| 1. |
- Ardonne, Eddy, et al.
(author)
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Classification of Metaplectic Fusion Categories
- 2021
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In: Symmetry. - : MDPI AG. - 2073-8994. ; 13:11
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Journal article (peer-reviewed)abstract
- In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for so(2p+1)2. These categories describe non-abelian anyons dubbed ‘metaplectic anyons’. We obtain explicit expressions for all the F- and R-symbols. Based on these, we conjecture a classification for their monoidal equivalence classes from an analysis of their gauge invariants and define a function which gives us the number of classes.
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| 2. |
- Ardonne, Eddy, et al.
(author)
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Classification of metaplectic modular categories
- 2016
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In: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 466, s. 141-146
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Journal article (peer-reviewed)abstract
- We obtain a classification of metaplectic modular categories: every metaplectic modular category is a gauging of the particle hole symmetry of a cyclic modular category. Our classification suggests a conjecture that every weakly-integral modular category can be obtained by gauging a symmetry (including the fermion parity) of a pointed (super-)modular category.
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| 3. |
- Ardonne, Eddy, et al.
(author)
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Degeneracy of non-Abelian quantum Hall states on the torus : domain walls and conformal field theory
- 2008
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In: Journal of Statistical Mechanics. - 1742-5468. ; , s. P04016-
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Journal article (peer-reviewed)abstract
- We analyze the non-Abelian Read–Rezayi quantum Hall states on the torus, where it is natural to employ a mapping of the many-body problem onto a one-dimensional lattice model. On the thin torus—the Tao–Thouless (TT) limit—the interacting many-body problem is exactly solvable. The Read–Rezayi states at filling ν = k/(kM+2) are known to be exact ground states of a local repulsive k+1-body interaction, and in the TT limit this is manifested in that all states in the ground state manifold have exactly k particles on any kM+2 consecutive sites. For M \neq 0 the two-body correlations of these states also imply that there is no more than one particle on M adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domain walls between the ground states, and we show that the number of distinct domain wall patterns gives rise to the nontrivial degeneracies, required by the non-Abelian statistics of these states. In the second part of the paper we consider the quasihole degeneracies from a conformal field theory (CFT) perspective, and show that the counting of the domain wall patterns maps one to one on the CFT counting via the fusion rules. Moreover we extend the CFT analysis to topologies of higher genus
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| 4. |
- Budich, Jan Carl, et al.
(author)
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Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D
- 2013
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 88:7, s. 075419-
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Journal article (peer-reviewed)abstract
- Topological superconductors in one spatial dimension exhibiting a single Majorana bound state at each end are distinguished from trivial gapped systems by aZ(2) topological invariant. Originally, this invariant was calculated by Kitaev in terms of the Pfaffian of the Majorana representation of the Hamiltonian: The sign of this Pfaffian divides the set of all gapped quadratic forms of Majorana fermions into two inequivalent classes. In the more familiar Bogoliubov de Gennes mean-field description of superconductivity, an emergent particle-hole symmetry gives rise to a quantized Zak-Berry phase, the value of which is also a topological invariant. In this work, we explicitly show the equivalence of these two formulations by relating both of them to the phase winding of the transformation matrix that brings the Majorana representation matrix of the Hamiltonian into its Jordan normal form.
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| 5. |
- Budich, Jan Carl, et al.
(author)
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Fractional topological phase in one-dimensional flat bands with nontrivial topology
- 2013
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 88:3, s. 035139-
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Journal article (peer-reviewed)abstract
- We consider a topologically nontrivial flat-band structure in one spatial dimension in the presence of nearest-and next-nearest-neighbor Hubbard interaction. The noninteracting band structure is characterized by a symmetry-protected topologically quantized Berry phase. At certain fractional fillings, a gapped phase with a filling-dependent ground-state degeneracy and fractionally charged quasiparticles emerges. At filling 1/3, the ground states carry a fractional Berry phase in the momentum basis. These features at first glance suggest a certain analogy to the fractional quantum Hall scenario in two dimensions. We solve the interacting model analytically in the physically relevant limit of a large band gap in the underlying band structure, the analog of a lowest Landau level projection. Our solution affords a simple physical understanding of the properties of the gapped interacting phase. We pinpoint crucial differences to the fractional quantum Hall case by studying the Berry phase and the entanglement entropy associated with the degenerate ground states. In particular, we conclude that the fractional topological phase in one-dimensional flat bands is not a one-dimensional analog of the two-dimensional fractional quantum Hall states, but rather a charge density wave with a nontrivial Berry phase. Finally, the symmetry-protected nature of the Berry phase of the interacting phase is demonstrated by explicitly constructing a gapped interpolation to a state with a trivial Berry phase.
