Inozemtsev model

For a chain with ${\displaystyle L}$ spin 1/2 sites, the quantum phase space is described by the Hilbert space ${\displaystyle {\mathcal {H}}=(\mathbb {C} ^{2})^{\otimes L}}$. The (elliptic) Inozemtsev model is given by the (unnormalised) Hamiltonian[2]

$${\displaystyle H_{u}=\sum _{j<k}^{L}\wp (j-k){\frac {1-{\vec {\sigma }}_{j}\otimes {\vec {\sigma }}_{k}}{2}}}$$

where ${\displaystyle \wp (z)}$ is the Weierstrass elliptic function and ${\displaystyle {\vec {\sigma }}_{j}}$ is the Pauli vector acting on the ${\displaystyle j}$th copy of the tensor product Hilbert space.

More precisely, the parameters used for the Weierstrass p are

$${\displaystyle \wp (z)=\wp (z;L,i\omega )={\frac {1}{z^{2}}}+\sum _{m,n}'\left({\frac {1}{(z-mL-in\omega )^{2}}}-{\frac {1}{(mL+in\omega )^{2}}}\right)}$$

for ${\displaystyle \omega }$ a positive real number. The notation ${\displaystyle \sum '}$ means sum over all pairs of integers except ${\displaystyle (m,n)=(0,0)}$. This ensures that the function is ${\displaystyle L}$-periodic in the real direction, and is real for real values of ${\displaystyle z}$.

The system has been exactly solved by means of a Bethe ansatz method. The Bethe ansatz equations were found by Inozemtsev.[3][4] In fact, the model was first solved in the infinite size limit.[5]

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 ${\displaystyle \mathrm {AdS} _{5}\times \mathrm {S} ^{5}}$, the product of five-dimensional Anti-de Sitter space with the five-dimensional sphere. 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. In the so-called 'planar limit' or 'large N' limit of N = 4 SYM, in which the number of colors ${\displaystyle N}$, which parametrizes the gauge group ${\displaystyle SU(N)}$, is sent to infinity, determining one-loop anomalous dimensions becomes equivalent to the problem of diagonalizing an appropriate spin chain. The Inozemtsev model is one such model which has been useful in determining these quantities.[6]

• Quantum Heisenberg model
• Haldane–Shastry model