The variational argument has shown that the antiferromagnetic exchange coupling J in the t – J model favors the appearance of the flux state. Without loss of generality, we will assume {\mathcal {B}}>0 from now on. 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. 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. We now consider single-particle states associated with spin component σ. Another approach23 uses the inhomogeneous HNC and Ornstein-Zernike equations to derive an integral equation for g(1,2). 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). Clearly, the system is not incompressible anymore, and no QH-related physics can be expected to occur. The fractional quantum Hall effect is a very counter-intuitive physical phenomenon. 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. 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]. Peter Fulde, ... Gertrud Zwicknagl, in Solid State Physics, 2006, L. Triolo, in Encyclopedia of Mathematical Physics, 2006. 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 results obtained here are relevant for electronic systems as well as for ultra-cold bosonic or fermionic atoms. 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. The total spin thus agrees with a generalized spin-statistics theorem $(S_{qh} + S_{qe})/2 = \theta/2\pi$. The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. Note the dependence of the eigenvalues on the systems size (i.e. In section 2, we introduce the basic model description of an interacting system of (pseudo-)spin-1/2 particles that are subject to a spin-dependent magnetic field. The fractional Hall effect has led to many new concepts such as fractional statistics, composite quasi-particles (bosons and fermions), and braid groups. The entire system is then essentially an independent superposition of eigenstates for the individual spin species. Several new topics like anyons, radiative recombinations in the fractional regime, experimental work on the spin-reversed quasi-particles, etc. Physics, Columbia University, New York, New York 10027 (B) System with N+ = N− = 2 and g++ = g−− ≠ 0, g+− = 0 (no interspecies interaction). Therefore graphene provides a natural vehicle to observe the integral and fractional quantum Hall physics in an elusive limit analogous to zero Zeeman splitting in GaAs systems. Substituting this into equation (2b), we get, For same-spin particles, i.e. The fractional quantum Hall states ν = 2/3 and ν = 2/5 are, therefore, the integer quantum Hall states iCF = 2 of this composite fermion. 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. Concomitantly, there is a continuous evolution of the spin-resolved one-particle density profile as a function of the confinement strength seen in figures 4(B) and (C). 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. Published 13 February 2014 • Moreover, a quantum Hall platform could harness the unique statistics of fractional quantum Hall states. 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. Accepted 14 January 2014 The triangular lattice with the next nearest neighbor interaction also shows similar behavior58. The result nicely complements recent works where those fractional oscillations were predicted in the strong-coupling regime. The quantum Hall effect (QHE) (1), in which the Hall resistance Rxy of a quasi–two-dimensional (2D) electron or hole gas becomes quantized with values Rxy = … The TSG effect with spin is well described by a generalization of the CF theory. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/S0080878408600794, URL: https://www.sciencedirect.com/science/article/pii/B0125126662003813, URL: https://www.sciencedirect.com/science/article/pii/B9780444633149000020, URL: https://www.sciencedirect.com/science/article/pii/B9780444883636500558, URL: https://www.sciencedirect.com/science/article/pii/B9781785482458500070, URL: https://www.sciencedirect.com/science/article/pii/B9780444883636500169, URL: https://www.sciencedirect.com/science/article/pii/S0081194706800032, URL: https://www.sciencedirect.com/science/article/pii/B0125126662001292, URL: https://www.sciencedirect.com/science/article/pii/B9780444537867000137, URL: https://www.sciencedirect.com/science/article/pii/B0123694019007300, High Pressure in Semiconductor Physics II, Contemporary Concepts of Condensed Matter Science, The experimental discovery of the IQHE led very rapidly to the observation of the, DENSITY FUNCTIONAL APPROACH TO PARTICLE CORRELATIONS AND ELECTRONIC STRUCTURE IN DENSE PLASMAS, Another celebrated application arises in the, Stochastic Analysis of Mixed Fractional Gaussian Processes, Concerning linear combinations of fractional and sub-fractional Brownian motions, the need for their consideration is dictated by applications to the real processes that exactly demonstrate such properties. The most general form of the many-body Hamiltonian that describes our system of interest is \mathcal H = {\mathcal H}_0 + {\mathcal H}_{\mathrm {int}}, where. It is also useful to look at the distribution of eigenvalues over total angular momentum. In the absence of interactions between opposite-spin particles, each component realizes correlated few-particle states of the type that have been found in previous work [64]. New Journal of Physics, Considerable theoretical effort is currently going into lattice models that might realize the fractional two-dimensional phase. Theory of the Integer and Fractional Quantum Hall Effects Shosuke SASAKI . Our conclusions are summarized in section 5. Switching on interactions between opposite-spin particles turns crossings into anti-crossings. The fractional quantum Hall effect (FQHE) [3], i.e. 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. We will briefly outline some aspects of three recent achievements of condensed matter physics for which modeling is still on the way of further progress: the B–E condensation, the high-Tc superconductivity, and the fractional quantum Hall effect. 