[Paper Review] Non-Liquid Cellular States

Juven Wang|arXiv (Cornell University)|Feb 28, 2020

Opinion Dynamics and Social Influence19 citations

TL;DR

This paper introduces a generalized framework for constructing non-liquid cellular topological states by gluing gauge-symmetry-breaking and gauge-symmetry-extension interfaces as extended defects in a tensor network, incorporating higher-symmetries and sub-dimensional symmetries. The approach uses modified group cohomology and cobordism theory to ensure topological, geometrical, and renormalization consistency, enabling the classification of gapped and gapless phases beyond standard TQFTs, including fracton orders.

ABSTRACT

The existence of quantum non-liquid states and fracton orders, both gapped and gapless states, challenges our understanding of phases of entangled matter. We generalize the cellular topological states to liquid or non-liquid cellular states. We propose a mechanism to construct more general non-abelian states by gluing gauge-symmetry-breaking vs gauge-symmetry-extension interfaces as extended defects in a cellular network, including defects of higher-symmetries, in any dimension. Our approach also naturally incorporates the anyonic particle/string condensations and composite string (related to particle-string or p-string)/membrane condensations. This approach shows gluing the familiar extended topological quantum field theory or conformal field theory data via topology, geometry, and renormalization consistency criteria (via certain modified group cohomology or cobordism theory data) in a tensor network can still guide us to analyze the non-liquid states. (Part of the abelian construction can be understood from the K-matrix Chern-Simons theory approach and the coupled-layer-by-junction constructions.) This approach may also lead us toward a unifying framework for quantum systems of both higher-symmetries and sub-system/sub-dimensional symmetries.

Motivation & Objective

• To extend the concept of cellular topological states beyond liquid phases to include non-liquid, gapped, and gapless topological orders.
• To address the limitations of standard TQFTs in describing fracton orders and other non-liquid quantum phases.
• To unify the description of higher-symmetries, sub-system symmetries, and extended defects (e.g., strings, membranes) in topological quantum matter.
• To provide a consistent, topologically and geometrically constrained method for constructing generalized topological phases using interface tensors and renormalization criteria.

Proposed method

• Uses modified group cohomology and cobordism theory to define topological consistency criteria for interface gluing in cellular networks.
• Introduces a generalized tunneling matrix W as the interface tensor to encode the coupling between different gauge theories or anyon models.
• Applies renormalization consistency criteria via topological and geometrical constraints, including crossing symmetry in lattice structures like hexagonal and square columns.
• Employs tensor network constructions that incorporate both anyonic particle/string condensations and composite string/membrane condensations.
• Generalizes the K-matrix Chern-Simons and coupled-layer-by-junction approaches to non-abelian and non-liquid settings.
• Applies the framework to Z2-toric code, double-Fibonacci, Ising anyon models, and 2-form gauge theories to demonstrate consistency across dimensions and symmetry types.

Experimental results

Research questions

• RQ1 Can non-liquid topological states be systematically constructed using interface gluing between gauge-symmetry-breaking and gauge-symmetry-extension phases?
• RQ2 How can higher-symmetries and sub-dimensional symmetries be consistently incorporated into a unified topological framework?
• RQ3 To what extent can TQFT axioms be generalized to describe fracton orders that do not admit standard TQFT formulations?
• RQ4 What are the topological and geometrical consistency conditions for gluing extended defects (e.g., strings, membranes) in tensor networks?
• RQ5 Can the proposed framework unify abelian and non-abelian topological orders, including gapless and gapped phases, under a single formalism?

Key findings

• The paper constructs non-liquid cellular states by gluing Z2-gauge theories, twisted Z2-gauge theories, and anyon models (e.g., double-Fibonacci, Ising) across interfaces with distinct symmetry-breaking and extension properties.
• For three Z2 gauge theories, the framework yields both liquid and non-liquid cellular states, demonstrating the existence of stable non-liquid topological phases.
• The method successfully incorporates higher-symmetries, such as 1-form and 2-form gauge symmetries, and time-reversal symmetry, extending the scope beyond conventional TQFTs.
• The generalized tunneling matrix W and its renormalized versions W′ and ˜W satisfy topological and geometrical consistency criteria, ensuring stability under coarse-graining.
• The framework explains the emergence of composite string and membrane condensations as natural outcomes of interface gluing in higher-dimensional cellular networks.
• The approach provides a consistent path to describe fracton orders that do not admit standard TQFT formulations, suggesting a broader classification scheme beyond Atiyah’s axioms.