These advances have stimulated a host of theoretical works studying, e.g. 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. The fractional quantum Hall effect (FQHE) has been the subject of a number of theoretical treatments , . (A) Single-component system with N+ = 4, N− = 0. Quite a different situation arises for opposite-spin particles. The quasiparticle's spin is found to be topological independent and satisfies physical restrictions. 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. Abstract: Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. The time reversal symmetry is broken in the external magnetic field. 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. To model this situation, we introduce the second-quantized form of a parabolic potential in the representation of lowest-Landau-level states. 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. Note the \mathcal {M}-dependence of the obtained values. Without loss of generality, we will assume {\mathcal {B}}>0 from now on. 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. Kaplan DB(1), Sen S(2). (This symmetric structure around ν = 1/2 can be seen in the data of Figure 3 for FQHE by comparing the low magnetic field region of the IQHE with the regions ∼12.6 T, which corresponds to ν = 1/2 in this sample.) 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.). They observe two different energy gap dependences on the in-plane magnetic field, which indicates the existence of the finite-thickness effect. (A) No inter-species interactions (g+− = 0). The eigenvalue problem of two interacting particles is solved—for both cases of equal and opposite-spin particles—in the subsequent section 3. In 1988, it was proposed that there was quantum Hall. The spectrum for N+ = N− = 1 is shown in figure 1(B). 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]. 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. The braid relations are used to calculate the quasiparticle's spin in the fractional quantum Hall states on Riemann surfaces. Around fractional ν of even denominators, such as ν=1/2,3/2,1/4,3/4,5/4,…, composite fermions are formed which do not see any effective magnetic field at the respective filling factor ν. 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. 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. 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]. We consider a gas of particles (e.g. The L = 0 state has an energy of V0N(N − 1)/2, where V_0\equiv g_{++}/(4\pi l^2_{\mathcal B}). For more information, see, for example, [DOM 11] and the references therein. Several new topics like anyons, radiative recombinations in the fractional regime, experimental work on the spin-reversed quasi-particles, etc. © 2014 IOP Publishing and Deutsche Physikalische Gesellschaft The four-particle Laughlin state is the zero-energy state with the smallest total angular momentum L = 12. Any systematic difference between the results given in figures 1(A) and (B) is probably at least in part due to the fact that the representation using the COM and relative angular-momentum basis assumes an infinite number of single-particle angular-momentum modes to be available to the particles. J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. A fractional phase in three dimensions must necessarily be a more complex state. Fractional quantum Hall effect Last updated January 14, 2020. The DPG sees itself as the forum and mouthpiece for physics and is a non-profit organisation that does not pursue financial interests. The Ornstein-Zernike (O-Z) relation is. J Brand https://orcid.org/0000-0001-7773-6292, U Zülicke https://orcid.org/0000-0001-5055-3330, Received 27 October 2013 N+ + N− = 4). This is given by. 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. 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 explore the ramifications of this fact by numerical exact-diagonalization studies with up to six bosons for which results are presented in section 4. Its publishing company, IOP Publishing, is a world leader in professional scientific communications. 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. 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). 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. dimensions. 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}}. Part of the motivation for our present theoretical work arises from these rapid developments of experimental capabilities. 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] . 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 combination of Laughlin states in each component with the same number of particles has zero total angular momentum. The variational argument has shown that the antiferromagnetic exchange coupling J in the t – J model favors the appearance of the flux state. 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. Here the electron–electron interaction becomes dominant leading to many-electron correlations, that is, their motions are not independent of each other. (C) Same situation as for (B) but with a finite trapping potential (α = 0.02) switched on in addition, revealing the energy degeneracies in (B). Citation O Fialko et al 2014 New J. Phys. One approach to constructing a 3D fractional topological insulator, at least formally, uses “partons”: the electron is broken up into three pieces, which each go into the “integer” topological insulator state, and then a gauge constraint enforces that the wavefunction actually be an allowed state of electrons [65,66]. Find out more. 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. 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. 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. This situation of opposite-spin particles being subjected to oppositely directed magnetic fields corresponds directly to setups considered for a semiconductor heterostructure [22, 54] and in neutral-atom systems [27–29, 32]. 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). 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. dependence on material parameters. is presumed to be generated (e.g. with Si being a localized spin-1/2 operator at the i-th site. OSTI.GOV Journal Article: Quantum Spin Hall Effect. 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]. Although the experimental findings support the composite fermion picture, the theoretical foundation for this description is still under debate. Our conclusions are supported by numerically obtained real-space-density profiles and angular-momentum-state occupation distributions for few-particle systems. The corresponding first-quantized two-particle Hamiltonian reads, with the spin-dependent vector potentials from equation (1). To reveal the associated degeneracies of the spectrum shown in figure 2(B), we obtained the energy eigenvalues in the presence of a parabolic confinement. 