Inphysics, theGaudin model, sometimes known as thequantum Gaudin model, is a model, or a large class of models, instatistical mechanics first described in its simplest case byMichel Gaudin.^[1] They areexactly solvable models, and are also examples of quantumspin chains.

The simplest case was first described by Michel Gaudin in 1976,^[1] with the associatedLie algebra taken to be${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$, the two-dimensionalspecial linear group.

Let${\displaystyle \mathfrak{g}}$ be asemi-simple Lie algebra of finite dimension${\displaystyle d}$.

Let${\displaystyle N}$ be a positive integer. On the complex plane${\displaystyle \mathbb{C}}$, choose${\displaystyle N}$ different points,${\displaystyle z_i}$.

Denote by${\displaystyle V_{\lambda }}$ the finite-dimensional irreducible representation of${\displaystyle \mathfrak{g}}$ corresponding to thedominant integral element${\displaystyle \lambda }$. Let${\displaystyle (\mathit{\lambda }):=({\lambda }_1,\cdots ,{\lambda }_N)}$ be a set of dominant integral weights of${\displaystyle \mathfrak{g}}$. Define thetensor product${\displaystyle V_{(\mathit{\lambda })}:=V_{{\lambda }_1}\otimes \cdots \otimes V_{{\lambda }_N}}$.

The model is then specified by a set of operators${\displaystyle H_i}$ acting on${\displaystyle V_{(\mathit{\lambda })}}$, known as theGaudin Hamiltonians.^[2] They are described as follows.

Denote by${\displaystyle ⟨\cdot ,\cdot ⟩}$ the invariantscalar product on${\displaystyle \mathfrak{g}}$ (this is often taken to be theKilling form). Let${\displaystyle \{I_a\}}$ be a basis of${\displaystyle \mathfrak{g}}$ and${\displaystyle \{I^a\}}$ be the dual basis given through the scalar product. For an element${\displaystyle A\in \mathfrak{g}}$, denote by${\displaystyle A^{(i)}}$ the operator${\displaystyle 1\otimes \cdots \otimes A\otimes \cdots \otimes 1}$ which acts as${\displaystyle A}$ on the${\displaystyle i}$th factor of${\displaystyle V_{(\mathit{\lambda })}}$ and as identity on the other factors. Then

$${\displaystyle H_i=\sum _{j\ne i}\sum _{a=1}^d\frac{I_a^{(i)}{I}^{a(j)}}{z_i-z_j}.}$$

These operators are mutually commuting. One problem of interest in the theory of Gaudin models is finding simultaneouseigenvectors and eigenvalues of these operators.

Instead of working with the multiple Gaudin Hamiltonians, there is another operator${\displaystyle S(u)}$, sometimes referred to as theGaudin Hamiltonian. It depends on a complex parameter${\displaystyle u}$, and also on thequadratic Casimir, which is an element of theuniversal enveloping algebra${\displaystyle U(\mathfrak{g})}$, defined as$${\displaystyle \mathrm{\Delta }=\frac{1}{2}\sum _{a=1}^d{I}_a{I}^a.}$$This acts on representations${\displaystyle V_{(\mathit{\lambda })}}$ by multiplying by a number dependent on the representation, denoted${\displaystyle \mathrm{\Delta }(\lambda )}$. This is sometimes referred to as the index of the representation. The Gaudin Hamiltonian is then defined$${\displaystyle S(u)=\sum _{i=1}^N\left[\frac{H_i}{u-z_i}+\frac{\mathrm{\Delta }({\lambda }_i)}{(u-z_i{)}^2}\right].}$$Commutativity of${\displaystyle S(u)}$ for different values of${\displaystyle u}$ follows from the commutativity of the${\displaystyle H_i}$.

When${\displaystyle \mathfrak{g}}$ has rank greater than 1, the commuting algebra spanned by the Gaudin Hamiltonians and the identity can be expanded to a larger commuting algebra, known as theGaudin algebra. Similarly to theHarish-Chandra isomorphism, these commuting elements have associated degrees, and in particular the Gaudin Hamiltonians form the degree 2 part of the algebra. For${\displaystyle \mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2}$, the Gaudin Hamiltonians and the identity span the Gaudin algebra. There is another commuting algebra which is 'universal', underlying the Gaudin algebra for any choice of sites and weights, called the Feigin–Frenkel center. Seehere.

Then eigenvectors of the Gaudin algebra define linear functionals on the algebra. If${\displaystyle X}$ is an element of the Gaudin algebra${\displaystyle \mathfrak{G}}$, and${\displaystyle v}$ an eigenvector of the Gaudin algebra, one obtains a linear functional${\displaystyle {\chi }_v:\mathfrak{G}\to \mathbb{C}}$ given by$${\displaystyle Xv={\chi }_v(X)v.}$$The linear functional${\displaystyle {\chi }_v}$ is called acharacter of the Gaudin algebra. Thespectral problem, that is, determining eigenvalues and simultaneous eigenvectors of the Gaudin algebra, then becomes a matter of determining characters on the Gaudin algebra.

A solution to a Gaudin model often means determining thespectrum of the Gaudin Hamiltonian or Gaudin Hamiltonians. There are several methods of solution, including

• Algebraic Bethe ansatz, used by Gaudin
• Separation of variables, used bySklyanin^[3]
• Correlation functions/opers, using a method described byFeigin,Frenkel andReshetikhin.^[2]

For${\displaystyle \mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2}$, let${\displaystyle \{E,H,F\}}$ be the standard basis. For any${\displaystyle X\in \mathfrak{g}}$, one can define the operator-valuedmeromorphic function$${\displaystyle X(z)=\sum _{i=1}^N\frac{X^{(i)}}{z-z_i}.}$$Its residue at${\displaystyle z=z_i}$ is${\displaystyle X^{(i)}}$, while${\displaystyle \underset{z\to \mathrm{\infty }}{lim}zX(z)=\sum _{i=1}^N{X}^{(i)}=:X^{(\mathrm{\infty })},}$ the 'full' tensor representation.

