A. B. Balantekin, T. Dereli, Y. Pehlivan
Three well-known solutions of the Gaudin equation are obtained under a set of standard assumptions. By relaxing one of these assumptions we introduce a class of mutually commuting Hamiltonians based on a different solution of the Gaudin equation. Application of the algebraic Bethe ansatz technique to diagonalize these Hamiltonians reveals a new infinite dimensional complex Lie algebra.
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We identify the Bethe algebra of the Gaudin model associated to gl(N) acting on a suitable representation with the center of the rational Cherednik algebra and with the algebra of regular functions on the Calogero-Moser space.
We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated.
The generating function for elements of the Bethe subalgebra of Hecke algebra is constructed as Sklyanin's transfer-matrix operator for Hecke chain. We show that in a special classical limit q -> 1 the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of Hecke chain. We consruct a non-local analogue of the Gaudin Hamiltonians for the case of Hecke algebras.
Eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian have recently been employed as a variational wavefunction ansatz in quantum chemistry. This wavefunction is a mean-field of pairs of electrons (geminals). In this contribution we report optimal expressions for their reduced density matrices in both the original physical basis and the basis of the Richardson-Gaudin pairs. Physical basis expressions were originally reported by Gorohovsky and Bettelheim. In each case, the expressions scale like $\mathcal{O}(N^4)$, with the most expensive step the solution of linear equations. Analy…
We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of N-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. A new object is the Bethe vector averaging maps.
The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this curious property. These models have the notable feature that they do not preserve U(1) symmetry, which is typically associated to a non-conservation of particle number. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtained via the algebraic Bethe ansatz formalism.
We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.
A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.
Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras
本文证明了具有不规则奇点的Gaudin模型中$\mathfrak{gl}_d$与$\mathfrak{gl}_{p+m|q+n}$的双重性，并展示了其在无限维模空间上的作用特性。
Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras
本文利用超对偶性，将经典李（超）代数上酉不可约最高权模张量积的（超）Gaudin Hamiltonians的奇异向量归结为对应李代数有限维模情形，进而证明其在奇异向量空间及整个模上可对角化，并特别建立了正交与辛李代数在无限维模上的对角化。
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic B…
We propose an extension of the numerical approach for integrable Richardson-Gaudin models based on a new set of eigenvalue-based variables. Starting solely from the Gaudin algebra, the approach is generalized towards the full class of XXZ Richardson-Gaudin models. This allows for a fast and robust numerical determination of the spectral properties of these models, avoiding the singularities usually arising at the so-called singular points. We also provide different determinant expressions for the normalization of the Bethe Ansatz states and form factors of local spin operators, opening up poss…
Many exactly solvable models are based on Lie algebras. The pairing interaction is important in nuclear physics and its exact solution for identical particles in non-degenerate single-particle levels was first given by Richardson in 1963. His solution and its generalization to Richardson-Gaudin quasi-exactly solvable models have attracted the attention of many contemporary researchers and resulted in the exact solution of the isovector pn-pairing within the so(5) RG-model and the equal strength spin-isospin pn-pairing within the so(8) RG-model. Basic properties of the RG-models are summarized …
A theorem of Feigin, Frenkel and Reshetikhin provides expressions for the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in terms of elements of the center of the affine vertex algebra at the critical level. In our recent work, explicit Harish-Chandra images of generators of the center were calculated in all classical types. We combine these results to calculate the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in an explicit form. The Harish-Chandra images can be interpreted as elements of classical $W$-algebras. We provide a direct connection between …
In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield st…
Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master …
We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we conf…
Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe Ansatz wavefunction, provided the free variables in the Ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe Ansatz is a simple product state of generalised pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a non-linear way, making them prone to singularities. Unfortunately, it is non-trivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical …
Gaudin hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini-Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes.
The Gaudin models based on the face-type elliptic quantum groups and the $XYZ$ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding face-type Knizhnik-Zamolodchikov equations and their solutions are given.
This thesis deals with a class of integrable field theories called models with twist function. The main examples of such models are integrable non-linear sigma models, such as the Principal Chiral Model, and their deformations. A first obtained result is the proof that the so-called Bi-Yang-Baxter model, which is a two-parameter deformation of the Principal Chiral Model, is also a model with twist function. It is then shown that inhomogeneous Yang-Baxter type deformations modify certain global symmetries of the undeformed model into Poisson-Lie symmetries. Another chapter concerns the construc…
We consider the XXX homogeneous Gaudin system with $N$ sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new "Gaudin" algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties …
We present a method to construct a basis of singular and non-singular common eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie algebra SL(2). The subset of singular vectors is completely described by analogy with covariant differential operators. The relation between singular eigenvectors and the Bethe Ansatz is discussed. In each weight subspace the set of singular eigenvectors is completed to a basis, by a family of non-singular eigenvectors. We discuss also the generalization of this method to the case of an arbitrary Lie algebra.
We show that the system of Gaudin's equations for quasimomenta k_{j}, which describes a one-dimensional system of spinless point bosons with zero boundary conditions, has the unique real solution for each set of quantum numbers n_{j}.
