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)). A fractional phase in three dimensions must necessarily be a more complex state. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. 4. Although the experimental findings support the composite fermion picture, the theoretical foundation for this description is still under debate. 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. 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. They observe two different energy gap dependences on the in-plane magnetic field, which indicates the existence of the finite-thickness effect. Switching on moderate repulsive (attractive) interaction strength between opposite-spin particles smoothens the transitions and also shifts the critical values of α to larger (smaller) values. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. 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. The time reversal symmetry is broken in the external magnetic field. The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of /. 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. New Journal of 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 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 … In figure 3, the interplay between interactions and confinement is elucidated. \left | 0 \right \rangle. We investigate the issue of whether quasiparticles in the fractional quantum Hall effect possess a fractional intrinsic spin. Switching on interactions between opposite-spin particles turns crossings into anti-crossings. where \alpha = M \Omega ^2 l_{\mathcal B}^2 in terms of the harmonic-trap frequency Ω. 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. Finite size calculations (Makysm, 1989) were in agreement with the experimental assignment for the spin polarization of the fractions. The two-dimensional topological insulator mercury telluride can be described by an effective Hamiltonian that is essentially a Taylor expansion in the wave vector k of the interactions between the lowest conduction band and the highest valence band: 2 2. Joel E. Moore, in Contemporary Concepts of Condensed Matter Science, 2013. There are some subtleties in this description, especially in 3D; in 2D it is understood how different compactification conditions determine whether BF theory has a gapless edge, as in the paired Chern-Simons form relevant to topological insulators, or no gapless edge, as in the Z2 spin liquid phase [69]. 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. 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. 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. Stronger interactions strengths between the spin components significantly change the character of the few-particle state at small α (panel (D)). Such an absence of global self-similarity is a problem, and the variability of scales can be well analyzed by the simple use of a multi-scalable fractional Brownian motion (in other words, mixed fractional Brownian motion). in terms of the Euler Gamma function Γ(x). 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. Here σz denotes the diagonal Pauli matrix, and the {\skew3\hat {\boldsymbol {\jmath }}} are Cartesian unit vectors in real space. Investigation of the one-particle angular-momentum-state distribution for the few-particle ground states discussed so far further solidifies our conclusions. When particles occupy states in both components, the situation becomes complex. 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. It has been expected [22, 38, 42] that such systems exhibit the fractional QSH effect, but we find that interactions between particles with opposite spin weaken or destroy features associated with fractional-QSH physics. (D) Same situation as for (B) but with finite interspecies interaction g+− = g++ in addition. 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. In the calculation, lowest-Landau-level states with m ≤ 18 have been included. Our conclusions are summarized in section 5. Its publishing company, IOP Publishing, is a world leader in professional scientific communications. with Si being a localized spin-1/2 operator at the i-th site. Theoretically, when electron–electron interaction is omitted, electronic and thermal transport properties in systems with confined geometries are often well understood. 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. 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. 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. An integer filling factor νCF=ν/1−2ν is reached for 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,…. Different panels correspond to different interspecies-interaction strengths. Let the homogeneous plasma density ρ¯ be explicitly denoted by ρ0, with N particles in Ωc. Some of the essential differences in the calculated excitation energies in the FQHE are probably related to such inconsistencies. However, we do not have sufficient data to draw a conclusion on this problem at the moment. Anyons, Fractional Charge and Fractional Statistics. (Details are given in the following section.) Quantum Hall Hierarchy and Composite Fermions. 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. The second issue, that is, the high-temperature superconductivity, certainly deserves much attention. Following the familiar approach [34], we define the harmonic-oscillator Landau-level ladder operator for states with spin σ via, Similarly, the ladder operator operating within a Landau level for spin component σ is. 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). Composite fermions experience an effective magnetic field and form Landau-like levels called Λ levels (ΛLs). Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. 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. Maude, J.C. Portal, in Semiconductors and Semimetals, 1998. However, in contrast to ordinary multi-component QH states discussed, e.g. Moreover, a quantum Hall platform could harness the unique statistics of fractional quantum Hall states. Figure 1(C) illustrates the different density-of-states behavior for interacting two-particle systems for the two cases of particles having the same and opposite spin, respectively. The corresponding first-quantized two-particle Hamiltonian reads, with the spin-dependent vector potentials from equation (1). Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. It is now possible to simulate magnetic fields by inducing spatially varying U(1) (i.e. 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. Results obtained for systems with N+ + N− = 4 are shown in figure 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). 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 data for \mathcal {M}=10 are also shown as the magenta data points in panel (A) and exhibit excellent agreement with the power-law-type distribution predicted from the solution in COM and relative angular-momentum space. 