C. Nancarrow: Bootstrapping the Many-Body Problem

The solution of strongly-coupled quantum systems with large numbers of con- stituents remains an active area of mathematical and computational research 100 years since the foundations of theoretical quantum physics were laid. Adapting mod- ern numerical bootstrap and tensor-network techniques to this long-standing problem, we are able to circumvent the challenges posed by the large (formally infinite) number of interacting bodies in these systems. This Dissertation details two new techniques for computing rigorous bounds on quantum observables in the many-body problem: 1. The Spectral Bootstrap 2. The Coarse-Grained Quantum Mechanics Bootstrap. In each case, the technique leverages positivity of a Hilbert Space inner product, quantum commutation relations, and equations of motion to numerically constrain (“bootstrap”) the allowed values of observable data. The spectral bootstrap addi- tionally makes use of a quantum “crossing equation” to constrain off-diagonal matrix elements as well as the energy gap. The coarse-grained bootstrap systematically viicompresses the constraint equations using Matrix Product States (MPS) as coarse- graining maps, allowing us to incorporate the pertinent relations between a much larger number of bodies in our numerical optimization routines. We test these techniques on the (1+1)-dimensional Transverse Field Ising Model (TFIM) as well as a related spin chain model where the Z2 symmetry of the TFIM is explicitly broken. The spectral bootstrap obtains bounds on the gap across the TFIM phase diagram, while the coarse-grained bootstrap makes significant improvements on earlier methods in bounding nearest-neighbor correlation functions. In the deformed model we exhibit a particular parameter region in which the coarse-graining technique is required in order to resolve the ground state physics against a near ground state degeneracy.