THEORETICAL ASPECTS OF QUANTUM HALL EFFECT AND TWO-DIMENSIONAL CFT

1992 ◽  
Vol 06 (11n12) ◽  
pp. 2217-2239 ◽  
Author(s):  
G. CRISTOFANO ◽  
G. MAIELLA ◽  
R. MUSTO ◽  
F. NICODEMI
Keyword(s):  
Quantum Hall ◽  
Coulomb Gas ◽  
Topological Order ◽  
Two Dimensional ◽  
Edge States ◽  
Conformal Field ◽  
Highest Weight ◽  
Finite Set ◽  
Magnetic Translation

A review of the QHE is presented where the emphasis is placed on the role of the magnetic translation group and of the related topological properties. After a presentation of general experimental and theoretical features we briefly summarize known results of two-dimensional Conformal Field Theory relevant for the QHE. Then we show how to evaluate groundstate wave functions on the plane and on the torus by the use of CFT techniques. In the latter case it is shown how for filling ν=1/m a consistent description is achieved by means of a finite set of Coulomb Gas Vertex Operators. They describe (fractional) charged particles with the associated quantized magnetic flux (“anyons"). Furthermore, from these vertices and relative highest weight states, one does find that the g.s.w.f. for the torus should be m-fold degenerate showing, also, the role of a new kind of long-range topological order recently advocated. Then we show that the presence of m sectors of edge states for a cylinder is strictly related to such a degeneracy, and their relation with a Kac-Moody algebra is discussed.

HALL CONDUCTANCE AND EDGE STATES IN THE COULOMB GAS VERTEX OPERATOR FORMALISM

1991 ◽  
Vol 06 (32) ◽  
pp. 2985-2993 ◽  
Author(s):  
G. CRISTOFANO ◽  
G. MAIELLA ◽  
R. MUSTO ◽  
F. NICODEMI
Keyword(s):  
Quantum Hall ◽  
Coulomb Gas ◽  
Vertex Operators ◽  
Topological Order ◽  
Edge States ◽  
Operator Formalism ◽  
Fractional Quantum ◽  
Hall Conductance ◽  
Fractional Quantum Hall ◽  
Conformal Field

The fractional quantum Hall effect is discussed in terms of a c = 1 conformal field theory and the associated U(1) Kac–Moody current algebra, using the Coulomb gas vertex operators. A geometrical derivation of the Hall conductance is given and the possible topological order is considered. The consistency requires that only at filling ν = 1/m one of the "particles" described by the vertices can be associated with the electron.

scholarly journals From Chern–Simons to Tomonaga–Luttinger

2018 ◽  
Vol 33 (02) ◽  
pp. 1850013 ◽  
Author(s):  
Nicola Maggiore
Keyword(s):  
Degrees Of Freedom ◽  
Quantum Hall ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Edge States ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Chern Simons ◽  
Weyl Fermion

A single-sided boundary is introduced in the three-dimensional Chern–Simons model. It is shown that only one boundary condition for the gauge fields is possible, which plays the twofold role of chirality condition and bosonization rule for the two-dimensional Weyl fermion describing the degrees of freedom of the edge states of the Fractional Quantum Hall Effect. The symmetry on the boundary is derived, which determines the effective two-dimensional action, whose equation of motion coincides with the continuity equation of the Tomonaga–Luttinger theory. The role of Lorentz symmetry and of discrete symmetries on the boundary is also discussed.

THE QUANTUM HALL EFFECT AT ARBITRARY RATIONAL FILLING: A PROPOSAL

1992 ◽  
Vol 07 (28) ◽  
pp. 2583-2591 ◽  
Author(s):  
G. CRISTOFANO ◽  
G. MAIELLA ◽  
R. MUSTO ◽  
F. NICODEMI
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Hierarchical Model ◽  
Quantum Hall ◽  
Coulomb Gas ◽  
Vertex Operators ◽  
Two Dimensional ◽  
Conformal Field

A description of the quantum Hall effect, already proposed for the fractional filling ν=1/m, based on the introduction of Coulomb gas-like vertex operators typical of a two-dimensional conformal field theory, is extended to the case ν=p/m. The resulting physical picture is compared with the hierarchical model.

