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arXiv:0806.3645

Multiboson Logical Operators, Laughlin States, and Virasoro Algebra

Francesco Antonino Raffa, Mario Rasetti

Published Jun 23, 2008• 1 citation
arXivHTMLPDF

Abstract

Leading idea of this manuscript is to discuss the structure and the deep correlations among different quantum physical systems, and to explore how such correlations bear on the capacity of the systems to encode and manipulate information. In particular, the role of logical operators constructed out of multi-boson (and possibly fractionary-boson) operators is analyzed in its relation on the one hand with the algebra of diffeomorphisms of the circle, on the other with the physical properties of the fractional quantum Hall effect (described in terms of Laughlin states) and of its many electron representation in terms of the Chern-Simons topological quantum field theory.

DOI
10.48550/arxiv.0806.3645

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Leading idea of this manuscript is to discuss the structure and the deep correlations among different quantum physical systems, and to explore how such correlations bear on the capacity of the systems to encode and manipulate information. In particular, the role of logical operators constructed out of multi-boson (and possibly fractionary-boson) operators is analyzed in its relation on the one hand with the algebra of diffeomorphisms of the circle, on the other with the physical properties of the fractional quantum Hall effect (described in terms of Laughlin states) and of its many electron representation in terms of the Chern-Simons topological quantum field theory.
Abstract The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The effective formalism used in many-body quantum Hall theories — the Chern–Simons theory is also presented. The Chern–Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall analyses, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. An introduction to multi-component quantum Hall systems and spin quantum Hall systems is sketched.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.
Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.
Over the years, many theoretical frameworks have been developed to understand the remarkable physics of the quantum Hall system. In this work we discuss the interplay among quantum wires, Chern-Simons theory, bosonization, and particle-vortex duality, which enable a unified approach for the Laughlin states of the fractional quantum Hall system. Starting from the so-called quantum wires system, which is a semimicroscopic description in terms of (1+1)(1+1)(1+1)-dimensional theories, we discuss the emergence of (2+1)(2+1)(2+1)-dimensional low-energy effective field theories by using different maps connecting the microscopic degrees of freedom with the macroscopic ones. We show the origin of these maps by embedding the bosonized version of the original quantum wires model in a gauge-invariant theory and using particle vortex duality.
The Kalmeyer-Laughlin state, which is a lattice version of the bosonic Laughlin state at filling factor one half, has attracted much attention due to its topological and chiral spin liquid properties. Here we show that the Kalmeyer-Laughlin state on the torus can be expressed in terms of a correlator of conformal fields from the SU(2)1SU(2)_1SU(2)1​ Wess-Zumino-Witten model. This reveals an interesting underlying mathematical structure and provides a natural way to generalize the Kalmeyer-Laughlin state to arbitrary lattices on the torus. We find that the many-body Chern number of the states is unity for more different lattices, which suggests that the topological properties of the states are preserved when the lattice is changed. Finally, we analyze the symmetry properties of the states on square lattices.