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57
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 35 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
Existence of a stable polarized vacuum in the BogoliubovDiracFock approximation
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Charge Deficiency, Charge Transport and Comparison of Dimensions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. ..."
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Cited by 30 (0 self)
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We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
Spectral shift function in strong magnetic fields, Algebra i Analiz 16 (2004), 207  238; see also St
 Petersburg Math. Journal
"... Dedicated to Professor Mikhail Birman on the occasion of his 75th birthday Abstract. The threedimensional Schrödinger operator H with constant magnetic field of strength b> 0 is considered under the assumption that the electric potential V ∈ L1(R3) admits certain powerlike estimates at infinit ..."
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Cited by 18 (11 self)
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Dedicated to Professor Mikhail Birman on the occasion of his 75th birthday Abstract. The threedimensional Schrödinger operator H with constant magnetic field of strength b> 0 is considered under the assumption that the electric potential V ∈ L1(R3) admits certain powerlike estimates at infinity. The asymptotic behavior as b→ ∞ of the spectral shift function ξ(E;H,H0) is studied for the pair of operators (H, H0) at the energies E = Eb + λ, E> 0 and λ ∈ R being fixed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels, we pick E/2 ∈ Z+ and λ ∈ R; then the main term is always of order b, and is independent of λ. In the second asymptotic regime, called asymptotics near a Landau level, we choose E = 2q0, q0 ∈ Z+, and λ = 0; in this case the leading term of the SSF could be of order b or b for different λ. The main object of investigation in the present paper is the spectral shift function
Equality of bulk and edge Hall conductance revisited
 Comm. Math. Phys
, 2002
"... The integral quantum Hall eect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to dierent de nitions of Hall conductance, which agree under appropriate hypotheses, as shown by SchulzBaldes et al. by means o ..."
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Cited by 17 (1 self)
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The integral quantum Hall eect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to dierent de nitions of Hall conductance, which agree under appropriate hypotheses, as shown by SchulzBaldes et al. by means of Ktheory. We propose an alternative proof based on a generalization of the index of a pair of projections to more general operators. The equality of conductances is an expression of the stability of that index as a ux tube is moved from within the bulk across the boundary of a sample.
A Minimization Method for Relativistic Electrons in a MeanField Approximation of Quantum Electrodynamics
 Phys. Rev. A
"... Abstract. We study a meanfield relativistic model which is able to describe both the behavior of finitely many spin1/2 particles like electrons and of the Dirac sea which is selfconsistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb ..."
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Cited by 16 (9 self)
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Abstract. We study a meanfield relativistic model which is able to describe both the behavior of finitely many spin1/2 particles like electrons and of the Dirac sea which is selfconsistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the photon field. All our results are nonperturbative and mathematically rigorous. Contents
Selfconsistent solution for the polarized vacuum in a nophoton QED model
, 2005
"... We study the BogoliubovDiracFock model introduced by Chaix and Iracane (J. Phys. B., 22, 3791–3814, 1989) which is a meanfield theory deduced from nophoton QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as t ..."
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Cited by 15 (11 self)
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We study the BogoliubovDiracFock model introduced by Chaix and Iracane (J. Phys. B., 22, 3791–3814, 1989) which is a meanfield theory deduced from nophoton QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a selfconsistent equation. In a recent paper, we proved the convergence of the iterative fixedpoint scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cutoff Λ and the bare fine structure constant α. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cutoff Λ and without any constraint on the external field. We also study the behaviour of the minimizer as Λ goes to infinity and show that the theory is “nullified ” in that limit, as predicted first by Landau: the vacuum totally cancels the external potential. Therefore the limit case of an infinite cutoff makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant α, on a simplified model where the exchange term is neglected.
An analytic approach to spectral flow in von Neumann algebras
, 2005
"... The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by BreuerFredholm operators in a semifinite von Neumann algebra. The latter have ..."
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Cited by 14 (7 self)
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The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by BreuerFredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow first analysed in the classical setting by Getzler and extended to BreuerFredholm operators by some of the current authors. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (noncompact) covering spaces of a compact spin manifold using the adiabatic method. This answers a question of Mathai connecting Atiyah’s L 2index theorem to our analytic spectral flow. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on R N using some ideas of Shubin. This provides us with an opportunity to describe the deepest part of the theory namely the semifinite local index theorem in noncommutative geometry. This theorem, which gives a formula for spectral flow was recently proved by some of the present authors. It provides a farreaching generalisation of the original 1995 result of Connes and Moscovici.