[Paper Review] Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness

Arkya Chatterjee, Xiao-Gang Wen|arXiv (Cornell University)|May 12, 2022

Molecular spectroscopy and chiralityChemistry18 citations

TL;DR

This paper introduces a holographic framework based on symmetry/topological-order (Symm/TO) correspondence to unify the description of continuous phase transitions, gapless critical points, and symmetry-protected gapless phases. By classifying gapped and gapless states via condensable algebras in higher-dimensional topological orders, it establishes that gaplessness emerges when the reduced symmetry topological order M/A is nontrivial—providing a symmetry-protected mechanism for criticality applicable to anomalous, higher-form, and non-invertible symmetries.

ABSTRACT

Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that: (1) For systems with a symmetry described by symmetry TO $M$, their gapped and gapless states are classified by condensable algebras $A$, formed by elementary excitations in $M$ with trivial self/mutual statistics. Such classified states (called $A$-states) can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc., in a unified way. (2) The local low-energy properties of an $A$-state can be calculated from its reduced symmetry TO $M_{/A}$, using holographic modular bootstrap (holoMB) which takes $M_{/A}$ as an input. Here $M_{/A}$ is obtained from $M$ by condensing excitations in $A$. Notably, an $A$-state must be gapless if $M_{/A}$ is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or non-invertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. (4) 1+1D bosonic systems with $S_3$ symmetry have four gapped phases with unbroken symmetries $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, and $\mathbb{Z}_1$. We find a duality between two transitions $S_3 \leftrightarrow \mathbb{Z}_1$ and $\mathbb{Z}_3 \leftrightarrow \mathbb{Z}_2$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT).

Motivation & Objective

• To develop a unified framework for classifying gapped and gapless quantum phases beyond Landau symmetry breaking.
• To establish a holographic duality between bulk topological orders and boundary criticality.
• To explain the emergence and symmetry protection of gaplessness in terms of condensable algebras in higher-dimensional topological orders.
• To analyze continuous phase transitions in systems with mixed anomalies and non-invertible symmetries.
• To classify gapped and gapless phases in 1+1D systems with S3 and Z2×Z′2 symmetries, including anomalous cases.

Proposed method

• Uses the symmetry/topological-order (Symm/TO) correspondence to classify global symmetries via gappable-boundary topological orders in one higher dimension.
• Defines holomorphic equivalence of global symmetries through isomorphism of their local symmetric operator algebras.
• Applies holographic modular bootstrap (holoMB) to compute low-energy properties from the reduced symmetry topological order M/A.
• Identifies gapped and gapless states via condensable algebras A in the symmetry topological order M, where A consists of anyons with trivial statistics.
• Constructs phase diagrams by analyzing the structure of condensable algebras and their relations in M.
• Applies the framework to 1+1D systems with Z2×Z′2 and S3 symmetries, including anomalous and mixed-anomaly cases.

Experimental results

Research questions

• RQ1 How can continuous phase transitions be systematically classified when Landau theory fails due to non-subgroup symmetry relations?
• RQ2 What is the role of topological order in protecting gaplessness in critical phases?
• RQ3 How do anomalous, higher-form, and non-invertible symmetries influence the emergence of gapless criticality?
• RQ4 Can a unified framework describe symmetry breaking, SPT, and topological order phases in a single formalism?
• RQ5 What determines whether a phase transition is first-order or continuous in systems with non-invertible or anomalous symmetries?

Key findings

• For 1+1D Z2×Z′2 symmetry with mixed anomaly, a stable continuous transition (deconfined quantum critical point) exists between Z2-breaking and Z′2-breaking phases, described by the same critical theory as a Z4 symmetry-breaking transition.
• In 1+1D S3 symmetry, four gapped phases exist with unbroken symmetries S3, Z3, Z2, and Z1, and transitions S3↔Z1 and Z3↔Z2 are either both first-order or both continuous, with the latter described by the same conformal field theory.
• Anomalous S(1)3 and S(2)3 symmetries support stable chiral gapless phases protected by symmetry, demonstrating symmetry-protected gaplessness in anomalous systems.
• The critical point of the Z2×Z′2 mixed-anomaly transition is shown to be equivalent to a Z4 symmetry-breaking critical point, indicating a universal CFT description.
• The framework predicts that an A-state is necessarily gapless if the reduced symmetry topological order M/A is nontrivial, providing a general mechanism for symmetry-protected gaplessness.
• The method successfully classifies gapped and gapless phases in anomalous S3 systems, revealing dualities and universal CFTs for continuous transitions.