Physicists have developed a new computational method that allows far more precise measurements of quantum entanglement at quantum critical points, the special conditions where materials undergo fundamental changes in their quantum state. The work, published in Reports on Progress in Physics, provides a powerful tool for studying strongly interacting quantum systems and for testing theoretical predictions about two-dimensional quantum materials.
A quantum critical point occurs at absolute zero temperature and, unlike ordinary phase transitions driven by heat, is triggered by changing another property such as a magnetic field, pressure, or interaction strength. One example is the transition between an ordered magnetic state and a disordered quantum state. Understanding what happens at these points is essential for describing the behaviour of quantum materials, but the mathematics involved quickly becomes intractable.
The key quantity in this research is entanglement entropy, a measure of how much quantum information two parts of a system share. If a system is split into two regions, low entanglement entropy means that knowing about one half tells you little about the other, while high entanglement entropy means the two halves are strongly connected through quantum mechanics. Entanglement entropy carries important information about quantum phases of matter, quantum critical points, and universal properties of quantum systems.
Calculating entanglement entropy accurately at quantum critical points in two-dimensional quantum systems has long been a major challenge. These systems are commonly described as (2+1)-dimensional because their critical theories involve two spatial dimensions and one time dimension. The difficulty arises because the systems are too complex to solve exactly, and standard numerical approaches struggle to capture the subtle quantum correlations that define the critical behaviour.
The research team, led by scientists at The University of Hong Kong, developed a novel Quantum Monte Carlo algorithm to address this problem. Quantum Monte Carlo methods use random sampling to study quantum systems that cannot be solved exactly. The researchers started with a standard quantum magnet model, the transverse-field Ising model, and added extra interactions that allowed them to study different types of phase transitions, including ordinary Ising critical points and a tricritical point that, in (2+1) dimensions, is described by a Gaussian free theory.
The crucial innovation was in how the team compared their measurements. They calculated the second Rényi entanglement entropies of two specially chosen regions with the same boundary length. This approach directly cancelled the dominant area-law contribution, allowing the much smaller universal corner term to become the leading signal. This subtraction technique effectively isolates the subtle universal signature that had previously been buried under larger, less informative contributions.
Using this method, the researchers obtained a precise value for the entanglement entropy at the Ising critical point. Their results showed that the Ising and tricritical/Gaussian critical points have different universal entanglement fingerprints, demonstrating that entanglement can distinguish between different types of quantum critical behaviour. This is a significant finding because it confirms that entanglement carries information about the nature of the phase transition itself, not just its presence.
The broader significance of the work lies in its applicability. The new algorithm offers a powerful and general method for studying entanglement in strongly interacting quantum systems, a class of materials that includes high-temperature superconductors and other exotic states of matter. It also provides a way to test theoretical predictions about two-dimensional quantum materials, where experimental probes of entanglement remain extremely difficult.
The research opens the door to more detailed investigations of quantum critical phenomena and could eventually aid in the design of quantum materials and devices that exploit entanglement for practical applications.
