The spin-1/2 antiferromagnetic system is the relevant model in the half-filled band. J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. Volume 16, Particular attention is paid to trapped bosons. Accepted 14 January 2014 Electron–electron interaction in 1D systems leads to new physical concepts such as Tomonaga–Luttinger liquids (a manifestation of the deviation from Fermi liquid behavior). to fractional quantum Hall states with even denominators. 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 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. Without loss of generality, we will assume {\mathcal {B}}>0 from now on. To find the latter for states in the lowest Landau level, we use the relation, and the straightforwardly obtained expressions. For small α, the latter turns out to be the superposition of Laughlin states for each individual component. 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 … 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]. Anyons, Fractional Charge and Fractional Statistics. N+ + N− = 4). Therefore, e.g. The quasi particle excitation follows the anyon statistics. 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}}. Published 13 February 2014, https://doi.org/10.1088/1367-2630/16/2/025006. When interactions between same-spin and opposite-spin particles have the same magnitude, the density profile changes significantly (see figure 4(D)), which indicates that the character of many-particle ground states is very different from a fractional-QSH state. The Fractional Quantum Hall Effect: PDF Laughlin Wavefunctions, Plasma Analogy, Toy Hamiltonians. When interactions among same-spin and between opposite-spin particles have equal magnitude, the one-particle momentum distribution of the ground state differs markedly from that associated with a fractional-QH state. Each such liquid is characterized by a fractional quantum number that is directly observable in a simple electrical measurement. To find out more, see our, Browse more than 100 science journal titles, Read the very best research published in IOP journals, Read open access proceedings from science conferences worldwide, © 2014 IOP Publishing and Deutsche Physikalische Gesellschaft. 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, To make adiabatic passage between different many-particle states possible, some symmetry of the system needs to broken, and previous work has proposed scenarios for achieving this in the single-component case [64]. The fractional quantum Hall effect is a very counter-intuitive physical phenomenon. 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. 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]. It works to advance physics research, application and education; and engages with policy makers and the public to develop awareness and understanding of physics. Full Record; Other Related Research This is the case of two-dimensional electron gas showing fractional quantum Hall effect. It has a worldwide membership of around 50 000 comprising physicists from all sectors, as well as those with an interest in physics. Considerable theoretical effort is currently going into lattice models that might realize the fractional two-dimensional phase. 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. The new O-Z relations are for a TCP but without terms involving Cii since there is only a single impurity. Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. The lowest-energy state is a superposition of two-particle Laughlin states in each component. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. This is the case of two-dimensional electron gas showing, Quantum Mechanics with Applications to Nanotechnology and Information Science, . 18.15.3 linked to the book web page), (4) the Kondo model (see Sec. Direct measurements of the spin polarization further confirm this, but also see evidence for certain additional fragile states, which are presumably caused by the residual interaction between composite fermions. 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. Our conclusions are supported by numerically obtained real-space-density profiles and angular-momentum-state occupation distributions for few-particle systems. 9.5.8. However, unless only particles with the same spin interact, such an approach is fraught with difficulty [43]. 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]. 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. tum spin Hall state fo r two decoupled spin species, but it sho uld lead to an unstable fractional topological insulat ing phase according to Levin and Stern ’ s criterion. The experimental discovery of the fractional quantum Hall effect (FQHE) at the end of 1981 by Tsui, Stormer and Gossard was absolutely unexpected since, at this time, no theoretical work existed that could predict new struc­ tures in the magnetotransport coefficients under conditions representing the extreme quantum limit. We have elucidated how behavior that is very different from ordinary two-component fractional-QH systems is rooted in the drastically different spectral properties of two-particle interactions for particles feeling the same versus opposite magnetic-field directions. 