arXiv subjects

A. Karlhede

Publications and source records attributed to A. Karlhede.

At least 19 recordsLinked to original sources

arXiv2012

Topological Field Theory for p-wave Superconductors

We propose a topological field theory for a spin-less two-dimensional chiral superconductor that contains fundamental Majorana fields. Due to a fermionic gauge symmetry, the Majorana modes survive as dynamical degrees of freedom only at magnetic vortex cores, and on edges. We argue that these modes have the topological properties pertinent to a p-wave superconductor including the non-abelian braiding statistics, and support this claim by calculating the ground state degeneracy on a torus. We also briefly discuss the connection to the Moore-Read Pfaffian quantum Hall state, and extensions to the spinful case and to three-dimensonal topological superconductors.

cond-mat.supr-con↗

arXiv2009

Tao-Thouless Revisited

It is now established that the state proposed by Tao and Thouless for the fractional quantum Hall effect in 1983, shortly after Laughlin's work, is the ground state in the so-called Tao-Thouless limit, and that it is adiabatically connected to the Laughlin state. We review the interesting history of the Tao-Thouless state, and its generalizations, and discuss its relevance and shortcomings. In particular, we calculate the exclusion statistics of quasiparticles in the Tao-Thouless limit and point out a principal difficulty which prevents calculation of the fractional exchange statistics in this limit.

cond-mat.str-el↗

arXiv2009

Quantum Hall Circle

We consider spin-polarized electrons in a single Landau level on a cylinder as the circumference of the cylinder goes to infinity. This gives a model of interacting electrons on a circle where the momenta of the particles are restricted and there is no kinetic energy. Quantum Hall states are exact ground states for appropriate short range interactions, and there is a gap to excitations. These states develop adiabatically from this one-dimensional quantum Hall circle to the bulk quantum Hall states and further on into the Tao-Thouless states as the circumference goes to zero. For low filling fractions a gapless state is formed which we suggest is connected to the Wigner crystal expected in the bulk.

cond-mat.mes-hall↗

arXiv2008

Quantum Hall system in Tao-Thouless limit

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2π/L_1$, where $L_1$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit, $L_1\to 0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $ν=p/q\le 1$. For odd $q$, the ground state has the same qualitative properties as a bulk ($L_1 \to \infty$) quantum Hall hierarchy state and the lowest energy quasiparticle exitations have the same fractional charges as in the bulk. These states are the $L_1 \to 0$ limits of the Laughlin/Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd $q$, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states, {\it ie} that there is no phase transition as $L_1 \to \infty$ for filling factors where such states can be observed. For even denominator fractions, a phase transition occurs as $L_1$ increases. For $ν=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite $L_1$, this then develops continuously into the composite fermion wave function that is believed to describe the bulk $ν=1/2$ system. The analysis generalizes to non-abelian quantum Hall states.

cond-mat.mes-hall↗

arXiv2008

Quantum Hall wave functions on the torus

We present explicit expressions for a large set of hierarchy wave functions on the torus. Included are the Laughlin states, the states in the positive Jain series, and recently observed states at e.g. $ν= 4/11$. The techniques we use constitute a nontrivial extension of the conformal field theory methods developed earlier to construct the corresponding wave functions in disc geometry.

cond-mat.mes-hall↗

arXiv2008

Microscopic theory of the quantum Hall hierarchy

We solve the quantum Hall problem exactly in a limit and show that the ground states can be organized in a fractal pattern consistent with the Haldane-Halperin hierarchy, and with the global phase diagram. We present wave functions for a large family of states, including those of Laughlin and Jain and also for states recently observed by Pan {\it et. al.}, and show that they coincide with the exact ones in the solvable limit. We submit that they establish an adiabatic continuation of our exact results to the experimentally accessible regime, thus providing a unified approach to the hierarchy states.

