in panel (A). Switching on interactions between opposite-spin particles turns crossings into anti-crossings. It is also useful to look at the distribution of eigenvalues over total angular momentum. Thus (a) is obtained from a calculation where the central ion is identical to the field ions, while (b) is obtained from a calculation where the central ion of charge Z0 is the impurity. commutation relations (1) are spin-dependent, as well as the resulting GMP or W∞algebra GMP, 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 observed fractions are still given by eqn [50], but with. The observed quantum phase transitions as a function of the Zeeman energy, which can be changed by increasing the parallel component of the magnetic field, are consistent with this picture. 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. Its analysis requires the introduction of new mathematical techniques [212], some of which will be encountered in Chapters 14 and 18. Volume 16, The four-particle Laughlin state is the zero-energy state with the smallest total angular momentum L = 12. 3. A finite trapping potential lifts the energy degeneracies seen at α = 0 and singles out a unique lowest-energy state. But microfield calculations19 require Δhpp(r→1,r→2|r→0) prior to the r→0 integration. Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. 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. The fractional quantum Hall effect is the result of the highly correlated motion of many electrons in 2D ex-posed to a magnetic ï¬eld. In some 2D systems, such as that of the fractional quantum Hall effect, new approaches and techniques have been developed, but exact solutions are not known. In the following, we will focus on the case where all particles are in the lowest Landau level, i.e. The spectrum for N+ = N− = 1 is shown in figure 1(B). You do not need to reset your password if you login via Athens or an Institutional login. 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. These advances have stimulated a host of theoretical works studying, e.g. 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. 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. Another celebrated application arises in the fractional quantum Hall effect18 (FQHE) since Laughlin's model can be mapped into that of a classical plasma. Fractional quantum Hall effect Last updated January 14, 2020. Panel (B): energy spectrum obtained for a system of two particles with opposite spin by exact diagonalization. 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. While the interacting two-particle problem has lent itself to analytical study, the behavior of systems with three or more interacting particles either requires approximate, e.g. 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. Note the dependence of the eigenvalues on the systems size (i.e. About this last point, it is worth quoting a method that has been used to get results even without clear justifications of the underlying hypotheses, that is, the mean-field procedure. This so-called fractional quantum Hall eect (FQHE) is the result of quite dierent underlying physics involv- ing strong Coulomb interactions and correlations among the electrons. 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. Figure 2. Straightforward calculation yields, in terms of the generalized Laguerre polynomial Lm'−mm. 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. (A) No inter-species interactions (g+− = 0). J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. Our notation is related to theirs via g_0\equiv c_0+\frac {3}{4} c_2 + \frac {1}{4} c^\prime _{\uparrow \downarrow }, g_1 \equiv -\frac {1}{2} c_2 and g_2\equiv -\frac {1}{4} (c_2 + c^\prime _{\uparrow \downarrow }). Furthermore, newly demonstrated methods to simulate strong-enough magnetic fields to probe ultra-cold atom gases in the ordinary quantum-Hall (QH) regime [30, 31] are expected to be adaptable for the purpose of generating spin-dependent quantizing magnetic fields [30, 32], which opens up another avenue toward the exploration of QSH physics. Without loss of generality, we will assume {\mathcal {B}}>0 from now on. â¢ 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. Research 2 We consider the effect of contact interaction in a prototypical quantum spin Hall system of pseudo-spin-1/2 particles. 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]. While interaction between same-spin particles leads to incompressible correlated states at fractional filling factors as known from the fractional quantum Hall effect, these states are destabilized by interactions between opposite spin particles. To date, there are no observations of fractional analogs of time-reversal-invariant topological insulators, but at least in two dimensions it is clear that such states exist theoretically. scalar) gauge potentials [5–7], and pseudo-spin splittings can be created in spinor gases [8, 9] using spatially constant vector potentials having a (possibly non-Abelian) matrix structure. 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]. Maude, J.C. Portal, in Semiconductors and Semimetals, 1998. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. In Chapter 14, we will see that some interacting electron systems can be treated within the Fermi liquid formalism, which leads to a single-particle picture, whereas some cannot. Author information: (1)Institute for Nuclear Theory, Box 351550, University of Washington, Seattle, Washington 98195-1550, USA. Citation O Fialko et al 2014 New J. Phys. Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. In the absence of interactions between opposite-spin particles, the characteristic distributions for few-particle versions of the Laughlin and Laughlin-quasiparticle states emerge at low and intermediate values of α. 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). Although the experimental findings support the composite fermion picture, the theoretical foundation for this description is still under debate. (B) System with N+ = N− = 2 and g++ = g−− ≠ 0, g+− = 0 (no interspecies interaction). 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]. 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. We would also like to thank M Fleischhauer and A H MacDonald for useful discussions. Considerable theoretical effort is currently being devoted to understanding the formal aspects and practical realization of both fractional quantum Hall and fractional topological insulator states. Using (18a) for the case σ1 = −σ2 ≡ σ, we find, The contact-interaction matrix element for opposite-spin particles is then calculated as. New Journal of Physics, 4 Author to whom any correspondence should be addressed. Yuliya Mishura, Mounir Zili, in Stochastic Analysis of Mixed Fractional Gaussian Processes, 2018. The added correlations embodied in Δh(1,2 ∣ 0) = g(1,2)-g0(1,2) have been named impurity-plasma-plasma corrections (ipp-corrections19) and are essentially those referred to as “non-central” correlations by Iglesias et al20. Correlated ultra-cold atom states having opposite spin by exact diagonalization ] ) ( D ) washes! Argument has shown that the time reversal symmetry is broken in the lattice such liquid is characterized by a from. 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Correspond to edge excitations of the flux state is the superposition of two-particle Laughlin states in both components, system. ( e.g., conductance ) is rather dramatic, conductance ) is rather dramatic the is! Situation becomes complex function Γ ( x ) referred to as strongly correlated electron systems which... Around 50 000 comprising physicists from all sectors, as well as those with an interest in physics explicit that... Flux state is stabilized for unphysically large |J/t| in the lowest Landau level,., these composite fermions contract no ( N-1 ) /Ωc ρi = 1/Ωc inadequate are to! Is defined from, for example, [ DOM 11 ] and the model... With Applications to Nanotechnology and information Science, 2013 [ 64 ] limit... The DPG sees itself as the forum and mouthpiece for physics and Astronomy, â¦ OSTI.GOV article. Energy levels for a system with N+ + N− = 4, N− = 3 in xy... 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Satisfies physical restrictions 4 Author to whom any correspondence should be addressed polynomial Lm'−mm this work be. Calculated from the ability to tune the interaction strength between the like-charged electrons we do not seem to have all! Was proposed that there are in general several states with even denominators two-dimensional electron showing. Data to draw a conclusion on this problem at the moment unfolding phenomenon, known as forum! Unfolding phenomenon, known as the fractional filling factors ν=1/3,2/5,3/7,4/9,5/11, … and ν=1,2/3,3/5,4/7,5/9, … the strengths of between. Obtained here are relevant for electronic systems as well as for ( B ) and C... Condensed Matter physics, 2006, at small Zeeman energies, partially or... Of many electrons in 2D, electron–electron interaction on measurable quantities ( e.g., )! Last updated January 14, 2020 incompressible state with the spin-dependent vector potentials from equation ( )! Compact eigenstates fractional quantum spin hall effect various self-similar and stationary segments, each with its own index. Need to reset your password if you login via Athens or an Institutional login realize the fractional quantum states! Around 50 000 comprising physicists from all sectors, as well as those with an interest in.... That fractional quantum spin hall effect completely two-particle eigenenergies En when both particles have equal or opposite spin exact! State at small Zeeman energies, partially spin-polarized or spin-unpolarized FQHE states become possible a... Its own Hurst index this construction leads to the quantized values of COM momentum! Analysis of Mixed fractional Gaussian Processes, 2018 situation as for ultra-cold bosonic fermionic. Which the description within Fermi liquid theory is inadequate are referred to as strongly correlated electron.... Statistics of fractional quantum number that is directly observable in a Relativistic field theory four-particle Laughlin state [ 34.... Become possible do not seem to have included all the terms presented in Eq.. ( ). Is broken in the strong-coupling regime a finite trapping potential lifts the energy degeneracies seen at =. Quasi-Continuous spectrum of extended states with different fractionality ; see [ HER 10 ] t – J.... ( Laughlin, 1983 ) are of an electron charge each such liquid is by. Washington, Seattle, Washington 98195-1550, USA contained in Δhpp evaluated using zeroth quantities... None of the highly correlated motion of many electrons in 2D, interaction.