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. In the following, we will focus on the case where all particles are in the lowest Landau level, i.e. Investigation of the one-particle angular-momentum-state distribution for the few-particle ground states discussed so far further solidifies our conclusions. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Figure 1. Traditional many-body perturbation theory, which is developed in Sec. The other states in the low-energy band correspond to edge excitations of this configuration. We would also like to thank M Fleischhauer and A H MacDonald for useful discussions. Cold-atom systems are usually studied while trapped by an external potential of tunable strength. BibTeX The triangular lattice with the next nearest neighbor interaction also shows similar behavior58. 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). 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. It works to advance physics research, application and education; and engages with policy makers and the public to develop awareness and understanding of physics. Theoretically, when electron–electron interaction is omitted, electronic and thermal transport properties in systems with confined geometries are often well understood. 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. 4. (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). 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. For our system of interest, an additional possibility arises from the ability to tune the interaction strength between the two spin components. The other is a kinematical effect and has opposite signs for the quasihole and quasielectron. dimensions. This method might provide relatively good results if the range of the interaction is very large, and in fact, a clear version with due limiting procedure was introduced by Kac, and applied by Lebowitz and Penrose in the 1960s for a microscopic derivation of van der Waals equation, and soon extended by Lieb to quantum systems. Fractional quantum Hall effect: Experimental progress and quantum computing applications ( Nanowerk News ) The Hall effect, discovered in 1879, is observable when a Hall voltage perpendicular to the current is produced across a conductor under a magnetic field. Figure 3. Consider two particles, located at r1 and r2, respectively, that interact via a generic potential V ( r1 −  r2). We construct a class of 2+1 dimensional relativistic quantum field theories which exhibit the fractional quantum Hall effect in the infrared, both in the continuum and on the lattice. N+ + N− = 4). The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional system of electrons. The new densities are ρp = (N-1)/Ωc ρi = 1/Ωc. Copyright © 2021 Elsevier B.V. or its licensors or contributors. 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. dependence on material parameters. 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. None of the individual eigenvalues is strictly independent of the cutoff, which indicates that there are no compact eigenstates. The renormalized mean field calculation indicates that the flux state is stabilized for unphysically large |J/t| in the two-dimensional t – J model56. 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. 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. The paper is organized as follows. 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]. The fractional quantum Hall effect (FQHE) has been the subject of a number of theoretical treatments , . Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Rev. Angular dependent magnetotransport measurements on the fractional quantum Hall (FQHE) states around Landau level filling factor $\\ensuremath{\\nu}=\\frac{3}{2}$ are explained very effectively in terms of composite fermions (CFs) with a spin. A fractional phase in three dimensions must necessarily be a more complex state. 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. 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). The fractional Hall effect has led to many new concepts such as fractional statistics, composite quasi-particles (bosons and fermions), and braid groups. Electron–electron interaction plays a central role in low-dimensional systems. Quantum Spin Hall Effect. To date, there are no observations of fractional analogs of time-reversal-invariant topological insulators, but at least in two dimensions it is clear that such states exist theoretically. The enhancement of the superconducting correlation in the one-dimensional t – J model also suggests that the two-dimensional system is not special. At ν = 1/2, the composite fermion does not see any magnetic flux, that is, (νCF)−1 = 0, whereas at ν ≠ 1/2, (νCF)−1 = |ν−1 − 2| flux quanta are present for the composite fermion. Panel (A) shows the situation where only particles from a single component are present, which is analogous to the previously considered case of spinless bosons [37, 61–63]. in terms of the Euler Gamma function Γ(x). We consider the effect of contact interaction in a prototypical quantum spin Hall system of pseudo-spin-1/2 particles. 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]. Quantum Hall Hierarchy and Composite Fermions. The spin polarization of fractional states was measured experimentally by varying the Zeeman energy by rotating the magnetic field away from the normal (Clarke et al., 1989; Eisenstein et al., 1989) or by applying hydrostatic pressure (Morawicz et al., 1993). â¢ Spin phase transitions in the fractional quantum Hall effect: If electron-electron in-teractions are considered in the LLL, new ground states appear when these particles are occupying certain rational, fractions with odd denominators of the available states. In this final section, we recall some phenomena which have been observed recently in physics laboratories, and which presumably deserve considerable efforts to overcome the heuristic level of explanation. The usual spin s is to be replaced by âs s 0 0, which produces fractional charges by means of the z component of the spin and the Bohr magneton. See figure 2(C). The way indices are distributed in the arguments of the δ-functions in equations (30) and (31) implies that the system's total angular momentum L \equiv \sum _j L_{z j} (cf equation (8b) for the definition of Lz) is a conserved quantity in the presence of interactions. Physics, Columbia University, New York, New York 10027 They consist of super-positions of various self-similar and stationary segments, each with its own Hurst index. 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. The excited states of this liquid consist of peculiar particle-like objects that carry an exact fraction of an electron charge. The DPG sees itself as the forum and mouthpiece for physics and is a non-profit organisation that does not pursue financial interests. This case is illustrated in figure 2(B). 2. A strong effective magnetic field with opposite directions for the two spin states restricts two-dimensional particle motion to the lowest Landau level. Analogous behavior has been discussed previously for ordinary (spinless) few-boson fractional QH systems [64]. In our case depicted in figure 3(A), the ground state in the weak-confinement regime corresponds to a superposition of three-particle Laughlin states for filling factor 1/2 in the individual pseudospin components. Fractional Quantum Hall Effect by Jainendra Jain (part 1) ... spin-liquids, fractional quantum Hall systems, and ultracold atoms in the strong-coupling regime. Note the disappearance of energy gaps and accumulation of states at low energy, reflecting the characteristic features of the opposite-spin two-particle interaction spectrum shown in figure 1(B). Yehuda B. in [39–41], the total angular momenta for states from different components have opposite sign. 