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

Here we evaluate the many-body entanglement properties of a generalized SU($n$) valence-bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for an elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and Rényi entropies of a block, localizable entanglement, and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU($2$) case.

Figure 1

(a) Structure of the SU($n$) VBS state $|\Psi \rangle$ for $N$ sites and open boundary conditions. (b) MPS structure of the state (up to an overall normalization constant). At the boundaries there are SU($n$) degrees of freedom $a,b=0,1,\dots ,n-1$. (c) MPS transfer matrix $A(1)$. (d) Transfer matrix of a block $A(L)=A(1){}^L$.

Figure 2

Diagrammatic representation of several reduced density matrices sharing the same spectrum (up to normalization constants). (a) Of a block of length $L$, called ${\rho }_L$. (b) Of the environment of a block of length $L$ for the environment as in (a). (c) Of a block of length $L$ once the environment is computed [dominant left and right eigenvectors of $A(1)$, which are singlets]. (d) Of the environment of a block of length $L$ for the environment as in (c), equal to $A(L)$.