Particle-hole mirror symmetries around the half-filled shell: Quantum numbers and algebraic structure of composite fermions

Article

关键词: HIERARCHY;    STATES;   

DOI  :  10.1103/PhysRevB.98.115140

来源: SCIE

【 摘 要 】

Composite fermions (CFs) of the fractional quantum Hall effect (FQHE) are described as spherical products of electron and vortex spinors, built from underlying L = 1/2 ladder operators aligned so that the spinor angular momenta, L-e and L-v, are maximal. We identify the CF's quantum numbers as the angular momentum L in (LeLv)L, its magnetic projection m(L), the electron number N [L-v = (N - 1)/2], and magnetic v spin, m(v) = L-e - L-v. Translationally invariant FQHE states are formed by fully filling p subshells with their respective CFs, in order of ascending L for fixed L-e and L-v, beginning with the lowest allowed value, L = vertical bar m(v)vertical bar. CF subshells are contained entirely within the first Landau level (FLL). Alternatively, we provide an equivalent hierarchical wave function in which the underlying objects are vortices with L-v = p/2, correlated pairwise via (L) over right arrow (vi) center dot (L) over right arrow (vj). We show that CFs can be written as a valence operator carrying the angular momentum quantum numbers L, m(v), m(L) acting on a scalar half-filled intrinsic state. This scalar state serves as the vacuum for the valence electron (b(dagger), (b) over tilde) and vortex (v(dagger),(v) over tilde) ladder operators that create FQHE states. With respect to this vacuum, FQHE states can be grouped into v-spin multiplets mirror symmetric around m(v) = 0, in which N is held constant. m(v) not equal 0 states have a net electron particle or hole number. Particle-hole conjugation with respect to this vacuum is identified as the mirror symmetry relating FQHE states of the same N but distinct fillings v = p/(2p + 1) and (v) over bar = p/(2p - 1), e.g., 2/5 <-> 2/3. Alternatively, mirror symmetric v-spin multiplets can be constructed in which the magnetic field strength is held fixed: the valence states are electron particle-vortex hole excitations relative to the half-filled vacuum (m(v) > 0) and their mirror conjugates (m(v) < 0). Multiplet members are linked by the v-spin raising/lowering operators, <(S)over cap>(v)(+/-). Particle-hole (PH) symmetry-relating the N-particle FQHE state Psi of filling v = p/(2p + 1) to the (N) over bar -particle state (Psi) over bar of filling (v) over bar = (p + 1)/(2p + 1), e.g., 2/5 <-> 3/5-is shown to be equivalent to electron-vortex exchange, b(vertical bar) <-> v(dagger) and (b) over tilde <-> (v) over tilde The N-particle states Psi and (S) over cap (Psi) over bar are connected by this mirror symmetry. In this construction, (N) over bar - N CFs of the state (Psi) over bar occupy an extra zero-mode subshell that is annihilated by (S) over cap (v)(-). We link this structure, familiar from supersymmetric quantum mechanics, to the CF Pauli Hamiltonian, which we show is isospectral, quadratic in the v-spin raising and lowering operators (S) over cap (v)(+/-), and fourfold degenerate in Psi, (S) over cap (v)(-) Psi, (Psi) over bar, and (S) over cap (v)(-) (Psi) over bar. On linearization, it takes a Dirac form similar to that found in the integer quantum Hall effect (IQHE).

【 授权许可】

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