We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on suggested quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.
We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on suggested quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.
We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. All known examples of Topological Quantum Order display Gauge-Like Symmetries. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and, most importantly, show the insufficiency of the energy spectrum, (recently defined) entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. Duality mappings illustrate that not withstanding the existence of spectral gaps, thermal fluctuations may impose restrictions on suggested topological quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer new systems with Topological Quantum Order.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature-it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic pointlike excitations. The fermionic toric code, importantly, possesses an anomalous two-form symmetry, associated with the spacelike Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to a phase of matter that, in the context of equilibrium phases of matter, only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous two-form symmetries. Our Letter therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.
Symmetry is a fundamental organising principle in theoretical physics, underpinning the conservation of physical quantities and the characterisation of phases of matter. It is closely related to topology, leading to generalisations such as higher-form and non-invertible symmetries, with applications in electromagnetism, the standard model of particle physics, condensed matter, and quantum computation. This thesis harnesses generalised symmetries to derive universal results in gapped and gapless phases of matter, focusing particularly on topologically ordered systems and conformal field theories (CFTs). Topological order is characterised by long-range entangled gapped phases, namely ground states that cannot be disentangled by finite-depth quantum circuits. At low energies it is described via topological quantum field theories (TQFTs). One part of this thesis focusses on extracting patterns of long-range entanglement from the low-energy TQFT for a class of topological orders described by abelian BF theory. By considering algebras of topological operators restricted to subregions, a new measure of entanglement, termed essential topological entanglement, is introduced to refine topological entanglement entropy. This is complemented by a detailed analysis of the edge-mode spectrum of abelian BF theory and its contribution to entanglement entropy across different topologies. A key mathematical result, enabling the above analysis, is the generalisation of Kac–Moody current algebras and their representation theory to higher dimensions. On the gapless front, the thesis considers CFTs with continuous higher-form symmetries, invertible or non-invertible. It is shown that they also possess infinite-dimensional current algebras akin to those described above. Their representation theory allows for a complete characterisation of the quantum states of these theories across arbitrary spatial topologies. In four dimensions, CFTs with continuous one-form symmetries can be described in terms of free photons. Combining this with the previous analysis leads to a one-to-one correspondence between states on particular topologies and line operators, i.e. wires carrying distributions of electric and magnetic charge.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.