Spin Hamiltonian for two singlet states

Simon C. Davenport, Eddy Ardonne, Nicolas Regnault +1 more

We characterize in detail a wave function conceivable in fractional quantum Hall systems where a spin or equivalent degree of freedom is present. This wave function combines the properties of two previously proposed quantum Hall wave functions, namely the non-Abelian spin-singlet state and the nonunitary Gaffnian wave function. This is a spin-singlet generalization of the spin-polarized Gaffnian, which we call the ``spin-singlet Gaffnian'' (SSG). In this paper we present evidence demonstrating that the SSG corresponds to the ground state of a certain local Hamiltonian, which we explicitly construct, and, further, we provide a relatively simple analytic expression for the unique ground-state wave functions, which we define as the zero energy eigenstates of that local Hamiltonian. In addition, we have determined a certain nonunitary, rational conformal field theory which provides an underlying description of the SSG and we thus conclude that the SSG is ungapped in the thermodynamic limit. In order to verify our construction, we implement two recently proposed techniques for the analysis of fractional quantum Hall trial states: The ``spin dressed squeezing algorithm,'' and the ``generalized Pauli principle.''

Muzaffar Qadir Lone, Amit Dey, Sreenija Yarlagadda

Rogers, Nick

In this thesis I present a theory of a macroscopic singlet-triplet qubit in quantum dots embedded in nanowires, each containing 4 electrons and together simulating an artficial Haldane gap material. A Haldane gap material exhibits a 4-fold degenerate ground state separated by an energy gap from excitations. The ground state is equivalent to a degenerate spin-singlet and -triplet state. The 4 degenerate states exhibit the characteristics of spins-1/2 localized on either end of the chain. These states may be used as a coded qubit for quantum information processing. Using the effective mass approximation, I calculate single-particle energy levels of one and two quantum dots in a quantum wire. Using these energy levels I compute the Coulomb matrix elements of the interacting Hamiltonian. Using configuration interaction I demonstrate that the ground state of a quantum dot with 4 electrons is a spin-1 state. I then show that the two dot system behaves approximately like two spin-1 objects interacting via an antiferromagnetic Heisenberg Hamiltonian. While the Heisenberg model is approximate, the two dots have a spin-0 ground-state, indicating antiferromagnetic coupling. I then present a simpler spin model to illustrate the physical parameters which control this interaction. Finally, I present a brief solution to the Heisenberg Hamiltonian for finite spin-chains, and show how one can manipulate the singlet-triplet combined ground state of the spin-chain via localized magnetic field, realizing a singlet-triplet qubit in a macroscopic semiconductor device.

W.J. Caspers

Michael Janas, Michael E. Cuffaro, Michel Janssen B

Per-Olof Westlund, Håkan Wennerström

We analyze the spin coincidence experiment considered by Bell in the derivation of Bells theorem. We solve the equation of motion for the spin system with a spin Hamiltonian, Hz, where the magnetic field is only in the z-direction. For the specific case of the coincidence experiment where the two magnets have the same orientation the Hamiltonian Hz commutes with the total spin Iz, which thus emerges as a constant of the motion. Bells argument is then that an observation of spin up at one magnet A necessarily implies spin down at the other B. For an isolated spin system A-B with classical translational degrees of freedom and an initial spin singlet state there is no force on the spin particles A and B. The spins are fully entangled but none of the spin particles A or B are deflected by the Stern-Gerlach magnets. This result is not compatible with Bells assumption that spin 1/2 particles are deected in a Stern-Gerlach device. Assuming a more realistic Hamiltonian Hz + Hx including a gradient in x direction the total Iz is not conserved and fully entanglement is not expected in this case. The conclusion is that Bells theorem is not applicable to spin coincidence measurement originally discussed by Bell.