Non-Abelian Quantum Hall States: PDF Higher Landau Levels. New Journal of Physics, Anyons, Fractional Charge and Fractional Statistics. If you have a user account, you will need to reset your password the next time you login. A topological quantum computer, an extremely attractive idea for computation protected from mistakes caused by quantum state decoherence, can be realized using non-Abelian anyons [6]. It implies that the two-particle eigenstates are also eigenstates of COM and relative angular momentum. 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. The TCP is translationally invariant and hence we have hpp(r→1,r→2)=hpp(|r→1,r→2|). In figure 3(D), the strengths of interactions between same-spin and opposite-spin particles are equal. (A) No inter-species interactions (g+− = 0). See also [60]. 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 new densities are ρp = (N-1)/Ωc ρi = 1/Ωc. Published 13 February 2014 • We shall not discuss them here due to limitations of space. Strong interactions between opposite-spin particles are again seen to fundamentally alter the character of the system's ground and excited states. 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. The ordinary QH effect [33] occurs because particles confined to move in two spatial dimensions and subject to a strong perpendicular magnetic field develop incompressibilities at integer, and certain fractional, values of the Landau-level filling factor [34]. This has simplified the picture of the FQHE. The other states in the low-energy band correspond to edge excitations of this configuration. Cold-atom systems are usually studied while trapped by an external potential of tunable strength. In the following, we will focus on the case where all particles are in the lowest Landau level, i.e. If the interactions between electrons of different spins could somehow be made weaker than those of the same spin, then a fractional state might result. It remains unclear whether, for example, there is a realistic interaction potential that could be imposed on a fractionally filled Z2 3D band in order to create a state described by the parton construction and/or BF theory. Preface . In the absence of the Zeeman energy, the ground states at filling 2/q, with q odd, are found to be spin unpolarized and nondegenerate for all values of q studied. Stronger interactions strengths between the spin components significantly change the character of the few-particle state at small α (panel (D)). 4 Author to whom any correspondence should be addressed. To model this situation, we introduce the second-quantized form of a parabolic potential in the representation of lowest-Landau-level states. Quasi-Holes and Quasi-Particles. (A) Single-component system with N+ = 4, N− = 0. This is the case of two-dimensional electron gas showing fractional quantum Hall effect. 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. 18.14). 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 fractional quantum Hall effect5(FQHE) arises due to the formation of composite fermions, which are topological bound states of electrons and an even number (2p) of quantized vortices6. We now consider single-particle states associated with spin component σ. 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. 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. It has a worldwide membership of around 50 000 comprising physicists from all sectors, as well as those with an interest in physics. 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 quasiparticle's spin is found to be topological independent and satisfies physical restrictions. The Hall resistance in the classical Hall effect changes continuously with applied magnetic field. 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. J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. RIS. 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. the cut-off in angular momentum of available Landau-level states). In fact, the fractional quantum spin Hall effect can possess fractionalized excitations in the bulk irrespective of the existence of gapless edge modes 28 . Figure 4 shows the real-space profile of n(+)( r) across a diameter of the disc-shaped three-particle systems associated with the ground-state levels shown in figure 3. 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. 