[Paper Review] Conformal Field Theories as Scaling Limit of Anyonic Chains

Modjtaba Shokrian Zini, Zhenghan Wang|arXiv (Cornell University)|Jun 26, 2017

Algebraic structures and combinatorial models31 references3 citations

TL;DR

This paper proposes a mathematical framework for the low-energy scaling limit of non-relativistic quantum theories, specifically applying it to anyonic chains to show that chiral unitary rational conformal field theories (CFTs), including the Ising minimal model $M(4,3)$, emerge as scaling limits. It establishes a precise algebraic correspondence between Temperley-Lieb generators and Virasoro generators, verifying this for the Ising model and conjecturing its validity for all unitary minimal models $M(k+2,k+1)$, $k \geq 3$, under a proposed convergence condition.

ABSTRACT

We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate Conjecture 4.3 on conditions when a chiral unitary rational (1+1)-conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model $M(4,3)$ using Ising anyonic chains. Part of the conjecture is a precise relation between Temperley-Lieb generators $\{e_i\}$ and some finite stage operators of the Virasoro generators $\{L_m+L_{-m}\}$ and $\{i(L_m-L_{-m})\}$ for unitary minimal models $M(k+2,k+1)$ in Conjecture 5.5. A similar earlier relation is known as the Koo-Saleur formula in the physics literature [39]. Assuming Conjecture 4.3, most of our main results for the Ising minimal model $M(4,3)$ hold for unitary minimal models $M(k+2,k+1), k\geq 3$ as well. Our approach is inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers, and supported by extensive numerical simulation and physical proofs in the physics literature.

Motivation & Objective

• To develop a rigorous mathematical definition of the low-energy scaling limit for general non-relativistic quantum theories in any dimension.
• To investigate whether chiral unitary rational (1+1)-dimensional conformal field theories arise as scaling limits of anyonic chains.
• To verify a conjecture linking Temperley-Lieb algebra generators to Virasoro algebra generators for unitary minimal models.
• To provide a foundation for the efficient quantum simulation of conformal field theories using quantum computers.
• To recover the algebra of local observables in vertex operator algebras and local conformal nets from finite anyonic chains.

Proposed method

• Introduces a formalism for the low-energy scaling limit of quantum theories, focusing on convergence of observables as an algebra.
• Applies the formalism to Ising anyonic chains, analyzing the limit of local observables and their algebraic structure.
• Establishes a precise relation between Temperley-Lieb generators $\{e_i\}$ and finite-stage operators $\{L_m + L_{-m}\}$ and $\{i(L_m - L_{-m})\}$ of the Virasoro algebra.
• Uses smeared and point-like vertex operators to define correlation functions in the scaling limit, with convergence under distributional limits of smearing functions.
• Leverages Wightman’s observables and local conformal nets to characterize the continuum limit algebraically.
• Relies on numerical simulations and physical proofs from the physics literature to support the mathematical conjectures.

Experimental results

Research questions

• RQ1 Under what conditions does a chiral unitary rational (1+1)-dimensional conformal field theory emerge as the scaling limit of an anyonic chain?
• RQ2 Can a precise algebraic correspondence be established between Temperley-Lieb generators and Virasoro generators in the scaling limit?
• RQ3 How do smeared and point-like correlation functions in the anyonic chain converge to their continuum counterparts in the scaling limit?
• RQ4 To what extent can the algebra of local observables in vertex operator algebras and local conformal nets be recovered from finite anyonic chains?
• RQ5 What is the role of hidden locality in both space and energy in the scaling limit, and how does it support quantum simulation of CFTs?

Key findings

• The Ising minimal model $M(4,3)$ is shown to arise as the scaling limit of Ising anyonic chains, verifying Conjecture 4.3 for this case.
• A precise relation is established between Temperley-Lieb generators $e_i$ and the Virasoro generators $L_m + L_{-m}$ and $i(L_m - L_{-m})$, generalizing the Koo-Saleur formula.
• The convergence of smeared correlation functions to point-like correlation functions is rigorously justified in the limit of delta-distribution smearing functions with disjoint support.
• The algebra of local observables in the vertex operator algebra and local conformal net for $M(4,3)$ is recovered from the anyonic chain formalism.
• Assuming Conjecture 4.3, the results for the Ising model extend to all unitary minimal models $M(k+2,k+1)$ with $k \geq 3$, suggesting a general mechanism for CFT emergence.
• The framework supports the eventual goal of simulating conformal field theories on quantum computers by establishing a finite, computable model via anyonic chains.