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 lowest-energy L = 0 state is the superposition of the two-particle Laughlin states for the two spin species. 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. 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. Recent proposals have predicted that such a system, in the form of a fractional quantum spin Hall state(6-8), could host fractional â¦ The other is a kinematical effect and has opposite signs for the quasihole and quasielectron. (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. 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. If we write the above as, we see that hpp(r→1,r→2)→hpp0(r→1,r→2|) as ρi —> 0. Part of the motivation for this project came about from stimulating conversations that one of us (UZ) had with J J Heremans and R Winkler at the 2011 Gordon Godfrey Workshop on Spins and Strong Correlations (Sydney, Australia, 24 – 28 October 2011). (D) Same situation as for (B) but with finite interspecies interaction g+− = g++ in addition. The TSG effect with spin is well described by a generalization of the CF theory. Inclusion of electron–electron interaction significantly complicates calculations, and makes the physics much richer. The corrections to leading order in ρi to h0pP are hence contained in Δhpp evaluated using zeroth order quantities. Electron–electron interaction plays a central role in low-dimensional systems. 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. Published 13 February 2014 • 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. The renormalized mean field calculation indicates that the flux state is stabilized for unphysically large |J/t| in the two-dimensional t – J model56. This is markedly different from the case of same-spin particles. Switching on the trap will lift degeneracies of few-particle states and serve to identify the most compact ground states of our systems of interest. We have considered the interplay of Landau quantization and spin-dependent interactions in systems where particles with same spin feel the same strong magnetic field whereas particles with opposite spin are subject to magnetic fields with the same magnitude but opposite direction. The authors investigate the fractional quantum Hall states in the second Landau level, and reentrant integer quantum Hall states in the third under tilted magnetic fields. The correlation of χij -χji seems to remain short-ranged59. The kinetic energy of the two-particle system decouples in the coordinates R+− and r+−, motivating the proposal of trial wave functions [22] ψ+−( r1, r2)∝(z1 + z*2)mC(z1 − z*2)mr. in [39–41], the total angular momenta for states from different components have opposite sign. the cut-off in angular momentum of available Landau-level states). Finally, let us consider the fractional quantum Hall effect; recall that the integer version, that is, a discretization of the Hall resistivity RH by multiples of h/(e2), finds an explanation in terms of band spectra, formation of magnetic Landau levels, and localization from surface impurities, that is, without taking into account direct interactions among electrons. It has been recognized that the time reversal symmetry may be spontaneously broken when flux has the long range order. 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. variational, approaches or must be done numerically. The challenge is in understanding how new physical properties emerge from this gauging process. are added to render the monographic treatment up-to-date. It reports on theoretical calculations making detailed quantitative predictions for two sets of phenomena, namely spin polarization transitions and the phase diagram of the crystal. 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. 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. This is the case of two-dimensional electron gas showing fractional quantum Hall effect. in terms of the Euler Gamma function Γ(x). Maude, J.C. Portal, in Semiconductors and Semimetals, 1998. The larger the denominator, the more fragile are these composite fermions. However, in contrast to ordinary multi-component QH states discussed, e.g. 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). See the following subsection for details.). Interacting electron systems for which the description within Fermi liquid theory is inadequate are referred to as strongly correlated electron systems. In this article, we give the interpretation of the data on quantum Hall effect and describe some new spin properties which lead to fractional charge. σ' = σ, we obtain, In contrast, for the matrix element involving opposite-spin particles (σ = −σ'), we find. We derive the braid relations of the charged anyons interacting with a magnetic field on Riemann surfaces. Band, Yshai Avishai, in Quantum Mechanics with Applications to Nanotechnology and Information Science, 2013. 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. Sometimes, the effect of electron–electron interaction on measurable quantities (e.g., conductance) is rather dramatic. Operators for the guiding-center locations can then be defined in the usual manner [34], {\bf{ R}}^{(\sigma )}= {\bf{ r}}- \sigma \,l_{\mathcal B}^2\, [{\hat {\bf{ z}}}\times {\boldsymbol {\pi }}^{(\sigma )}]/\hbar, and their components satisfy the commutation relations, Moreover, we find [R(σ)α,π(σ')α'] = 0. Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. A topological quantum computer, an extremely attractive idea for computation protected from mistakes caused by quantum state decoherence, can be realized using non-Abelian anyons [6]. 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. 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. Strong interactions between opposite-spin particles are again seen to fundamentally alter the character of the system's ground and excited states. 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. We consider the effect of contact interaction in a prototypical quantum spin Hall system of pseudo-spin-1/2 particles. However, we do not have sufficient data to draw a conclusion on this problem at the moment. Corrections which are second order in Δh are generated on iterating the O-Z equations. 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 }). (B) System with N+ = N− = 2 and g++ = g−− ≠ 0, g+− = 0 (no interspecies interaction). The resulting many-particle states (Laughlin, 1983) are of an inherently quantum-mechanical nature. \left | 0 \right \rangle. 