scholarly journals Two-dimensional topological order and operator algebras

2021 ◽  
pp. 2130003
Author(s):  
Yasuyuki Kawahigashi
Keyword(s):  
Field Theory ◽  
Mathematical Theory ◽  
Domain Walls ◽  
Operator Algebras ◽  
Matrix Product ◽  
Topological Order ◽  
Two Dimensional ◽  
Tensor Categories ◽  
Conformal Field

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to two-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

scholarly journals Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Sungmin Kim ◽  
Johannes Schwenk ◽  
Daniel Walkup ◽  
Yihang Zeng ◽  
Fereshte Ghahari ◽  
...  
Keyword(s):  
Atomic Force Microscopy ◽  
Chemical Potential ◽  
Quantum Hall ◽  
Broken Symmetry ◽  
Topological Order ◽  
Edge States ◽  
Force Microscopy ◽  
Atomic Force ◽  
Intimate Relation ◽  
Boundary Measurements

AbstractThe quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded the concept of topological order in physics bringing into focus the intimate relation between the “bulk” topology and the edge states. The QH effect in graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials. However, the broken-symmetry edge states have eluded spatial measurements. In this article, we spatially map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of $${{\nu }}={{0}},\pm {{1}}$$ ν = 0 , ± 1 across the quantum Hall edge boundary using high-resolution atomic force microscopy (AFM) and show a gapped ground state proceeding from the bulk through to the QH edge boundary. Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moiré superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.

A COULOMB GAS DESCRIPTION OF THE COLLECTIVE STATES FOR THE FRACTIONAL QUANTUM HALL EFFECT

1991 ◽  
Vol 06 (19) ◽  
pp. 1779-1786 ◽  
Author(s):  
G. CRISTOFANO ◽  
G. MAIELLA ◽  
R. MUSTO ◽  
F. NICODEMI
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Coulomb Gas ◽  
Fractional Quantum Hall Effect ◽  
Irreducible Representations ◽  
Vertex Operators ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Magnetic Translation

Many anyons wavefunctions relevant for the fractional Quantum Hall Effect at filling ν = 1/m are obtained by using Coulomb gas conformal Vertex operators. They provide irreducible representations of a subgroup of the magnetic translation group on the torus and their degeneracy is related to the allowed set of anyonic charges.

scholarly journals MULTIPLE EDGE PARTITION FUNCTIONS FOR FRACTIONAL QUANTUM HALL STATES

1999 ◽  
Vol 14 (23) ◽  
pp. 3745-3759
Author(s):  
KAZUSUMI INO
Keyword(s):  
Quantum Hall ◽  
Multiple Edge ◽  
Partition Functions ◽  
Edge States ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Verlinde Formula ◽  
Quantum Hall States ◽  
Edge Partition

We consider the multiple edge states of the Laughlin state and the Pfaffian state. These edge states are globally constrained through the operator algebra of conformal field theory in the bulk. We analyze these constraints by introducing an expression of quantum Hall state by chiral vertex operators and obtain the multiple edge partition functions by using the Verlinde formula.

scholarly journals Quantum Hall phases emerging from atom–photon interactions

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Alexander V. Poshakinskiy ◽  
Janet Zhong ◽  
Yongguan Ke ◽  
Nikita A. Olekhno ◽  
Chaohong Lee ◽  
...  
Keyword(s):  
Quantum Hall Effect ◽  
Quantum Electrodynamics ◽  
Quantum Hall ◽  
Landau Levels ◽  
Topological Order ◽  
Edge States ◽  
Quantum Structure ◽  
Many Body ◽  
Photon Interactions ◽  
Synthetic Magnetic Field

AbstractWe reveal the emergence of quantum Hall phases, topological edge states, spectral Landau levels, and Hofstadter butterfly spectra in the two-particle Hilbert space of an array of periodically spaced two-level atoms coupled to a waveguide (waveguide quantum electrodynamics). While the topological edge states of photons require fine-tuned spatial or temporal modulations of the parameters to generate synthetic magnetic fields and the quantum Hall effect, here we demonstrate that a synthetic magnetic field can be self-induced solely by atom–photon interactions. The fact that topological order can be self-induced in what is arguably the simplest possible quantum structure shows the richness of these waveguide quantum electrodynamics systems. We believe that our findings will advance several research disciplines including quantum optics, many-body physics, and nonlinear topological photonics, and that it will set an important reference point for the future experiments on qubit arrays and quantum simulators.

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