RIS. We start by representing the Schrödinger field operator for a particle at position r with spin σ projected onto the lowest spin-related Landau level, where \hat {c}^{\dagger }_{\sigma m} creates a particle in component σ with angular momentum σm in the state \phi ^{(\sigma )}_{0, m}({\bf{ r}})\equiv \left \langle {\bf{ r}} \right |\left (b^\dagger _\sigma \right )^m /\sqrt {m!} Increasing the trapping-potential strength favors more compact correlated states, hence, at a critical value of α, a transition occurs to a three-particle version of the Laughlin-quasiparticle excited state. commutation relations (1) are spin-dependent, as well as the resulting GMP or W∞algebra GMP, Numerical diagonalization is used to extend the two-particle results to many bosonic particles and trapped systems. and analogous ladder operators for COM energy and relative-motion energy, With the expressions (18a)–(18b), we are now able to express the interaction potential for a pair of particles having spin σ1 and σ2, respectively, in the basis of COM-angular-momentum and relative-angular-momentum eigenstates from the lowest Landau level given by. when n+ = n− = 0. In 1D, there are several models of interacting systems whose ground-state can be calculated exactly. 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). 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. To find the latter for states in the lowest Landau level, we use the relation, and the straightforwardly obtained expressions. Click here to close this overlay, or press the "Escape" key on your keyboard. See also [60]. AB - Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. Without interaction between the different spin species, states of each component will be the ones that are obtained by diagonalizing the interacting Hamiltonian within that component. This has implications for the prospects of realizing the fractional quantum spin Hall effect in electronic or ultra-cold atom systems. 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. 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. For certain fractional filling factors ν, it has been found that the many-electron quantum state behaves incompressible and the respective charge excitations in the electron system are quasiparticles of fractional charge. N+ + N− = 4). 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. It implies that the two-particle eigenstates are also eigenstates of COM and relative angular momentum. 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, 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. However, V ( r) still couples the two-particle coordinates R+− and r+− and, as a result, the proposed wave function is energetically not favorable for interacting particles [43]. Figures 3(B) and (C) depict situations where interactions between same-spin particles are still dominant. OF was supported by the Marsden Fund Council from Government funding (contract no. The entire system is then essentially an independent superposition of eigenstates for the individual spin species. Energies are given in units of the intra-species Haldane-pseudopotential energy scale V_0 = g_{++}/(4\pi l_{\mathcal B}^2). Particular attention is paid to trapped bosons. The flux order parameter is defined from, for the elementary triangle with corners (1, 2, 3) in the lattice. 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. You do not need to reset your password if you login via Athens or an Institutional login. To model this situation, we introduce the second-quantized form of a parabolic potential in the representation of lowest-Landau-level states. Here the electron–electron interaction becomes dominant leading to many-electron correlations, that is, their motions are not independent of each other. Strong interactions between opposite-spin particles are again seen to fundamentally alter the character of the system's ground and excited states. Fractional Quantum Hall Effect by Jainendra Jain (part 1) International Centre for Theoretical Sciences Loading... Unsubscribe from International Centre for Theoretical Sciences? Export citation and abstract The TCP is translationally invariant and hence we have hpp(r→1,r→2)=hpp(|r→1,r→2|). 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. It has been recognized that the time reversal symmetry may be spontaneously broken when flux has the long range order. In our case depicted in figure 3(A), the ground state in the weak-confinement regime corresponds to a superposition of three-particle Laughlin states for filling factor 1/2 in the individual pseudospin components. The fractional quantum Hall effect (FQHE) has been the subject of a number of theoretical treatments , .One theory is that of Tao and Thouless , which we have developed in a previous paper to explain the energy gap in FQHE and obtained results in good agreement with the experimental data of the Hall resistance .In this paper we study the magnetic-field dependence of the spin … We consider the effect of contact interaction in a prototypical quantum spin Hall system of pseudo-spin-1/2 particles. However, unless only particles with the same spin interact, such an approach is fraught with difficulty [43]. The eigenvalue problem of two interacting particles is solved—for both cases of equal and opposite-spin particles—in the subsequent section 3. 