Discovering Fixed Points: A Major Advancement in Theoretical Physics Using Tensor Networks

Three researchers, Slava Rychkov, Permanent Professor at IHES, his PhD student Nikolay Ebel, and Tom Kennedy, Professor of mathematics at the University of Arizona, have recently published an article titled “Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group” in the prestigious Physical Review X.

Their work represents a significant advancement in understanding fixed points in statistical lattice models, combining mathematical innovation with unprecedented numerical precision to explore critical phenomena that have long remained theoretical.

In statistical physics, spin models on a lattice, such as the well-known Ising model or the three-state Potts model, serve as laboratories for exploring phase transitions—radical changes in the behavior of a system, like water transitioning from liquid to gas. True critical points, however, occur only under very specific temperature or pressure conditions, where the system remains fundamentally the same when observed at different scales. These configurations correspond to fixed points of a renormalization transformation. “A fixed point is a transformation that leaves the system unchanged,” the researchers explain. “For a critical phenomenon, this means the system remains exactly the same after a change in resolution. Studying these fixed points allows us to directly analyze the fundamental properties of the critical phenomenon.”

The novelty of their approach lies in using tensor networks, which represent the statistical weights of local configurations rather than the total energy of spin interactions. This local and stable representation allows precise tracking of the system’s transformation toward its fixed point. A key element of their method is introducing a 90° rotation of the lattice, which flips the sign of the marginal eigenvalues—directions of transformation that otherwise form a continuous family of fixed points difficult to isolate. This modification makes it possible to apply Newton’s method directly, a classical mathematical tool for solving complex equations, achieving an exceptional numerical precision of 10^-9, far beyond what had been previously reached.

To validate their method, the researchers chose the Ising model and, following a reviewer’s suggestion, the three-state Potts model. “These models are well-studied, simple enough to test new methods, yet physically rich,” they explain. The results confirmed the effectiveness of their approach and pave the way for applications to more complex models. According to the team, there was no single “eureka” moment but rather a series of small confirmations at each step, each success reinforcing their confidence in the method.

Although this research is highly theoretical, it lays the groundwork for potential advances in mathematical physics and, indirectly, other fields. The method could one day enable rigorous proofs of the existence of fixed points, a major result that would strengthen the foundations of statistical physics. “These critical phenomena are rare and difficult to observe in everyday life,” they note. “But understanding these fundamental transitions could eventually inspire technologies related to material control or quantum devices.”

This publication demonstrates that it is possible to determine fixed points in statistical lattice models with unprecedented numerical precision. The approach combines classical numerical techniques, such as Newton’s method, with tensor network representation and the lattice rotation, allowing the isolation of fixed points and overcoming difficulties associated with marginal deformations, which otherwise form a continuous family that is hard to treat. The results obtained for the Ising and Potts models show that this method is effective and opens the way for more rigorous analyses of fixed points in other complex statistical systems.