Gaudin Spin Chains Research Articles

On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras

Using the technique of the quasigraded Lie algebras, we construct general spectral-parameter dependent solutions r12(u, v) of the permuted classical Yang–Baxter equation with the values in the tensor square of simple Lie algebra g. We show that they are connected with infinite-dimensional Lie algebras with Adler–Kostant–Symmes decompositions and are labeled by solutions of a constant quadratic equation on the linear space g⊕N, N ≥ 1. We formulate the conditions when the corresponding r-matrices are skew-symmetric, i.e., they are equivalent to the ones described by Belavin–Drinfeld classification. We illustrate the developed theory by the example of the elliptic r-matrix of Sklyanin. We apply the obtained result to the explicit construction of the generalized quantum and classical Gaudin spin chains.

Classical r-matrices, “elliptic” BCS and Gaudin-type hamiltonians and spectral problem

In this article we make a brief review of the generalized Gaudin spin chains with [21] and without [18], [19] external magnetic field in the simplest case of g=so(3) and provide the r-matrix interpretation and generalization of the recent results of [38], [39], [40], on their exact solution. We present the classification theorem for the r-matrices that are diagonal in the natural basis and for the “shift elements” playing the role of the interaction strengths and external magnetic fields in the corresponding Gaudin-type models. We apply these results to the description, classification and solution of BCS-Richardson-type models. In particular, we introduce “elliptic” BCS-Richardson hamiltonian corresponding to non-skew-symmetric elliptic r-matrix and calculate its spectrum. We show that the “closed” [23], [30] and the “open” [36] BCS-Richardson hamiltonian of p+ip-type coincide with its trigonometric degenerations.

Exact solutions and critical chaos in dilaton gravity with a boundary

We consider (1 + 1)-dimensional dilaton gravity with a reflecting dynamical boundary. The boundary cuts off the region of strong coupling and makes our model causally similar to the spherically-symmetric sector of multidimensional gravity. We demonstrate that this model is exactly solvable at the classical level and possesses an on-shell SL(2, ℝ) symmetry. After introducing general classical solution of the model, we study a large subset of soliton solutions. The latter describe reflection of matter waves off the boundary at low energies and formation of black holes at energies above critical. They can be related to the eigenstates of the auxiliary integrable system, the Gaudin spin chain. We argue that despite being exactly solvable, the model in the critical regime, i.e. at the verge of black hole formation, displays dynamical instabilities specific to chaotic systems. We believe that this model will be useful for studying black holes and gravitational scattering.

Decompositions of quasigraded Lie algebras, non-skew-symmetric classical [formula omitted]-matrices and generalized Gaudin models

We construct a special family of quasigraded Lie algebras that generalize loop algebras in different gradings and admit Adler–Kostant–Symes decomposition into a sum of two subalgebras. We analyze the special cases when the constructed Lie algebras admit additionally other types of Adler–Kostant–Symes decompositions. Based on the proposed Lie algebras and their decompositions we explicitly construct several new classes of non-skew-symmetric classical r-matrices r(u,v) with spectral parameters. Using them we obtain new types of the generalized quantum and classical Gaudin spin chains.

Generalized Gaudin spin chains, nonskew symmetric r-matrices, and reflection equation algebras

Using a one-parametric family of automorphisms σ(u) of finite-dimensional simple Lie algebras g and classical skew-symmetric r-matrices r12(u−v), we construct “twisted” nonskew symmetric classical r-matrices r12σ(u,v) with spectral parameters. We show that in the special case σ(u)=AdK(u) and classical matrix Lie algebras g, they are connected with the quasiclassical limit of the reflection equation algebras. Using them and results of our previous papers, we construct new integrable quantum spin chains of the “generalized Gaudin” type with an arbitrary value of spin living at each site.

Generalized Gaudin systems in a magnetic field and non-skew-symmetric r-matrices

We construct integrable cases of generalized classical and quantum Gaudin spin chains in an external magnetic field. For this purpose, we generalize the ‘shift of argument method’ onto the case of classical and quantum integrable systems governed by an arbitrary -valued, non-dynamical classical r-matrix with spectral parameters. We consider several examples of the obtained construction for the cases of skew-symmetric, ‘twisted’ non-skew-symmetric and ‘anisotropic’ non-skew-symmetric classical r-matrices. We show, in particular, that in a general case in order for the Gaudin system in a magnetic field to be integrable, the corresponding magnetic field should be non-homogeneous.

Integrable quantum spin chains, non-skew symmetric [formula omitted]-matrices and quasigraded Lie algebras

Using general non-skew symmetric classical r -matrices with a spectral parameter, we construct new quantum spin chains that generalize famous Gaudin spin chains. With the help of the special quasigraded Lie algebras, we construct new examples of the non-skew symmetric classical r -matrices with a spectral parameter and explicit examples of new commuting Gaudin-type hamiltonians.

Generalized quantum Gaudin spin chains, involutive automorphisms and “twisted” classical r-matrices

Using automorphisms σ of finite-dimensional Lie algebras and classical skew-symmetric r-matrices we construct “twisted” nonskew symmetric r-matrices with the spectral parameters. Using them and results of our previous papers we construct new quantum spin chains that generalize famous Gaudin spin chains. We consider several examples of the twisted nonskew symmetric r-matrices and the corresponding Gaudin-type systems.

Harmonics on hyperspheres, separation of variables and the bethe ansatz

The relation between solutions to Helmholtz's equation on the sphere $S^{n-1}$ and the $[{\gr sl}(2)]^n$ Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian and a complete set of commuting second order operators suggested by the $R$--matrix approach to integrable systems, based on the loop algebra $\wt{sl}(2)_R$, are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions on $S^{n-1}$ to the Bethe ansatz approach to integrable systems is explained.

Knot invariants from four-dimensional gauge theory

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the $M$-theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks.