n-cluster models in a transverse magnetic field

Abstract

In this paper we analyze a family of one dimensional fully analytically solvable models, named the n-cluster models in a transverse magnetic field, in which a many-body cluster interaction competes with a uniform transverse magnetic field. These models, independently by the cluster size n + 2, exhibit a quantum phase transition, that separates a paramagnetic phase from a cluster one, that corresponds to a nematic ordered phase or a symmetry-protected topological ordered phase for even or odd n respectively. Due to the symmetries of the spin correlation functions, we prove that these models have no genuine multipartite entanglement. On the contrary, for a magnetic field strong enough, a non vanishing concurrence arises between spins at the endpoints of the cluster. Due to their integrability and entanglement properties, the n-cluster models in a transverse magnetic field may serve as a prototype for studying non trivial-spin orderings and as a potential reference system for the applications of quantum information tasks.

Resilience of the topological phases to frustration

“…Our results on the Cluster-Ising model are even more interesting if we observe that we can directly transfer our general arguments to the m -cluster Ising model, studied in Refs. 43 , 44 , which consists of m -body cluster interaction competing with the antiferromagnetic Ising pairing. It is known that for any odd m , the model is characterized by a symmetry protected topologically ordered phase, which is, by our general arguments, expected not to be affected by geometrical frustration.…”

“…Since our goal is to study the effect of topological frustration, we assume periodic boundary conditions and that N is an odd number ( ). Due to the existence of an analytical solution, the family of spin-1/2 cluster models was intensively studied in the past years 31 , 32 , 41 – 44 . For the Cluster-Ising model in Eq.…”

“…As a consequence, the three-body cluster interaction is not expected to show any geometrical frustration. This should be contrasted with the case of cluster interaction with an even number of sites, such as the 2-Cluster-Ising chain 43 , 44 , which do not have topological order. According to the argument above, in these cases we expect that frustration will be effective and indeed we have proven that to be the case 17 , 18 .…”

Quantum phase transition induced by topological frustration

“…Nonetheless, after a few years, it has become clear that the richness of quantum many-body systems goes beyond the standard Landau paradigm. Indeed, topologically ordered phases 5,6 , which have no equivalent in the classical regime, as well as nematic ones 7 , represent instances in which violation of the same symmetry is associated with different (typically non-local) and non-equivalent order parameters [8][9][10] , depending on the model under analysis. This implied that Landau's theory had to be extended to incorporate more general concepts of order, which include the non-local effects that come along with the quantum regime and have no classical counterpart.…”

Topological Frustration can modify the nature of a Quantum Phase Transition

“…Here and in the following σ α j (α = x, y, z) stand for Pauli's operators on the j-th spin, O j = σ y j−1 σ x j σ y j+1 is the cluster operator [20] that allows to rewrite the cluster interaction term in a form resembling a two body one, φ is a parameter that allows to tune the relative weight between the two terms and the periodic boundary conditions imply that σ α j+N = σ α j , as well as O j+N = O j . In Fig.…”