Abstract
Symmetry is at the heart of modern physics. Phases of matter are classified by symmetry breaking, topological phases are characterized by non-local 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 a novel 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 (see github.com/ClarkResearchGroup/qosy for our open-source code). This approach builds on the slow operator method [PRE 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 non-integrability. This result demonstrates how our approach can automatically generate new spin liquid Hamiltonians with interesting properties not often seen in solvable models.
Highlights
Symmetry is central to our understanding of the phases of matter seen in nature
To construct new Z2 quantum spin-liquid Hamiltonians on the Kagome lattice, we provided as input to symmetric Hamiltonian construction (SHC): (1) the four straight-line Wilson loop operators XLa1, ZLa1, XLa2, ZLa2 [see Fig. 7(c)] and (2) the symmetry group of the Kagome lattice generated by translations of lattice vectors a1 and a2, 60◦ rotation, and reflection
We have introduced a new approach, the symmetric Hamiltonian construction (SHC), for constructing Hamiltonians with desired symmetries
Summary
Symmetry is central to our understanding of the phases of matter seen in nature. Many phase transitions, such as those of liquids, magnets, or superconductors, can be described by the spontaneous breaking of symmetry according to Landau’s theory [1]. We use the SHC to construct new Hamiltonians for two topological systems: superconductors with Majorana zero modes and Z2 quantum spin liquids. The Hamiltonians that we discover are not sums of commuting operators, are not frustration free, and can possess local and nonlocal integrals of motion We find that these Hamiltonians, perturbed in an appropriate way, exhibit GOE level-spacing statistics in particular quantum number sectors, suggesting that they could be nonintegrable. These models provide new, interesting examples of Z2 topological order in spin systems.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have