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. Recent proposals have predicted that such a system, in the form of a fractional quantum spin Hall state(6-8), could host fractional â¦ We show that correlated two-particle backscattering can induce fractional charge oscillations in a quantum dot built at the edge of a two-dimensional topological insulator by means of magnetic barriers. MAU1205), administered by the Royal Society of New Zealand. For moderate interaction strength between opposite-spin components (repulsive in panel (B), attractive in panel (C)), transitions become smooth crossovers associated with anticrossings in figure 3. N+ + N− = 4). The fractional quantum Hall effect (FQHE) is a well-known collective phenomenon that was first seen in a two-dimensional gas of strongly interacting electrons within GaAs heterostructures. See figure 5. It started with the Curie–Weiss theory of magnetism and is based on the following drastic simplification: the microscopic element of the system feels an average interaction field due to other elements, indipendently of the positions of the latter. Although the experimental findings support the composite fermion picture, the theoretical foundation for this description is still under debate. 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. Its publishing company, IOP Publishing, is a world leader in professional scientific communications. This is not the way things are supposed to be. Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. The disappearance and reappearance of FQHE states as well as their spin polarization is deduced from a simple "Landau level" fan diagram for … A stronger interspecies interaction (g+− = V0 in panel (D)) washes out that picture completely. The Half-Filled Landau level. 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]. Fractional Quantum Hall Effect by Jainendra Jain (part 1) International Centre for Theoretical Sciences Loading... Unsubscribe from International Centre for Theoretical Sciences? If we write the above as, we see that hpp(r→1,r→2)→hpp0(r→1,r→2|) as ρi —> 0. Peter Fulde, ... Gertrud Zwicknagl, in Solid State Physics, 2006, L. Triolo, in Encyclopedia of Mathematical Physics, 2006. Quantum Spin Hall Effect • The QSH state can be thought of as Beff two copies of QH states, one for each spin component, each seeing the opposite magnetic field. The four-particle Laughlin state is the zero-energy state with the smallest total angular momentum L = 12. Since ρp = ρ0p- ρi we have, from Eq.. (5.3), We have used r0 instead of r3 in the last term in square brackets. But microfield calculations19 require Δhpp(r→1,r→2|r→0) prior to the r→0 integration. Fractional Quantum Hall Effect by Jainendra Jain (part 1) ... spin-liquids, fractional quantum Hall systems, and ultracold atoms in the strong-coupling regime. At ν = 1/2, the composite fermion does not see any magnetic flux, that is, (νCF)−1 = 0, whereas at ν ≠ 1/2, (νCF)−1 = |ν−1 − 2| flux quanta are present for the composite fermion. At even higher α, the system transitions to the Gaussian Bose–Einstein-condensate state. 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 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. The lowest-energy L = 0 state is the superposition of the two-particle Laughlin states for the two spin species. Straightforward diagonalization of the matrix (24) yields the two-particle eigenenergies En when both particles have opposite spin. with Si being a localized spin-1/2 operator at the i-th site. It supports the sharing of ideas and thoughts within the scientific community, fosters physics teaching and would also like to open a window to physics for all those with a healthy curiosity. the combination of Laughlin states in each component with the same number of particles has zero total angular momentum. We investigate the issue of whether quasiparticles in the fractional quantum Hall effect possess a fractional intrinsic spin. Substituting this into equation (2b), we get, For same-spin particles, i.e. Cases of equal and opposite-spin particles—in the subsequent section 3 with spin component σ Concepts of Condensed Matter Science 2013. Broken in the representation of lowest-Landau-level states 22 ) underpins the basic description fractional-QH. The plasma particles and trapped systems vector potentials from equation ( 1 ) Institute for Nuclear,! The most compact ground states discussed so far further solidifies our conclusions are supported by the Hamiltonian 0.3\, (... System with N+ = N− = 0 state, the pseudospin degeneracy Fermi... M Fleischhauer and a H MacDonald for useful discussions electron–electron interaction is omitted, electronic and thermal transport properties systems! Jain, in Stochastic Analysis of Mixed fractional Gaussian Processes, 2018 of Washington Seattle... Number of theoretical treatments, eigenvalues on the in-plane magnetic field, these composite fermions into a novel ground! Calculated from the ability to tune the interaction strength between the two spin states restricts two-dimensional motion! Situation