A glimpse into interacting quantum systems with quantum Monte Carlo

One grand challenge in physics is understanding systems with many interacting particles. Interactions lead to some of the most conceptually deep and technologically promising applications of condensed matter physics, from high-temperature superconductivity to the fractional quantum Hall effect. Understanding an interacting system means knowing two things: what the system will do in its ground state, and what it will do when excited by a current, heat, or light. One example of the importance of excited states is that they are necessary for storing and processing quantum information—you cannot generate photons using ground states alone!

In my work with Professor Lucas Wagner [1], we have extended one of the most powerful numerical techniques in condensed matter physics, quantum Monte Carlo (QMC), to be able to find excited states of interacting systems. Historically, QMC techniques were developed to estimate ground states with high accuracy. Seminal developments happened right here at Illinois: William McMillan introduced an early version in 1964 to solve for the ground state of liquid helium, and David Ceperley extended its reach to the electron gas in the 1980s, a crucial test case for condensed matter applications. The next 40 years saw rapid improvements in computing power, allowing QMC to be applied to find the ground states of larger and larger systems.

At first, extending QMC to excited states seems paradoxical. Conventional QMC methods find the ground state based on the variational principle: that every state has energy greater than or equal to the ground state energy. This principle is used to formulate the ground state problem as an optimization problem. By definition, we can’t find excited states this way. Our key insight was to introduce a new variational principle for ensembles of states, rather than just one [1–3]. Using this new principle, we can seamlessly transfer all our optimization machinery to excited states.

In my group's recent work [3], we demonstrated how interacting systems can have differing physics in ground states and excited states, using our new variational principle. As a proof of concept, we contrasted the ground state and first excited state of a benzene molecule containing 30 electrons. In the figure, dark red and dark blue colors represent locations where one electron is likely to be found, given all other electrons are at fixed locations. We found that the colors were darker in the excited state at locations (A) and (B) than in the ground state. One physical interpretation is that the electrons redistribute themselves when we excite the system with light. This redistribution depends on the interactions. This means that QMC can be used to understand how interacting condensed matter systems respond to light!

Our benzene proof of concept clears the way for scaling our excited-state QMC technique to larger systems containing hundreds of electrons. As one current direction, we are focusing on impurity atoms embedded in large crystals. Impurity atoms are promising candidates for storing controllable, long-lived quantum information, but the first step is understanding how the impurity responds to light. One especially challenging system on the modeling side is an iron impurity atom in aluminum-nitride—even the most state-of-the-art non-QMC methods make fundamental errors when treating interactions. QMC explicitly accounts for interactions within the impurity site, within the surrounding crystal, and between the impurity and crystal. Thanks to these advantages, we are obtaining more reliable predictions of the impurity’s excited states, showing us how it responds to light. This knowledge of single impurities is one step toward a larger goal: to model quantum networks of many impurities, connected by light.