cond-mat.mes-hall↗

arXiv2007

Hierarchy wave functions--from conformal correlators to Tao-Thouless states

Laughlin's wave functions, describing the fractional quantum Hall effect at filling factors $ν=1/(2k+1)$, can be obtained as correlation functions in conformal field theory, and recently this construction was extended to Jain's composite fermion wave functions at filling factors $ν=n/(2kn+1)$. Here we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multi-hole states, make the connection to Wen's general classification of abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally we discuss to what extent our wave functions can be described in the language of composite fermions.

cond-mat.mes-hall↗

arXiv2006

The Pfaffian quantum Hall state made simple--multiple vacua and domain walls on a thin torus

We analyze the Moore-Read Pfaffian state on a thin torus. The known six-fold degeneracy is realized by two inequivalent crystalline states with a four- and two-fold degeneracy respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension $2^{n-1}$ for $2n-k$ holes and $k$ particles at fixed positions and assign each a charge $\pm e/4$. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model hamiltonian and a small number of particles supports the presence of a stable phase with degenerate vacua and quarter charged domain walls also away from the thin torus limit. A spin chain hamiltonian encodes the degenerate vacua and the various domain walls.

cond-mat.mes-hall↗

arXiv2006

Phase space topology of a switching current detector

We examine in theory and by numerical simulation, the dynamic process of switching from a zero voltage to a finite voltage state in a Josephson junction circuit. The theoretical model describes small capacitance Josephson junctions which are overdamped at high frequencies, and can be applied to detection of the quantum state of a qubit circuit. We show that the speed and fidelity of the readout are strongly influenced by the topology of the phase space attractors. The readout will be close to optimal when choosing the circuit parameters so as to avoid having an unstable limiting cycle which separates the two basins of attraction.

cond-mat.supr-con↗

arXiv2004

Properties of the doped spin 3/2 Mott insulator near half filling

We develop an exact generalized Bogoliubov transformation for the spin 3/2 Hubbard model with large anti-Hunds rule coupling near half filling. Since our transformation is unitary, we can thereafter employ standard approximate mean field theory methods in the full Hilbert space to analyze the doped Mott insulator, in contrast to a conventional approach based on truncated Hilbert spaces complemented with hard core constraints. The ground state at exactly half filling is an insulating (Mott) singlet, and according to our analysis a non-Fermi liquid order parameter $ Δ$ usually associated with extended s-wave superconductivity, will appear self-consistently as soon as a finite density $ n $ holes are introduced. The non-Fermi liquid behavior is a consequence of the nonlinear nature of the unitary transformation mapping the Mott singlet state to a Fock vacuum which introduces anomalous terms such as $Δn$ in the effective Lagrangian. Our analysis uses an approach that generalizes readily to multi-band Hubbard models and could provide a mechanism whereby a non-Fermi liquid order parameter proportional to density is developed in Mott insulators with locally entangled ground states. For more complicated systems, such an order parameter could coexist naturally with a variety of other order parameters.

cond-mat.str-el↗

arXiv2004

Solitons and Quasielectrons in the Quantum Hall Matrix Model

We show how to incorporate fractionally charged quasielectrons in the finite quantum Hall matrix model.The quasielectrons emerge as combinations of BPS solitons and quasiholes in a finite matrix version of the noncommutative $ϕ^4$ theory coupled to a noncommutative Chern-Simons gauge field. We also discuss how to properly define the charge density in the classical matrix model, and calculate density profiles for droplets, quasiholes and quasielectrons.

cond-mat.str-el↗

arXiv2004

Charge and Current in the Quantum Hall Matrix Model

We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even at the classical level. We calculate the linear response at low momenta -- this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. These results show that the classical quantum Hall matrix models describe important aspects of the dynamics of electrons in the lowest Landau level. In the quantum theory the ordering ambiguities are more severe; we discuss possible strategies, but we have not been able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra.

