Abstract
Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output. However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians. We introduce an alternative computational "inverse method," the Eigenstate-to-Hamiltonian Construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems. EHC takes as input a wave function $|\psi_T\rangle$ and produces as output Hamiltonians for which $|\psi_T\rangle$ is an eigenstate. This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix. EHC is widely applicable to a number of models and in this work we consider seven different examples. Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model.
Highlights
Our understanding of quantum many-body physics comes primarily from the use of “forward methods.” In the forward method approach, shown in Fig. 1(a), a quantum model describing a material, e.g., a model Hamiltonian, is solved
Ground-state methods were used to map out the ground-state manifold, but often could only provide a lower bound on the dimensionality of the manifold (Dim. g.s. manifold stands for dimension of ground state manifold)
The eigenstate-to-Hamiltonian construction (EHC) approach fits into a broader class of techniques, such as machine-learning approaches, for automating physical understanding that previously required significant insight
Summary
Our understanding of quantum many-body physics comes primarily from the use of “forward methods.” In the forward method approach, shown in Fig. 1(a), a quantum model describing a material, e.g., a model Hamiltonian, is solved. Our understanding of quantum many-body physics comes primarily from the use of “forward methods.”. In the forward method approach, shown, a quantum model describing a material, e.g., a model Hamiltonian, is solved. Often, solving each Hamiltonian is difficult, requiring expensive numerics or complex analytic approaches. This restricts our attention to a few representative Hamiltonians or materials that support particular properties or interesting physics. The space of quantum models is vast and high dimensional. The forward approach provides a limited perspective by restricting our focus to a small fraction of this space. The entire space, though, almost certainly contains a myriad of interesting physical Hamiltonians corresponding to undiscovered phases, unknown exactly solvable points, and Hamiltonians with desirable properties
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