(Bernevig and Zhang, PRL, 2006) • The QSH state does not break This is given by. 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. This site uses cookies. The renormalized mean field calculation indicates that the flux state is stabilized for unphysically large |J/t| in the two-dimensional t – J model56. Modest interspecies-interaction strengths (g_{\sigma \bar {\sigma }}=0.2\, V_0 in panel (B) and g_{\sigma \bar {\sigma }}=-0.2\, V_0 in panel (C)) cause avoided crossings but preserve the incompressible nature of the states seen in panel (A). But microfield calculations19 require Δhpp(r→1,r→2|r→0) prior to the r→0 integration. The TSG effect with spin is well described by a generalization of the CF theory. Self-consistent solutions of the KS equations demonstrate that our f … The lowest-energy state is a superposition of two-particle Laughlin states in each 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). Quite a different situation arises for opposite-spin particles. 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 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. 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. This is the case of two-dimensional electron gas showing fractional quantum Hall effect. At even higher α, the system transitions to the Gaussian Bose–Einstein-condensate state. Strong interactions between opposite-spin particles are again seen to fundamentally alter the character of the system's ground and excited states. 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. When particles occupy states in both components, the situation becomes complex. They are also conveniently calculable from the O-Z equations of an inhomogeneous system. 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. It is also useful to look at the distribution of eigenvalues over total angular momentum. The enhancement of the superconducting correlation in the one-dimensional t – J model also suggests that the two-dimensional system is not special. The spectrum for N+ = N− = 1 is shown in figure 1(B). Note, however, the different parameterization used in [8] where c0,2 are interaction constants associated with the atomic spin-1 degree of freedom from which the pseudo-spin-1/2 components are derived. This case is illustrated in figure 2(B). 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. Energies are given in units of the intra-species Haldane-pseudopotential energy scale V_0 = g_{++}/(4\pi l_{\mathcal B}^2). The fractional quantum Hall effect results in deep minima in the diagonal resistance, accompanied by exact quantization of the Hall plateaux at fractional filling factors (Tsui et al., 1982). 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. Figure 4 shows the real-space profile of n(+)( r) across a diameter of the disc-shaped three-particle systems associated with the ground-state levels shown in figure 3. The Fractional Quantum Hall Effect: PDF Laughlin Wavefunctions, Plasma Analogy, Toy Hamiltonians. 9.5.8) in which the Hall conductance is quantized as σH=νe2∕h where the filling factor ν are rational numbers. Preface . Similar behavior has been seen in numerical studies of lattice realizations of fractional-QSH systems [47]. See also [60]. The other is a kinematical effect and has opposite signs for the quasihole and quasielectron. Stronger interactions strengths between the spin components significantly change the character of the few-particle state at small α (panel (D)). 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. Each such liquid is characterized by a fractional quantum number that is directly observable in a simple electrical measurement. The larger the denominator, the more fragile are these composite fermions. 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). the cut-off in angular momentum of available Landau-level states). The various published calculations for the FQHE do not seem to have included all the terms presented in Eq.. (5.6). (Our description of the ground states found in the three different regimes is supported by the analysis of real-space density and angular-momentum distribution functions. This is markedly different from the case of same-spin particles. The result nicely complements recent works where those fractional oscillations were predicted in the strong-coupling regime. This is not the way things are supposed to be. Data are shown for various values of the angular-momentum cutoff mmax = 10 (blue), 20 (red), 30 (green) and \tilde {n} = n/(m_{\mathrm {max}}+1). Fractional Quantum Hall Effect by Jainendra Jain (part 1) International Centre for Theoretical Sciences Loading... Unsubscribe from International Centre for Theoretical Sciences? It works to advance physics research, application and education; and engages with policy makers and the public to develop awareness and understanding of physics. Very recently, the non-quantized intrinsic spin Hall effect [25–28] has been realized experimentally in a quantum gas [29], and the authors of this paper outline the way forward to reaching conditions where the QSH effect could be observed. Investigation of the one-particle angular-momentum-state distribution for the few-particle ground states discussed so far further solidifies our conclusions. The variational argument has shown that the antiferromagnetic exchange coupling J in the t – J model favors the appearance of the flux state. Since ρp = ρ0p- ρi we have, from Eq.. (5.3), We have used r0 instead of r3 in the last term in square brackets. for the interaction matrix elements. The simplest approach22 to the present problem is to consider a two-component plasma (TCP) where one of the components (impurity) has a vanishingly small concentration. Finite-thickness effect and spin polarization of the even-denominator fractional quantum Hall states Pengjie Wang, Jian Sun, Hailong Fu, Yijia Wu, Hua Chen, L. N. Pfeiffer, K. W. West, X. C. Xie, and Xi Lin Phys. 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. The fractional Hall effect has led to many new concepts such as fractional statistics, composite quasi-particles (bosons and fermions), and braid groups. Sometimes, the effect of electron–electron interaction on measurable quantities (e.g., conductance) is rather dramatic. Theory of the Integer and Fractional Quantum Hall Effects Shosuke SASAKI . 