We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In …
Let $\mathfrak q$ be a Lie algebra over a field $\mathbb K$ and $p,\tilde p\in\mathbb K[t]$ two different normalised polynomials of degree at least 2. As vector spaces both quotient Lie algebras $\mathfrak q[t]/(p)$ and $\mathfrak q[t]/(\tilde p)$ can be identified with $W=\mathfrak q{\cdot}1\oplus\mathfrak q\bar t\oplus\ldots\oplus\mathfrak q\bar t^{n-1}$. If $\mathrm{deg}\,(p-\tilde p)$ is at most 1, then the Lie brackets $[\,\,,\,]_p$, $[\,\,,\,]_{\tilde p}$ induced on $W$ by $p$ and $\tilde p$, respectively, are compatible. By a general method, known as the Lenard-Magri scheme, we construc…
We consider the isotropic spin$-\frac{1}{2}$ Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin-Korepin formula, \textit{i.e.}, the square of the norm, is also obtained.
The three-point Gaudin model and branched coverings of the Riemann sphere
本文研究三点量子sl_2-Gaudin模型，将Gaudin Hamilton量的联合谱视为代数曲线，分析其对Riemann球面的覆盖，并用三对角矩阵行列式刻画曲线方程，揭示覆盖的几何结构。
In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz usin…
Gaudin from Cherednik
本文针对复反射群W，在De Concini-Procesi奇妙模型X上的切列德尼克代数层中定义子层，推测t=0时其相关像为高丁代数层，并对W=S₃证明了该猜想。
The Gaudin model and the super Bethe ansatz for unitarizable modules over the general linear Lie (super)algebra
本文研究一般线性李（超）代数 Gaudin 代数在酉模奇异权空间上的作用，证明其为循环模且 Gaudin 代数为 Frobenius 代数，并建立超 Bethe 拟设。
The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows for its eigenstates to be constructed algebraically. In this work we show that quantum group theory provides a possibility to deform the Hamiltonian while preserving integrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the Richardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term into the auxiliary L-operator which generates integrals of motion of the…
Classical $N$-Reflection Equation and Gaudin Models
本文引入经典 $N$-反射方程推广可积边界条件，构造非反对称 $r$-矩阵的 Poisson 代数，并由此生成新的 Gaudin 型哈密顿量，包括 $BC_L$-型特例。
We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnovs determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propo…
Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras
该研究证明一般线性李超代数的二次及三次Gaudin哈密顿量在奇异权空间上可对角化，并建立了其与一般线性李代数相应哈密顿量及Knizhnik-Zamolodchikov方程解的联系。
Lax equations for relativistic ${\rm GL}(NM,{\mathbb C})$ Gaudin models on elliptic curve
本文描述了椭圆曲线上具有$n$个单极点的${\rm GL}_{NM}$经典椭圆可积系统，对相对论高丁模型分类，还给出了非均匀 Ruijsenaars 链的相关结果及模型的三角、有理类比。
In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson-Gaudin models which, in addition to $N-1$ spins $\frac{1}{2}$, contains one arbitrarily large spin $S$. The proposed determinant representation is written in terms of a set of variables which, from previous work, are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic Bethe equations. Such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the us…
The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz
本文研究一般线性李超代数Gaudin模型，证明奇异空间上的Gaudin代数是循环模和Frobenius代数，并对一般参数证明其可对角化且具有单谱，从而给出Bethe ansatz完备性的新解释。
We study 1+1 field-generalizations of the rational and elliptic Gaudin models. For ${\rm sl}(N)$ case we introduce equations of motion and L-A pair with spectral parameter on the Riemann sphere and elliptic curve. In ${\rm sl}(2)$ case we study the equations in detail and find the corresponding Hamiltonian densities. The $n$-site model describes $n$ interacting Landau-Lifshitz models of magnets. The interaction depends on position of the sites (marked points on the curve). We also analyze the 2-site case in its own right and describe its relation to the principal chiral model. We emphasize tha…
In this work we demonstrate how one can, in a generic approach, derive a set of $N$ simple quadratic Bethe equations for integrable Richardson-Gaudin (RG) models built out of $N$ spins-1/2. These equations depend only on the $N$ eigenvalues of the various conserved charges so that any solution of these equations defines, indirectly through the corresponding set of eigenvalues, one particular eigenstate. The proposed construction covers the full class of integrable RG models of the XYZ (including the subclasses of XXZ and XXX models) type realised in terms of spins-1/2, coupled with one another…
The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We show also that these Bethe roots are intimately associated to the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of $O(3)$ and the computation of the entanglement …
The affine Gaudin model, associated with an untwisted affine Kac-Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological-holomorphic Chern-Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.
We solve the non-stationary Schrodinger equation for several time-dependent Hamiltonians, such as the BCS Hamiltonian with an interaction strength inversely proportional to time, periodically driven BCS and linearly driven inhomogeneous Dicke models as well as various multi-level Landau-Zener tunneling models. The latter are Demkov-Osherov, bow-tie, and generalized bow-tie models. We show that these Landau-Zener problems and their certain interacting many-body generalizations map to Gaudin magnets in a magnetic field. Moreover, we demonstrate that the time-dependent Schrodinger equation for th…
A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the iterations of the coproduct map on the generators of the algebra. In this way several examples of N-body dynamical systems obtained from q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2) Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of Ruijsenaars type arising from the same (non co-boundary) q-deformatio…
We describe a reproduction procedure which, given a solution of the $\mathfrak{gl}_{M|N}$ Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family $P$ of other solutions called the population. To a population we associate a rational pseudodifferential operator $R$ and a superspace $W$ of rational functions. We show that if at least one module is typical then the population $P$ is canonically identified with the set of minimal factorizations of $R$ and with the space of full superflags in $W$. We conjecture that the singular eigenvectors (up to rescal…
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