Jul 30 2018

**Researchers at the University of Illinois at Urbana-Champaign have put together an algorithm that could provide significant answers to condensed matter physicists in their pursuits for novel and up-and-coming properties in materials. The algorithm, invented by physics professor Bryan Clark and his graduate student Eli Chertkov, inverts the characteristic mathematical process condensed matter physicists use to hunt for interesting physics. Their new technique starts with the answer - what kinds of physical properties would be stimulating to find—and works backward to the question - what group of materials would host such properties.**

Inverse problem solving is not a new method in classical physics, but this algorithm signifies one of the first fruitful examples of an inverse problem-solving technique with quantum materials. And it could make hunting for interesting physics a more efficient and deliberate process for many researchers. The number of physicists working in condensed matter is more than any other subfield of physics - the abundant diversity of condensed matter systems and phenomena provide sufficient unsolved problems to study, from superfluidity and superconductivity to magnetism and topology.

Experimentalists examine the macro-and microscopic properties of materials to study the behavior and interactions of particles in materials under a stringent set of controls. Theoretical condensed matter physicists, on the other hand, work to create mathematical models that predict or explain the central laws that dictate these behaviors and interactions.

The field of theoretical condensed matter physics has the well-deserved reputation for being esoteric and difficult for the layperson to decode, with its focus on comprehending the quantum mechanics of materials. The process of writing and solving condensed matter equations is very intricate and methodical. That process normally begins with a Hamiltonian - a mathematical model that totals up the energies of all the particles in the system.

Clark explains, “*For a typical condensed matter problem, you start with a model, which comes out as a Hamiltonian, then you solve it, and you end up with a wave function - and you can see the properties of that wave function and see whether there is anything interesting. This algorithm inverts that process. Now, if you know the desired type of physics you would like to study, you can represent that in a wave function, and the algorithm will generate all of the Hamiltonians - or the specific models—for which we would get that set of properties. To be more exact, the algorithm gives us Hamiltonians with that wave function as an energy eigenstate.” *

Clark says the algorithm gives a new method to examine physical phenomena such as superconductivity.

*“Typically, you would guess Hamiltonians that are likely to be superconducting and then try to solve them. What this algorithm - in theory - will allow us to do is to write down a wave function that we know superconducts and then automatically generate all of the Hamiltonians or the specific models that give that wave function as their solution. Once you have the Hamiltonians, in some sense, that gives you all the other properties of the system - the excitation spectrum, all the finite temperature properties. That requires some more steps once you have the Hamiltonian, so we didn’t improve that part of the research process. But what we did, we found a way to find interesting models, interesting Hamiltonians.”*

Chertkov adds, *“There are lots of wave functions people have written down for which there are no known Hamiltonians—maybe 50 years’ worth. Now we can take any of these wave functions and ask if any Hamiltonians give those as eigenstates and you may end up with one model, no models, or many. For example, we are interested in spin-liquid wave functions, highly entangled quantum states with interesting topological properties. Theorists have constructed many spin-liquid wave functions, but don’t know which Hamiltonians give them. In the future, our algorithm should let us find these Hamiltonians.”*

Clark and Chertkov tried out the algorithm on wave functions associated with frustrated magnetism, a subject that presents interesting physics with a number of open questions. Frustrated magnetism takes place in a group of materials that is insulating, so the electrons do not travel around, but their spins interact. Clark elucidates one such wave function they analyzed, *“The electron spins in a frustrated magnet want to be anti-aligned, like the north and south on a magnet, but can’t because they live on triangles. So we make a wave function out of a linear-superposition of all of these frustrated states and we turn the crank of this algorithm, and ask, given this wavefunction, which is an interesting quantum state on a frustrated magnet, are there Hamiltonians that would give it. And we found some.”*

Chertkov says the outcomes of the algorithm could direct experimentalists in the right direction to discover interesting new physics: *“That would hopefully be one way it would be used. You pick a wave function that has some kind of physics that you care about and you see what sort of interactions can give you that sort of physics, and hopefully then the models you find through this method can be looked for in experiments. And it turns out you find many models with our method.”*

Clark summarizes, *“This has inverted the part of the process where we were sort of hunting in the dark. Before, you could say, we’re going to try lots of models until we find something interesting. Now you can say, this is the interesting thing we want, let’s turn the crank on this algorithm and find a model that gives that.”*

These findings were reported online in the July 27^{th} issue of the Physical Review X (PRX), in an article titled “Computational inverse method for constructing spaces of quantum models from wave functions.”

This research was aided by the US Department of Energy Office of Science’s SciDAC program, and is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation and the State of Illinois. The conclusions offered are those of the scientists and not necessarily those of the funding agencies.