| Summary: | Symmetry is at the heart of modern physics. Phases of matter are classified by symmetry breaking, topological phases are characterized by nonlocal symmetries, and point group symmetries are critical to our understanding of crystalline materials. Symmetries could then be used as a criterion to engineer quantum systems with targeted properties. Toward that end, we have developed an approach, the symmetric Hamiltonian construction (SHC), that takes as input symmetries, specified by integrals of motion or discrete symmetry transformations, and produces as output all local Hamiltonians consistent with these symmetries. This approach builds on the slow operator method [H. Kim, M. C. Bañuls, J. I. Cirac, M. B. Hastings, and D. A. Huse, Phys. Rev. E 92, 012128 (2015)PLEEE81539-375510.1103/PhysRevE.92.012128]. We use our new approach to construct new Hamiltonians for topological phases of matter. Topological phases of matter are exotic quantum phases with potential applications in quantum computation. In this work, we focus on two types of topological phases of matter: superconductors with Majorana zero modes and Z_{2} quantum spin liquids. In our first application of the SHC approach, we analytically construct a large and highly tunable class of superconducting Hamiltonians with Majorana zero modes with a given targeted spatial distribution. This result lays the foundation for potential new experimental routes to realizing Majorana fermions. In our second application, we find new Z_{2} spin-liquid Hamiltonians on the square and Kagome lattices. These new Hamiltonians are not sums of commuting operators nor frustration free and, when perturbed appropriately (in a way that preserves their Z_{2} spin-liquid behavior), exhibit level-spacing statistics that suggest nonintegrability. This result demonstrates how our approach can automatically generate new spin-liquid Hamiltonians with interesting properties not often seen in solvable models.
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