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A view of the honeycomb network, with a triangular region indicated with black locations (spheres). Electrons in the model studied by the researchers can jump between locations in this network, and the team calculated the average amount of entanglement between triangular regions like this and their surroundings. Credit: D’Emidio et al. (PRL, 2024).

Entanglement is a widely studied quantum physics phenomenon, in which two particles are linked in such a way that the state of one affects the state of the other, regardless of the distance between them. When studying systems composed of multiple strongly interacting particles (i.e., many body systems) in two or more dimensions, numerically predicting the amount of information shared between these particles, a measure known as entanglement entropy (EE), becomes highly challenging.

Researchers at the Donostia International Physics Center recently introduced a new method for calculating an EE measurement, namely the Rényi EE, for many-body systems beyond the scope of previous numerical methods. This method, described in *Physical Review Letters*was effectively used to extract the universal features of EE in a 2D model of interacting fermions, focusing on the half-filled honeycomb Hubbard model.

“My previous research dealt with simple network models of quantum magnets, where I developed a highly efficient way of calculating entanglement entropies at very large scales,” Jonathan D’Emidio, lead author of the paper, told Phys.org. “Several years ago, an expert in the field asked me if it would be possible to apply this technique to more complicated models of fermions (electrons), where no suitable techniques were available.”

D’Emidio began examining interactive fermion models in collaboration with his colleagues Román Orús, Nicolas Laflorencie and Fernando de Juan. Soon after starting to collaborate on this project, the researchers realized that the computational method previously developed by D’Emidio could also be applied effectively in this new context.

“The goal of our study was simple: calculate the Rényi EE in a model of interacting fermions with enough precision to see something interesting,” said D’Emidio. “In particular, to observe features that can identify the various phases and phase transitions of fermions. These features were predicted to exist but have never been observed directly in numerical simulations.”

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The two different types of triangular regions investigated in this work. The triangle that cuts the most links in the honeycomb network (right) has a ‘bearded’ edge and produces the expected phase transition behavior, whilst this feature is lost in the triangle on the left which has a ‘zigzag’ edge. . More theoretical work is needed to understand these types of differences. Credit: D’Emidio et al.

The method used by D’Emidio and his colleagues to calculate the Rényi EE is based on basic concepts rooted in thermodynamics and statistical mechanics. Essentially, this method identifies the Rényi EE with a free energy difference between two different sets of fermions.

“For example, free energy differences indicate whether proteins will fold in a certain way or whether a certain reaction will occur naturally or not,” explained D’Emidio. “To make these processes go in the opposite direction, work must be done on the system. The original formulation I used exactly matched the calculation of the work required to partially fuse two copies of the quantum wave function.”

The main advantage of the computational technique proposed by this research team is that it naturally captures the most important configurations that dominate the overall EE value. This is in stark contrast to previous formulations, which suffered from massive contributions from extremely rare events, making the associated calculations virtually impossible to perform.

“One of the biggest surprises for us was that sometimes the results can depend on how the entanglement region is defined, although theoretically there is no explanation why this should happen,” said D’Emidio.

“For example, when calculating the EE of a triangle with the rest of the system, it shouldn’t matter how the triangle is placed in the lattice; yet, we found that the phase transition fingerprint was lost when the triangle had a zigzag edge as opposed to a bearded edge. We hope this result helps gain a theoretical understanding of why the Rényi EE may depend on such definitions.”

This recent study by D’Emidio and his collaborators demonstrates the feasibility of calculating the Rényi EE with satisfactory accuracy, high enough to gather valuable new insights into the collective physics of systems composed of interacting fermions. In their future work, the researchers plan to continue using their computational approach to study complex models of interacting many-body systems.

“Personally, I am very interested in studying spin-liquids, which are quantum phases that appear completely magnetically disordered, but actually have an intricate topological structure that can be revealed with EE properties,” D’Emidio added.

“There are several spin-liquid candidates based on models of interacting fermions, similar to the iconic Hubbard model that we investigated in this work. I would soon like to investigate these models with the new method.”

**More information:**

Jonathan D’Emidio et al, Universal Characteristics of Entanglement Entropy in the Hubbard Honeycomb Model, *Physical Review Letters* (2024). DOI: 10.1103/PhysRevLett.132.076502. About *arXiv*: DOI: 10.48550/arxiv.2211.04334

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