Departmental Seminar

Corner Transfer Matrix Approach to the Yang-Lee Singularity in the 2D Ising Model

Mr Bryte Hagan
PhD Candidate, Fundamental and Theoretical Physics

The Ising model is a toy model with rich and fascinating mathematical structure. Over the near century since its inception, the model has expanded from a humble statistical model for magnets and lattice gases to being a prime candidate for testing the Universality hypothesis, scaling theories and probing the link between statistical mechanics and conformal field theory. Much of the success of the model is owed to its integrability in restricted regimes, which is used as a base to explore realms beyond integrability.

In this talk I give a recount of the important developments in the 2D Ising Model's career, cover the modern investigations in the field and present new results regarding the free energy scaling function near the Yang-Lee singularity. The model has three main regimes of interest. Near the critical point for the lattice variables temperature and external magnetic field (τ, H) = (0, 0), the direction τ ≠ 0 gives the Onsager solution which reduces to free fermions in the scaling limit, the direction of H ≠ 0 gives the integrable Ising Field Theory with 8 types of particles with different masses in the scaling limit, while a combination of the two makes the model non-integrable. The partition function Z(τ, H) has complex zeros when the associated scaling variable takes on the imaginary value ξ(τ, H) ≈ i0.1893506, and around this point the model exhibits a single kind of particles with the same mass. One major difficulty of this work is that the exact location of this singularity is not even fully known. This work presents a high accuracy estimate of the singularity location.

The main numerical technique of this work is the Corner Transfer Matrix Renormalisation Group method, and this technique is detailed in the talk. This is a powerful method that can be broadly applied to a variety of statistical models. It boasts strong convergence when the eigenvalues of an associated transfer matrix exponentially decay at a sufficient rate. It is not commonly applied to complex models; when it is then the SVD approach is suggested. When the SVD approach failed, we developed a modification on the real method that is suitable for complex models.

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Date & time

Thu 29 Sep 2022, 11am–12pm



Members of RSPE welcome