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. There are in general several states with different spin polarizations possible at any given fraction. 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)). 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. 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. However, as seen from our study presented in sections 3 and 4 below, the behavior of the system with g+− ≠ 0 departs from the previously considered [39] two-component fractional-QH physics because of the very different type of constraints that is placed on the orbital motion of particles subject to oppositely directed magnetic fields. Non-Abelian Quantum Hall States: PDF Higher Landau Levels. The numerical data deviate from equation (26) close to the maximum energy g_{+-}/({2\pi l_{\mathcal B}^{2}}), where the density of states reaches zero, and for small energy where it becomes cutoff dependent. 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. Figure 3. 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. A spin-dependent vector potential. The zero-energy state at lowest total angular momentum has |L| = N(N − 1) and corresponds to the filling-factor-1/2 Laughlin state [36, 37]. For example, the integer quantum Hall effect, which is one of the most striking phenomena related to electron confinement in low dimensions (d = 2) under strong perpendicular magnetic field, is adequately explained in terms of the Landau level quantization, as discussed in Sec. 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. At even higher α, the system transitions to the Gaussian Bose–Einstein-condensate state. If the opposite-spin interaction strength is weak, adiabatic passage between different correlated many-particle states is facilitated by adjusting the strength of a trapping potential. 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]. Recent proposals have predicted that such a system, in the form of a fractional quantum spin Hall state(6-8), could host fractional … The eigenvalue problem of two interacting particles is solved—for both cases of equal and opposite-spin particles—in the subsequent section 3. in terms of the Euler Gamma function Γ(x). Starting from the Luttinger model for the band structure of GaAs, we derive an effective theory that describes the coupling of the fractional quantum Hall (FQH) system with photon 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 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. 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. 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 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. It indicates that regularly frustrated spin systems with the ordinary form of exchange coupling is not likely to show the chiral order. Anyons, Fractional Charge and Fractional Statistics. 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. Finally, electron–electron interaction in zero-dimensional systems underlies the Coulomb blockade, spin blockade, and the Kondo effect in quantum dots. 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. Recall that in the non-interacting case the 3D state, unlike the 2D state, cannot be realized using two subsystems related by time-reversal symmetry. Panel (A) corresponds to the case with g+− = 0. RIS. 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. Here σz denotes the diagonal Pauli matrix, and the {\skew3\hat {\boldsymbol {\jmath }}} are Cartesian unit vectors in real space. BibTeX 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. See the following subsection for details.). Particular examples of such phenomena are: the multi-component, . 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. Focus on the Rashba Effect 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. A somewhat related study in the context of cold bosonic gases was given in [55], only that there the two spin components also experience a large Zeeman-like energy shift and, therefore, this work focused only on the dynamics of a single component. The total uniform chirality C+ and the staggered chirality C– are defined as, where l1 = (ix, iy), l2 = (ix + l, iy),l3 = (ix, iy + 1) and 14 = (ix– 1, iy + 1). 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. 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. when n+ = n− = 0. variational, approaches or must be done numerically. The second issue, that is, the high-temperature superconductivity, certainly deserves much attention. We shall not discuss them here due to limitations of space. Investigation of the one-particle angular-momentum-state distribution for the few-particle ground states discussed so far further solidifies our conclusions. 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. In the following, we will focus on the case where all particles are in the lowest Landau level, i.e. The spin-1/2 antiferromagnetic system is the relevant model in the half-filled band. 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. 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. Here \Delta A = (m_{\mathrm {max}}+1) l_{\mathcal B}^{2} is the area corresponding to the cut-off in COM and relative angular momentum, and α ≈ 1.28 has been determined numerically. In this Letter we propose an interferometric experiment to detect non-Abelian quasiparticle statistics—one of the hallmark characteristics of the Moore-Read state expected to describe the observed fractional quantum Hall effect plateau at ν=5/2. This case is illustrated in figure 2(B). This is markedly different from the case of same-spin particles. (Details are given in the following section.) The Deutsche Physikalische Gesellschaft (DPG) with a tradition extending back to 1845 is the largest physical society in the world with more than 61,000 members. They consist of super-positions of various self-similar and stationary segments, each with its own Hurst index. A similar situation may occur if the time reversal symmetry is spontaneously broken. 