Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models

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2021

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In statistical physics, the exact partition function of simple (quasi)-one-dimensional models can be obtained from the numerical transfer matrix (TM) method. This method involves solving for the leading eigenvalues of a matrix representing all possible interactions between the states that a unit of the system can take. Because the size of this matrix grows exponentially with the number of those units, the TM method is ideally suited for models that have a finite state space and short-range interactions. Its success nevertheless relies on the use of efficient iterative eigensolvers and on leveraging system symmetry, whenever possible.

By careful finite-size extrapolation of sufficiently large systems, the TM method can also be used to examine two-dimensional models. A particularly interesting series of such systems are Ising models modified with next-nearest-neighbor frustration, which recapitulate the formation of equilibrium modulated phases in systems as varied as magnetic alloys, lipid surfactants, and cell morphogenesis. In these models, frustration results in large mixing times for Markov chain Monte Carlo simulations, but the TM approach sidesteps this slowdown and thus provides a putatively well-controlled computational scheme. The effectiveness of TM approach for these models, however, had previously been obfuscated by the limited range of system sizes computationally available for the numerical analysis. In this thesis, I build on the sparse matrix decomposition and take advantage of the structure and symmetry of the TM to develop optimized algorithms for the method, and thereby overcome the computational challenge. The resulting algorithm is implemented in various canonical frustrated next-nearest-neighbor Ising models, aiming to solve long-standing physical problems in these models. The approach provides benchmark results for related statistical physics models. It could also inspire the development of adapted eigensolver for similarly structured matrices.

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Hu, Yi (2021). Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models. Master's thesis, Duke University. Retrieved from https://hdl.handle.net/10161/23828.

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