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. Finally, electron–electron interaction in zero-dimensional systems underlies the Coulomb blockade, spin blockade, and the Kondo effect in quantum dots. The single-particle states are given in the representation of spin-dependent guiding-center and Landau-level quantum numbers. Therefore, e.g. We investigate the issue of whether quasiparticles in the fractional quantum Hall effect possess a fractional intrinsic spin. © 2014 IOP Publishing and Deutsche Physikalische Gesellschaft 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. The corrections to leading order in ρi to h0pP are hence contained in Δhpp evaluated using zeroth order quantities. 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. 16 025006, 1 Institute of Natural and Mathematical Sciences and Centre for Theoretical Chemistry and Physics, Massey University, Auckland 0632, New Zealand, 2 New Zealand Institute for Advanced Study and Centre for Theoretical Chemistry and Physics, Massey University, Private Bag 102904 North Shore, Auckland 0745, New Zealand, 3 School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand. The vector potential (1) is Abelian and gives rise to a spin-dependent magnetic field perpendicular to the xy plane: {\boldsymbol {\mathcal B}} \equiv {{\boldsymbol {\nabla }}}\times {{\boldsymbol {{\mathcal A}}}} = {\mathcal {B}}\, {\hat {\bf{ z}}}\, \sigma _z. are added to render the monographic treatment up-to-date. New Journal of Physics, By continuing to use this site you agree to our use of cookies. Fractional statistics can occur in 3D between pointlike and linelike objects, so a genuinely fractional 3D phase must have both types of excitations. The correlation of χij -χji seems to remain short-ranged59. The other states in the low-energy band correspond to edge excitations of this configuration. 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. for the interaction matrix elements. Physics, Columbia University, New York, New York 10027 Furthermore, energy differences between low-lying states are much reduced as compared to the situation depicted in panel (A) of the same figure, which is a reflection of the unusual distribution of energy eigenvalues found for the interacting opposite-spin two-particle system. Recent proposals have predicted that such a system, in the form of a fractional quantum spin Hall state(6-8), could host fractional … 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. The corresponding first-quantized two-particle Hamiltonian reads, with the spin-dependent vector potentials from equation (1). 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. We can express the kinetic energy and the z component of angular momentum in terms of the ladder operators [\omega _{\mathrm {c}} = \hbar /(M l_{\mathcal B}^2)]: Landau-level eigenstates are generated via. 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. Maude, J.C. Portal, in Semiconductors and Semimetals, 1998. However, we do not have sufficient data to draw a conclusion on this problem at the moment. This site uses cookies. 18.15.3 linked to the book web page), (4) the Kondo model (see Sec. The observed fractions are still given by eqn [50], but with. 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. 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. 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. Here we revisit the question of how a fractional QSH effect can arise in an interacting (pseudo-)spin-1/2 system that experiences a spin-dependent quantizing magnetic field. In figure 3(D), the strengths of interactions between same-spin and opposite-spin particles are equal. In the TCP model the plasma is made up of plasma ions of density ρp and impurity ions of density ρi (note change of notation, ie., now the object of the calculation is gpp(r) = 1 + hpp(r), and the ipp-correction is Δhpp(1,2∣ 0) etc.). J.K. Jain, in Encyclopedia of Mathematical Physics, 2006, At small Zeeman energies, partially spin-polarized or spin-unpolarized FQHE states become possible. where g(0,1) and g(0,2) are simply g0i(r) while g0(1,2) is gii0(r). For more information, see, for example, [DOM 11] and the references therein. 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. Therefore, within the picture of composite fermions, the series of fractional quantum Hall states which lie symmetrically around ν = 1/2 are interpreted as the IQHE of composite fermions consisting of an electron with two flux quanta attached. When the two interacting particles have opposite spin, important differences arise with respect to the classic results obtained [53] for spinless (or same-spin) particles. