scholarly journals Ergodic edge modes in the 4D quantum Hall effect

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Benoit Estienne ◽  
Blagoje Oblak ◽  
Jean-Marie Stéphan
Keyword(s):  
Quantum Hall Effect ◽  
Invariant Torus ◽  
Quantum Hall ◽  
Quantum Correlations ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Chaotic Flows ◽  
Conformal Field ◽  
Three Sphere ◽  
Confining Potential

The gapless modes on the edge of four-dimensional (4D) quantum Hall droplets are known to be anisotropic: they only propagate in one direction, foliating the 3D boundary into independent 1D conduction channels. This foliation is extremely sensitive to the confining potential and generically yields chaotic flows. Here we study the quantum correlations and entanglement of such edge modes in 4D droplets confined by harmonic traps, whose boundary is a squashed three-sphere. Commensurable trapping frequencies lead to periodic trajectories of electronic guiding centers; the corresponding edge modes propagate independently along S^1S1 fibers, forming a bundle of 1D conformal field theories over a 2D base space. By contrast, incommensurable frequencies produce quasi-periodic, ergodic trajectories, each of which covers its invariant torus densely; the corresponding correlation function of edge modes has fractal features. This wealth of behaviors highlights the sharp differences between 4D Hall droplets and their 2D peers; it also exhibits the dependence of 4D edge modes on the choice of trap, suggesting the existence of observable bifurcations due to droplet deformations.

scholarly journals W1+∞ Field Theories for the Edge Extensions in the Quantum Hall Effect

1997 ◽  
Vol 12 (06) ◽  
pp. 1101-1111 ◽  
Author(s):  
Andrea Cappelli ◽  
Carlo A. Trugenberger ◽  
Guillermo R. Zemba
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
A Priori ◽  
Quantum Hall ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Universality Classes ◽  
Effective Theories

We briefly review these low-energy effective theories for the quantum Hall effect, with emphasis and language familiar to field theorists. Two models have been proposed for describing the most stable Hall plateaus (the Jain series): the multi-component Abelian theories and the minimal W1+∞ models. They both lead to a-priori classifications of quantum Hall universality classes. Some experiments already confirmed the basic predictions common to both effective theories, while other experiments will soon pin down their detailed properties and differences. Based on the study of partition functions, we show that the Abelian theories are rational conformal field theories while the minimal W1+∞ models are not.

scholarly journals NON-ABELIAN FRACTIONAL QUANTUM HALL STATES AND CHIRAL COSET CONFORMAL FIELD THEORIES

2000 ◽  
Vol 15 (30) ◽  
pp. 4857-4870 ◽  
Author(s):  
D. C. CABRA ◽  
E. FRADKIN ◽  
G. L. ROSSINI ◽  
F. A. SCHAPOSNIK
Keyword(s):  
Quantum Hall ◽  
Field Theories ◽  
Simons Theory ◽  
Conformal Field Theories ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Quantum Hall States ◽  
Effective Lagrangian ◽  
Chern Simons

We propose an effective Lagrangian for the low energy theory of the Pfaffian states of the fractional quantum Hall effect in the bulk in terms of non-Abelian Chern–Simons (CS) actions. Our approach exploits the connection between the topological Chern–Simons theory and chiral conformal field theories. This construction can be used to describe a large class of non-Abelian FQH states.

scholarly journals BITS AND PIECES IN LOGARITHMIC CONFORMAL FIELD THEORY

2003 ◽  
Vol 18 (25) ◽  
pp. 4497-4591 ◽  
Author(s):  
MICHAEL A. I. FLOHR
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Hall ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Logarithmic Conformal Field Theory ◽  
Verlinde Formula

These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories. The two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory.

scholarly journals $\bm{\widehat{{\rm U}(1)}\times\widehat{{\rm SU}({\lc m})}_{1}}$ THEORY AND c=m W1+∞ MINIMAL MODELS IN THE HIERARCHICAL QUANTUM HALL EFFECT

2000 ◽  
Vol 15 (06) ◽  
pp. 915-926 ◽  
Author(s):  
MARINA HUERTA
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
Degrees Of Freedom ◽  
Central Charge ◽  
Quantum Hall ◽  
Detailed Comparison ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Group Flow

Two classes of Conformal Field Theories have been proposed to describe the Hierarchical Quantum Hall Effect: the multicomponent bosonic theory, characterized by the symmetry [Formula: see text] and the W1+∞ minimal models with central charge c=m. In spite of having the same spectrum of edge excitations, they manifest differences in the degeneracy of the states and in the quantum statistics, which call for a more detailed comparison between them. Here, we describe their detailed relation for the general case, c=m and extend the methods previously published for c≤3. Specifically, we obtain the reduction in the number of degrees of freedom from the multicomponent Abelian theory to the minimal models by decomposing the characters of the [Formula: see text] representations into those of the c=mW1+∞ minimal models. Furthermore, we find the Hamiltonian whose renormalization group flow interpolates between the two models, having the W1+∞ minimal models as an infrared fixed point.

scholarly journals Defects and perturbation

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Enrico M. Brehm
Keyword(s):  
Entanglement Entropy ◽  
Topological Defects ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field

Abstract We investigate perturbatively tractable deformations of topological defects in two-dimensional conformal field theories. We perturbatively compute the change in the g-factor, the reflectivity, and the entanglement entropy of the conformal defect at the end of these short RG flows. We also give instances of such flows in the diagonal Virasoro and Super-Virasoro Minimal Models.

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