Gaudin model explained
In physics, the Gaudin model, sometimes known as the quantum Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin.[1] They are exactly solvable models, and are also examples of quantum spin chains.
History
The simplest case was first described by Michel Gaudin in 1976, with the associated Lie algebra taken to be
, the two-dimensional
special linear group.
Mathematical formulation
Let
be a
semi-simple Lie algebra of finite dimension
.
Let
be a positive integer. On the complex plane
, choose
different points,
.
Denote by
the finite-dimensional irreducible representation of
corresponding to the dominant integral element
. Let
(\boldsymbol{λ}):=(λ1, … ,λN)
be a set of dominant integral weights of
. Define the
tensor product
.
The model is then specified by a set of operators
acting on
, known as the
Gaudin Hamiltonians.
[2] They are described as follows.
Denote by
the invariant
scalar product on
(this is often taken to be the
Killing form). Let
be a basis of
and
be the dual basis given through the scalar product. For an element
, denote by
the operator
which acts as
on the
th factor of
and as identity on the other factors. Then
These operators are mutually commuting. One problem of interest in the theory of Gaudin models is finding simultaneous eigenvectors and eigenvalues of these operators.
Instead of working with the multiple Gaudin Hamiltonians, there is another operator
, sometimes referred to as the
Gaudin Hamiltonian. It depends on a complex parameter
, and also on the
quadratic Casimir, which is an element of the
universal enveloping algebra
, defined as
This acts on representations
by multiplying by a number dependent on the representation, denoted
. This is sometimes referred to as the index of the representation. The Gaudin Hamiltonian is then defined
Commutativity of
for different values of
follows from the commutativity of the
.
Higher Gaudin Hamiltonians
When
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 the
Gaudin algebra. Similarly to the
Harish-Chandra isomorphism, these commuting elements have associated degrees, and in particular the Gaudin Hamiltonians form the degree 2 part of the algebra. For
, 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. See here.
Then eigenvectors of the Gaudin algebra define linear functionals on the algebra. If
is an element of the Gaudin algebra
, and
an eigenvector of the Gaudin algebra, one obtains a linear functional
given by
The linear functional
is called a
character of the Gaudin algebra. The
spectral problem, that is, determining eigenvalues and simultaneous eigenvectors of the Gaudin algebra, then becomes a matter of determining characters on the Gaudin algebra.
Solutions
A solution to a Gaudin model often means determining the spectrum of the Gaudin Hamiltonian or Gaudin Hamiltonians. There are several methods of solution, including
Algebraic Bethe ansatz
For sl2
For
, let
be the standard basis. For any
, one can define the operator-valued
meromorphic functionIts residue at
is
, while
\limzzX(z)=
X(i)=:X(infty),
the 'full' tensor representation.
The
and
satisfy several useful properties
but the
do not form a representation:
. The third property is useful as it allows us to also diagonalize with respect to
, for which a diagonal (but degenerate) basis is known.
For an
Gaudin model specified by sites
and weights
, define the
vacuum vector to be the tensor product of the highest weight states from each representation:
.
A Bethe vector (of spin deviation
) is a vector of the form
for
. Guessing eigenvectors of the form of Bethe vectors is the
Bethe ansatz. It can be shown that a Bethe vector is an eigenvector of the Gaudin Hamiltonians if the set of equations
holds for each
between 1 and
. These are the
Bethe ansatz equations for spin deviation
. For
, this reduces to
Completeness
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 be complete 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
, for
in
general 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
, there exist specific configurations for which completeness fails due to the Bethe ansatz equations having no solutions.
[6] For general complex simple g
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
case. Furthermore, for
of rank greater than 1, that is, all others besides
, there are higher Gaudin Hamiltonians, for which it is unknown how to generalize the Bethe ansatz.
ODE/IM isomorphism
There is an ODE/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
.
Generalizations
There exist generalizations arising from weakening the restriction on
being a strictly semi-simple Lie algebra. For example, when
is allowed to be an
affine Lie algebra, the model is called an affine Gaudin model.
A different way to generalize is to pick out a preferred automorphism of a particular Lie algebra
. 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 of classical Gaudin model. Historically, the quantum Gaudin model was defined and studied first, unlike most physical systems. Certain classical integrable 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 principal chiral model, coset sigma models and affine Toda field theory.[8]
External links
Notes and References
- Gaudin . Michel . Diagonalisation d'une classe d'hamiltoniens de spin . Journal de Physique . 1976 . 37 . 10 . 1087–1098 . 10.1051/jphys:0197600370100108700 . 26 September 2022.
- Feigin . Boris . Frenkel . Edward . Reshetikhin . Nikolai . Gaudin Model, Bethe Ansatz and Critical Level . Commun. Math. Phys. . 3 Apr 1994 . 166 . 1 . 27–62 . 10.1007/BF02099300 . hep-th/9402022 . 1994CMaPh.166...27F . 17099900 .
- Sklyanin . Evgeny . Separation of variables in the Gaudin model . Journal of Soviet Mathematics . October 1989 . 47 . 2 . 2473–2488 . 10.1007/BF01840429 . 120267573 . free .
- Scherbak . I. . Varchenko . A. . Critical Points of Functions, sl2 Representations, and Fuchsian Differential Equations with only Univalued Solutions . Moscow Mathematical Journal . 2003 . 3 . 2 . 621–645 . 10.17323/1609-4514-2003-3-2-621-645 . 13 November 2023.
- Frenkel . Edward . Gaudin Model and Opers . Infinite Dimensional Algebras and Quantum Integrable Systems . 2005 . 1–58 . 10.1007/3-7643-7341-5_1 . 13 November 2023 . Birkhäuser . en.
- Mukhin . E. . Varchenko . A. . Multiple Orthogonal Polynomials and a Counterexample to the Gaudin Bethe Ansatz Conjecture . Transactions of the American Mathematical Society . 2007 . 359 . 11 . 5383–5418 . 13 November 2023 . 0002-9947.
- Vicedo . Benoît . Young . Charles . Cyclotomic Gaudin Models: Construction and Bethe Ansatz . Communications in Mathematical Physics . 1 May 2016 . 343 . 3 . 971–1024 . 10.1007/s00220-016-2601-3 . 1409.6937 . 2016CMaPh.343..971V . 253748577 .
- Vicedo . Benoit . On integrable field theories as dihedral affine Gaudin models . 2017 . hep-th . 1701.04856.