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Symmetry is at the heart of modern physics. Phases of matter are classifiedby symmetry breaking, topological phases are characterized by non-localsymmetries, and point group symmetries are critical to our understanding ofcrystalline materials. Symmetries could then be used as a criterion to engineerquantum systems with targeted properties. Toward that end, we have developed anovel approach, the symmetric Hamiltonian construction (SHC), that takes asinput symmetries, specified by integrals of motion or discrete symmetrytransformations, and produces as output all local Hamiltonians consistent withthese symmetries (see github.com/ClarkResearchGroup/qosy for our open-sourcecode). This approach builds on the slow operator method [PRE 92, 012128]. Weuse our new approach to construct new Hamiltonians for topological phases ofmatter.  Topological phases of matter are exotic quantum phases with potentialapplications in quantum computation. In this work, we focus on two types oftopological phases of matter: superconductors with Majorana zero modes and$Z_2$ quantum spin liquids. In our first application of the SHC approach, weanalytically construct a large and highly tunable class of superconductingHamiltonians with Majorana zero modes with a given targeted spatialdistribution. This result lays the foundation for potential new experimentalroutes to realizing Majorana fermions. In our second application, we find new$Z_2$ spin liquid Hamiltonians on the square and kagome lattices. These newHamiltonians are not sums of commuting operators nor frustration-free and, whenperturbed appropriately (in a way that preserves their $Z_2$ spin liquidbehavior), exhibit level-spacing statistics that suggest non-integrability.This result demonstrates how our approach can automatically generate new spinliquid Hamiltonians with interesting properties not often seen in solvablemodels.

Engineering Topological Models with a General-Purpose Symmetry-to-Hamiltonian Approach