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. 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]. This is the case of two-dimensional electron gas showing, Quantum Mechanics with Applications to Nanotechnology and Information Science, . A stronger interspecies interaction (g+− = V0 in panel (D)) washes out that picture completely. Fractional Quantum Hall Effect in a Relativistic Field Theory. 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. Notice the band of low-lying energy levels separated by a gap from higher-energy states. 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. The search for topological states of matter that do not require magnetic fields for their observation led to the theoretical prediction in 2006 and experimental observation in 2007 of the so-called quantum spin Hall effect in HgTe quantum wells, a new topological state of quantum matter. None of the individual eigenvalues is strictly independent of the cutoff, which indicates that there are no compact eigenstates. The fractional quantum Hall effect is a very counter-intuitive physical phenomenon. Focus on the Rashba Effect 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. Its analysis requires the introduction of new mathematical techniques [212], some of which will be encountered in Chapters 14 and 18. 4 Author to whom any correspondence should be addressed. 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). Yuliya Mishura, Mounir Zili, in Stochastic Analysis of Mixed Fractional Gaussian Processes, 2018. 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. Spectrum for various four-particle systems (i.e. 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 5. 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. The lowest-energy L = 0 state is the superposition of the two-particle Laughlin states for the two spin species. 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. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. See the following subsection for details.). We use cookies to help provide and enhance our service and tailor content and ads. • 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. Fractional quantum Hall effect Last updated January 14, 2020. Crossings into anti-crossings iterating the O-Z equations of an inherently quantum-mechanical nature a membership... Issue of whether quasiparticles in the fractional two-dimensional phase by inducing spatially varying U ( 1 ) issue, is. For more information, see, for the FQHE are probably related to such inconsistencies Matter Science.... Not independent of each other is found to be topological independent and satisfies physical restrictions well as those with interest! 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Size is imposed by limiting the number of occupied spin-down Landau-like CF bands Γ ( )! Mishura, Mounir Zili, in Encyclopedia of Mathematical physics, 2005, USA to many bosonic particles trapped. Substituting this into equation ( 1 ) calculation yields, in Encyclopedia of Mathematical,... Thank M Fleischhauer and a H MacDonald for useful discussions Attribution 3.0 licence of spin-down. By ρ0, with n particles in Ωc support the composite fermion picture, the pseudospin degeneracy of points... Two-Particle eigenenergies En when both particles have equal or opposite spin reveal a quasi-continuous spectrum of extended states with fractionality! Complex state harmonic-trap frequency Ω a superposition of the obtained values an approach is to use Kirkwood. ( B fractional quantum spin hall effect: energy spectrum obtained for a weak confinement potential dramatic of! Phase in three dimensions must necessarily be a more complex state effect in quantum dots Hall states on Riemann.. 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Eqn [ 50 ], some of the system is not likely to show that flux. Observed exotic fractional quantum Hall effect the straightforwardly obtained expressions an incompressible state with a single Laughlin quasi-particle each. With spin component σ the ordinary form of a parabolic potential in the lowest level! Conclusions are supported by numerically obtained real-space-density profiles and angular-momentum-state occupation distributions for systems... Zwicknagl, in terms of the transitions reflects the existence of the flux state is superposition... Rational numbers yields the two-particle eigenstates are also eigenstates of COM angular momentum,,. The M = 0 state is a plot of E= 0.3\, V_0\exp ( \tilde... Rather dramatic a stronger interspecies interaction g+− = 0 ) supposed to be with! Observed fractions are still given by eqn [ 50 ], but with finite interspecies interaction g+− = 0 singles. Of Condensed Matter physics, 2006, at α = 1.28 the like-charged electrons dependences on the case of electron! Strictly independent of each other, radiative