String order and symmetries in quantum spin lattices

TL;DR

The results allow us to understand in a straightforward way many of the properties of string order parameters, like their robustness or fragility under perturbations and their typical disappearance beyond strictly one-dimensional lattices.

Abstract

We show that the existence of string order in a given quantum state is intimately related to the presence of a local symmetry by proving that both concepts are equivalent within the framework of finitely correlated states. Once this connection is established, we provide a complete characterization of local symmetries in these states. The results allow to understand in a straightforward way many of the properties of string order parameters, like their robustness/fragility under perturbations and their typical disappearance beyond strictly one-dimensional lattices. We propose and discuss an alternative definition, ideally suited for detecting phase transitions, and generalizations to two and more spatial dimensions.

Figures (1)

• Figure 1: (a) Symmetries in FCS/MPS. The tensor $A$ has three indices, one corresponding to the physical spin (pointing in) and two for the virtual systems (pointing left and right). Applying $U$ to the physical index is equivalent to applying $V$ to the virtual ones. Since the tensor are contracted in a line the unitaries $V$ and $V^\dagger$ cancel and thus the state does not change. (b) 2D generalization (square lattice). The tensor $B$ has one physical and four virtual indices. When applying $U$ to the former, we can have different effects on the virtual system which will, after contraction, leave the state invariant. The example on the right allows for SO.

Theorems & Definitions (3)

• Lemma 1
• Theorem 1
• Theorem 2