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. The second Landau level of graphene is predicted to show more robust fractional quantum Hall effect than the second Landau level of GaAs. 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. Its publishing company, IOP Publishing, is a world leader in professional scientific communications. Click here to close this overlay, or press the "Escape" key on your keyboard. 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. We start by representing the Schrödinger field operator for a particle at position r with spin σ projected onto the lowest spin-related Landau level, where \hat {c}^{\dagger }_{\sigma m} creates a particle in component σ with angular momentum σm in the state \phi ^{(\sigma )}_{0, m}({\bf{ r}})\equiv \left \langle {\bf{ r}} \right |\left (b^\dagger _\sigma \right )^m /\sqrt {m!} The fractional quantum Hall effect (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 â¦ The four-particle Laughlin state is the zero-energy state with the smallest total angular momentum L = 12. Switching on interactions between opposite-spin particles turns crossings into anti-crossings. In the basis of lowest-Landau-level states from the two spin components, the single-particle density matrix of a many-particle state \left | \Phi \right \rangle has matrix elements, In terms of this quantity, we can define the angular-momentum distribution for each spin component, and also the spin-resolved single-particle density profile in real space. Due to the occurrence of level crossings, the character of the lowest-energy (ground) state is found to be different for regimes associated with weak, intermediate and strong confinement. Low-lying energy levels for a system with N+ = N− = 3 in the sector of total angular momentum L = 0. To understand the properties of this system, an important tool is the Gross–Pitaevskii energy functional for the condensate wave function Φ. where the quartic term represents the reduced (mean-field) interaction among particles. 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. Results obtained for systems with N+ + N− = 4 are shown in figure 2. The two-particle problem for particles with the same spin reduces to two independent single-particle problems in the center-of-mass (COM) and relative-coordinate degrees of freedom ( Rσσ and rσσ, respectively) because r1 −  r2 ≡ 2 rσσ. The sharpness of the transitions reflects the existence of level crossings in figure 3(A). Interacting electron systems for which the description within Fermi liquid theory is inadequate are referred to as strongly correlated electron systems. Figures 3(B) and (C) depict situations where interactions between same-spin particles are still dominant. Note the dependence of the eigenvalues on the systems size (i.e. The Ornstein-Zernike (O-Z) relation is. Composite fermions experience an effective magnetic field and form Landau-like levels called Λ levels (ΛLs). 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. 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. Fractional Quantum Hall Effect in a Relativistic Field Theory We construct a class of 2+1 dimensional relativistic quantum field theories which exhibit the fractional quantum Hall effect in the infrared, both in the continuum and on the lattice. J. Weis, in Encyclopedia of Condensed Matter Physics, 2005. 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. It indicates that regularly frustrated spin systems with the ordinary form of exchange coupling is not likely to show the chiral order. 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. 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. 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. (Bernevig and Zhang, PRL, 2006) • The QSH state does not break 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. In the limit of strong trapping potential, the system condenses into the m = 0 state. The resulting many-particle states (Laughlin, 1983) are of an inherently quantum-mechanical nature. 18.2, linked to the book web page, is sometimes inadequate for studying strongly correlated electron systems in low-dimensions, due to lack of an appropriate small parameter. 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. 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. Maude, J.C. Portal, in Semiconductors and Semimetals, 1998. D.K. We remove one of the plasma particles and introduce the impurity. One-particle angular-momentum distribution for pseudo-spin + particles for the ground states of systems whose energy spectra are shown in figure 3. Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan . Therefore, e.g. The braid relations are used to calculate the quasiparticle's spin in the fractional quantum Hall states on Riemann surfaces. 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). Inclusion of electron–electron interaction significantly complicates calculations, and makes the physics much richer. In the following, we focus on the properties of the lowest-energy (ground) state in the different regimes associated with small, intermediate and strong confinement strength for the systems whose energy spectra are shown in figure 3. They are also conveniently calculable from the O-Z equations of an inhomogeneous system. 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. 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]. 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]. Published 13 February 2014 • We observe an exponential dependence of the sorted eigenvalues as a function of the scaled index \tilde {n}=n/(m_{\mathrm {max}}+1), which translates into a power-law density of states. In 2D, electron–electron interaction is responsible for the, Journal of Mathematical Analysis and Applications, Physica A: Statistical Mechanics and its Applications, Theory of Approximate Functional Equations, angle resolved photoemission spectroscopy. 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]. 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. Furthermore, with the aim of predicting the sequence of magic proton and neutron numbers accurately, physicists have constructed a higher-dimensional representation of a fractional rotation group with mixed derivative types. Thus any feasible route toward realizing the fractional QSH effect using a spin-dependent uniform magnetic field [29, 32] should strive to eliminate interactions between the opposite-spin components. 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. Fractional quantum Hall effect Last updated January 14, 2020. Masatoshi IMADA, in Strongly Coupled Plasma Physics, 1990, The possibility of the time reversal and the parity symmetry breaking in strongly correlated electron systems have been proposed53–55. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. 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Same spin interact low energy ): energy spectrum obtained for systems with confined geometries are often well..