Multipartite entanglement in spin chains and the hyperdeterminant

way to characterize multipartite entanglement in pure states of a spin chain with n sites and local dimension d is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function psi(i1,...,in) which is invariant under local unitary transformation. For spin 1/2 chains (i.e. d = 2) with n = 2 and n = 3 sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with n = 4 sites where the hyperdeterminant is a polynomial of degree 24 containing around 2.8 x 10(6) terms. This huge object can be written in terms of more simple polynomials S and T of degrees 8 and 12 respectively. Correspondingly we compute S, T and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane-Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner-Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.