A symmetry principle for Topological Quantum Order

TL;DR

<3-5 sentence high-level summary>: The paper develops a unifying framework for Topological Quantum Order (TQO) grounded in Gauge-Like Symmetries ($d$-GLSs) and entanglement, and proves that systems with low-dimensional GLSs exhibit TQO at zero and finite temperature under broad conditions. It shows that TQO can be diagnosed via GLS selection rules, not necessarily requiring a spectral gap, and that dualities map complex topological orders to simpler, non-local operator languages (e.g., Kitaev’s toric code to a 1D Ising chain). The authors provide algorithmic constructions of long-range string/brane correlators, discuss dimensional reduction and thermal fragility in various models, and analyze the relationship between entanglement entropy and true TQO, concluding that spectra alone do not determine topological order. These results offer a versatile toolkit for engineering and recognizing TQO and illuminate connections to topology, fractionalization, and non-local order across multiple dimensions and platforms.

Abstract

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The guiding principle behind our approach is that of symmetries and entanglement. We introduce the concept of low-dimensional Gauge-Like Symmetries (GLSs), and the physical conservation laws (including topological terms and fractionalization) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The topological defects which are associated with the restoration of GLSs lead to TQO. Selection rules associated with the GLSs enable us to systematically construct states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. All currently known examples of TQO display GLSs. We analyze spectral structures and show that Kitaev's toric code model and Wen's plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain. Despite the spectral gap in these systems, the toric operator expectation values may vanish once thermal fluctuations are present. This mapping illustrates that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string orders in general systems with entangled ground states.

Figures (14)

• Figure 1: Schematic representation of potential energy spectra. Case $(a)$ may represent a Fermi liquid, case $(b)$ a Fractional Quantum Hall liquid, case $(c)$ a band insulator, and case $(d)$.
• Figure 2: Schematics of the interactions and symmetries involved in three of the known examples which display TQO. The lower panel represents Kitaev's Toric-code model, the middle one corresponds to the ${\mathbb{Z}}_{2}$ gauge theory and the upper panel corresponds to Wen's plaquette model. Dark lines (solid or dashed) represent symmetries while the brighter (red) lines or crosses correspond to the interaction terms. The hollow circles represent the spin locations. See text.
• Figure 3: From Refs. NBCvBCN. The symmetries of Eq. (\ref{['symorb']}) applied on the uniform state (at left).
• Figure 4: Schematics of the interactions and symmetries involved in the classical rendition of three $D=2$ examples. The left panel represents (i) the local ($d=0$) symmetries of the Ising gauge theory [Eq. (\ref{['gz2+']})]. The middle panel represents (ii) an orbital compass model [Eqs. (\ref{['orb']}), and (\ref{['compass1']})] with $d=1$ symmetries; here the symmetry operations span lines [see Eq. (\ref{['symorb']}). The right panel depicts (iii) an XY model with $d=2$ symmetries; the symmetry here spans the entire $D=2$ dimensional plane.
• Figure 5: From Ref. BN. The anisotropic hopping amplitudes leading to the Kugel-Khomskii (KK) Hamiltonian. Similar to Harris, the four lobed states denote the $3d$ orbitals of a transition metal while the intermediate small $p$ orbitals are oxygen orbital through which the super-exchange process occurs. The dark and bright shades denote positive and negative regions of the orbital wave-function. Due to orthogonality with intermediate oxygen $p$ states, in any orbital state $|\alpha \rangle$ (e.g. $| Z \rangle \equiv | d_{xy} \rangle$ above), hopping is forbidden between sites separated along the cubic $\alpha$ ($z$ above) axis. The ensuing super-exchange (KK) Hamiltonian exhibits a $d=2$$SU(2)$ symmetry corresponding to a uniform rotation of all spins whose orbital state is $|\alpha \rangle$ in any plane orthogonal to the cubic direction $\alpha$.