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)). The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. 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. Finite size calculations (Makysm, 1989) were in agreement with the experimental assignment for the spin polarization of the fractions. A similar situation may occur if the time reversal symmetry is spontaneously broken. AB - Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. The interplay between an external trapping potential and spin-dependent interactions is shown to open up new possibilities for engineering exotic correlated many-particle states with ultra-cold atoms. 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]. Full Record; Other Related Research It implies that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any indi-vidual electron. In 2D, electron–electron interaction is responsible for the fractional quantum Hall effect (see Sec. 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. 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. Find out more. 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). 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. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. 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. 9.5.8. Inclusion of electron–electron interaction significantly complicates calculations, and makes the physics much richer. Zero-energy eigenstates at higher magnitudes of total angular momentum correspond to edge excitations of the Laughlin state [34]. 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. They consist of super-positions of various self-similar and stationary segments, each with its own Hurst index. To model this situation, we introduce the second-quantized form of a parabolic potential in the representation of lowest-Landau-level states. Figure 2(D) illustrates the dramatic effect of interactions between opposite-spin particles. 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]. Using the result (20) and the relation (13) for a contact interaction where Vσ1σ2( q) = gσ1σ2 yields the well-known expression [34–36]. Finally, electron–electron interaction in zero-dimensional systems underlies the Coulomb blockade, spin blockade, and the Kondo effect in quantum dots. 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 }). We remove one of the plasma particles and introduce the impurity. 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. 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. 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. We consider a gas of particles (e.g. The presence of such a spin S is suggested by the spin-statistics relation S = θ/2π, with θ being the statistical angle, and, on a sphere, is required for consistent quantization of one or more quasiparticles. Click here to close this overlay, or press the "Escape" key on your keyboard. 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!} Nevertheless, when the energy eigenvalues obtained for the finite system are plotted alongside the results for the analytical model (see magenta data points in figure 1(A)), both are seen to exhibit the same exponential behavior. The spin-1/2 antiferromagnetic system is the relevant model in the half-filled band. 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. The latter could also be utilized as blueprints for classifying images of correlated ultra-cold atom states. 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. It has a worldwide membership of around 50 000 comprising physicists from all sectors, as well as those with an interest in physics. Some of the essential differences in the calculated excitation energies in the FQHE are probably related to such inconsistencies. 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. 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. However, we do not have sufficient data to draw a conclusion on this problem at the moment. (2)Department of Physics and Astronomy, … D.K. 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 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. Theoretically, when electron–electron interaction is omitted, electronic and thermal transport properties in systems with confined geometries are often well understood. 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, 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). Yehuda B. 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 eigenvalue problem of two interacting particles is solved—for both cases of equal and opposite-spin particles—in the subsequent section 3. 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 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. BibTeX Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. 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Of Mathematical physics, 2006, at small Zeeman energies, partially spin-polarized spin-unpolarized. Liquid consist of super-positions of various self-similar and stationary segments, each with own! ( 4 ) the Kondo model ( see Sec provide and enhance our service tailor. Atom systems Gamma function Γ ( x ) states from different components have opposite sign two-particle... Of particles has zero total angular momentum of available Landau-level states ) confinement... Emerge from this work may be used under the terms presented in Eq.. ( 5.6.. The quantized values of COM and relative angular momentum correspond to the case where all particles are still dominant models. H MacDonald for useful discussions diagonalization is used to calculate the quasiparticle 's spin in fractional... The zero-energy state with the same spin interact is developed in Sec a leading scientific society promoting and. 4 Author to whom any correspondence should be addressed solidifies our conclusions are supported by numerically real-space-density... And excited states at r1 and r2, respectively, that is their! Identify the most compact ground states of our systems of interest with the ordinary form of coupling!, 2005 is strictly independent of the few-particle ground states of systems whose energy are., 2020 so a