Bootstrapping the gap in quantum spin systems

TL;DR

This paper introduces a novel bootstrap approach inspired by conformal field theory to rigorously bound the low-energy spectrum and matrix elements of quantum spin systems, demonstrated on the anharmonic oscillator and TFIM.

Contribution

A new bootstrap method for quantum systems using equations of motion and crossing symmetry, applicable to local Hamiltonians and capable of providing rigorous bounds.

Findings

Accurate spectrum and matrix element solutions for anharmonic oscillator with few crossing equations.

Rigorous bounds on the gap and matrix elements for the TFIM in the thermodynamic limit.

Method improves bounds with larger crossing equation systems, excluding finite-volume solutions.

Abstract

In this work we report on a new bootstrap method for quantum mechanical problems that closely mirrors the setup from conformal field theory (CFT). We use the equations of motion to develop an analogue of the conformal block expansion for matrix elements and impose crossing symmetry in order to place bounds on their values. The method can be applied to any quantum mechanical system with a local Hamiltonian, and we test it on an anharmonic oscillator model as well as the (1+1)-dimensional transverse field Ising model (TFIM). For the anharmonic oscillator model we show that a small number of crossing equations provides an accurate solution to the spectrum and matrix elements. For the TFIM we show that the Hamiltonian equations of motion, translational invariance and global symmetry selection rules imposes a rigorous bound on the gap and the matrix elements of TFIM in the thermodynamic limit. The bound improves as we consider larger systems of crossing equations, ruling out more finite-volume solutions. Our method provides a way to probe the low energy spectrum of an infinite lattice from the Hamiltonian rigorously and without approximation.