Spectral duality in integrable systems from AGT conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl _N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the Alday-Gaiotto-Tachikawa relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous Adams-Harnad-Hurtubise duality.     

Nekrasov prepotential with fundamental matter from the quantum spin chain

Nekrasov functions were conjectured in Mironov and Morozov (2009) [1] to be related to exact Bohr-Sommerfeld periods of quantum integrable systems. This statement was thoroughly checked for the case of the pure SU(Nc) gauge theory in Mironov and Morozov (2009) [2] and Popolitov (2010) [3]. Here we successfully perform a set of checks in the case of gauge group SU(Nc) with additional Nf fundamental hypermultiplets. We show that the Baxter equation for the spin chain gives the same quantum periods as the one for the Gaudin system in this case.     

Construction of the elliptic Gaudin system based on Lie algebra

Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics. The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra. Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, A _n , B _n , C _n , D _n , and we calculate a classical r-matrix for the elliptic Gaudin system with spin.      

Bulk and boundary S-matrices for the SU(N) chain

We consider both closed and open integrable antiferromagnetic chains constructed with the SU(N)-invariant R-matrix. For the closed chain, we extend the analyses of Sutherland and Kulish — Reshetikhin by considering also complex “string” solutions of the Bethe ansatz equations. Such solutions are essential to describe general multiparticle excited states. We also explicitly determine the SU(N) quantum numbers of the states. In particular, the model has particle-like excitations in the fundamental representations [k] of SU(N), with k = 1, …, N − 1. We directly compute the complete two-particle S-matrices for the cases [1] ⊗ [1] and [1] ⊗ [N − 1 ]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) × SU(N−l) × U(1), as well as a new “duality” symmetry which maps l→N−l. With the help of these symmetries, we compute by means of the Bethe ansatz for particles of types [1] and [N − 1 ] the corresponding boundary S-matrices.     

Spin Chain Models with Spectral Curves from M Theory

We construct the integrable model corresponding to the ?= 2 supersymmetric SU(N) gauge theory with matter in the antisymmetric representation, using the spectral curve found by Landsteiner and Lopez through M Theory. The model turns out to be the Hamiltonian reduction of a N+2 periodic spin chain model, which is Hamiltonian with respect to the universal symplectic form we had constructed earlier for general soliton equations in the Lax or Zakharov–Shabat representation. Received: 22 December 1999 / Accepted: 3 March 2000     

Spin Excitations in the Field-Induced Phase of the Quasi-One-Dimensional S = 1 Heisenberg Antiferromagnet NDMAP

The S = 1 quasi-one-dimensional Heisenberg antiferromagnet [Ni(C₅H₁₄N₂)₂N₃](PF₆), abbreviated as NDMAP, has been studied by electron spin resonance in a magnetic field above the critical field (H _c). We studied angular and frequency dependences of spin excitations. The angular dependence of the spin excitations in the vicinity of H _c is explained well by a phenomenological field theory, but the agreement between the experiment and the calculation is not satisfactory above 10 T. In high magnetic fields above 15 T, we obtained some characteristic spin excitations which are well explained by conventional antiferromagnetic resonance modes. These results suggest that the spin excitations change from a quantum state to a classical one due to the suppression of quantum fluctuations by high magnetic fields.     

Entanglement and elementary excitations in quantum spin chain

The entanglement induced by elementary excitations in the XX spin chain is investigated by Bethe ansatz method. The reduced density matrix and correlation function between any pair of spins can be obtained for ground and all excited states with N qubits. Rely on them we show the explicit and general relations between entanglement and elementary excitations in XX spin chain. We further show our method can be extend to other integrable models.     