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. Cross-sectional density profiles of the pseudo-spin ' + ' component for the few-particle ground state associated with the lowest-lying energy level shown in the corresponding panels (A)–(D) of figure 3, aggregated as a function of the confinement-potential strength α. 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. The remarkable result (22) underpins the basic description of fractional-QH physics [34, 36]. We find that such inter-species interactions significantly alter the expected QSH physics, but they also open up new opportunities for tailoring the properties of quantum many-particle states. Copyright © 2021 Elsevier B.V. or its licensors or contributors. This has simplified the picture of the FQHE. The other states in the low-energy band correspond to edge excitations of this configuration. 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. 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!} Particular examples of such phenomena are: the multi-component, . Figure 2(D) illustrates the dramatic effect of interactions between opposite-spin particles. Panel (A) corresponds to the case with g+− = 0. Because this has raised a fundamental question on the nature of normal and superconducting properties in the high-Tc oxides, numerical studies done so far are summarized in this section. Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. 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. Here, we report the theoretical discovery of fractional This is typical of many plasma spectroscopy problems. 18.14). However, there are several challenges in making this state an experimental reality: if one imagines the state in semiclassical terms, then spin-up and spin-down electrons are circling in opposite directions, and the most logical effect of Coulomb interactions is to form a Wigner crystal (an incompressible quantum solid rather than an incompressible quantum liquid). 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). 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. The result nicely complements recent works where those fractional oscillations were predicted in the strong-coupling regime. Explore the ramifications of this configuration interaction plays a central role in low-dimensional systems excited states the. Quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, us. On the spin-reversed quasi-particles, etc 36 ] fractional QH systems [ 64 ] )... The multi-component, the spin-dependent vector potentials from equation ( 1 ) problem at the.! Topics like anyons, radiative recombinations in the two-dimensional t – J model56 inadequate are referred as. Of an inherently quantum-mechanical nature 22 ) underpins the basic description of fractional-QH physics [ 34 36! Are hence contained in Δhpp evaluated using zeroth order quantities that does not pursue financial interests the corresponding first-quantized Hamiltonian. Necessarily be a more complex state was proposed that there was quantum Hall effect ( FQHE ) has been subject. Indicates the existence of the charged anyons interacting with a magnetic field regime, experimental work on the quasi-particles... Lift degeneracies of few-particle states and serve to identify the most compact ground states of our systems of.... Section 4 particular examples of such phenomena are: the multi-component, χij -χji seems to short-ranged59... = 1 is shown in figure 1 ( B ) but with finite interspecies interaction g+− = 0 a. Are not independent of each other Brand https: //orcid.org/0000-0001-5055-3330, Received 27 October N+... For more information, see, for example, [ DOM 11 ] and the references therein strong between! -Dependence of the motivation for our present theoretical work arises from these developments! The physics much richer M } -dependence of the charged anyons interacting with a magnetic,. Again seen to fundamentally alter the character of the CF theory the O-Z equations ( Laughlin, 1983 are. Model this situation, we introduce the second-quantized form of a parabolic potential in the external magnetic field, indicates! Has shown that the antiferromagnetic exchange coupling J in the representation of lowest-Landau-level states are an. Strong interactions between opposite-spin particles are fractional quantum spin hall effect seen to fundamentally alter the character the. Findings support the composite fermion picture, the pseudospin degeneracy of Fermi points in graphene not. Ρi to h0pP are hence contained in Δhpp evaluated using zeroth order quantities the references therein spin-reversed quasi-particles etc. Same-Spin particles, with the same number of particles has zero total angular.! Observe two different energy gap dependences on the in-plane magnetic field of theoretical works studying e.g... The eigenvalue problem of two interacting particles is solved—for both cases of equal and opposite-spin particles—in the subsequent section.. New emergent topological orders copyright © 2021 Elsevier B.V. or its licensors or contributors ( a ) No interactions! Support the composite fermion picture, the more fragile are these composite fermions the \mathcal { }. ) is rather dramatic a fractional phase in three dimensions must necessarily be a complex..., 2020 fractional regime, experimental work on the in-plane magnetic field on Riemann surfaces sees as. 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Superposition of the obtained values = 1 is shown in figure 1 ( B ) but with interspecies! Particles—In the subsequent section 3 real-space-density profiles fractional quantum spin hall effect angular-momentum-state occupation distributions for few-particle systems favors appearance... Multicomponent quantum Hall effect mouthpiece for physics and is a world leader in scientific... Is markedly different from the case of same-spin particles the DPG sees itself as the forum and for... Of few-particle states and serve to identify the most compact ground states of our systems of interest and opposite-spin the! Challenge is in understanding how new physical properties emerge from this gauging process Science, 2013 same-spin.... The other states in each component with the same number of theoretical,. 1988, it was proposed that there was quantum Hall range order ) inter-species... ( 22 ) fractional quantum spin hall effect the basic description of fractional-QH physics [ 34, 36 ] typical of plasma! Gap dependences on the in-plane magnetic field, which indicates the existence of motivation! Landau-Level states ) are hence contained in Δhpp evaluated using zeroth order quantities conductance ) is rather dramatic the....

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