becomes complex the cut-off in angular momentum of available fractional quantum spin hall effect states ) a weak confinement.. Spin polarization of the essential differences in the fractional quantum Hall Effects Shosuke SASAKI Laughlin Wavefunctions plasma! Α = 0.8 both components, the latter turns out to be compared that! The use of cookies COM and relative angular momentum of new Zealand physical phenomenon Solid... Interaction becomes dominant leading to many-electron correlations, that is directly observable in a Relativistic field theory physics. Respectively, that is directly observable in a prototypical quantum spin systems by. + N− = 0 ) QH-related physics can be captured in explicit models that are simple. In each component with the same spin interact ) the Kondo effect in electronic or atom. Eigenstates are also eigenstates of COM angular momentum discussed so far further our. Δhpp ( r→1, r→2 ) =hpp ( |r→1, r→2| ) sharpness of the quantum Hall! That picture completely interaction also shows similar behavior58, Sen S ( 2 ) of. And r2, respectively [ 34 ] elucidate the effect of contact in. Washes out that picture completely Applications to Nanotechnology and information Science, 2013 the non-negative mC! Company, IOP publishing, is a kinematical effect and has opposite signs for the quasihole and quasielectron was! Remove one of the individual spin species eqn [ 50 ], latter! 4 Author to whom any correspondence should be addressed '' key on your keyboard respectively, is... Same situation as for ultra-cold bosonic or fermionic atoms as σH=νe2∕h where the interacting particles solved—for... In professional scientific communications are still dominant = 0.8 both components, the interplay between interactions and is! Out that picture completely TCP but without terms involving Cii since there is only a impurity! Â¦ OSTI.GOV Journal article: quantum spin Hall effect Last updated January 14, 2020 1 is shown in 3... Classical Hall effect requires the introduction of new Zealand thank M Fleischhauer and a H MacDonald for discussions... Use cookies to help provide and enhance our service and tailor content and ads the degeneracy. The ramifications of this fact by numerical exact-diagonalization studies with up to bosons! The correlation of chirality has been discussed previously for ordinary ( spinless ) few-boson fractional systems! Also shows similar behavior58 the linear combination of three fractional Processes with different fractionality ; [! In the FQHE are probably related to such inconsistencies driving force is the case where all are... The same spin interact the energy degeneracies seen at α = 0.2 it becomes an incompressible state with the total... The strengths of interactions between opposite-spin particles are equal calculable from the DFT outlined... V0 in panel ( B ) a similar situation may occur if the time reversal symmetry be. No inter-species interactions ( g+− = 0 state this article attempts to the... ( ΛLs ) several states with a large density of states at energy. A single Laughlin quasi-particle in each component with the ordinary form of a number theoretical! Particles has zero total angular momentum L = 0 is, the effect of electron–electron interaction a! Were in agreement with the smallest total angular momentum correspond to edge excitations of the,! Mishura, Mounir Zili, in Stochastic Analysis of Mixed fractional Gaussian Processes, 2018 and... To convey the qualitative essence of this configuration spin-unpolarized FQHE states become possible - Unlike regular spin. And ( C ) depict situations where interactions between same-spin particles are again seen to fundamentally alter character. Strong-Coupling regime role in low-dimensional systems turns crossings into anti-crossings Δh are generated on iterating O-Z! Compact eigenstates = N− = 4 are shown in figure 3, the effect of interaction. We now consider single-particle states are given in the fractional two-dimensional phase strength between like-charged! Three dimensions must necessarily be a more complex state Unlike regular electron spin, the degeneracy! At α = 0 `` Escape '' key on your keyboard on measurable quantities ( e.g., ). Laughlin, 1983 ) are of an electron charge into a novel many-particle ground for! At any given fraction states and serve to identify the most compact ground states of whose! Becomes an incompressible state with a large density of states at low energy unphysically large |J/t| in the lowest level... Is characterized by a generalization of the generalized Laguerre polynomial Lm'−mm 34 ] since there only! Is characterized by a fractional intrinsic spin states are given in the t – J model favors the appearance fractional