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. J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. 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). The challenge is in understanding how new physical properties emerge from this gauging process. 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]. We remove one of the plasma particles and introduce the impurity. 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. Green stars show the energy calculated for two-particle versions of trial states [22] ψ+−( r1, r2)∝(z1 + z*2)mC(z1 − z*2)mr with mC = 0 and mr = 2, 9, 14. Chandre DHARMA-WARDANA, in Strongly Coupled Plasma Physics, 1990, An important class of plasma problems arises where the properties of an impurity ion placed in the plasma become relevant. 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. Interacting electron systems for which the description within Fermi liquid theory is inadequate are referred to as strongly correlated electron systems. Cold-atom systems are usually studied while trapped by an external potential of tunable strength. In 3D the possible compactifications are less clear, but at the classical non-compact level 3D BF theory does allow a Dirac fermion surface state [68]. 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. Panel (C): comparison of two-particle densities of states for same-spin case (blue arrows indicating delta functions) and for opposite-spin case (red curve). The other states in the low-energy band correspond to edge excitations of this configuration. 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. This construction leads to the linear combination of three fractional processes with different fractionality; see [HER 10]. 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 has been shown that the flux state is nothing but the chiral spin state in the half-filled limit50, where the chirality order parameter is defined from the spin of fermions as, for the elementary triangle in the lattice. 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]. Quantum Spin Hall Effect. Research 2 The fractional Hall effect has led to many new concepts such as fractional statistics, composite quasi-particles (bosons and fermions), and braid groups. The Fractional Quantum Hall Effect presents a general survery of most of the theoretical work on the subject and briefly reviews the experimental results on the excitation gap. In the following, we will focus on the case where all particles are in the lowest Landau level, i.e. 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]. It appears that strong inter-component interactions favor a state with increased occupation of high-angular-momentum states, spreading out the particles more evenly across the accessible sample size and leading to an accumulation at the system's boundary. The sharpness of the transitions reflects the existence of level crossings in figure 3(A). The quasiparticle's spin is found to be topological independent and satisfies physical restrictions. The various published calculations for the FQHE do not seem to have included all the terms presented in Eq.. (5.6). 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.). In the limit of vanishingly small trapping-potential strength α, the latter is defined by the cut-off for single-particle angular momentum applied in our calculations. Rev. It indicates that regularly frustrated spin systems with the ordinary form of exchange coupling is not likely to show the chiral order. Traditional many-body perturbation theory, which is developed in Sec. Without loss of generality, we will assume {\mathcal {B}}>0 from now on. 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. Kaplan DB(1), Sen S(2). 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]. The triangular lattice with the next nearest neighbor interaction also shows similar behavior58. 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. 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 DPG sees itself as the forum and mouthpiece for physics and is a non-profit organisation that does not pursue financial interests. (2)Department of Physics and Astronomy, … We do this with a different numerical scheme using exact diagonalization of the two-particle Hilbert space on a disc, as it preserves the z component of angular momentum as a good quantum number. Author information: (1)Institute for Nuclear Theory, Box 351550, University of Washington, Seattle, Washington 98195-1550, USA. The UV completion consists of a perturbative U(1)×U(1) gauge theory with integer-charged fields, while the low energy spectrum consists of nontrivial topological phases supporting fractional currents, bulk anyonic excitations, and exotic phenomena such as a fractional quantum spin Hall effect. 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. 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!} The fractional quantum Hall effect is the result of the highly correlated motion of many electrons in 2D ex-posed to a magnetic field. MAU1205), administered by the Royal Society of 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. Figure 3. where g(0,1) and g(0,2) are simply g0i(r) while g0(1,2) is gii0(r). Notice the band of low-lying energy levels separated by a gap from higher-energy states. Since ρp = ρ0p- ρi we have, from Eq.. (5.3), We have used r0 instead of r3 in the last term in square brackets. We derive the braid relations of the charged anyons interacting with a magnetic field on Riemann surfaces. 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. But microfield calculations19 require Δhpp(r→1,r→2|r→0) prior to the r→0 integration. A standard approach is to use the Kirkwood decomposition. For more information, see, for example, [DOM 11] and the references therein. 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 }). One-particle angular-momentum distribution for pseudo-spin + particles for the ground states of systems whose energy spectra are shown in figure 3. The variation of few-particle states as a function of confinement strength is seen to be almost uniform, again pointing to the loss of distinctiveness for few-particle states in the presence of inter-species interactions. 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. Laughlin Wavefunctions, plasma Analogy, Toy Hamiltonians dramatic effect of electron–electron on! 0.3\, V_0\exp ( -\alpha \tilde { n } ) with α 0.8... Between same-spin particles are still dominant three fractional Processes with different spin polarizations possible any. Phenomena are: the multi-component, trap will lift degeneracies of few-particle states and serve identify. 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