Why are some things hard and others soft? Why do some materials conduct electricity? And can we make materials that conduct even better? The properties of any material are determined by the interactions between its electrons, and understanding the way such properties arise at the electron level is both a fundamental scientific problem and key to creating new materials with desirable properties. The Simons Collaboration on the Many Electron Problem, which launched in March 2014, brings together physicists and chemists to work to refine existing approaches to this problem, and to invent new approaches.
This drawing depicts a member of a class of mathematical objects called tensor networks, which are used to represent many-body wave functions. This particular tensor network has a dual interpretation: It describes a thermal state of a many-body system, but it may also be interpreted as representing the space-time geometry of a black hole.
A great deal of the challenge boils down to big numbers. The number of states available to a many-particle system grows exponentially as the number of electrons grows. The sheer size of the vector space that represents those states means that conventional computational approaches to managing electron-electron interactions fail. “A lot of our work consists of finding clever ways around that exponential barrier,” says Emanuel Gull of the University of Michigan, who is a member of the collaboration.
The collaboration brings together four main threads of research: cluster embedding theory, Monte Carlo methods, real material methods and tensor networks. The program hosts several conferences each year in addition to a summer school for graduate and postdoctoral students.
A variety of numerical methods have been developed to attack different aspects of the many electron problem, and the early stages of the collaboration have focused on taking stock of the strengths and weaknesses of those methods by using them on simplified model systems. “The ‘fruit fly’ of this type of physics is the so-called Hubbard model. That’s a model that has all of the physics stripped down, except for one local interaction term and one nearest-neighbor term,” says Gull. “It’s sort of the minimum model that gives you the physics of electron correlation.” In comparing various approaches, researchers try to identify the ways that the different methods might fail in a simple system, in order to understand how to get around the same problems in a more complicated one.
One promising approach is based on tensor networks, a reformulation of the wave function description of quantum mechanics. A breakthrough in that field about ten years ago radically changed the way researchers can attack the many electron problem; in essence, tensor networks provide a way to home in on a small subset of states characterized by certain quantum entanglement conditions. Guifre Vidal, a researcher at the Perimeter Institute for Theoretical Physics and one of the researchers behind the breakthrough, says, “What we have identified is that in this huge vector space, there are many vectors we should not care about, that are not relevant to the many electron problem, and a very small subset — a subset of measure zero — is actually the one we care about.” This means that in certain situations, the computational cost has been vastly reduced — it grows as a power of the number of electrons rather than exponentially.
The tensor network breakthrough, when coupled with increases in computing power, means that an understanding of larger and larger systems becomes more feasible. “We are greedy,” says Vidal. “There is never a computer that is big enough. We used to complain that we couldn’t go beyond 20 electrons. Now we can go to hundreds. Once we go to hundreds, we try to go to thousands. It is not unreasonable to be so greedy. We’d like to study millions. That’s what real systems are made of.”
The other collaboration research areas have made similar progress in recent years. The logical next steps will be to compare, contrast and combine the approaches, to determine which work best in which situations, and to use insights in one area to improve results in another. For example, the collaboration is exploring the implications of the recent discovery that it is possible to use stochastic (‘Monte Carlo’) methods to evaluate Feynman diagram series, in the hope that these methods can be used to improve the standard techniques used to provide first order approximations to the properties of molecules and solids.
“A couple of theoretical breakthroughs allowed us to access a new world of physics,” says Gull. “That’s the spirit of the collaboration.”
