Topological stability of the edge states is investigated for non-Hermitian
systems. We examine two classes of non-Hermitian Hamiltonians supporting real
bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians.
As an SU(1,1) Hamiltonian, the tight-binding model on the honeycomb lattice
with imaginary on-site potentials is examined. Edge states with ReE=0 and their
topological stability are discussed by the winding number and the index
theorem, based on the pseudo-anti-Hermiticity of the system. As a higher
symmetric generalization of SU(1,1) Hamiltonians, we also consider SO(3,2)
models. We investigate non-Hermitian generalization of the Luttinger
Hamiltonian on the square lattice, and that of the Kane-Mele model on the
honeycomb lattice, respectively. Using the generalized Kramers theorem for the
time-reversal operator Theta with Theta^2=+1 [M. Sato et al., arXiv:1106.1806],
we introduce a time-reversal invariant Chern number from which topological
stability of gapless edge modes is argued.Comment: 29 pages, 19 figures, typos fixe
For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy
when the system has a half-odd-integer spin and the time reversal operator
obeys \Theta^2=-1, but no such a degeneracy exists when \Theta^2=+1. Here we
point out that for non-hermitian systems, there exists a degeneracy similar to
Kramers even when \Theta^2=+1. It is found that the new degeneracy follows from
the mathematical structure of split-quaternion, instead of quaternion from
which the Kramers degeneracy follows in the usual hermitian cases. Furthermore,
we also show that particle/hole symmetry gives rise to a pair of states with
opposite energies on the basis of the split quaternion in a class of
non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN
Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2
Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.Comment: 40 pages, 2 figures; typos fixed, references adde
We construct higher dimensional quantum Hall systems based on fuzzy spheres. It is shown that fuzzy spheres are realized as spheres in colored monopole backgrounds. The space noncommutativity is related to higher spins which is originated from the internal structure of fuzzy spheres. In 2k-dimensional quantum Hall systems, Laughlin-like wave function supports fractionally charged excitations, q = m − 1 2 k(k+1) (m is odd). Topological objects are (2k − 2)-branes whose statistics are determined by the linking number related to the general Hopf map. Higher dimensional quantum Hall systems exhibit a dimensional hierarchy, where lower dimensional branes condense to make higher dimensional incompressible liquid.
It is well-known that coordinates of a charged particle in a monopole background become noncommutative. In this paper, we study the motion of a charged particle moving on a supersphere in the presence of a supermonopole. We construct a supermonopole by using a supersymmetric extension of the first Hopf map. We investigate algebras of angular momentum operators and supersymmetry generators. It is shown that coordinates of the particle are described by fuzzy supersphere in the lowest Landau level. We find that there exist two kinds of degenerate wavefunctions due to the supersymmetry. Ground state wavefunctions are given by the Hopf spinor and we discuss their several properties.
Supersymmetric quantum-Hall liquids are constructed on a supersphere in a supermonopole background. We derive a supersymmetric generalization of the Laughlin wave function, which is a ground state of a hard-core OSp(1/2) invariant Hamiltonian. We also present excited topological objects, which are fractionally charged deficits made by super Hall currents. Several relations between quantum-Hall systems and their supersymmetric extensions are discussed.
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