Inverse Problems in Quantum Materials: Work Backward from a Desired State - Yenra

Turn the forward question around

A forward calculation starts with a Hamiltonian and investigates the states and properties it produces. An inverse calculation starts with a target and constrains the Hamiltonians that could support it. The target might represent an interesting arrangement of spins or a state with a desired type of correlation.

Chertkov and Clark's 2018 paper on eigenstate-to-Hamiltonian construction introduced a computational method for this task. It takes a target wave function and a selected space of candidate operators, then uses a quantum covariance matrix to find combinations compatible with the target being an eigenstate. The Illinois research account describes the motivation: exploring models systematically from interesting states.

The chosen operator space is essential. It determines which interactions the search is allowed to combine. A result within a selected family says something about that family; it does not enumerate every possible physical system.

A two-state example makes the ambiguity visible

Consider an invented real symmetric two-by-two Hamiltonian with top row (a, c) and bottom row (c, b). Choose the target vector (1, 0). Matrix multiplication gives (a, c). For the output to equal an energy E times the target, we need c = 0 and E = a. The number b remains free.

This is an exact elementary example of the eigenstate condition, not a simulation of a quantum material or an implementation of the full construction algorithm.

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The eigenvector stayed the same, yet its relation to the rest of the spectrum changed. This is why “the target is an eigenstate” and “the target is the ground state” are distinct statements. Adding a multiple of the identity also shifts all energies without changing the eigenvectors, creating a simple form of nonuniqueness that a meaningful search must handle.

What the covariance calculation is checking

For a normalized state, the energy variance is the expectation of H² minus the square of the expectation of H. An exact eigenstate has zero energy variance. By expressing H as a linear combination of selected operators, the construction organizes this condition into a matrix problem for their coefficients.

In the published-version manuscript, zero modes of the quantum covariance matrix identify compatible combinations. In practical calculations, expectation values may be numerical estimates, so tolerances, convergence and numerical precision matter. A very small computed variance needs to be assessed against those errors.

Before running such a search, state the target, its representation, the allowed interactions, the system size and the numerical method used to obtain expectation values. If the space is too restrictive, it may contain no useful nontrivial solution. A larger space may reveal additional solutions but also introduce interactions that are difficult to realize.

From a compatible model to a materials proposal

Finding coefficients is the start of another investigation. Establish whether the target is the relevant low-energy state, examine competing states and excitations, and test how behavior changes with system size, temperature and perturbations. Then ask which physical ingredients could implement the required interactions.

A useful proposal connects a term in the model to a controllable interaction or a plausible microscopic description. It also identifies an observable that would distinguish the proposed state from alternatives. An attractive wave function and an elegant Hamiltonian can motivate that work without settling it.

1. Write the desired state and the property that makes it interesting.
2. List the allowed operator terms and explain the restriction.
3. Report solutions, numerical tolerances and any trivial freedom.
4. Check ground-state status and competing states independently.
5. Identify a physical realization and a measurement that could test it.

Keep failed or excluded candidate models with their reasons. The scientific value includes learning which assumptions prevent the target from being realized. That record also makes it possible to revisit the search when better numerical methods or new experimental platforms become available.