In quantum statistical physics, the Inozemtsev model is a spin chain, defined on a one-dimensional, periodic lattice.
Unlike the prototypical Heisenberg spin chain, which only includes interactions between neighboring sites of the lattice, the Inozemtsev model has long-range interactions, that is, interactions between any pair of sites, regardless of the distance between them.
[1] Like those spin chains, the Inozemtsev model is exactly solvable.
spin 1/2 sites, the quantum phase space is described by the tensor product Hilbert space
The (elliptic) Inozemtsev model is given by the (unnormalised) Hamiltonian[1][2]
th site (acting nontrivially on the
The periods of the Weierstrass elliptic function are the length
of the chain, to ensure periodic boundary conditions, together with an imaginary period that sets the interaction range and is traditionally parameterized as
The truly long-range Haldane-Shastry chain is obtained when the imaginary period is removed (
) while, upon renormalisation, the Heisenberg spin chain is recovered in the limit
, which is why the resulting spin chain is sometimes called the hyperbolic (as opposed to elliptic) Inozemtsev chain.
The system has been exactly solved by means of an 'extended' Bethe ansatz method.
The model was solved by Inozemtsev first in the infinite lattice size limit,[3] and later for finite size.
[4][5][2] The model can be used to understand certain aspects of the AdS/CFT correspondence proposed by Maldacena.
Specifically, integrability techniques have turned out to be useful for an 'integrable' instance of the correspondence.
On the string theory side of the correspondence, one has a type IIB superstring on
On the conformal field theory (CFT) side one has N = 4 supersymmetric Yang–Mills theory (N = 4 SYM) on four-dimensional space.
Spin chains have turned out to be useful for computing specific anomalous dimensions on the CFT side, which can then provide evidence for the correspondence if matching observables are computed on the string theory side.
' limit of N = 4 SYM, in which the number of colors
, is sent to infinity, determining one-loop anomalous dimensions becomes equivalent to the problem of diagonalizing an appropriate spin chain.
[6] While the match only holds up to three loops in perturbation theory, and its appearance might thus have been somewhat of a coincidence, this development has brought the Inozemtsev chain under the attention of a wider audience of researchers.