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arXiv:2312.14002 · New Journal of Physics

Tensor network finite-size scaling for two-dimensional 3-state clock model

Debasmita Maiti, Sing-Hong Chan, Pochung Chen

Published Apr 24, 2025• 1 citation
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Abstract

Abstract We benchmark recently proposed tensor network based finite-size scaling analysis in (Phys. Rev. B 2023 107 205123) against two-dimensional classical 3-state clock model. Due to the higher complexity of the model, more complicated crossover behavior is observed. We advocate that the crossover behavior can be understood from the perspective of finite bond dimension inducing relevant perturbation. This leads to a general strategy to best estimate the critical properties for a given set of control parameters. For the critical temperature T c , the relative error at the order of 10−7 can be reached with bond dimension D = 70. On the other hand, with bond dimension D = 60, the relative errors of the critical exponents ν , β , α are at the order of 10−2. Increasing the bond dimension to D = 90, these relative errors can be reduced at least to the order of 10−3. In all cases our results indicate that the errors can be systematically reduced by increasing the bond dimension and the stacking number.

DOI
10.1088/1367-2630/add048

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The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied to a model suffering the notorious quantum Monte Carlo sign problem---the orbital eg{e}_{g}eg​ model with spatially highly anisotropic orbital interactions. Coarse graining of the tensor network along the inverse temperature $\ensuremath{\beta}$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension DDD---limiting the amount of entanglement---is a natural refinement parameter. Increasing DDD we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature Tc{T}_{c}Tc​, provide a numerically exact estimate of Tc{T}_{c}Tc​, and give the critical exponents within 11%1 of the 2D Ising universality class.
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