Bootstrapping the Gross-Neveu-Yukawa Archipelago and Skydiving Algorithm

Author: Liu, Aike

Year: 2024

Degree: Dissertation (Ph.D.)

Advisor: Simmons-Duffin, David

Committee Members: Ooguri, Hirosi; Kapustin, Anton N.; Hsieh, David; Simmons-Duffin, David

Option: Physics

DOI: 10.7907/qxe6-3s08

Abstract

The goal of the conformal bootstrap is to solve conformal field theories (CFTs) by imposing physical constraints including symmetries and unitarity. It has been a powerful tool to rigorously constrain CFT data, especially for strongly-coupled theories where traditional perturbative methods fail. Based solely on unitarity, symmetry, and assumptions about the spectrum of scaling dimensions, the bootstrap method has produced stringent bounds on critical exponents of several universality classes describing real-world statistical and quantum phase transitions.

The numerical bootstrap method combines the physical constraints with convex optimization. Specifically, the physics problems are converted into semidefinite programs and solved numerically. Such methods have led to precise and rigorous predictions on critical exponents of condensed-matter systems, such as 3d Ising models and the $O(N)$ models. In my first research project, we perform a bootstrap analysis of a mixed system of four-point functions of bosonic and fermionic operators in parity-preserving 3d CFTs with O(N) global symmetry. Our results provide rigorous bounds on scaling dimensions and OPE coefficients of the O(N) symmetric Gross-Neveu-Yukawa (GNY) fixed-points, constraining these theories to live in isolated islands in the space of CFT data. We delivered the bounds on the critical points with N = 1, 2, 4, and 8, which have applications to phase transitions in condensed matter systems. We were also able to demonstrate the existence of the supercurrent when supersymmetry emerges at N=1 without prior assumptions of the symmetry.

On the other hand, as we progress towards larger systems to study and to obtain more precise bounds on various CFTs, the limits on computational resources cannot be overlooked. To tackle the numerical challenges and improve efficiency, my second research project studies families of semidefinite programs (SDPs) that depend nonlinearly on a small number of “external” parameters. Such families appear universally in numerical bootstrap computations. The traditional method for finding an optimal point in parameter space works by first solving an SDP with fixed external parameters, then moving to a new point in parameter space and repeating the process. Instead, we unify solving the SDP and moving in parameter space in a single algorithm that we call “skydiving”. We test skydiving on some representative problems in the conformal bootstrap, finding significant speedups compared to traditional methods.

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