Bootstrapping the Ising Model on the Lattice
Bootstrap is a strategy that does not calculate the physical quantities directly, but obtains them based on some basic constraints.The bootstrap idea was first proposed in the 1950s as an alternative to Feynman diagrams and split into two directions in the 1970s. The mathematical formulation of S-matrix theory, divorced from the bootstrap techniques, helped to develop string theory .Meanwhile, bootstrap techniques went on to become a powerful way to study individual correlation functions of conformal field theories.
In this talk,we will focus on an alternative bootstrap approach, based on positivity together with certain identities that constrain expection values. I will first demonstrate the bootstrap method using two models, the classical and quantum non-harmonic oscillator models[1],and show how boostrap give a two-sided bounds on momnents and energy. Then we will move on to statistical Ising model of spins[2] on the square lattice using a bootstrap method that combines spin-flip identities with positivity conditions, including reflection positivity and Griffiths inequalities, to derive rigorous two-sided bounds on spin correlators through semi-definite programming.
[1] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.041601