The Eight-Vertex Model and Lattice Supersymmetry

We show that the XYZ spin chain along the special line of couplings J _x J _y +J _x J _z +J _y J _z =0 possesses a hidden N=(2,2)mathcal{N}=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter’s Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain.     

Auxiliary Vertices Method for Kagomé-Lattice Eight-Vertex Model

We investigate a class of eight-vertex models on a Kagomé lattice. With the help of auxiliary vertices, the Kagomé-lattice eight-vertex model (KEVM) is related to an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. Using an equation for commuting transfer matrices, we determine their eigenvalues. From calculated eigenvalues the correlation length of the KEVM is derived with its full anisotropy. There are two cases: In the first case the anisotropic correlation length (ACL) is the same as that of the triangular/honeycomb-lattice Ising model. By the use of an algebraic curve, it is shown that the Kagomé-lattice Ising model, the diced-lattice Ising model, and the hard-hexagon model also have (essentially) the same ACL as the KEVM. In the second case we find that the ACL displays 12fold rotational symmetry.     

The Six and Eight-Vertex Models Revisited

Elliott Lieb's ice-type models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Qequations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the six-vertex and chiral Potts models, and between the eight-vertex and Kashiwara–Miwa models.     

The Differential Graded Odd NilHecke Algebra

We equip the odd nilHecke algebra and its associated thick calculus category with diagrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum \({{\mathfrak{sl}_2}}\) at a fourth root of unity.     

Odd Viscosity

When time reversal is broken, the viscosity tensor can have a nonvanishing odd part. In two dimensions, and only then, such odd viscosity is compatible with isotropy. Elementary and basic features of odd viscosity are examined by considering solutions of the wave and Navier–Stokes equations for hypothetical fluids where the stress is dominated by odd viscosity.     

Completeness of the Bethe Ansatz for the Six and Eight-Vertex Models

We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the beyond the equator, infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.     

Odd Invariant Semidensity and Divergence-like Operators on an Odd Symplectic Superspace

The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hypersurfaces in Euclidean space. Received: 19 August 1997 / Accepted: 27 March 1998     

New Developments in the Eight Vertex Model II. Chains of Odd Length

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites N. For systems at the root of unity pointsη=mK/L with m odd the transfer matrix is known to satisfy the famous ‘‘TQ’’ equation where Q(υ) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(υ) matrix is qualitatively different from the case of evenN and in particular they satisfy a previously unknown equation which is more general than what is often called ‘‘Bethe’s equation.’’ For the case of even m where no Q(υ) matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the TQ equation. The ground state for the particular case of η=2K/3 and N odd is investigated in detail.     

On Odd Laplace Operators

We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an 'orbit space' of volume forms. This includes earlier results for the odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin–Vilkovisky equations. We compare this situation with that of Riemannian and even Poisson manifolds. In particular, we show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein's 'modular class'.     

On Quantizable Odd Lie Bialgebras

Motivated by the obstruction to the deformation quantization of Poisson structures in infinite dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures.     

Odd tracks at hadron colliders

New physics that exhibits irregular tracks such as kinks, intermittent hits, or decay in flight may easily be missed at hadron colliders. We demonstrate this by studying viable models of light, O(10 GeV), colored particles that decay predominantly inside the tracker. Such particles can be produced at staggering rates, and yet, may not be identified or triggered on at the LHC, unless specifically searched for. In addition, the models we study provide an explanation for the original measurement of the anomalous charged track distribution by CDF. The presence of irregular tracks in these models reconcile that measurement with the subsequent reanalysis and the null results of ATLAS and CMS. Our study clearly illustrates the need for a comprehensive study of irregular tracks at the LHC.     

Semidensities on Odd Symplectic Supermanifolds

We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new -operator action on semidensities as the proper framework for the Batalin-Vilkovisky (BV) formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batalin-Vilkovisky geometry. Another application is to (1.1)-codimensional surfaces in E. We construct a kind of pull-back of semidensities to such surfaces. This operation and the -operator are used for obtaining integral invariants for (1.1)-codimensional surfaces.     