Abilities of Antioxidants to Eliminate the Peroxyl Radical Derived from 2,2′-Azobis(2-Amidinopropane) Dihydrochloride (AAPH)

Electron spin resonance (ESR) of 5-(2,2-dimethyl-1,3-propoxy cyclophosphoryl)-5-methyl-1-pyrroline N-oxide (CYPMPO) spin adducts of free radicals derived from the UV irradiation of 2,2′-azobis(2-amidinopropane) dihydrochloride (AAPH) under rapid scanning condition was examined. The ESR signal obtained was the superposition of two spectra, the spin adduct of the alkoxyl radical [CYPMPO-ORa, Ra = C(CH₃)₂-C(⁺NH₂Cl^?)NH₂] and the peroxyl radical (CYPMPO-OORa). The decay rate of CYPMPO-OORa was 8,400 times that of CYPMPO-ORa. The order of the k _SB/k _ST values (the slope of the Stern–Volmer’s plot) for the peroxyl radical (RaOO·) was l-ascorbic acid > caffeic acid > rutin ~ Trolox ~ (+)-catechin ~ glutathione (reduced), which was almost the same order as that for the alkoxyl radical (RaO·). Though the k _SB/k _ST value of each antioxidant for the peroxyl radical was about half of that for the alkoxyl radical, the ratios of the values of antioxidants to that of trolox (the relative ORAC values) were almost the same between the peroxyl and alkoxyl radicals. The relative oxygen radical absorbance capacity (ORAC) values of polyphenols estimated by the ORAC-ESR assay using both peroxyl and alkoxyl radicals were smaller than those by the ORAC-FL assay.     

Chern–Simons theory with vector fermion matter

We study three-dimensional conformal field theories described by U(N) Chern?CSimons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger?CDyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ?² as a function of the ??t?Hooft coupling ??=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |??|=1; the conformal theory does not exist for |??|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W _?? algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.     

Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials

Original proofs of the AGT relations with the help of the Hubbard–Stratanovich duality of the modified Dotsenko–Fateev matrix model did not work for β ≠ 1, because Nekrasov functions were not properly reproduced by Selberg–Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper, we describe a complete proof of AGT in the simple case of GL(2) (N = 2) Yang–Mills theory, i.e., the 4-point spherical conformal block of the Virasoro algebra.     

New Integrable Models from the Gauge/YBE Correspondence

We introduce a class of new integrable lattice models labeled by a pair of positive integers N and r. The integrable model is obtained from the Gauge/YBE correspondence, which states the equivalence of the 4d $\mathcal {N} =1$ $S^{1}\times S^{3}/ \mathbb {Z} _{r}$ index of a large class of SU(N) quiver gauge theories with the partition function of 2d classical integrable spin models. The integrability of the model (star-star relation) is equivalent with the invariance of the index under the Seiberg duality. Our solution to the Yang-Baxter equation is one of the most general known in the literature, and reproduces a number of known integrable models. Our analysis identifies the Yang-Baxter equation with a particular duality (called the Yang-Baxter duality) between two 4d $\mathcal {N} =1$ supersymmetric quiver gauge theories. This suggests that the integrability goes beyond 4d lens indices and can be extended to the full physical equivalence among the IR fixed points.     

Triplet Z(N) model in a triangular lattice

The phase structure of a class of two-dimensional spin models with three-body interactions defined on a triangular lattice is studied. This class of models, containing the Baxter-Wu model as a special case, is shown to share the duality properties of a wide class of spin theories in two and three dimensions and the Z(N) gauge theory in four dimensions. Like these models, our theory is shown to possess a massless, Kosterlitz-Thouless-like phase when the number of available spin states exceeds a critical value.     

Magnetic dynamics of dilute iron nano-clusters in silver films from Mössbauer spectroscopy and muon spin rotation

A silver film containing nanometer size clusters of iron (nominal conc. 1 at%) has been studied by Mössbauer spectroscopy and Low-Energy Muon Spin Rotation. Below about 20 K spin glass freezing due to interparticle interactions is found from both methods. Whereas Mössbauer spectra are insensitive to the fast fluctuations of cluster moments above spin glass freezing temperature, muon spin rotation in magnetic fields applied perpendicular to the polarized muon spins allows tracing the fluctuations of superparamagnetic moments. The temperature dependence of the damping of the muon spin rotation signal shows Arrhenius behavior between 10 to 100 K. Depending on the assumed shape of damping the activation energy of superparamagnetic fluctuations of cluster moments ranges between about 20 K ·k _B and 40 K ·k _B . Above about 120 K muon spin depolarization indicates diffusion and trapping of muons.     

