Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0012189

A Free Field Representation of the Osp(2|2) current algebra at level k=-2, and Dirac Fermions in a random SU(2) gauge potential

Andreas W. W. Ludwig

Published Dec 11, 2000• 19 citations
arXivHTMLPDF

Abstract

The Osp(2|2) current algebra at level k=-2 is known to describe the IR fixed point of 2D Dirac fermions, subject to a random SU(2) gauge potential. We show that this theory has a simple free-field representation in terms of a compact, and a non-compact free scalar field, as well as a free fermionic ghost, at c=-2. The fermionic twist fields are crucial for the construction. The logarithmic current-algebra primary field with vanishing scaling dimension, transforming in an indecomposable representation, appears as a consequence of familiar logarithmic operators at c=-2.

DOI
10.48550/arxiv.cond-mat/0012189

Related papers

The gl(1|1) super-current algebra: the rôle of twist and logarithmic fields

André LeClair

A free field representation of the gl(1|1) k current algebra at arbitrary level k is given in terms of two scalar fields and a symplectic fermion.The primary fields for all representations are explicitly constructed using the twist and logarithmic fields in the symplectic fermion sector.A closed operator algebra is described at integer level k.Using a new super spincharge separation involving gl(1|1) N and su(N ) 0 , we describe how the gl(1|1) N current algebra can describe a non-trivial critical point of disordered Dirac fermions.Local gl(1|1) invariant lagrangians are defined which generalize the Liouville and sine-Gordon theories.We apply these new tools to the spin quantum Hall transition and show that it can be described as a logarithmic perturbation of the osp(2|2) k current algebra at k = -2.

Advances in Theoretical and Mathematical Physics•Jan 1, 2009•10 citations

The gl(1|1) super-current algebra: the role of twist and logarithmic fields

André LeClair

A free field representation of the gl(1|1)_k current algebra at arbitrary level k is given in terms of two scalar fields and a symplectic fermion. The primary fields for all representations are explicitly constructed using the twist and logarithmic fields in the symplectic fermion sector. A closed operator algebra is described at integer level k. Using a new super spin charge separation involving gl(1|1)_N and su(N)_0, we describe how the gl(1|1)_N current algebra can describe a non-trivial critical point of disordered Dirac fermions. Local gl(1|1) invariant lagrangians are defined which generalize the Liouville and sine-Gordon theories. We apply these new tools to the spin quantum Hall transition and show that it can be described as a logarithmic perturbation of the osp(2|2)_k current algebra at k=-2.

arXiv:0710.2906•Oct 16, 2007•7 citations

Random Dirac Fermions: The su(N) Gauge Potential and Z_N Twists

M. J. Bhaseen

We obtain the disorder averaged (critical) four-point correlation functions for N species of (two-dimensional Euclidean) Dirac fermions subject to a (Gaussian) random su(N) gauge field. The replica approach and the strong disorder approach yield identical results, as one might expect in the light of recent developments in the N=2 case [1]. We establish a connection with `dressed' Z_N twist fields in the c=-2 logarithmic conformal field theory (LCFT), thereby extending the recent connection to `dressed' Z_2 twists [1,2]. We draw attention to the fact that `dressed' Z_2 twist operators have previously appeared in the description of half flux-quantum quasiholes in the bulk excitations of the Haldane--Rezayi Quantum Hall state [3].

arXiv:cond-mat/0012420•Dec 22, 2000•1 citation

Exact S-matrices and related algebraic structures in two- dimensional quantum systems

