Holographic Symmetries and Generalized Order Parameters for Topological Matter

TL;DR

The paper introduces a bond-algebraic duality framework to identify generalized, non-local order parameters for topological matter, anchored by the concept of holographic symmetry where global symmetries map to boundary symmetries under duality. It provides a systematic method to map Landau order parameters to generalized OPs in topologically ordered systems, enabling the diagnosis of deconfinement, confinement, and Higgs transitions through string correlators and edge-state descriptions. Through explicit dualities involving gauged Kitaev wires, Z2 and Zp gauge theories, Abelian Higgs models, extended toric codes, and XY models on frustrated lattices, the work demonstrates how to derive and interpret these generalized OPs in a variety of dimensions and boundary conditions. The discussion outlines future directions, including identifying sufficient conditions for mapping between Landau and topological orders and connecting generalized OPs to effective field theories.

Abstract

We introduce a universally applicable method, based on the bond-algebraic theory of dualities, to search for generalized order parameters in disparate systems including non-Landau systems with topological order. A key notion that we advance is that of {\em holographic symmetry}. It reflects situations wherein global symmetries become, under a duality mapping, symmetries that act solely on the system's boundary. Holographic symmetries are naturally related to edge modes and localization. The utility of our approach is illustrated by systematically deriving generalized order parameters for pure and matter-coupled Abelian gauge theories, and for some models of topological matter.

Figures (3)

• Figure 1: The $\mathds{Z}$ gauge theory exactly dual to the quantum XY model must satisfy special boundary conditions and possesses a boundary symmetry. The lattice corresponding to the XY model is shown in thick lines, for $N=4$.
• Figure 2: Dual sets of links $\Gamma^*$ and $\Gamma$.
• Figure 3: The quantum, $U(1)$ symmetric XY model on the kagome lattice is holographically dual to a solid-on-solid-like gauge theory on the dice lattice.