Zoo of quantum-topological phases of matter (1610.03911v3)

Abstract: What are topological phases of matter? First, they are phases of matter at zero temperature. Second, they have a non-zero energy gap for the excitations above the ground state. Third, they are disordered liquids that seem have no feature. But those disordered liquids actually can have rich patterns of many-body entanglement representing new kinds of order. This paper will give a simple introduction and a brief survey of topological phases of matter. We will first discuss topological phases that have topological order (ie with long range entanglement). Then we will cover topological phases that have no topological order (ie with only short-range entanglement).

• The paper introduces quantum-topological phases defined by many-body entanglement, challenging traditional symmetry-breaking theories.
• It details topological order with phenomena like ground-state degeneracy and fractionalized excitations that resist local perturbations.
• The work also explores symmetry-protected topological states with nontrivial edge behaviors, guiding future quantum computing applications.

Overview of "Zoo of Quantum-Topological Phases of Matter"

The paper "Zoo of quantum-topological phases of matter" authored by Xiao-Gang Wen provides a detailed exploration into the burgeoning field of quantum-topological phases. This paper serves as a comprehensive guide to various topological phases of matter which can be broadly categorized into those with and without topological order, delineating their characteristics and implications in condensed matter physics.

Key Concepts and Findings

Wen's research elucidates the limitations of Landau's symmetry-breaking framework, which has historically underpinned our understanding of phase transitions and symmetry classifications in materials. Contrary to this paradigm, quantum-topological phases of matter exhibit orders beyond symmetry-breaking, characterized by many-body entanglement patterns rather than traditional order parameters.

Topological Order: The paper explores topological orders, which pertain to long-range quantum entanglements intrinsic to certain gapped quantum states. Topologically ordered phases demonstrate phenomena such as quantum Hall effects and fractionalized excitations that are unexplained by symmetry-breaking theories. These phases are robust against local perturbations, with properties that include ground-state degeneracy dependent on the topology of the system and the entanglement of quasi-particles carrying fractional statistics.

Symmetry-Protected Topological (SPT) Order: The paper further explores SPT phases, which are distinct from traditional orders through their reliance on symmetry to protect the phase state. These phases, although not possessing intrinsic topological order, exhibit robust edge states when placed in the presence of symmetry-breaking at the boundaries. SPT states are particularly intriguing in their ability to manifest as trivial bulk states while supporting non-trivial boundary phenomena due to the presence of a protecting symmetry.

Notable Discussions and Theoretical Implications

The paper discusses the critical distinction and implications of topological insulators and superconductors, which represent SPT states that challenge the conventional wisdom of conducting surfaces. Contrary to common characterization, the surface conductivity in topological insulators is not universal, particularly under strong electron interactions. This complexity underscores the need for a nuanced understanding of SPT states in practical applications.

The theoretical framework extends to multidimensional topological orders, leveraging concepts from category theory to classify these phases. In 2+1 dimensions, topological orders are captured by modular tensor categories, while considerations for modular extensions in the context of fermionic systems present further layers for exploration.

Practical and Future Directions

The classification of all gapped quantum orders represents a significant research frontier, with implications for both practical and theoretical advancements. The paper emphasizes the need to unify the understanding of quantum phases across dimensions, acknowledging that many-body entanglement and quantum topology pose rich domains for future investigations.

The exploration of quantum liquids and string-net models as demonstrated in Wen's work points to potential applications in quantum computing, where the realization of robust topological qubits could revolutionize computational paradigms. Moreover, the quest to synthesize new materials exhibiting these quantum-topological characteristics could foster technological innovations beyond current capabilities.

In summary, Xiao-Gang Wen's "Zoo of quantum-topological phases of matter" offers profound insights into a novel class of states of matter, encouraging further research to expand our understanding of quantum order and leverage it for emerging applications.