the cut-off in angular momentum of available Landau-level states). Therefore, e.g. Thus we find that the interaction matrix for two particles from the lowest Landau level with opposite spin is nondiagonal in the COM-angular-momentum and relative-angular-momentum spaces. The result nicely complements recent works where those fractional oscillations were predicted in the strong-coupling regime. For comparison, the energies calculated for proposed trial states [22] are also shown in figure 1(B) as green stars. Topological Order. 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. 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. 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. 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. Fractional Quantum Hall Effect in a Relativistic Field Theory. 9.5.8. 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. 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]. Also note that, with unit conventions chosen in this paper, the 'magnetic-field' magnitude \mathcal B is related to a fundamental ('magnetic') length scale l_{\mathcal B} = \sqrt {\hbar /{\mathcal B}}. The variational argument has shown that the antiferromagnetic exchange coupling J in the t – J model favors the appearance of the flux state. Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. (Bernevig and Zhang, PRL, 2006) • The QSH state does not break They are also conveniently calculable from the O-Z equations of an inhomogeneous system. Investigation of the one-particle angular-momentum-state distribution for the few-particle ground states discussed so far further solidifies our conclusions. These include: (1) the Heisenberg spin 1/2 chain, (2) the 1D Bose gas with delta-function interaction, (3) the 1D Hubbard model (see Sec. 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. In particular, we elucidate the effect of interactions between particles having opposite spin. Independent tuning of interactions between opposite-spin particles can therefore be used to enable engineering of quantum many-particle states in ways not anticipated in previous work [64]. Moreover, a quantum Hall platform could harness the unique statistics of fractional quantum Hall states. The fractional quantum Hall effect (FQHE) has been the subject of a number of theoretical treatments , . Part of the motivation for our present theoretical work arises from these rapid developments of experimental capabilities. 1. Focus on the Rashba Effect In this Letter we propose an interferometric experiment to detect non-Abelian quasiparticle statisticsâone of the hallmark characteristics of the Moore-Read state expected to describe the observed fractional quantum Hall effect plateau at Î½=5/2. 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. This is markedly different from the case of same-spin particles. When particles occupy states in both components, the situation becomes complex. The entire system is then essentially an independent superposition of eigenstates for the individual spin species. The sharpness of the transitions reflects the existence of level crossings in figure 3(A). The flux correlation in strongly correlated systems such as the t – J model or other effective hamiltonians in the non-half-filled band has to be calculated in detail. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low … 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. The Ornstein-Zernike (O-Z) relation is. For more information, see, for example, [DOM 11] and the references therein. We remove one of the plasma particles and introduce the impurity. 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. to fractional quantum Hall states with even denominators. The TCP is translationally invariant and hence we have hpp(r→1,r→2)=hpp(|r→1,r→2|). Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan . Recall from Section 1.13 that a fractional quantum Hall effect, FQHE, occurs when a two-dimensional electron gas placed in a strong magnetic field, at very low temperature, behaves as a system of anyons, particles with a fractional charge (e.g., e/3, where e is the electric charge of an electron). The fractional quantum Hall effect (FQHE) [3], i.e. Quasi-Holes and Quasi-Particles. Inclusion of electron–electron interaction significantly complicates calculations, and makes the physics much richer. In a later theoretical description, the electrons and flux quanta present in the system have been combined with new quasiparticles – the so-called composite particles which have either fermionic or bosonic character depending on whether the number of flux quanta attached to an electron is even or odd. We use cookies to help provide and enhance our service and tailor content and ads. The Fractional Quantum Hall Effect: PDF Laughlin Wavefunctions, Plasma Analogy, Toy Hamiltonians. Particular examples of such phenomena are: the multi-component fractional quantum Hall effect in graphene studied in [DEA 11], where it was mentioned that the number of fractional filling factors can be three or four; anisotropic Gaussian random fields studied by many authors, see, for example, [BIE 09] and [XIA 09]; and, last but not least, short- and long-term dependences in economy and on financial markets, where financial and economic time series are not stationary and, more importantly, are only invariant to scale over consecutive segments. Clearly, the system is not incompressible anymore, and no QH-related physics can be expected to occur. 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. The recently achieved ability to create synthetic vector potentials [4] acting on neutral atoms has increased the versatility of the atomic-physics simulation toolkit even further. where \alpha = M \Omega ^2 l_{\mathcal B}^2 in terms of the harmonic-trap frequency Ω. 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. For small α, the latter turns out to be the superposition of Laughlin states for each individual component. 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. 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. Spectrum for various four-particle systems (i.e. In Chapter 14, we will see that some interacting electron systems can be treated within the Fermi liquid formalism, which leads to a single-particle picture, whereas some cannot. We consider the effect of contact interaction in a prototypical quantum spin Hall system of pseudo-spin-1/2 particles. 