Discriminating string order parameter for topological phases of gapped $SU\left(N\right)$ spin chains

One-dimensional gapped spin chains with symmetry $PSU\left(N\right)=SU\left(N\right)/{\mathbb{Z}}_N$ are known to possess $N$ different topological phases. In this paper, we introduce a nonlocal string order parameter which characterizes each of these $N$ phases unambiguously. Numerics confirm that our order parameter allows one to extract a quantized topological invariant from a given nondegenerate gapped ground state wave function. Discontinuous jumps in the discrete topological order that arise when varying physical couplings in the Hamiltonian may be used to detect quantum phase transitions between different topological phases.

Figure 1

Visualization of different congruence classes for $SU\left(2\right)$ and $SU\left(3\right)$ in terms of colors. The shaded blue boxes are possible representatives of topological classes.

Figure 2

Sketch of the one-site transfer matrix $E_X^{\left[k\right]}$ (upper diagram) and of the expectation value (20) (lower diagram). For the latter, periodic boundary conditions are assumed.

Figure 3

Construction of the two-site Hamiltonian.

Figure 4

A phase transition between two topological phases. The diagram shows the $c$ dependence of the parameters $t$ (blue, squares) and $R$ (red, circles). The inlet shows the deviation of the measured form of the string order matrix from its analytical form. We used adapted increments in the vicinity of the phase transition.