| 1. |
- Bergholtz, E. J., et al.
(author)
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Pfaffian quantum Hall state made simple : Multiple vacua and domain walls on a thin torus
- 2006
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 74:8, s. 081308-
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Journal article (peer-reviewed)abstract
- 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.
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| 2. |
- Bergholtz, Emil Johansson, et al.
(author)
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Quantum Hall hierarchy wave functions: From conformal correlators to Tao-Thouless states
- 2008
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In: Physical Review B Condensed Matter. - 0163-1829 .- 1095-3795. ; 77:16, s. 165325-1-165325-9
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Journal article (peer-reviewed)abstract
- 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.
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| 3. |
- Fremling, Mikael, 1985-, et al.
(author)
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Hall viscosity of hierarchical quantum Hall states
- 2014
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 89:12, s. 125303-
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Journal article (peer-reviewed)abstract
- Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter tau, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor nu = 2/5 Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex t plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.
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| 4. |
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| 5. |
- Hansson, Thors Hans, et al.
(author)
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Conformal Field Theory Approach to Abelian and Non-Abelian Quantum Hall Quasielectrons
- 2009
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In: Physical Review Letters. - : American Physical Society. - 0031-9007 .- 1079-7114. ; 102:16, s. 166805-1-166805-4
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Journal article (peer-reviewed)abstract
- The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.
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| 6. |
- Hansson, Thors Hans, et al.
(author)
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Effective field theories for topological states of matter
- 2020
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In: Springer Proceedings in Physics. - Cham : Springer. - 0930-8989 .- 1867-4941. ; 239, s. 1-68
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Journal article (peer-reviewed)abstract
- Since the discovery of the quantum Hall effect in the 1980s it has been clear that there exists states of matter characterized by subtle quantum mechanical effects that renders certain properties surprisingly stable against dirt and noise. The theoretical understanding of these topological quantum phases have continued to develop during the last few decades and it has really surged after the discovery of the time-reversal invariant topological insulators. There are many examples of topological phases that have been important for the theoretical understanding of topological states of matter as well as being of great physical relevance. In this chapter we will focus on some examples that we find particularly enlightening and relevant, but we will not make a complete classification. Some of the most important tools for the understanding of topological quantum matter are based on effective field theory methods. We shall employ two different types of effective field theories. The first, which is valid at intermediate length and time-scales, will not capture the physics at microscopic scales. Such theories are the analogs, for topological phases, of the Ginzburg–Landau theories used to describe the usual symmetry breaking non-topological phases. The second type of theories describe the physics on scales where non-topological gapped states would be very boring, namely at distances and times much larger than the correlation length and the time set by the inverse gap. On these scales everything is independent of any distance and the theories will be topological field theories, which do not describe any dynamics in the bulk, but do carry information about topological properties of the excitations, and also about excitations at the boundaries of the system. Finally, we will also study effective response actions. In a strict sense these are not effective theories, since they do not have any dynamical content, but encode the response of the system to external perturbations, typically an electromagnetic field. As we shall see, however, the effective response action for topological states can be used to extract parts of the dynamic theory through a method called functional bosonization.
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| 7. |
- Hansson, Thors Hans, et al.
(author)
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Effective field theory for a p-wave superconductor in the subgap regime
- 2015
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In: Physical Review B. Condensed Matter and Materials Physics. - 1098-0121 .- 1550-235X. ; 91:7
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Journal article (peer-reviewed)abstract
- We construct an effective field theory for the 2d spinless p-wave paired superconductor that faithfully describes the topological properties of the bulk state, and also provides a model for the subgap states at vortex cores and edges. In particular, it captures the topologically protected zero modes and has the correct ground-state degeneracy on the torus. We also show that our effective field theory becomes a topological field theory in a well defined scaling limit and that the vortices have the expected non-Abelian braiding statistics.
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| 8. |
- Hansson, Thors Hans, et al.
(author)
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Quantum Hall physics : Hierarchies and conformal field theory techniques
- 2017
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In: Reviews of Modern Physics. - : American Physical Society. - 0034-6861 .- 1539-0756. ; 89:2
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Journal article (peer-reviewed)abstract
- The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past 30 years, has generated many ground-breaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the 30 years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of non-Abelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.
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| 9. |
- Hansson, Thors Hans, et al.
(author)
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Quantum Hall quasielectron operators in conformal field theory
- 2009
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In: Physical Review B Condensed Matter. - : American Physical Society. - 0163-1829 .- 1095-3795. ; 80:16, s. 165330-1-165330-22
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Journal article (peer-reviewed)abstract
- In the conformal field theory (CFT) approach to the quantum Hall effect, the multielectron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires nonlocal operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, braiding, and fusion. Here we construct a quasielectron operator satisfying these demands and show that it reproduces known good quasiparticle wave functions, as well as predicts additional ones. In particular, we propose explicit wave functions for quasielectron excitations of the Moore-Read Pfaffian state. Further, this operator allows us to explicitly express the composite fermion wave functions in the positive Jain series in hierarchical form, thus settling a long-time controversy. We also critically discuss the status of the fractional statistics of quasiparticles in the Abelian hierarchical quantum Hall states and argue that our construction of localized quasielectron states sheds new light on their statistics. At the technical level we introduce a generalized normal ordering that allows us to “fuse” an electron operator with the inverse of an hole operator and also an alternative approach to the background charge needed to neutralize CFT correlators. As a result we get a fully holomorphic CFT representation of a large set of quantum Hall wave functions.
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| 10. |
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