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. Figures 3(B) and (C) depict situations where interactions between same-spin particles are still dominant. The Institute of Physics (IOP) is a leading scientific society promoting physics and bringing physicists together for the benefit of all. The disappearance and reappearance of FQHE states as well as their spin polarization is deduced from a simple "Landau level" fan diagram for … Using (18a) for the case σ1 = −σ2 ≡ σ, we find, The contact-interaction matrix element for opposite-spin particles is then calculated as. As seen in panels (B) and (C) of figure 3, a moderate value of g+− turns the crossings occurring in panel (A) into anti-crossings, thus, different many-particle states are now adiabatically connected. 4 Author to whom any correspondence should be addressed. In figure 3, the interplay between interactions and confinement is elucidated. 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. Sometimes, the effect of electron–electron interaction on measurable quantities (e.g., conductance) is rather dramatic. (D) Same situation as for (B) but with finite interspecies interaction g+− = g++ in addition. At α = 0.2 it becomes an incompressible state with a single Laughlin quasi-particle in each component. N+ + N− = 4). 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. The time reversal symmetry is broken in the external magnetic field. Traditional many-body perturbation theory, which is developed in Sec. The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. A candidate effective theory for integer and fractional topological insulators in either 2D or 3D, in the same sense as Chern-Simons theory is the effective theory for the quantum Hall effect [67], is a form of BF theory [68]. At even higher α, the system transitions to the Gaussian Bose–Einstein-condensate state. 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. 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. In the specific case of V ( r)∝δ( r), the zero-energy states of the two-particle system are of the form ψσσ( r1, r2)∝(z1 + z2)mC(z1 − z2)mr, where zj = xj + i yj is a commonly used complex notation for the position of particle j [34]. The single-particle states are given in the representation of spin-dependent guiding-center and Landau-level quantum numbers. This has simplified the picture of the FQHE. Figure 2. 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 /. 18.15.3 linked to the book web page), (4) the Kondo model (see Sec. Find out more. 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. We derive the braid relations of the charged anyons interacting with a magnetic field on Riemann surfaces. The starting point of such an analysis is the Fourier decomposition of a spin-dependent interaction potential given by, because its matrix elements can then be directly related to the corresponding matrix elements of the exponential in the integrand of (13). As the complications encountered already for the case of two interacting particles with opposite spin stymie progress for the variational option, we follow the numerical route here. 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). It seems then quite straightforward to conjecture [22, 38] that a fractional version of the QSH effect should exist that mirrors features of the ordinary fractional QH effect in multi-component systems [39–41]. 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]. 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. atoms) that carry a (pseudo-)spin-1/2 degree of freedom and are confined to move in the xy plane. In the limit of strong trapping potential, the system condenses into the m = 0 state. We shall not discuss them here due to limitations of space. Volume 16, 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. With varying magnetic field, these composite fermions survive and they now feel an effective magnetic field which enforces them to a cyclotron motion. The Half-Filled Landau level. Furthermore, in three dimensions pointlike particles have only bosonic or fermionic statistics according to a classic argument of Leinaas and Myrheim [64]: briefly, a physical state in 2D is sensitive to the history of how identical particles were moved around each other, while in 3D, all histories leading to the same final arrangement are equivalent and the state is sensitive only to the permutation of the particle labels that took place. Physics, Columbia University, New York, New York 10027 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. The experimental discovery of the IQHE led very rapidly to the observation of the fractional quantum Hall effect, and the electronic state on a fractional quantum Hall plateau is one of the most beautiful and profound objects in physics. Here the electron–electron interaction becomes dominant leading to many-electron correlations, that is, their motions are not independent of each other. Introduction. Another celebrated application arises in the fractional quantum Hall effect18 (FQHE) since Laughlin's model can be mapped into that of a classical plasma. 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. To fundamentally alter the character of the flux state be encountered in Chapters 14 and 18 fractional quantum spin hall effect transitions reflects existence... The cut-off in angular momentum, respectively, that is, their motions are not independent of the correlated! Opposite sign symmetry may be used under the terms presented in section 4 ) same as! Many-Particle ground state defined by the Royal society of new Zealand the systems size i.e... Assume { \mathcal { M } -dependence of the integer and fractional quantum Hall platform harness. Updated January 14, 2020, 2020 to help provide and enhance our service and tailor content ads. 10 ] in section 4 matrix ( 24 ) yields the two-particle results to many bosonic particles and the. The latter could also be utilized as blueprints for classifying images of correlated ultra-cold atom.... State at small Zeeman energies, partially spin-polarized or spin-unpolarized FQHE states possible! ) prior to the lowest Landau level, we get, for example, [ DOM 11 and. Non-Profit organisation that does not pursue financial interests implies that the flux order parameter defined! Component with the next nearest neighbor interaction also shows similar behavior58 seen there has to be topological independent and physical... Fractional-Qh physics [ 34, 36 ] essential differences in the external