Zorawar S. Bassi

This thesis studies exact S-matrices and integrability aspects for several two-dimensional quantum systems. The emphasis is on the algebraic structures that underlie exact solvability in two dimensions. The first project details the construction of an optics model for n-level atoms interacting with an electromagnetic field. In the slowly varying envelope and rotating wave approximations, the equations of motion are shown to have a zero-curvature representation, implying an infinite set of quantum conservation laws. The second system investigated is a disordered field theory of Dirac fermions, which after averaging over the potential can be expressed as an osp(2|2) current- current model. The current-current model in turn can be viewed as the fully renormalized version of an osp(2]2) (1) (marginal) Toda system. Non-local charges satisfying the Uq[osp(2|2)(1)] quantum superalgebra are constructed. The corresponding quantum affine symmetry is used to build the Toda S-matrix for the vector representation. The exact S-matrix for the fundamental “solitons” of the osp(2|2) current-current model is then obtained by taking the marginal limit of the Toda S-matrix. The remaining two projects shift the focus to boundary theories. In the first boundary study, two massive versions of the anisotropic spin 1/2 Kondo model are introduced. The two models have the same bulk sine-Gordon interactions but differ in their boundary terms. At the free fermion point, each of the models can be understood as two decoupled Ising models in boundary magnetic fields. Integrability of the models is discussed and boundary S-matrices at and away from the free fermion point are constructed. The final system studied describes an Anderson impurity in a BCS superconductor. This model can be mapped onto an Ising boundary field theory. The boundary S - matrix is shown to exhibit Andreev scattering and the existence of bound states lying inside the superconducting gap.

•Oct 1, 2000•0 citations

Strong-coupling fixed points of current interactions and disordered fermions in two dimensions

André LeClair

The all-orders $\ensuremath{\beta}$ function is used to study disordered Dirac fermions in two dimensions. The generic strong coupling fixed ``points'' of anisotropic current-current interactions at large distances are actually isotropic manifolds corresponding to subalgebras of the maximal current algebra at short distances. The argue that IR fixed point theories are generally current algebra cosets. We illustrate this with the simple example of anisotropic $\mathrm{su}(2),$ which is the physics of Kosterlitz-Thouless transitions. We propose a phase diagram for the Chalker-Coddington network model which is in the universality class of the integer quantum Hall transition. One phase is in the universality class of dense polymers.

arXiv:cond-mat/0011413•Physical review. B, Condensed matter•Jul 9, 2001•14 citations

Critical points of 2d disordered Dirac fermions: the Quantum Hall Transitions revisited

André LeClair

We propose a resolution of the renormalization group flow for the disordered Dirac fermion theories describing the quantum Hall transition (QHT) and spin Quantum Hall transition (SQHT), which previously revealed no perturbative fixed points at 1-loop and higher. The approach involves carrying out the flow in 2 stages, the first stage utilizing a new form of super spin-charge separation to flow to gl(1|1)_N and osp(2|2)_{-2N} supercurrent algebra theories, where N is the number of copies. This leads to the unconventional feature that at the critical point the exponents depend on the original number of copies N. In the second stage, additional forms of disorder are incorporated as dimension zero logarithmic operators, and the resulting actions have explicit forms in terms of two scalar fields and a symplectic fermion. Under some assumptions, the multi-fractal exponents are computed with the result q(1-q)/4 and q(1-q)/8 for the QHT and SQHT respectively, and are within a few percent of numerical estimates.

