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26
Lectures on non perturbative field theory and quantum impurity problems, in Topological aspects of low dimensional systems (Les Houches
, 1998
"... They are a sequel to the notes I wrote two years ago for the Summer School, “Topological Aspects ..."
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Cited by 47 (1 self)
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They are a sequel to the notes I wrote two years ago for the Summer School, “Topological Aspects
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 25 (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.
Heisenberg honeycombs solve Veneziano puzzle
 International Mathematical Forum (2008) in press, arxiv: hepth/0608117
"... In this (expository) paper and its (more technical) companion we reformulate some results of the Nobel Prize winning paper by Werner Heisenberg into modern mathematical language of honeycombs. This language was recently developed in connection with complete solution of the Horn problem (to be define ..."
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Cited by 5 (5 self)
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In this (expository) paper and its (more technical) companion we reformulate some results of the Nobel Prize winning paper by Werner Heisenberg into modern mathematical language of honeycombs. This language was recently developed in connection with complete solution of the Horn problem (to be defined and explained in the text). Such a reformulation is done with the purpose of posing and solving the following problem: is by analyzing the (spectroscopic) experimental data it possible to recreate the underlying microscopic model generating these data? Although in the case of Hydrogen atom positive answer is known, to obtain an affirmative answer for spectra of other quantum mechanical systems is much harder task. Development of Heisenberg’s ideas happens to be the most useful for this purpose. It supplies needed tools to solve the Veneziano and other puzzles. The meaning of the word ”puzzle ” is twofold. On one hand, it means (in the case of Veneziano amplitudes) to find a physical model reproducing these amplitudes. On another, from the point of view of combinatorics of honeycombs, it means to find explicitly fusion rules for such amplitudes. Solution of these tasks is facilitated by our earlier developed stringtheoretic formalism. In this paper only qualitative arguments are presented (with few exceptions). These arguments provide enough evidence that the underlying model compatible with Veneziano amplitudes is the standard (i.e. non supersymmetric!) QCD. In addition, usefulness of the proposed formalism is illustrated on numerous examples such as physically motivated solution of the saturation conjecture (to be defined in the text), derivation of the YangBaxter and KnizhnikZamolodchikov equations,Verlinde and Hecke algebras, computation of the GromovWitten invariants for small quantum cohomology ring, etc.
Anomalous BCS equation for a Luttinger superconductor
, 1999
"... In the context of the Anderson theory of high T c cuprates, we develop a BCS theory for Luttinger liquids. If the Luttinger interaction is much stronger than the BCS potential we find that the BCS equation is quite modified compared to usual BCS equation for Fermi liquids. In particular T c predict ..."
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In the context of the Anderson theory of high T c cuprates, we develop a BCS theory for Luttinger liquids. If the Luttinger interaction is much stronger than the BCS potential we find that the BCS equation is quite modified compared to usual BCS equation for Fermi liquids. In particular T c predicted by the BCS equation for Luttinger liquids is quite higher than the usual T c for Fermi liquids. 1 The anomalous BCS equation The equivalent of BCS theory for a Luttinger liquid has not formally worked out, despite the relevance of such theory for the problem of superconductivity in the highT c cuprates, see [1] or the discussion in the following section. Such theory should describe superconductors which in their normal state are Luttinger liquids. In this paper we develop such theory using constructive quantum field theory techniques, applied in many other Luttinger liquid problems, see [2], [3]. In particular, we compute in a rigorous way the BCS selfconsistence equation for a spinni...
The nested SU(N) offshell Bethe ansatz and exact form factors
, 611
"... This work is dedicated to the 75th anniversary of H. Bethe’s foundational work on the Heisenberg chain The form factor equations are solved for an SU(N) invariant Smatrix under the assumption that the antiparticle is identified with the bound state of N − 1 particles. The solution is obtained expl ..."
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Cited by 3 (3 self)
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This work is dedicated to the 75th anniversary of H. Bethe’s foundational work on the Heisenberg chain The form factor equations are solved for an SU(N) invariant Smatrix under the assumption that the antiparticle is identified with the bound state of N − 1 particles. The solution is obtained explicitly in terms of the nested offshell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. PACS: 11.10.z; 11.10.Kk; 11.55.Ds
Fusion hierarchies for N = 4 superYangMills theory, arXiv:0803.2035 [hepth
"... We employ the analytic Bethe Anzats to construct eigenvalues of transfer matrices with finitedimensional atypical representations in the auxiliary space for the putative longrange spin chain encoding anomalous dimensions of all composite singletrace gauge invariant operators of the maximally supe ..."
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We employ the analytic Bethe Anzats to construct eigenvalues of transfer matrices with finitedimensional atypical representations in the auxiliary space for the putative longrange spin chain encoding anomalous dimensions of all composite singletrace gauge invariant operators of the maximally supersymmetric YangMills theory. They obey an infinite fusion hierarchy which can be reduced to a finite set of integral relations for a minimal set of transfer matrices. This set is used to derive a finite systems of functional equations for eigenvalues of nested Baxter polynomials. 1
Theory of spin excitations in undoped and underdoped cuprates
, 1998
"... We consider the magnetic properties of high Tc cuprates from a gauge theory point of view, with emphasis on the underdoped regime. Underdoped cuprates possess certain antiferromagnetic correlations, as evidenced, for example, by different temperature dependence of the Cu and O site NMR relaxation ra ..."
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Cited by 2 (0 self)
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We consider the magnetic properties of high Tc cuprates from a gauge theory point of view, with emphasis on the underdoped regime. Underdoped cuprates possess certain antiferromagnetic correlations, as evidenced, for example, by different temperature dependence of the Cu and O site NMR relaxation rates, that are not captured well by slave boson mean field theories of the tJ model. We show that the inclusion of gauge fluctuations will remedy the deficiencies of the mean field theories. As a concrete illustration of the gaugefluctuation restoration of the antiferromangetic correlation and the feasibility of the 1/N perturbation theory, the Heisenberg spin chain is analyzed in terms of a 1+1D U(1) gauge theory with massless Dirac fermions. The 1/Nperturbative treatment of the same gauge theory in 2+1D (which can be motivated from the mean field πflux phase of the Heisenberg model) leads to a dynamical mass generation corresponding to an antiferromagnetic ordering. On the other hand, it is argued that in a similar gauge theory with an additional coupling to a Bose (holon) field, symmetry breaking does not occur, but antiferromagnetic correlations are enhanced, which is the situation in the underdoped cuprates.
Universality of onedimensional Fermi systems, II. The Luttinger liquid structure
, 2013
"... The critical behavior of onedimensional interacting Fermi systems is expected to display universality features, called Luttinger liquid behavior. Critical exponents and certain thermodynamic quantities are expected to be related among each others by modelindependent formulas. We establish such rel ..."
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The critical behavior of onedimensional interacting Fermi systems is expected to display universality features, called Luttinger liquid behavior. Critical exponents and certain thermodynamic quantities are expected to be related among each others by modelindependent formulas. We establish such relations, the proof of which has represented a challenging mathematical problem, for a general model of spinning fermions on a one dimensional lattice; interactions are short ranged and satisfy a positivity condition which makes the model critical at zero temperature. Proofs are reported in two papers: in the present one, we demonstrate that the zero temperature response functions in the thermodynamic limit are Borel summable and have anomalous powerlaw decay with multiplicative logarithmic corrections. Critical exponents are expressed in terms of convergent expansions and depend on all the model details. All results are valid for the special case of the Hubbard model. 1 Main Results 1.1