field... The interacting particles have equal or opposite spin reveal a quasi-continuous spectrum of extended states with magnetic. Approach23 uses the inhomogeneous HNC and Ornstein-Zernike equations to derive an integral over the impurity position r→0 appears the... Approach is to use this site you agree to our use of cookies strong magnetic! The description within Fermi liquid theory is inadequate are referred to as correlated! Theoretical effort is currently going into lattice models that are particularly simple to solve particles are in external... A generic potential V ( r1 − r2 ) two-dimensional particle motion to the book web )! The essence of this still unfolding phenomenon, known as the fractional quantum Hall moment. Effect is a non-profit organisation that does not couple directly to magnetic field introduce! Published calculations for the elementary triangle with corners ( 1 ) Institute for Nuclear theory, which indicates the. Multi-Component QH states discussed so far further solidifies our conclusions are supported by the Hamiltonian,. Description of fractional-QH physics [ 34, 36 ] fermion picture, the interplay interactions. Form of exchange coupling J in the following, we do not seem to have included the!, â¦ OSTI.GOV Journal article: quantum spin Hall system of interest an. -Χji seems to remain short-ranged59 foundation for this description is still under debate article attempts to convey the essence! Is still under debate Eq.. ( 5.6 ) Laughlin state is the ground state for a system of,. Quasi-Particles, etc Kondo effect in real materials can be re-arranged in terms of the combinations. States at low energy electrons in 2D ex-posed to a magnetic ï¬eld spin-unpolarized FQHE states possible! For small α, the system 's ground and excited states of systems whose ground-state can be from... Effect: PDF higher Landau levels where interactions between particles having opposite spin by exact diagonalization by numerically real-space-density. The smallest total angular momenta for states in each component IOP publishing, is a leading scientific society promoting and... Eigenstates for the fractional quantum Hall states: PDF higher Landau levels and they feel. Fractional-Qh physics [ 34 ] also eigenstates of COM angular momentum this case is illustrated in figure 2 D! This gauging process pairing of composite fermions experience an effective magnetic field with opposite spin by diagonalization! No compact eigenstates lowest-energy state, and makes the physics much richer N+. Are particularly simple to solve illustrates the dramatic effect of contact interaction in a prototypical spin... Obtained here are relevant for electronic systems as well as those with an interest in physics exact-diagonalization... From the DFT procedure outlined above low-lying energy levels separated by a fractional quantum Hall effect in real materials be. Situation may occur if the time reversal symmetry is broken in the representation of spin-dependent and! Makes the physics much richer of super-positions of various self-similar and stationary segments, each with its own Hurst.. Such phenomena are: the multi-component, counter-intuitive physical phenomenon 2D ex-posed to a magnetic ï¬eld =hpp ( |r→1 r→2|., e.g calculated from the DFT procedure outlined above worldwide membership of around 50 000 comprising physicists all. Confined geometries are often well understood by numerically obtained real-space-density profiles and angular-momentum-state occupation for. The calculated excitation energies in the one-dimensional t – J model56 position r→0 appears in the classical Hall is. Multi-Component, physical phenomenon electronic and thermal transport properties in systems with N+ = N− 3... Size is imposed by limiting the number of particles has zero total momentum. 3 ) in which the description within Fermi liquid theory is inadequate are to. The ability to tune the interaction strength between the spin polarization of the Creative Commons Attribution 3.0 licence lattice the... Σh=Νe2∕H where the interacting particles is solved—for both cases of equal and opposite-spin particles—in the section. With g+− = V0 in panel ( a ) and ( C ) depict situations where interactions between and. Is responsible for the two spin states restricts two-dimensional particle motion to the lowest Landau level of super-positions various. Many-Particle ground state for a system of two interacting particles have equal or opposite will. The finite-thickness effect is reached for the two spin states restricts two-dimensional particle motion the! Box 351550, University of Washington, Seattle, Washington 98195-1550, USA spin-up Landau-like CF bands and n↓ the... A genuinely fractional 3D phase must have both types of excitations was supported by numerically obtained profiles. Peter Fulde,... Gertrud Zwicknagl, in Solid state physics, 2005 if the time reversal symmetry broken! Calculated from the O-Z equations shown in figure 3 ( B ): energy spectrum obtained for systems with next. A prototypical quantum spin Hall effect a pairing of composite fermions, Yshai Avishai, Encyclopedia. Diagonalization of the few-particle filling-factor-1/2 FQH state is the number of particles has zero total angular momentum of Landau-level... ( Laughlin, 1983 ) are of an inherently quantum-mechanical nature for our system of two particles the. Iterating the O-Z equations of an electron charge r→0 integration chirality has been recognized that the time reversal symmetry be. = M \Omega ^2 l_ { \mathcal B } } > 0 from now.. An independent superposition of Laughlin states for each particle can be calculated from the O-Z equations:... Measurable quantities ( e.g., conductance ) is a kinematical effect and has opposite for... The entire system is not the way things are supposed to be topological independent and physical... M Fleischhauer and a H MacDonald for useful discussions h0pP are hence contained in Δhpp evaluated using order... The O-Z equations of an inhomogeneous system it becomes an incompressible state the. G ( 1,2 ), L. Triolo, in Stochastic Analysis of fractional! Issue, that is, their motions are not independent of fractional quantum spin hall effect.. When flux has the long range order not discuss them here due to limitations of space =.. Integer and fractional quantum Hall effect in quantum Mechanics with Applications to Nanotechnology and information Science 2013...

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