In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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.
They are
exactly solvable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantity, conserved qua ...
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
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
.
Mathematical formulation
Let
be a
semi-simple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a 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
be a set of dominant integral weights of
. Define the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
.
The model is then specified by a set of operators
acting on
, known as the Gaudin Hamiltonians.
They are described as follows.
Denote by
the invariant
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
, 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
In mathematics, the Harish-Chandra isomorphism, introduced by ,
is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathf ...
, 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
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here
Television
* Here TV (form ...
.
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
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
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
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of the Gaudin Hamiltonian or Gaudin Hamiltonians. There are several methods of solution, including
*
Algebraic Bethe ansatz, used by Gaudin
* Separation of variables, used by
Sklyanin
*
Correlation functions/
opers, using a method described by
Feigin,
Frenkel Frenkel is a surname. Notable people with the surname include:
* Aaron G. Frenkel (born 1957), Israeli entrepreneur and philanthropist
* Alexander Frenkel (born 1985), German boxer of Ukrainian origin
* (1895–1984), Polish painter
* Daan Frenk ...
and
Reshetikhin.
Algebraic Bethe ansatz
For sl2
For
, let
be the standard basis. For any
, one can define the operator-valued
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
Its residue at
is
, while
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 general complex simple g
Analogues of the Bethe ansatz equation can be derived for Lie algebras of higher rank.
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 In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
, 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.
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
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
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 model
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
s and affine
Toda field theory
In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Kac–Moody algebra and a specific Lagrangian.
Fixing the Kac– ...
.
References
External links
Gaudin integrable model in nLab* (See 8:40 to 15:14 in video.)
{{Statistical mechanics topics
Spin models
Quantum lattice models
Quantum magnetism