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Emergent Spacetime in Quantum Lattice Models

Matthew D. Horner
Dec 2022
Many quantum lattice models have an emergent relativistic description intheir continuum limit. The celebrated example is graphene, whose continuumlimit is described by the Dirac equation on a Minkowski spacetime. Not onlydoes the continuum limit provide us with a dictionary of geometric observablesto describe the models with, but it also allows one to solve models that wereotherwise analytically intractable. In this thesis, we investigate novelfeatures of this relativistic description for a range of quantum latticemodels. In particular, we demonstrate how to generate emergent curvedspacetimes and identify observables at the lattice level which reveal thisemergent behaviour, allowing one to simulate relativistic effects in thelaboratory. We first study carbon nanotubes, a system with an edge, whichallows us to test the interesting feature of the Dirac equation that it allowsfor bulk states with support on the edges of the system. We then study Kitaev'shoneycomb model which has a continuum limit describing Majorana spinors on aMinkowski spacetime. We show how to generate a non-trivial metric in thecontinuum limit of this model and how to observe the effects of this metric andits corresponding curvature in the lattice observables, such as Majoranacorrelators, Majorana zero modes and the spin densities. We also discuss howlattice defects and $\mathbb{Z}_2$ gauge fields at the lattice level cangenerate chiral gauge fields in the continuum limit and we reveal theiradiabatic equivalence. Finally, we discuss a chiral modification of the 1D XYmodel which makes the model interacting and introduces a non-trivial phasediagram. We see that this generates a black hole metric in the continuum limit,where the inside and outside of the black hole are in different phases. We thendemonstrate that by quenching this model we can simulate Hawking radiation.