Exactly Solvable Magnet of Conformal Spins in Four Dimensions

Area of Science:

• Quantum Field Theory
• Mathematical Physics
• String Theory

Background:

• Investigating quantum spin chains with nearest-neighbor interactions and open boundary conditions is crucial for understanding complex quantum systems.
• Conformal field theories (CFTs) describe critical phenomena and are fundamental in high-energy physics and condensed matter.
• The SO(1,5) group and its representations are key in various areas of theoretical physics, including supergravity and CFT.

Purpose of the Study:

• To provide eigenfunctions for a quantum chain of N conformal spins.
• To develop a method for computing conformal data in the fishnet limit of supersymmetric N=4 Yang-Mills theory.
• To explore the connection between quantum spin chains and conformal Feynman diagrams.

Main Methods:

• Derivation of eigenfunctions for a quantum chain with specific interactions and boundary conditions.
• Utilizing a new star-triangle identity to compute the spectral measure.
• Applying bootstrap techniques to reproduce all-loop fishnet integrals.
• Employing a cutting-and-gluing procedure on a square lattice.

Main Results:

• The eigenfunctions for the quantum chain in the SO(1,5) irreducible representation are provided.
• The spectrum is separated into N contributions, each labeled by a quantum number Y_{a}=[ν_{a},n_{a}].
• Orthogonal eigenfunctions and the spectral measure were computed.
• Conformal Feynman diagrams with square lattice topology are represented using separated variables.
• All-loop fishnet integrals are reproduced.

Conclusions:

• The derived eigenfunctions are conjectured to form a complete set.
• The eigenfunctions offer a tool for direct computation of conformal data in the fishnet limit.
• This work connects quantum spin chains to the study of supersymmetric Yang-Mills theory at finite coupling.