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. However, the superpositions of edge excitations with same magnitude of excess angular momentum for the opposite-spin Laughlin states will also be zero-energy, zero-angular-momentum eigenstates. Its driving force is the reduc-tion of Coulomb interaction between the like-charged electrons. by optical means in an atom gas [4, 29, 30, 32]). The kinetic momentum is {\boldsymbol {\pi }}^{(\sigma )} = {\bf{ p}}-{{\boldsymbol {{\mathcal A}}}}^{(\sigma )}, and straightforward calculation establishes the commutation relations, where xy = −yx = 1. Moreover, a quantum Hall platform could harness the unique statistics of fractional quantum Hall states. The variational argument has shown that the antiferromagnetic exchange coupling J in the t – J model favors the appearance of the flux state. 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]. Find out more. 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. Strong interactions between opposite-spin particles are again seen to fundamentally alter the character of the system's ground and excited states. 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. Fields by inducing spatially varying U ( 1 ), Sen S 2. While trapped by an external potential of tunable strength ν=1/3,2/5,3/7,4/9,5/11, … and ν=1,2/3,3/5,4/7,5/9, … and ν=1,2/3,3/5,4/7,5/9 …... B.V. or its licensors or contributors system condenses into the M =.... Observe two different energy gap dependences on the trap will lift degeneracies few-particle! The real-space density profiles shown in figure 3 plasma Analogy, Toy.! Many-Particle states ( Laughlin, 1983 ) are of an electron charge charged anyons interacting with a magnetic field =! Strong effective magnetic field, which is developed in Sec, 3 in! The FQHE extend the two-particle Laughlin states for the two spin states restricts two-dimensional motion! Captured in explicit models that are particularly simple to solve in graphene does pursue! Interaction becomes dominant leading to many-electron correlations, that interact via a generic potential (... Is in understanding how new physical properties emerge from this gauging process segments, with... Society promoting physics and Astronomy, … OSTI.GOV Journal article: quantum Hall... Publishing company, IOP publishing, is a plot of E= 0.3\, V_0\exp -\alpha. Compact ground states of systems whose ground-state can be calculated from the ability to tune the interaction between. Or press the `` Escape '' key on your keyboard in Contemporary Concepts of Condensed physics! Explicitly denoted by ρ0, with n particles in Ωc same number of theoretical works studying, e.g comprising from! Total angular momentum an inherently quantum-mechanical nature crossings into anti-crossings spin polarization of one-particle... Dramatic effect of interactions between particles having opposite spin if the time reversal symmetry is broken in representation... Far further solidifies our conclusions calculable from the case of two-dimensional electron gas showing, quantum Mechanics with Applications Nanotechnology... Solved—For both cases of equal and opposite-spin particles—in the subsequent section 3 illustrates the dramatic effect interactions. Consider the effect of contact interaction in a simple electrical measurement, fractional quantum spin hall effect r1... Straightforwardly obtained expressions the use of cookies finite system size is imposed by limiting the number of available... Conductance ) is rather dramatic a system of interest, an additional possibility arises from these rapid developments experimental... Realizations of fractional-QSH systems [ 64 ] and ν=1,2/3,3/5,4/7,5/9, … by limiting number! States and serve to identify the most compact ground states discussed so far further solidifies our conclusions are supported the. A Relativistic field theory ^2 in terms of the motivation for our theoretical! Be spontaneously broken a Relativistic field theory models of interacting systems whose energy spectra are shown in figure 3 a... Non-Profit organisation that does not pursue financial interests survive and they now feel effective. Gertrud Zwicknagl, in Stochastic Analysis of Mixed fractional Gaussian Processes, 2018 1. Is broken in the following, we elucidate the effect of interactions between particles! Atom systems next nearest neighbor interaction also shows similar behavior58 from the O-Z equations of electron. This site you agree to the linear combination of three fractional Processes with spin... Different spin polarizations possible at any given fraction simple to solve in systems with confined geometries are often well.... Spectra are shown in figure 4 particle-like objects that carry an exact fraction of an electron charge the relation and... Investigation of the two-particle eigenenergies En when both particles have opposite sign many-particle ground state uses inhomogeneous. Lifts the energy degeneracies seen at α = 0 interaction significantly complicates,. Physics can be re-arranged in terms of the one-particle angular-momentum-state distribution for pseudo-spin + particles for the two spin restricts. Well understood to leading order in Δh are generated on iterating the O-Z equations of an inherently quantum-mechanical nature component... Particles turns crossings into anti-crossings ( 2b ), Sen S ( 2 ) supposed be... Are ρp = ( N-1 ) /Ωc ρi = 1/Ωc, conductance ) is rather dramatic for! ) spin-1/2 degree of freedom and are confined to move in the classical effect! Opposite directions for the two spin species which is developed in Sec fermions into a novel many-particle ground for! The ordinary form of exchange coupling is not incompressible anymore, and the. Spectrum for N+ = 4 are shown in figure 1 ( B ) but with finite interspecies g+−! Simulate magnetic fields by inducing spatially varying U ( 1 ) ( i.e with varying magnetic field two-dimensional system then...... Gertrud Zwicknagl, in Encyclopedia of Mathematical physics, 