as. In that case, only the relative-coordinate degree of freedom feels the interaction potential V ( rσσ), and it can be minimized by placing two particles away from each other. 9.5.8. In 2D, electron–electron interaction is responsible for the, Journal of Mathematical Analysis and Applications, Physica A: Statistical Mechanics and its Applications, Theory of Approximate Functional Equations, angle resolved photoemission spectroscopy. Straightforward calculation yields, in terms of the generalized Laguerre polynomial Lm'−mm. Note, however, the different parameterization used in [8] where c0,2 are interaction constants associated with the atomic spin-1 degree of freedom from which the pseudo-spin-1/2 components are derived. Note the disappearance of energy gaps and accumulation of states at low energy, reflecting the characteristic features of the opposite-spin two-particle interaction spectrum shown in figure 1(B). 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]. dimensions. 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] . We would also like to thank M Fleischhauer and A H MacDonald for useful discussions. 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