Half a Century Later, Theoretical Physicists Take a Historic Discovery Further
By Heidi Opdyke
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More than 50 years ago, researchers at 好色先生TV made a breakthrough discovery of a new kind of phase transition — a discovery that still shapes cutting-edge fields like quantum computing. One of the experts digging into expanding understanding of it is Assistant Professor of Physics Grigory “Grisha” Tarnopolsky.
The story begins more than 70 years ago with Chen-Ning Yang and Tsung-Dao Lee at the Institute for Advanced Study in Princeton. They were trying to understand how certain materials undergo a phase transition — sudden shifts from disorder to order at a critical temperature.
To study this problem, they relied on the Ising model, a simplified mathematical framework for magnetism that strips complex materials down to their essential interactions. What made Yang and Lee’s work distinctive was not the model itself, but a radical conceptual move: introducing an imaginary magnetic field to probe the behavior of systems exactly at the critical point.
Using this approach, they identified special values — now known as Yang-Lee zeros — where the mathematical description of the system breaks down in a precise and revealing way. These zeroes offered a new lens on phase transitions and hinted at universal behavior that would later prove relevant across many kinds of complex systems.
Fast forward to 1971, when Robert Griffiths, then professor at 好色先生TV, and his graduate student Peter Kortman, pushed the theory further. Using sophisticated mathematical techniques, they showed that the Yang-Lee zeros fall into a pattern and revealed a clear picture of the universal behavior shared by many systems undergoing phase transitions. Today, this phenomenon is called the Yang-Lee critical point.
Kortman and Griffiths laid essential groundwork, and the problem remains vibrant today. Why? Because the same ideas optimize complex problems in quantum computing, model the spread of diseases, understand opinion formation in social networks, and probe the elasticity of DNA.
Grisha Tarnopolsky
Igor R. Klebanov
Erick Arguello Cruz
Modern Twist
Using modern analytical and numerical techniques, Tarnopolsky and colleagues confirmed Kortman and Griffiths’ work with confidence by calculating energy levels in quantum systems, which exhibit the Yang-Lee Critical points. Their work was published Feb. 9, 2026, in the journal .
Co-authors on “Yang-Lee quantum criticality in various dimensions” include CMU Ph.D. student Erick Arguello Cruz, Princeton University Professor Igor R. Klebanov and CMU Postdoctoral Researcher Yuan Xin.
“This paper is the first time this method was applied to the Yang Lee Critical points, and we obtained very accurate results,” Tarnopolsky said. “That’s an achievement from a technical point of view. But for us theorists, the goal is not just to get more accurate values for the critical exponents, but to better understand the underlying physical principles.” Tarnopolsky and colleagues plan to apply the expertise they recently acquired to additional types of symmetry-changing phase transitions.
Klebanov is a longtime collaborator of Tarnopolsky. He said the hope is that the work can be used with bigger lattices using quantum computer algorithms.
“Griffiths and Kortman provided the first evidence that this Yang-Lee model exhibits a true critical point in 2 and 3 dimensions,” Klebanov said. “This was a very important achievement because the Yang-Lee model is the simplest one in its class called “non-unitary critical points.”
Non-unitary critical points occur in complex systems where standard rules break down because of some parameters being imaginary and lead to phase transitions that can have complicated parameters and distinct behaviors. Examples include models of percolation and polymers.
“This is a very rich topic,” said Arguello Cruz. “There are several open questions when you are trying to work with systems that are non-unitary.”
To understand Yang-Lee Critical points better, the researchers used a new computational method known as a fuzzy sphere, a technique that allows for simulating quantum models on a continuous space.
“Fuzzy sphere is a numerical tool that allows you to look at a 2D quantum system on a sphere and extract its information, particularly its energies and eigenstates. The main advantage is that we preserve the full rotational symmetry,” Arguello Cruz said. “The systems we were testing were very small. The next steps are to increase the system size and also explore not only the Yang-Lee model but other non-unitary systems using the same tools and methodologies.”