Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions

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 $|{\psi }_T⟩$ and produces as output Hamiltonians for which $|{\psi }_T⟩$ 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. EHC gives an alternative inverse approach for studying quantum many-body phenomena.

Popular Summary

Physicists are accustomed to learning about the world by solving models. Usually, in the field of quantum materials, we start with a quantum model of a material and solve it to find wave functions that tell us about the material’s properties. Here, we propose an alternative “inverse method” for studying quantum materials in which we work backwards: We start with a wave function with a desired property and find many new models with that wave function as a solution. This approach fits into a broader class of techniques, such as machine learning approaches, for automating physical understanding, which previously required significant insight.

We develop a novel algorithm that carries out this inverse method automatically and efficiently for general quantum systems and that can be readily implemented with existing numerical tools. The key step of the algorithm is the evaluation and analysis of the quantum covariance matrix, a quantum mechanical generalization of a statistical covariance matrix. We test our algorithm on numerous wave functions and find a myriad of new models.

Because quantum models are essential for building our understanding of quantum mechanics, our algorithm could be an important tool for future advances in quantum physics. For example, new models discovered by our algorithm could potentially aid in the synthesis of new materials or the simulation of new types of quantum systems.

Figure 1

(a) A typical forward method used in quantum mechanics finds the ground state $|{\psi }_0⟩$ of a single Hamiltonian $\widehat{H}$. (b) We introduce a new inverse method, EHC, that finds Hamiltonian(s) ${\widehat{H}}_1,{\widehat{H}}_2,\dots$ from a target state $|{\psi }_T⟩$, with the property that $|{\psi }_T⟩$ is an energy eigenstate of these Hamiltonian(s).

Figure 2

All Hamiltonians of a finite-dimensional quantum system form a real vector space of Hermitian operators (shown in red). EHC is performed in a target space of Hamiltonians, a physically meaningful subspace of the entire vector space chosen by the user (shown in blue). The output of EHC is the eigenstate space, a subspace of the target space consisting of Hamiltonians that contain a target wave function $|{\psi }_T⟩$ as an energy eigenstate (shown in green). Using ground-state methods, one can further map out the ground-state manifold, the region in eigenstate space where the target state $|{\psi }_T⟩$ is a ground state (shown in white).

Figure 3

The steps of EHC. (1) Represent the target state $|{\psi }_T⟩$ numerically using, for example, MPS, VMC, or ED techniques. (2) Compute the QCM $C_T$ given by Eq. (1). (3) Diagonalize the QCM numerically to obtain the eigendecomposition $C_T=UD{U}^{†}$. The columns of $U$ are eigenvectors of $C_T$ and the diagonal entries of $D$ are their corresponding eigenvalues. The null vectors, i.e., eigenvectors with zero eigenvalue, correspond to the coupling constants of Hamiltonians with $|{\psi }_T⟩$ as an energy eigenstate. The $C_T$ matrix depicted is the QCM for the XX chain ground state used in our phase expansion example.

Figure 4

(a) For a classical Ising antiferromagnetic triangle, all three bond energies cannot be simultaneously minimized, leading to a sixfold ground-state degeneracy. (b) The uniform frustrated Ising (UFI) state is a uniform superposition of the ground states of an antiferromagnetic Ising model on a lattice of triangles. In this case, we consider the UFI state on a triangular two-leg ladder.

Figure 5

State collision example. Collision of the singlet dimer states $|{\psi }_{SD}^{\pm }⟩$ and the projected three-coloring states $|{\psi }_{P3C}^{m,l}⟩$ from (a) the perspective of the singlet dimer eigenstate space, which is given by Eq. (7) with $J_{xy}=1$ and is shown in green, and (b) the perspective of the projected three-coloring eigenstate space, which is given by Eq. (8) and is shown in teal. The space where all of the states are degenerate eigenstates is given by Eq. (6) and is shown in purple; it appears as a line in (a) and a plane in (b). The collision region, where all states are degenerate ground states, occurs on the indicated line segment in (a) and in the indicated triangular region in (b).

Figure 6

Phase expansion schematic and example. (a) Schematic representing the expansion of a phase diagram about a known Hamiltonian $\widehat{H_0}$, which has a known ground state $|{\psi }_T⟩$. Shown is the target space (blue) provided as input to the EHC method, the eigenstate space (green) of $|{\psi }_T⟩$ produced as the output of EHC and the expanded ground-state manifold (white) of $|{\psi }_T⟩$. (b) Numerical results for the phase expansion about the Hamiltonian ${\widehat{H}}_{XX}$ with ground state $|{\psi }_{XX}⟩$. Shown is a three-dimensional projection (green) of the six-dimensional eigenstate space of $|{\psi }_{XX}⟩$, given by Eq. (11) with $J_{0,+}=1$ and $J_{2,\pm }=0$ and the ground-state manifold of $|{\psi }_{XX}⟩$ (white), which almost has the shape of a tetrahedron. For simplicity of visualization, here, we plot a tetrahedron, which contains most of the ground-state manifold, though it ignores a small curved region that extends slightly beyond the tetrahedron.

Figure 7

A summary of phase expansion results obtained in this work. In each row, we show the target state(s) provided as input to EHC (and DEHC), the QCMs $C_T$ [and DQCMs $D_T$ (see Supplemental Material [65])] we calculated, and their spectra on a log scale. The eigenvectors of a QCM (or DQCM) are vectors of coupling constants that correspond to Hamiltonians with variance ${\sigma }_T^2$ under the target state $|{\psi }_T⟩$. The QCM for $|{\psi }_{\mathrm{BCS}}⟩$ in the bottom row was statistically estimated using variational Monte Carlo, while the others were computed with matrix product states.