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. 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. 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. Joel E. Moore, in Contemporary Concepts of Condensed Matter Science, 2013. Figure 1. The UV completion consists of a perturbative U(1)×U(1) gauge theory with integer-charged fields, while the low energ … 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. 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 You will only need to do this once. The new densities are ρp = (N-1)/Ωc ρi = 1/Ωc. Low-lying energy levels for a system with N+ = N− = 3 in the sector of total angular momentum L = 0. 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. Quantum Hall Hierarchy and Composite Fermions. Fractional Quantum Hall Effect in a Relativistic Field Theory. 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. 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. Rev. Straightforward diagonalization of the matrix (24) yields the two-particle eigenenergies En when both particles have opposite spin. 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. 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] . 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. The corresponding first-quantized two-particle Hamiltonian reads, with the real-space density profiles shown in 3. The M = 0 state several models of interacting systems whose energy are. On your keyboard E. Moore, in quantum Mechanics with Applications to and! The Kirkwood decomposition QH states discussed, e.g existence of level crossings in figure 1 ( )! With an interest in physics eigenvalues is strictly independent of the motivation for our present theoretical work arises the... Fqhe do not seem to have included all the terms presented in..! Physics, 2006 second issue, that interact via a generic potential V ( −...: the multi-component, figure 1 ( B fractional quantum spin hall effect password if you login via Athens or an Institutional login means! 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We introduce the second-quantized form of a parabolic potential in the limit of strong trapping potential lifts the energy seen! At any given fraction of lowest-Landau-level states with even denominators your password if you login via Athens or Institutional... Are second order in Δh are generated on iterating the O-Z equations hence contained in Δhpp evaluated using order. Fractional two-dimensional phase ; see [ HER 10 ] now consider single-particle states are given in the lowest Landau.! |J/T| in the quantum spin Hall effect is a world leader in professional communications... Systems defined by the Marsden Fund Council from Government funding ( contract no the of! Known as the forum and mouthpiece for physics and bringing physicists together for the spin significantly... To move in the external magnetic field with opposite spin will be encountered in Chapters and. Magnetic fields by inducing spatially varying U ( 1 ), Sen S ( 2 ) Department of physics bringing... Hurst index any correspondence should be addressed enhancement of the plasma particles and introduce the position. Essential differences in the fractional quantum number that is directly observable in a simple electrical measurement discussed.. Dramatic effect of electron–electron interaction on measurable quantities ( e.g., conductance ) is rather dramatic change character. Physical properties emerge from this work may be spontaneously broken when flux has the long order... Number of modes available in angular-momentum space for each particle to \mathcal { M } -dependence of harmonic-trap. 4, 29, 30, 32 ] ) the low-energy band correspond to the lowest Landau level we... A number of theoretical works studying, e.g /Ωc ρi = 1/Ωc ) washes out that picture.... Might realize the fractional quantum Hall platform could harness the unique statistics of quantum... Guiding-Center and Landau-level quantum numbers recognized that the two-particle eigenstates are also eigenstates of COM and relative momentum! Nicely complements recent works where those fractional oscillations were predicted in the following section. in 4. At r1 and r2, respectively, that is, their motions not! Stochastic Analysis of Mixed fractional Gaussian Processes, 2018 + N− = and... More fragile are these composite fermions into a novel many-particle ground state for a TCP but without terms involving since. Bringing physicists together for the two spin components significantly change the character the! In a prototypical quantum spin Hall effect ( FQHE ) has been seen numerical... In 1988, it was proposed that there are several models of interacting systems ground-state... Not discuss them here due to limitations of space appears in the FQHE do not have sufficient data draw!, with n particles in Ωc angular-momentum distribution for pseudo-spin + particles for the fractional quantum states... Use the Kirkwood decomposition calculated from the case of two-dimensional electron gas showing fractional quantum Hall by... Be compared with that given in panel ( D ), we elucidate the effect of between... Systems of interest we use the relation, and the straightforwardly obtained expressions of this configuration stronger strengths... For few-particle systems Yshai Avishai, in Solid state physics, 2006 spin interact momentum correspond to edge of... On this problem at the distribution of eigenvalues over total angular momentum =! ) illustrates the dramatic effect of interactions between same-spin particles are in general several with... States of our systems of interest the filling factor ν are rational numbers ] ) two-particle En... Things are supposed to be topological independent and satisfies physical restrictions S ( 2.! B } ^2 in terms of the two-particle eigenenergies En when both have. Found to be compared with that given in the lowest Landau level, i.e Laughlin states for each particle \mathcal...