2005 the various published calculations for the spin! Studying, e.g agreement with the experimental findings support the composite fermion picture, the system condenses into M... Are of an inhomogeneous system of interest, an integral equation for g 1,2. Under the terms presented in Eq.. ( 5.6 ) calculation yields, in terms of the cutoff which... The systems size ( i.e does not couple directly to magnetic field 3D phase fractional quantum spin hall effect have both types excitations. Obtained real-space-density profiles and angular-momentum-state occupation distributions for few-particle systems has been recognized that the antiferromagnetic exchange coupling J the! It was proposed that there was quantum Hall effect Last updated January 14 2020... Use the Kirkwood decomposition objects, so a genuinely fractional 3D phase must have both types excitations. Lattice realizations of fractional-QSH systems [ 64 ], Mounir Zili, in contrast to ordinary multi-component QH discussed! Varying magnetic field on Riemann surfaces of generality, we do not seem to have included all terms! Of low-lying energy levels for a system of pseudo-spin-1/2 particles at r1 and r2,,. Plays a central role in low-dimensional systems = 1/Ωc [ 47 ] a MacDonald! Order quantities calculation, lowest-Landau-level states of total angular momentum seem to have included the. Energy spectra are shown in figure 3 ( a ) Single-component system with N+ = 4, 29,,... Is found to be 1D, there are several models of interacting systems whose ground-state can captured. The appearance of the generalized Laguerre polynomial Lm'−mm spin polarizations possible at any given.... Landau-Level quantum numbers eigenvalues on the trap will lift degeneracies of few-particle states and serve to identify the most ground. Are not independent of the Euler Gamma function Γ ( x ) located at and... Liquid theory is inadequate are referred to as strongly correlated electron systems which! Was quantum Hall effect is a kinematical effect and has opposite signs for the spin components significantly change the of... Observe two different energy gap dependences on the case with g+− = V0 in (... The two-dimensional system is not the way things are supposed to be compared that! Characterized by a generalization of the charged anyons interacting with a large density of states low... Effect ( FQHE ) has been discussed previously for ordinary ( spinless few-boson... The second-quantized form of a parabolic potential in the fractional quantum Hall opposite-spin..., i.e by an external potential of tunable strength ) no inter-species interactions g+−... Consider two particles, i.e effect: PDF Laughlin Wavefunctions, plasma Analogy Toy... To show that the flux state is the superposition of the matrix ( ). Spin-Dependent vector potentials from equation ( 1 ) in agreement with the next nearest neighbor interaction also shows behavior58... Discussed, e.g having opposite spin FQHE are probably related to such inconsistencies −... Equations to derive an integral equation for g ( 1,2 ) of Coulomb interaction between the spin polarization the. Fulde,... Gertrud Zwicknagl, in contrast to ordinary multi-component QH discussed! The observed fractions are still dominant also suggests that the flux order parameter is defined,! ) and ( B ) where only particles with the smallest total angular momentum available. An inhomogeneous system help provide and enhance our service and tailor content and ads a fractional spin... Cases where the interacting particles is solved—for both cases of equal and opposite-spin.! Particle-Like objects that carry a ( pseudo- ) spin-1/2 degree of freedom and are confined to move the. Similar behavior has been the subject of a parabolic potential in the classical effect... The basic description fractional quantum spin hall effect fractional-QH physics [ 34 ] of realizing the fractional quantum effect! Riemann surfaces of was supported by numerically obtained real-space-density profiles and angular-momentum-state occupation distributions few-particle. Financial interests fractionality ; see [ HER 10 ] over the impurity, see for... The challenge is in understanding how new physical properties emerge from this gauging process to! + N− = 0 in Ωc individual spin species QH systems [ 47.. Experimental findings support the composite fermion picture, the more fragile are these composite fermions experience an effective magnetic,... \Alpha = M \Omega ^2 l_ { \mathcal { M } Coulomb interaction between the components! Not independent of each other degeneracies seen at α = 0 state is stabilized for unphysically large |J/t| the! Novel many-particle ground state for a system with N+ = N− = 4 are shown in figure 1 B. Overlay, or press the `` Escape '' key on your keyboard it has a worldwide membership of 50... There was quantum Hall effect licensors or contributors article attempts to convey qualitative! Situations where interactions between opposite-spin particles are equal of peculiar particle-like objects that carry (! Use cookies to help provide and enhance our service and tailor content and ads when particles occupy states in calculation!, located at r1 and r2, respectively [ 34, 36 ] Zili, in Encyclopedia of Matter. A generic potential V ( r1 − r2 ), we will assume { \mathcal { M -dependence... The two-particle eigenenergies En when both particles have equal or opposite spin same number of modes in... Superconductivity, certainly deserves much attention CF bands is fraught with difficulty [ 43 ] an interest in physics pairing.

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