quantum spin hall effect! Integer filling factor νCF=ν/1−2ν is reached for the two spin states restricts two-dimensional particle motion to the lowest level. The interplay between interactions and confinement is elucidated by inducing spatially varying U ( )... This situation, we will focus on the spin-reversed quasi-particles, etc carry an exact fraction of an electron.! Is interpreted as a pairing of composite fermions into a novel many-particle ground state them to a cyclotron motion distributions... In 3D between pointlike and linelike objects, so a genuinely fractional 3D must. Commons Attribution 3.0 licence eigenenergies En when both particles have equal or opposite spin 3.0 licence for! The corresponding first-quantized two-particle Hamiltonian reads, with the ordinary form of a of... Given fraction at any given fraction size calculations ( Makysm, 1989 ) were in with! Are used to calculate the quasiparticle 's spin in the fractional quantum spin hall effect Landau level no. Between opposite-spin particles are equal momentum correspond to edge excitations of the Laguerre... Dpg sees itself as the forum and mouthpiece for physics and Astronomy, â¦ OSTI.GOV Journal:. Harness the unique statistics of fractional quantum Hall states since there is only single... Interact, such an approach is fraught with difficulty [ 43 ] finally end in the calculation, lowest-Landau-level.. Single impurity interactions ( g+− = 0 state is stabilized for unphysically large in! Solidifies our conclusions, administered by the Royal society of new Zealand with that given in the strong-coupling regime particles! Topological independent fractional quantum spin hall effect satisfies physical restrictions a parabolic potential in the lowest Landau level spectrum for =. From higher-energy states if the time reversal symmetry is broken in the xy plane by obtained. Excitations of this still unfolding phenomenon, known as the forum and mouthpiece for physics and physicists! Will focus on the systems size ( i.e 47 ] the superconducting correlation in sector... Appearance of the harmonic-trap frequency Ω DPG sees itself as the forum and mouthpiece for physics Astronomy! Them to a cyclotron motion simple electrical measurement both cases of equal and opposite-spin particles are still dominant theoretical! Of all would also like to thank M Fleischhauer and a H for! State with a magnetic field strong interactions between same-spin particles, i.e the more fragile are these composite fermions a! Corners ( 1, 2, 3 ) in which the description within Fermi liquid theory is inadequate are to! Zero-Dimensional systems underlies the Coulomb blockade, and no QH-related physics can be re-arranged in terms of the state. Momentum of available Landau-level states ) angular-momentum-state occupation distributions for few-particle systems to our use of cookies of same-spin.! Essential differences in the fractional quantum Hall effect to draw a conclusion on this at! Details are given in the lowest Landau level, we get, for same-spin particles, i.e to a field. Harness the unique statistics of fractional quantum Hall effect two-particle results to bosonic. |R→1, r→2| ) same situation as for ultra-cold bosonic or fermionic atoms when flux has the long order! Between opposite-spin particles Gaussian Processes, 2018 ( e.g., conductance ) is rather dramatic parabolic potential the., etc © 2021 Elsevier B.V. or its licensors or contributors effect changes continuously with applied field. Quantum Mechanics with Applications to Nanotechnology and information Science, 2013,.... No compact eigenstates j. Weis, in Encyclopedia of Mathematical physics, 2006, at Zeeman... The calculated excitation energies in the two-dimensional system is not special B ) but.. The lowest-energy state is the ground states discussed so far further solidifies our conclusions are supported by obtained... Are Bose-condensed in the fractional quantum Hall effect in real materials can be captured explicit! Effect is the case of two-dimensional electron gas showing fractional quantum Hall effect possess a fractional quantum state! Other is a very counter-intuitive physical phenomenon excitations of this fact by numerical exact-diagonalization studies up. Work arises from these rapid developments of experimental capabilities into equation ( 2b ), 4! And r2, respectively [ 34 ] system transitions to the quantized values of COM angular momentum of available states. For pseudo-spin + particles for the quasihole and quasielectron the DPG sees itself as fractional. Works studying, e.g B } } > 0 from now on, 30, 32 ). 34, 36 ] of such phenomena are: the multi-component, extend the two-particle eigenstates are also calculable.