The${\displaystyle X(z)}$ and${\displaystyle X^{(\mathrm{\infty })}}$ satisfy several useful properties

• ${\displaystyle [X(z),Y^{(\mathrm{\infty })}]=[X,Y](z)}$
• ${\displaystyle S(u)=\frac{1}{2}\sum _a{I}_a(z)I^a(z)}$
• ${\displaystyle [H_i,X^{(\mathrm{\infty })}]=0}$

but the${\displaystyle X(z)}$ do not form a representation:${\displaystyle [X(z),Y(z)]=-[X,Y{]}^{\prime }(z)}$. The third property is useful as it allows us to also diagonalize with respect to${\displaystyle H^{\mathrm{\infty }}}$, for which a diagonal (but degenerate) basis is known.

For an${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$ Gaudin model specified by sites${\displaystyle z_1,\cdots ,z_N\in \mathbb{C}}$ and weights${\displaystyle {\lambda }_1,\cdots ,{\lambda }_N\in \mathbb{N}}$, define thevacuum vector to be the tensor product of the highest weight states from each representation:${\displaystyle v_0:=v_{{\lambda }_1}\otimes \cdots \otimes v_{{\lambda }_N}}$.

ABethe vector (of spin deviation${\displaystyle m}$) is a vector of the form$${\displaystyle F(w_1)\cdots F(w_m)v_0}$$for${\displaystyle w_i\in \mathbb{C}}$. Guessing eigenvectors of the form of Bethe vectors is theBethe ansatz. It can be shown that a Bethe vector is an eigenvector of the Gaudin Hamiltonians if the set of equations$${\displaystyle \sum _{i=1}^N\frac{{\lambda }_i}{w_k-z_i}-2\sum _{j\ne k}\frac{1}{w_k-w_j}=0}$$holds for each${\displaystyle k}$ between 1 and${\displaystyle m}$. These are theBethe ansatz equations for spin deviation${\displaystyle m}$. For${\displaystyle m=1}$, this reduces to$${\displaystyle \mathit{\lambda }(w):=\sum _{i=1}^N\frac{{\lambda }_i}{w-z_i}=0.}$$

In theory, the Bethe ansatz equations can be solved to give the eigenvectors and eigenvalues of the Gaudin Hamiltonian. In practice, if the equations are to completely solve the spectral problem, one must also check

• The number of solutions predicted by the Bethe equations
• The multiplicity of solutions

If, for a specific configuration of sites and weights, the Bethe ansatz generates all eigenvectors, then it is said to becomplete for that configuration of Gaudin model. It is possible to construct examples of Gaudin models which are incomplete. One problem in the theory of Gaudin models is then to determine when a given configuration is complete or not, or at least characterize the 'space of models' for which the Bethe ansatz is complete.

For${\displaystyle \mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2}$, for${\displaystyle z_i}$ ingeneral position the Bethe ansatz is known to be complete.^[4] Even when the Bethe ansatz is not complete, in this case it is due to the multiplicity of a root being greater than one in the Bethe ansatz equations, and it is possible to find a complete basis by defining generalized Bethe vectors.^[5]

Conversely, for${\displaystyle \mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3}$, there exist specific configurations for which completeness fails due to the Bethe ansatz equations having no solutions.^[6]

Analogues of the Bethe ansatz equation can be derived for Lie algebras of higher rank.^[2] However, these are much more difficult to derive and solve than the${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$ case. Furthermore, for${\displaystyle \mathfrak{g}}$ of rank greater than 1, that is, all others besides${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$, there are higher Gaudin Hamiltonians, for which it is unknown how to generalize the Bethe ansatz.

There is anODE/IM isomorphism between the Gaudin algebra (or the universal Feigin–Frenkel center), which are the 'integrals of motion' for the theory, and opers, which are ordinary differential operators, in this case on${\displaystyle {\mathbb{P}}^1}$.

There exist generalizations arising from weakening the restriction on${\displaystyle \mathfrak{g}}$ being a strictly semi-simple Lie algebra. For example, when${\displaystyle \mathfrak{g}}$ is allowed to be anaffine Lie algebra, the model is called an affine Gaudin model.

A different way to generalize is to pick out a preferredautomorphism of a particular Lie algebra${\displaystyle \mathfrak{g}}$. One can then define Hamiltonians which transform nicely under the action of the automorphism. One class of such models are cyclotomic Gaudin models.^[7]

There is also a notion ofclassical Gaudin model. Historically, the quantum Gaudin model was defined and studied first, unlike most physical systems. Certain classicalintegrable field theories can be viewed as classical dihedral affine Gaudin models. Therefore, understanding quantum affine Gaudin models may allow understanding of the integrable structure of quantum integrable field theories.

Such classical field theories include the principalchiral model, cosetsigma models and affineToda field theory.^[8]

• Gaudin integrable model in nLab
• Frenkel, Edward (July 25, 2023)."Challenge Talk 2 - Feynman's Last Blackboard: From Bethe Ansatz to Langlands Duality".Perimeter Institute Recorded Seminar Archive (pirsa.org).doi:10.48660/23070020. (See 8:40 to 15:14 in video.)

• Spin models
• Quantum lattice models
• Quantum magnetism

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