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Advances in Theoretical and Mathematical Physics

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

André LeClair

Published Jan 1, 2009• 10 citations
Publisher

Abstract

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.

DOI
10.4310/atmp.2009.v13.n1.a8

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Cambridge University Press eBooks•Aug 6, 2009•0 citations
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.
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.
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.
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.
We now describe the properties of the SU( >K ) chain in more algebraic terms. The relevant symmetry is the SU( K ) Yangian, which is usually referred to as Y ( sl K ). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y ( gl 2 ). The difference between Y ( sl K ) and Y ( gl K ) is that the charge degrees of freedom U(1) is absent in Y ( sl K ), while the spin degrees of freedom is included in both Y ( sl K ) and Y ( sl K ). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y ( sl K ), which makes the relation between Y ( gl K ) and Y ( sl K ) clearer. In Section 9.1, we begin with the construction of the SU( K ) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU( K ) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU( K ) Yangian. In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU( K ) Yangian states.