Numerical exploration of the bootstrap in spin chain models

TL;DR

This paper explores the use of semidefinite programming in the bootstrap approach to analyze one-dimensional spin chains, assessing its effectiveness in extracting physical data and identifying practical challenges.

Contribution

It demonstrates how semidefinite programming can be applied to spin chains to extract conformal data and highlights the method's limitations and complexities.

Findings

Correlation functions converge at and away from criticality

Semidefinite methods can extract physical data like central charges

Practical challenges include convergence issues and exponential complexity

Abstract

We analyze the bootstrap approach (a dual optimization method to the variational approach) to one-dimensional spin chains, leveraging semidefinite programming to extract numerical results. We study how correlation functions in the ground state converge to their true values at and away from criticality and at relaxed optimality. We consider the transverse Ising model, the three state Potts model, and other non-integrable spin chains and investigate to what extent semidefinite methods can reliably extract numerical emergent physical data, including conformal central charges, correlation lengths and scaling dimensions. We demonstrate procedures to extract these data and show preliminary results in the various models considered. We compare to exact analytical results and to exact diagonalization when the system volume is small enough. When we attempt to go to the thermodynamic limit, the semidefinite numerical method with translation invariance imposed as a constraint finds the solution with periodic boundary conditions even if these have not been specified. This implies that the determination of all conformal data in correlators has to be handled at finite volume. Our investigation reveals that the approach has practical challenges. In particular, the correlation functions extracted from the optimal solution, which function as slack variables in the optimization, have convergence issues that suggest an underlying exponential complexity in the system size.