Valence bond solids forSU(n)spin chains: Exact models, spinon confinement, and the Haldane gap

Abstract

To begin with, we introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a three-fold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation3 of SU (3) if the original spins of the model transform under rep. 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin reps. 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and hence a Haldane gap in the excitation spectrum. We generalize some of our models to SU(n). Finally, we use the emerging rules for the construction of VBS states to argue that models of antiferromagnetic chains of SU(n) spins in general possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes λ which is divisible by n. If λ and n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If λ and n have a common divisor different from n, it will depend on the specifics of the model including the range of the interaction.

Bipartite fluctuations as a probe of many-body entanglement

“…These exact results might be extended for various generalized AKLT states [126,132,133] where the SU(2) symmetry of the electron spin is replaced by SU(n), SP(n), or SO(n) symmetry.…”

Spin-spin correlation functions of the q-valence-bond-solid state of an integer spin model

“…Moreover, it lead to a deeper understanding for integer spin chains such as the discovery of the special type of long-range order [4,5]. The AKLT model has been generalized to higher-spin models, anisotropic models, etc [6,7,8,9,10,11,12,13,14,15,16,17]. The Hamiltonians are essentially linear combinations of projection operators with nonnegative coefficients, and their ground states are called valence-bond-solid (VBS) state.…”

Entanglement and SU(n) symmetry in one-dimensional valence-bond solid states

“…Moreover, its ground state is a valence bond solid (VBS) that admits a representation in terms of a Matrix Product State (MPS) [4], and is closely related to the Laughlin state [5] and the fractional quantum Hall effect [6]. This scenario has been recently generalized to other symmetry groups such as SO(n), SU(n) and Sp(2n) [7][8][9][10][11][12]. As for the behaviour of entanglement in these generalizatons, not too much is known.…”

“…As for the behaviour of entanglement in these generalizatons, not too much is known. Derivations have been carried out for the correlation length [8,9] as well as von Neumann and Rényi entropies [10,11] of SU(n) VBS states on a chain, but these involve a number of technicalities that make them quite lengthy.In this paper we provide an elegant and straightforward evaluation of the many-body entanglement properties of the above SU(n) valence bond solid state on a chain. In particular, we derive unknown quantities such as the geometric entanglement per block [13,14], but also re-derive other quantities such as the correlation length, von Neumann and Rényi entropies of a block in a significantly simpler way than previous derivations [10,11].…”

“…As for the behaviour of entanglement in these generalizatons, not too much is known. Derivations have been carried out for the correlation length [8,9] as well as von Neumann and Rényi entropies [10,11] of SU(n) VBS states on a chain, but these involve a number of technicalities that make them quite lengthy.…”