cond-mat.str-el↗

arXiv2001

Charges and Currents in the Noncommutative Chern-Simons Theory of the QHE

We couple the noncommutative Chern-Simons theory describing the fractional quantum Hall effect to external magnetic and electric potentials, and derive expressions for charge and current densities. To lowest non-trivial order the density operator satisfies the commutator algebra characteristic of the lowest Landau level. We take this as a strong indication that the theory correctly describes point particles in a strong magnetic field. We also obtain the correct QH response to a constant electric field, and density modulations in response to a weak variable magnetic field.

cond-mat.mes-hall↗

arXiv1999

Dynamics of the Compact, Ferromagnetic ν=1 Edge

We consider the edge dynamics of a compact, fully spin polarized state at filling factor $ν=1$. We show that there are two sets of collective excitations localized near the edge: the much studied, gapless, edge magnetoplasmon but also an additional edge spin wave that splits off below the bulk spin wave continuum. We show that both of these excitations can soften at finite wave-vectors as the potential confining the system is softened, thereby leading to edge reconstruction by spin texture or charge density wave formation. We note that a commonly employed model of the edge confining potential is non-generic in that it systematically underestimates the texturing instability.

cond-mat.mes-hall↗

arXiv1998

SU(N) Quantum Hall Skyrmions

We have investigated skyrmions in N-component quantum Hall systems. We find that SU(N) skyrmions are the lowest energy charged excitations for filling factors ν= 1,2,...,N-1 for small enough symmetry breaking terms. N>2 skyrmions can be realized in Si QH systems based on the (110) or (111) interfaces of Si, or perhaps in Si (100) systems, where the spin and valley isospin together provide an SU(4)-symmetry, or in multilayer QH systems. We also present Hartree-Fock results for a phenomenological easy-axis SU(2)-breaking model appropriate to valley degeneracy.

cond-mat.mes-hall↗

arXiv1998

A Field Theory for Partially Polarized Quantum Hall States

We propose a new effective field theory for partially polarized quantum Hall states. The density and polarization for the mean field ground states are determined by couplings to two Chern-Simons gauge fields. In addition there is a $σ$-model field, $\mh$, which is necessary both to preserve the Chern-Simons gauge symmetry that determines the correlations in the ground state, and the global SU(2) invariance related to spin rotations. For states with non zero polarization, the low energy dynamics is that of a ferromagnet. In addition to spin waves, the spectrum contains topological solitons, or skyrmions, just as in the fully polarized case. The electric charge of the skyrmions is given by $Q_{el}=νP Q_{top}$, where $ν$ is the filling fraction, $P$ the magnitude of the polarization, and $Q_{top}$ the topological charge. For the special case of full polarization, the theory involves a single scalar field and a single Chern-Simons field in addition to the $σ$-model field, $\mh$. We also give a heuristic derivation of the model lagrangians for both full and partial polarization, and show that in a mean field picture, the field $\mh$ is necessary in order to take into account the Berry phases originating from rotations of the electron spins.

cond-mat.mes-hall↗

arXiv1998

Effective Action Studies of Quantum Hall Spin Textures

We report on analytic and numerical studies of spin textures in quantum Hall systems using a long-wavelength effective action for the magnetic degrees of freedom derived previously. The majority of our results concern skyrmions or solitons of this action. We have constructed approximate analytic solutions for skyrmions of arbitrary topological and electric charge and derived expressions for their energies and charge and spin radii. We describe a combined shooting/relaxational technique for numerical determination of the skyrmion profiles and present results that compare favorably with the analytic treatment as well as with Hartree-Fock studies of these objects. In addition, we describe a treatment of textures at the edges of quantum Hall systems within this approach and provide details not reported previously.

cond-mat.mes-hall↗

arXiv1997

Reconstruction of the $ν=1$ Quantum Hall Edge

The sharp ν=1 quantum Hall edge present for hard confinement is shown to have two modes that go soft as the confining potential softens. This signals a second order transition to a reconstructed edge that is either a depolarized spin-texture edge or a polarized charge density wave edge.

cond-mat.mes-hall↗