N = 2 Supersymmetric Unconstrained Matrix GNLS Hierarchies are Consistent

We develop a pseudo-differential approach to the N = 2 supersymmetric unconstrained matrix (k|n, m)-generalized nonlinear Schrödinger hierarchies and prove consistency of the corresponding Lax-pair representation (nlin.SI/0201026). Furthermore, we establish their equivalence to the integrable hierarchies derived in the super-algebraic approach of the homogeneously-graded loop superalgebra ${sl(2k{+}n\vert 2k{+}m)\otimes C[{\lambda},{\lambda}^{-1}]}$ (nlin.SI/0206037). We introduce an unconventional definition of N = 2 supersymmetric strictly pseudo-differential operators so as to close their algebra among themselves.     

Spin chains in magnetic field,non-skew-symmetric classical r-matrices and BCS-type integrable systems

We construct generalized Gaudin systems in an external magnetic field corresponding to arbitrary so(3)$\mathit{so}(3)$-valued non-skew-symmetric r-matrices with spectral parameters and non-homogeneous external magnetic fields. In the case of r  -matrices diagonal in the sl(2)$\mathit{sl}(2)$ basis we calculate the spectrum and the eigen-values of the corresponding generalized Gaudin hamiltonians using the algebraic Bethe ansatz. We explicitly consider several one-parametric families of non-skew-symmetric classical r-matrices and the corresponding generalized Gaudin systems in a magnetic field. We apply these results to fermionic systems and obtain a wide class of new integrable fermionic BCS-type hamiltonians.     

Mössbauer spectroscopic studies of spin crossover in tris(N, N′-dialkyldithiocarbamato) iron (III) complexes: effect of alkyl substituents

Spin crossover behavior in tris(N,N′-dialkyldithiocarbamato) iron(III) complexes with varying alkyl groups has been studied by variable temperature magnetic moment and Mössbauer spectral studies. All the complexes may be divided into three broad groups; high spin (μ _eff > 4.8 BM), intermediate spin (μ _eff?=?3.5???4.6 BM) and low spin (μ _eff?< 3.2 B.M). Room temperature (RT) Mössbauer spectra exhibit an asymmetric doublet resolved into two doublets corresponding to high and low spin states. Estimated % contributions of HS and LS states and calculated μ _eff were comparable with the experimentally determined values. It has been shown that some complexes undergo spin crossover, ⁶A₁g→²T₂g whereas others exhibit spin transitions ⁶A₁g →⁴T₁g or ⁴T₁g → ²T₂g. IR spectra show characteristic ν _(Fe???S) bands in the region 208–285 (HS) and 311–380 cm^???1 (LS). Nature of alkyl groups affects the spin state.     

On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

We investigate (d + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d ?? 3 we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi- dimensional integrable systems in question. In particular, our results imply the non- existence of 2-component integrable systems of hydrodynamic type for d ?? 6. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2 + 1 dimensions.     

A Class of Integrable Spin Calogero-Moser Systems

 We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of A _n -type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids. Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research partially supported by NSF grant DMS00-72171.     

Off-shell Bethe ansatz equation for osp(2∣1) Gaudin magnets

The semi-classical limit of the algebraic Bethe ansatz method is used to solve the theory of Gaudin models. Via off-shell Bethe ansatz method we find the spectra and eigenvectors of the N−1 independents Gaudin Hamiltonians with symmetry osp(2∣1). We also show how the off-shell Gaudin equation solves the trigonometric Knizhnik–Zamolodchikov equation.