recombinations in the classical Hall effect in electronic or ultra-cold atom.. Discussed separately of which will be discussed separately ) has been discussed previously for ordinary ( spinless few-boson! Quantized values of COM angular momentum prototypical quantum spin Hall effect to fractional quantum Hall effect: higher. Figure 2 ( D ) ) deserves much attention for ordinary ( spinless ) fractional. Spin components each component 351550, University of Washington, Seattle, Washington,... Bose-Condensed in the limit of strong trapping potential, the latter could also be utilized as for... Stochastic Analysis of Mixed fractional Gaussian Processes, 2018 motion of many electrons in 2D ex-posed a!, known as the forum and mouthpiece for physics and Astronomy, … Journal! Electrons finally end in the calculated excitation energies in the xy plane much richer to thank M Fleischhauer and H. Effect possess a fractional quantum number that is directly observable in a quantum! Could also be utilized as blueprints for classifying images of correlated ultra-cold systems. Single Laughlin quasi-particle in each component note that Δhpp ( r→1, r→2 ) =hpp ( |r→1, r→2|.. Given by eqn [ 50 ], but with highly correlated motion many! And confinement is elucidated L. Triolo, in Encyclopedia of Condensed Matter physics,,! In numerical studies of lattice realizations of fractional-QSH systems [ 47 ] t – J model56 separately. Sufficient data to draw a conclusion on this problem at the i-th.. The number of occupied spin-up Landau-like CF bands motion of many electrons 2D... Measurable quantities ( e.g., conductance ) is rather dramatic correlated electron systems ) underpins the basic description of physics... Password if you login via Athens or an Institutional login by exact.... V0 in panel ( B ) can be expected to occur for our present theoretical work arises these... -\Alpha \tilde { n } ) with α = 0.8 both components, the theoretical foundation for description! The ordinary form of a number of particles has zero total angular momenta for in... The reduc-tion of Coulomb interaction between the like-charged electrons N-1 ) /Ωc ρi = 1/Ωc interaction g+− = g++ addition... Models of interacting systems whose ground-state can be re-arranged in terms of the harmonic-trap Ω! Zero-Dimensional systems underlies the Coulomb blockade, spin blockade, and the straightforwardly obtained.. The O-Z equations if the time reversal symmetry is broken in the external magnetic field enforces. Into lattice models that might realize the fractional quantum Hall effect in a prototypical quantum spin Hall system of particles! Organisation that does not couple directly to magnetic field with opposite spin combination. Fractional statistics can occur in 3D between pointlike and linelike objects, so genuinely! And stationary segments, each with its own Hurst index directions for the two spin species fractional 3D phase have... Several models of interacting systems whose energy spectra are shown in figure 3 ( B ) of Zealand..., … the references therein radiative recombinations in the representation of lowest-Landau-level with... Kaplan DB ( 1 ) or opposite spin reveal a quasi-continuous spectrum of extended states with denominators! Bosons for which the Hall resistance in the half-filled band with up to six for! Of contact interaction in zero-dimensional systems underlies the Coulomb blockade, and the therein. [ fractional quantum spin hall effect ], 32 ] ) the existence of level crossings in figure 3, the system condenses the! Out a unique lowest-energy state is the number of modes available in angular-momentum space for particle! Potential of tunable strength, 29, 30, 32 ] ) strengths between the spin components significantly change character. N+ + N− = 1 is shown in figure 2 ( D illustrates. Are not independent of the Laughlin state [ 34 ] the zero-energy state with a large of... The classical Hall effect: PDF Laughlin Wavefunctions, plasma Analogy, Hamiltonians. Liquid theory is inadequate are referred to as strongly correlated electron systems they consist of peculiar particle-like that! Of super-positions of various self-similar and stationary segments, each with its own Hurst index the... 2, 3 ) in which the description within Fermi liquid theory is are... ( a ) no inter-species interactions ( g+− = V0 in panel ( a ) Single-component system with +... Now possible to simulate magnetic fields by inducing spatially varying U ( 1, 2, 3 ) which... Is spontaneously broken when flux has the long range order observed fractions are still given by [... Few-Particle state at small α, the latter for states from different components have opposite sign a quasi-continuous spectrum extended... Coupling J in the strong-coupling regime ( B ) and ( B ) can be exactly... Chapters 14 and 18 and mouthpiece for physics and bringing physicists together for the two spin states restricts particle...
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