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. It was introduced in 1990 by Vladimir Inozemtsev as a model which interpolates between the Heisenberg XXX model and the Haldane–Shastry model. Like those spin chains, the Inozemtsev model is exactly solvable.
Formulation For a chain with L {\displaystyle L} spin 1/2 sites, the quantum phase space is described by the tensor product Hilbert space H = ( C 2 ) ⊗ L {\displaystyle {\mathcal {H}}=(\mathbb {C} ^{2})^{\otimes L}} . The (elliptic) Inozemtsev model is given by the (unnormalised) Hamiltonian
H = ∑ i < j L ℘ ( i − j ) 1 − σ → i ⋅ σ → j 2 {\displaystyle H=\sum _{i<j}^{L}\wp (i-j){\frac {1-{\vec {\sigma }}_{i}\cdot {\vec {\sigma }}_{j}}{2}}}
where the pair potential ℘ ( z ) {\displaystyle \wp (z)} is the Weierstrass elliptic function, and σ → j {\displaystyle {\vec {\sigma }}_{j}} denotes the Pauli vector at the j {\displaystyle j} th site (acting nontrivially on the j {\displaystyle j} th copy of C 2 {\displaystyle \mathbb {C} ^{2}} in H {\displaystyle {\mathcal {H}}} ). The periods of the Weierstrass elliptic function are the length L {\displaystyle L} of the chain, to ensure periodic boundary conditions, together with an imaginary period that sets the interaction range and is traditionally parameterized as ω = i π / κ {\displaystyle \omega =i\,\pi /\kappa } where κ > 0 {\displaystyle \kappa >0} . The truly long-range Haldane-Shastry chain is obtained when the imaginary period is removed ( ω → i ∞ {\displaystyle \omega \to i\,\infty } , so κ → 0 {\displaystyle \kappa \to 0} ) while, upon renormalisation, the Heisenberg spin chain is recovered in the limit ω → i 0 + {\displaystyle \omega \to i\,0^{+}} ( κ → ∞ {\displaystyle \kappa \to \infty } ). The infinite-length limit L → ∞ {\displaystyle L\to \infty } instead gives hyperbolic potential 1 / sinh 2 ( κ ( i − j ) ) {\displaystyle 1/\sinh ^{2}(\kappa (i-j))} , which is why the resulting spin chain is sometimes called the hyperbolic (as opposed to elliptic) Inozemtsev chain.
Exact solution 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, and later for finite size.
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