Odd crystal field in scheelite

Russian Physics Journal -     

Dirac Superfield Quantization in Odd Superspace

We generalize the superfield method of Dirac quantization to the odd sector of superspace for N = 2 extended models. We discuss the mechanism of generating constraints in the odd sector of supersymmetric classical mechanics and then Dirac quantization in the odd superspace in superfield version. An example of supersymmetric system, defined by odd superfields, and the application of the method in the odd superspace is given.     

The Odd SO(3,3) Fusions and the Gauge Principle

The origin of the gauge principle generating electroweak and gravistrong interactions is shown to lie already in the quaternionic vacuum. The leptonic Dirac equations gauged in respect of the electroweak and the linearized gravistrong field are derived from the SO(3,3) fusion 6 × 4 = 20 + 4 as a part of 4 × 4 × 4 . The procedure leading to respective equations including self-interactions and gauge coupling to the full nonlinear gravistrong field via 20 ″ × 4 = 60 + 20 as a part of 4 × 4 × 4 × 4 × 4 is outlined.     

Odd symplectic geometry types on supermanifolds

Generalization of symplectic geometry on manifolds in a supersymmetric case is examined in the present work. In the even case, this leads either to even symplectic geometry, that is, the geometry on supermanifolds with the nondegenerate Poisson bracket, or to the geometry on the Fedosov even supermanifolds. In the odd case, two different scalar symplectic structures exist (namely, the odd closed differential 2-form and antibracket), which can be used to construct various symplectic geometry types on supermanifolds. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 52–57, February, 2008.     

减少光子奇(偶)对相干态的非经典特性

构造了减少光子奇对相干态与减少光子偶对相干态,并用数值模拟方法研究了它们的非经典特性。结果表明,从奇对相干态与偶对相干态的第一个场模取走光子,可以显著地增强第二个场模的光子统计亚泊松分布特性;从奇对相干态的第一个场模取走光子后,在双模光子消灭算符本征值的模较小时可以增强第一个场模的光子反聚束性质。从偶对相干态的第一个场模取走光子,在双模光子消灭算符本征值的模较小时第二个场模的光子反聚束效应得到明显增强。     

Gauge Supergravities for All Odd Dimensions

Recently proposed supergravity theories in odddimensions whose fields are connection one-forms for theminimal supersymmetric extensions of anti-de Sittergravity are discussed. Two essential ingredients are required for this construction: (1) Thesuperalgebras, which extend the adS algebra fordifferent dimensions, and (2) the Lagrangians, which areChern-Simons (2n - 1)-forms. The first item completes the analysis of van Holten and Van Proeyen,which was valid for N = 1 only. The second ensures thatthe actions are invariant by construction under thegauge supergroup and, in particular, under localsupersymmetry. Thus, unlike standard supergravity, the localsupersymmetry algebra closes off-shell and withoutrequiring auxiliary fields. The superalgebras areconstructed for all dimensions and they fall into three families: osp (m|N) for D = 2, 3, 4, mod 8, osp(N|m) for D = 6, 7, 8, mod 8, and su(m - 2, 2|N) for D= 5 mod 4, with m = 2^[D/2]. The Lagrangian isconstructed for D = 5, 7, and 11. In all cases the field content includes the vielbein(e ^a ), the spin connection( ^ab ), N gravitini( ⁱ ), and some extrabosonic "matter" fields which vary from onedimension to another.     

The Fermi Golden Rule and its Form at Thresholds in Odd Dimensions

Let H be a Schrödinger operator on a Hilbert space , such that zero is a nondegenerate threshold eigenvalue of H with eigenfunction Ψ₀. Let W be a bounded selfadjoint operator satisfying 〈 Ψ₀, WΨ₀〈>0. Assume that the resolvent (H?z)^?1 has an asymptotic expansion around z=0 of the form typical for Schrödinger operators on odd-dimensional spaces. Let H(?) =H+?W for ?>0 and small. We show under some additional assumptions that the eigenvalue at zero becomes a resonance for H(?), in the time-dependent sense introduced by A. Orth. No analytic continuation is needed. We show that the imaginary part of the resonance has a dependence on ? of the form ?^2+(ν/2) with the integer ν≥?1 and odd. This shows how the Fermi Golden Rule has to be modified in the case of perturbation of a threshold eigenvalue. We give a number of explicit examples, where we compute the ``location'' of the resonance to leading order in ?. We also give results, in the case where the eigenvalue is embedded in the continuum, sharpening the existing ones.