Continuum Limits of Homogeneous Binary Trees and the Thompson Group

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Mathematical Physics

Background:

• Tree tensor networks (TTNs) approximate critical quantum spin chains, matching conformal field theory (CFT) predictions for correlation functions in the continuum limit.
• Incorporating dynamics into TTNs is crucial for a complete correspondence with CFT, which relies on representations of the circle diffeomorphism group.

Purpose of the Study:

• To investigate the generic behavior of tree tensor networks when incorporating dynamics, specifically addressing the construction of representations of the Thompson group T.
• To extend previous findings on discontinuous representations in discrete settings to a broader class of TTNs and tensor planar algebras.

Main Methods:

• Analyzing coarse-graining maps in tree tensor networks and their relation to representations of the Thompson group T.
• Investigating elements of the tensor planar algebra and their continuum limits.
• Identifying necessary conditions for the existence of a continuous limit in TTN constructions.

Main Results:

• The set of coarse-graining maps leading to discontinuous representations has full measure in the space of isometries for generic tree tensor networks.
• This finding generalizes a previously identified 'no-go' example to typical elements of the tensor planar algebra.
• A simple, verifiable condition is identified that singles out tree tensor networks admitting a continuous limit.

Conclusions:

• The generic discontinuity issue in TTNs poses a significant challenge for incorporating dynamics in a physically meaningful way.
• The identified condition provides a pathway to construct TTNs that yield continuous representations, relevant for holographic codes and quantum simulations.