Simplifying long-range quantum interactions in many-body systems

Calculations for some quantum systems whose parts interact over long distances will be much easier to perform thanks to the work of a RIKEN physicist and his collaborator, who have extended a valid hypothesis for materials with short-range interactions.

Notoriously dubbed “spectral action at a distance” by Albert Einstein, entanglement is one of the most fascinating aspects of quantum physics. It is an invisible connection between quantum systems which means that a system cannot be fully described without including the states of others, a connection that cannot be understood using classical mechanics.

Entanglement plays a central role in the physics of quantum systems consisting of many parts necessary to understand materials at low temperatures. One of the most rigorous ways to quantify entanglement is to use the entanglement entropy, which characterizes the complexity of a material’s lowest energy state. A zero entropy state is classical and exhibits no quantum properties. States with small but non-zero entanglement entropy can be described using a relatively simple quantum theory. But states with higher entanglement entropy become very difficult to model mathematically.

In many materials, entanglement occurs over a short range, existing only between the closest neighbors. These systems have been shown to have low entanglement entropy. Known as the area law conjecture, this hypothesis greatly simplifies modeling.

But some materials can exhibit unusual states of matter in which interactions between atoms can be maintained over greater distances. And then the question arises: is the area law still valid in materials with non-local quantum connections? This is the question being investigated by Tomotaka Kuwahara of the RIKEN Center for Advanced Intelligence Project and Keiji Saito of Keio University.

“Several numerical and theoretical studies have indicated that area law is violated in interacting long-range systems,” explains Kuwahara. “Our result is mathematically rigorous and solves the debate on the area law conjecture in long range one-dimensional interacting systems.”

Providing detailed evidence of an area law is extremely challenging. Kuwahara and Saito simplified the mathematics of the problem by modeling a one-dimensional chain. They observed a chain of long-range interacting magnetic particles. They decomposed the total system into two subsystems, left and right, and simulated the boundary as a series of discrete points. In this way, the pair proved that the entanglement entropy has a maximum possible value, which is a signature of an area law.

“The next step for us is to prove the area law conjecture in systems with more than one dimension,” says Kuwahara. “We have exploited several new mathematical techniques in our present study and hope to apply them to larger cases.”