arXiv:0710.3778•Oct 22, 2007•0 citations
The Osp(2|2) current algebra at level k=-2 is known to describe the IR fixed point of 2D Dirac fermions, subject to a random SU(2) gauge potential. We show that this theory has a simple free-field representation in terms of a compact, and a non-compact free scalar field, as well as a free fermionic ghost, at c=-2. The fermionic twist fields are crucial for the construction. The logarithmic current-algebra primary field with vanishing scaling dimension, transforming in an indecomposable representation, appears as a consequence of familiar logarithmic operators at c=-2.
A free field representation of the gl(1|1) k current algebra at arbitrary level k is given in terms of two scalar fields and a symplectic fermion.The primary fields for all representations are explicitly constructed using the twist and logarithmic fields in the symplectic fermion sector.A closed operator algebra is described at integer level k.Using a new super spincharge separation involving gl(1|1) N and su(N ) 0 , we describe how the gl(1|1) N current algebra can describe a non-trivial critical point of disordered Dirac fermions.Local gl(1|1) invariant lagrangians are defined which generalize the Liouville and sine-Gordon theories.We apply these new tools to the spin quantum Hall transition and show that it can be described as a logarithmic perturbation of the osp(2|2) k current algebra at k = -2.
A free field representation of the gl(1|1)_k current algebra at arbitrary level k is given in terms of two scalar fields and a symplectic fermion. The primary fields for all representations are explicitly constructed using the twist and logarithmic fields in the symplectic fermion sector. A closed operator algebra is described at integer level k. Using a new super spin charge separation involving gl(1|1)_N and su(N)_0, we describe how the gl(1|1)_N current algebra can describe a non-trivial critical point of disordered Dirac fermions. Local gl(1|1) invariant lagrangians are defined which generalize the Liouville and sine-Gordon theories. We apply these new tools to the spin quantum Hall transition and show that it can be described as a logarithmic perturbation of the osp(2|2)_k current algebra at k=-2.
We obtain the disorder averaged (critical) four-point correlation functions for N species of (two-dimensional Euclidean) Dirac fermions subject to a (Gaussian) random su(N) gauge field. The replica approach and the strong disorder approach yield identical results, as one might expect in the light of recent developments in the N=2 case [1]. We establish a connection with `dressed' Z_N twist fields in the c=-2 logarithmic conformal field theory (LCFT), thereby extending the recent connection to `dressed' Z_2 twists [1,2]. We draw attention to the fact that `dressed' Z_2 twist operators have previously appeared in the description of half flux-quantum quasiholes in the bulk excitations of the Haldane--Rezayi Quantum Hall state [3].
This thesis studies exact S-matrices and integrability aspects for several two-dimensional quantum systems. The emphasis is on the algebraic structures that underlie exact solvability in two dimensions. The first project details the construction of an optics model for n-level atoms interacting with an electromagnetic field. In the slowly varying envelope and rotating wave approximations, the equations of motion are shown to have a zero-curvature representation, implying an infinite set of quantum conservation laws. The second system investigated is a disordered field theory of Dirac fermions, which after averaging over the potential can be expressed as an osp(2|2) current- current model. The current-current model in turn can be viewed as the fully renormalized version of an osp(2]2) (1) (marginal) Toda system. Non-local charges satisfying the Uq[osp(2|2)(1)] quantum superalgebra are constructed. The corresponding quantum affine symmetry is used to build the Toda S-matrix for the vector representation. The exact S-matrix for the fundamental “solitons” of the osp(2|2) current-current model is then obtained by taking the marginal limit of the Toda S-matrix. The remaining two projects shift the focus to boundary theories. In the first boundary study, two massive versions of the anisotropic spin 1/2 Kondo model are introduced. The two models have the same bulk sine-Gordon interactions but differ in their boundary terms. At the free fermion point, each of the models can be understood as two decoupled Ising models in boundary magnetic fields. Integrability of the models is discussed and boundary S-matrices at and away from the free fermion point are constructed. The final system studied describes an Anderson impurity in a BCS superconductor. This model can be mapped onto an Ising boundary field theory. The boundary S - matrix is shown to exhibit Andreev scattering and the existence of bound states lying inside the superconducting gap.
The all-orders $\ensuremath{\beta}$ function is used to study disordered Dirac fermions in two dimensions. The generic strong coupling fixed ``points'' of anisotropic current-current interactions at large distances are actually isotropic manifolds corresponding to subalgebras of the maximal current algebra at short distances. The argue that IR fixed point theories are generally current algebra cosets. We illustrate this with the simple example of anisotropic su(2),\mathrm{su}(2),su(2), which is the physics of Kosterlitz-Thouless transitions. We propose a phase diagram for the Chalker-Coddington network model which is in the universality class of the integer quantum Hall transition. One phase is in the universality class of dense polymers.
We propose a resolution of the renormalization group flow for the disordered Dirac fermion theories describing the quantum Hall transition (QHT) and spin Quantum Hall transition (SQHT), which previously revealed no perturbative fixed points at 1-loop and higher. The approach involves carrying out the flow in 2 stages, the first stage utilizing a new form of super spin-charge separation to flow to gl(1|1)_N and osp(2|2)_{-2N} supercurrent algebra theories, where N is the number of copies. This leads to the unconventional feature that at the critical point the exponents depend on the original number of copies N. In the second stage, additional forms of disorder are incorporated as dimension zero logarithmic operators, and the resulting actions have explicit forms in terms of two scalar fields and a symplectic fermion. Under some assumptions, the multi-fractal exponents are computed with the result q(1-q)/4 and q(1-q)/8 for the QHT and SQHT respectively, and are within a few percent of numerical estimates.