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| 6. |
- Budich, Jan Carl, et al.
(author)
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Topological invariant for generic one-dimensional time-reversal-symmetric superconductors in class DIII
- 2013
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 88:13, s. 134523-
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Journal article (peer-reviewed)abstract
- A one-dimensional time-reversal-symmetric topological superconductor (symmetry class DIII) features a single Kramers pair of Majorana bound states at each of its ends. These holographic quasiparticles are non-Abelian anyons that obey Ising-type braiding statistics. In the special case where an additional U (1) spin rotation symmetry is present, this state can be understood as two copies of a Majorana wire in symmetry class D, one copy for each spin block. We present a manifestly gauge invariant construction of the topological invariant for the generic case, i.e., in the absence of any additional symmetries like spin rotation symmetry. Furthermore, we show how the presence of inversion symmetry simplifies the calculation of the topological invariant. The proposed scheme is suitable for the classification of both interacting and disordered systems and allows for a straightforward numerical evaluation of the invariant since it does not rely on fixing a continuous phase relation between Bloch functions. Finally, we apply our method to compute the topological phase diagram of a Rashba wire with competing s-wave and p-wave superconducting pairing terms.
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| 7. |
- Davenport, Simon C., et al.
(author)
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Spin-singlet Gaffnian wave function for fractional quantum Hall systems
- 2013
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 87:4, s. 045310-
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Journal article (peer-reviewed)abstract
- 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."
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| 8. |
- Edvardsson, Elisabet, et al.
(author)
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Sensitivity of non-Hermitian systems
- 2022
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In: Physical Review B. - 2469-9950 .- 2469-9969. ; 106:11
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Journal article (peer-reviewed)abstract
- Understanding the extreme sensitivity of the eigenvalues of non-Hermitian Hamiltonians to the boundary conditions is of great importance when analyzing non-Hermitian systems, as it appears generically and is intimately connected to the skin effect and the breakdown of the conventional bulk boundary correspondence. Here we describe a method to find the eigenvalues of one-dimensional one-band models with arbitrary boundary conditions. We use this method on several systems to find analytical expressions for the eigenvalues, which give us conditions on the parameter values in the system for when we can expect the spectrum to be insensitive to a change in boundary conditions. By stacking one-dimensional chains, we use the derived results to find corresponding conditions for insensitivity for some two-dimensional systems with periodic boundary conditions in one direction. This would be hard by using other methods to detect skin effect, such as the winding of the determinant of the Bloch Hamiltonian. Finally, we use these results to make predictions about the (dis)appearance of the skin effect in purely two-dimensional systems with open boundary conditions in both directions.
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| 9. |
- Gils, C., et al.
(author)
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Anyonic quantum spin chains : Spin-1 generalizations and topological stability
- 2013
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In: Physical Review B. Condensed Matter and Materials Physics. - : American Physical Society. - 1098-0121 .- 1550-235X. ; 87:23, s. 235120-
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Journal article (peer-reviewed)abstract
- There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called SU(2)(k) anyons, well-known deformations of SU(2), in which only the first k + 1 angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S = 1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter k when considering su(2)(k) anyonic theories with k >= 5, as well as for the special case of the su(2)(4) theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-1/2 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.
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| 10. |
- Kjäll, Jonas, et al.
(author)
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Matrix product state representation of quasielectron wave functions
- 2018
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In: Journal of Statistical Mechanics-Theory and Experiment. - : IOP Publishing. - 1742-5468.
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Journal article (peer-reviewed)abstract
- Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.
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