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  • 1.
    Bergholtz, E. J.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Kailasvuori, J.
    Stockholm University, Faculty of Science, Department of Physics.
    Wikberg, Emma
    Stockholm University, Faculty of Science, Department of Physics.
    Hansson, Thors Hans
    Stockholm University, Faculty of Science, Department of Physics.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus2006In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 74, no 8, 081308- p.Article in journal (Refereed)
    Abstract [en]

    We analyze the Moore-Read Pfaffian state on a thin torus. The known sixfold degeneracy is realized by two inequivalent crystalline states with a four- and twofold degeneracy, respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension 2n−1 for 2n−k holes and k particles at fixed positions and assign each a charge ±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 support 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.

  • 2.
    Bergholtz, Emil Johansson
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Hansson, Hans
    Stockholm University, Faculty of Science, Department of Physics.
    Hermanns, Maria
    Stockholm University, Faculty of Science, Department of Physics.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Microscopic theory of the quantum Hall hierarchy2007In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 99, no 25, 256803-1-256803-4 p.Article in journal (Refereed)
    Abstract [en]

    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 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.

  • 3.
    Bergholtz, Emil Johansson
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Hansson, Thors Hans
    Stockholm University, Faculty of Science, Department of Physics.
    Hermanns, Maria
    Stockholm University, Faculty of Science, Department of Physics.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Viefers, Susanne
    Quantum Hall hierarchy wave functions: From conformal correlators to Tao-Thouless states2008In: Physical Review B Condensed Matter, ISSN 0163-1829, E-ISSN 1095-3795, Vol. 77, no 16, 165325-1-165325-9 p.Article in journal (Refereed)
    Abstract [en]

    Laughlin’s wave functions, which describe the fractional quantum Hall effect at filling factorsν=1/(2k+1), can be obtained as correlation functions in a 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, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole 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.

  • 4.
    Bergholtz, Emil
    et al.
    Max Planck Institute for the Physics of Complex Systems, Dresden.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Quantum Hall Circle2009In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, Vol. P04015Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Hansson, T. H.
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Sato, Masatoshi
    Topological field theory for p-wave superconductors2012In: New Journal of Physics, ISSN 1367-2630, E-ISSN 1367-2630, Vol. 14, 063017- p.Article in journal (Refereed)
    Abstract [en]

    We propose a topological field theory for a spinless two-dimensional (2D) chiral superconductor (SC) 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 SC including the non-Abelian braiding statistics, and we 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 spinfull case and to 3D topological SCs.

  • 6.
    Hermanns, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Suorsa, Juha
    Bergholtz, Emil Johansson
    Stockholm University, Faculty of Science, Department of Physics.
    Hansson, Thors Hans
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Quantum Hall wave functions on the torus2008In: Physical Review B Condensed Matter, ISSN 0163-1829, E-ISSN 1095-3795, Vol. 77, no 12, 125321-1-125321-16 p.Article in journal (Refereed)
    Abstract [en]

    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 disk geometry.

  • 7.
    J. Bergholtz, Emil
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Quantum Hall system in Tao-Thouless limit2008In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 77, no 15, 155308- p.Article in journal (Refereed)
    Abstract [en]

    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/L1, where L1 is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit L1→0, the interacting many-electron problem is exactly diagonalized at any rational filling factor ν=p/q≤1. For odd q, the ground state has the same qualitative properties as a bulk (L1→∞) quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the L1→0 limits of the Laughlin and 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—i.e., that there is no phase transition as L1→∞ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1 increases. For ν=1/2 this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1; 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.

  • 8.
    Kardell, Mikael
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Exclusion statistics for quantum Hall states in the Tao-Thouless limit2011In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, E-ISSN 1742-5468, P02037Article in journal (Refereed)
    Abstract [en]

    We consider spin-polarized Abelian quantum Hall states in the Tao-Thouless limit, i.e. on a thin torus. For any filling factor nu = p/q a well-defined sector of low energy states is identified and the exclusion statistics of the excitations is determined. We study numerically, at and near nu = 1/3 and 2/5, how the low energy states develop as one moves away from the Tao-Thouless limit towards the physical regime. We find that the lowest energy states in the physical regime develop from states in the low energy sector but that the exclusion statistics is modified.

  • 9.
    Wikberg, Emma
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Bergholtz, Emil J.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Spin chain description of rotating bosons at v = 12009In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, E-ISSN 1742-5468, P07038- p.Article in journal (Refereed)
    Abstract [en]

    We consider bosons at Landau level filling ν = 1 on a thin torus. In analogy with previous work on fermions at filling ν = 1/2, we map the low-energy sector onto a spin-1/2 chain. While the fermionic system may realize the gapless XY phase, we show that typically this does not happen for the bosonic system. Instead, both delta function and Coulomb interaction lead to gapped phases in the bosonic system, and in particular we identify a phase corresponding to the non-Abelian Moore–Read state. In the spin language, the Hamiltonian is dominated by a ferromagnetic next-nearest-neighbor interaction, which leads to a description consistent with the non-trivial degeneracies of the ground and excited states of this phase of matter. In addition we comment on the similarities and differences of the two systems mentioned above and fermions at ν = 5/2.

  • 10.
    Wikberg, Emma
    et al.
    Stockholm University, Faculty of Science, Department of Physics.
    Larson, Jonas
    Stockholm University, Faculty of Science, Department of Physics.
    Bergholtz, Emil J.
    Karlhede, Anders
    Stockholm University, Faculty of Science, Department of Physics.
    Fractional domain walls from on-site softening in dipolar bosons2012In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 85, no 3, 033607- p.Article in journal (Refereed)
    Abstract [en]

    We study dipolar bosons in a 1D optical lattice and identify a region in parameter space-strong coupling but relatively weak on-site repulsion-hosting a series of stable charge-density-wave (CDW) states whose low-energy excitations, built from fractional domain walls, have remarkable similarities to those of non-Abelian fractional quantum Hall states. Here, a conventional domain wall between translated CDW's may be split by inserting strings of degenerate, but inequivalent, CDW states. Outside these insulating regions, we find numerous supersolids as well as a superfluid regime. The mentioned phases should be accessible experimentally and, in particular, the fractional domain walls can be created in the ground state using single-site addressing, i.e., by locally changing the chemical potential.

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