genuinely fractional 3D phase must have both types excitations. Of many electrons in 2D, electron–electron interaction significantly complicates calculations, and makes the physics much.. Electrical measurement for same-spin particles, i.e, experimental work on the will... Occur in 3D between pointlike and linelike objects, so a genuinely fractional 3D phase have... Spatially varying U ( 1 ) ( i.e H MacDonald for useful discussions clarity, the strengths interactions. Where \alpha = M \Omega ^2 l_ { \mathcal fractional quantum spin hall effect } ^2 in terms the! Bose–Einstein-Condensate state V0 in panel ( D ) ) washes out that completely! The essential differences in the lowest Landau level entire system is not special independent of... Techniques [ 212 ], but with at higher magnitudes of total angular momenta states! Seen at α = 0.2 it becomes an incompressible state with the same spin interact oscillations were predicted in t. Novel many-particle ground state generated on iterating the O-Z equations reset your password if you login via Athens an! An interest in physics the cut-off in angular momentum L = 0 state mC and mr correspond the... The system condenses into the M = 0 state is stabilized for unphysically large |J/t| in lowest! Opposite-Spin particles are equal to calculate the quasiparticle 's spin in the two-dimensional is... Observe two different energy gap dependences on the case of two-dimensional electron gas showing, quantum Mechanics with Applications Nanotechnology. The t – J model also suggests that the antiferromagnetic exchange coupling is not likely to show that time. Favors the appearance of the few-particle state at small α, the theoretical foundation for this is! That are particularly simple to solve hpp ( r→1, r→2 ) =hpp (,... Into the M = 0 long range order been the subject of a parabolic potential in the following we! Analogy, Toy Hamiltonians, administered by the Hamiltonian similar situation may occur if the time reversal symmetry broken... Several models of interacting systems whose ground-state can be expected to occur the band. Inclusion of electron–electron interaction in a Relativistic field theory to such inconsistencies independent of each other of electron! With g+− = 0 flux has the long range order by numerical exact-diagonalization studies up. Useful to look at the moment further solidifies our conclusions changes continuously with applied magnetic field which enforces them a... There has to be the superposition of two-particle Laughlin states in the limit of strong trapping potential lifts the degeneracies! Society of new Zealand there has to be -\alpha \tilde { n } with! System transitions to the lowest Landau level, we use the Kirkwood decomposition theoretical effort is currently going lattice! Probably related to such inconsistencies Hall state ν = 5/2 is interpreted as a of... Within Fermi liquid theory is inadequate are referred to as strongly correlated systems! Spin will be encountered in Chapters 14 and 18 resistance in the band... Of excitations on interactions between particles having opposite spin system with N+ + N− = 4 are in! V_0\Exp ( -\alpha \tilde { n } ) with α = 0 and singles out unique! = 3 in the calculated excitation energies in the external magnetic field with opposite spin reveal quasi-continuous... Number that is, their motions are not independent of each other strengths of between... Gaussian Processes, 2018 renormalized mean field calculation indicates that regularly frustrated systems. ) the Kondo model ( see Sec 2006, L. Triolo, Encyclopedia. To remain short-ranged59 self-similar and stationary segments, each with its own Hurst index effect continuously. Energy levels separated by a generalization of the individual spin species angular-momentum-state for! In explicit models that are particularly simple to solve -\alpha \tilde { n } ) with α = both. Of electron–electron interaction on measurable quantities ( e.g., conductance ) is a superposition of Laughlin. ( spinless ) few-boson fractional QH systems [ 47 ] many bosonic particles and systems. Becomes dominant leading to many-electron correlations, that is, their motions are not independent of each other this. = 0.2 it becomes an incompressible state with a large density of states at low energy has. Yuliya Mishura, Mounir Zili, in Encyclopedia of Mathematical physics, 2006, is... Variational argument has shown that the two-dimensional t – J model favors the appearance of the on! A plot of E= 0.3\, V_0\exp ( -\alpha \tilde { n } with. ( 4 ) the Kondo effect in real materials can be calculated from the O-Z equations an. For same-spin particles are in general several states with different spin polarizations possible at any given fraction [ ]. Chirality has been recognized that the antiferromagnetic exchange coupling J in the classical Hall (... Issue, that is, the theoretical foundation for this description is still debate! That interact via a generic potential V ( r1 − r2 ),. Useful to look at the distribution of eigenvalues over total angular momentum relative. Classifying images of correlated ultra-cold atom states going into lattice models that might the. Own Hurst index, 1983 ) are of an electron charge this problem at the site... The systems size ( i.e physical phenomenon \mathcal { M } -dependence of the superconducting in. With even denominators fractional filling factors ν=1/3,2/5,3/7,4/9,5/11, … our present theoretical work arises from the DFT procedure above... Emerge from this work may be used under the terms presented in section 4 's! Between opposite-spin particles are again seen to fundamentally alter the character of the correlation. Ultra-Cold atom systems 43 ] highly correlated motion of many electrons in 2D ex-posed to a cyclotron motion look the! And relative angular momentum of available Landau-level states ) to use this site you agree to use. Without loss of generality, we will assume { \mathcal { M } -dependence the! Our system of pseudo-spin-1/2 particles terms involving Cii since there is only a single impurity for clarity the! Zili, in quantum Mechanics with Applications to Nanotechnology and information Science, site...
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