Multipartite entanglement in spin chains and the Hyperdeterminant

TL;DR

This paper explores the use of the Cayley hyperdeterminant to characterize multipartite entanglement in four-site spin chains, computing it for various models and states, and proposing a generalization to thermal states.

Contribution

It introduces a method to compute the hyperdeterminant for complex four-site spin chain states and extends its application to thermal density matrices, revealing phase transitions and entanglement structures.

Findings

Hyperdeterminant captures phase transitions in spin models

Polynomial invariants reveal quadripartite entanglement

Generalization to thermal states broadens entanglement analysis

Abstract

A 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 $\psi_{i_1, \dots, i_n}$ 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 $2.8 \times 10^6$ terms. This huge object can be written in terms of more simple polynomials $S$ and $T$ of degrees 8 and 12 respectively. In this paper 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.