Convergent renormalization trajectories of Hopf-algebra boundary MPDOs under on-site noise are classified by finite *-quantum hypergroups via a new quantum Goursat lemma.
Tensor Renormalization Group Meets Computer Assistance
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abstract
Tensor renormalization group, originally devised as a numerical technique, is emerging as a rigorous analytical framework for studying lattice models in statistical physics. Here we introduce a new renormalization map - the 2x1 map - which coarse-grains the lattice anisotropically by a factor of two in one direction followed by a 90-degree rotation. We develop a novel graphical language that translates the action of the 2x1 map into a system of inequalities on tensor components, with rigorous estimates in the Hilbert-Schmidt norm. We define a finite-dimensional "bounding box" called the hat-tensor, and a master function governing its RG flow. Iterating this function numerically, we establish convergence to the high-temperature fixed point for tensors lying within a quantifiable neighborhood. Our main theorem shows that tensors with deviations bounded by 0.02 in 63 orthogonal sectors flow to the fixed point. We also apply the method to specific models - the 2D Ising and XY models - obtaining explicit bounds on their high-temperature phase. This work brings the Tensor RG program closer towards a rigorous, computer-assisted construction of critical fixed points.
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2026 2representative citing papers
A framework identifies self-consistent finite-size windows in tensor-network flows to extract conformal data from critical 2D Ising and clock models across multiple renormalization schemes.
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Renormalization flows for 1D mixed states and a quantum Goursat lemma
Convergent renormalization trajectories of Hopf-algebra boundary MPDOs under on-site noise are classified by finite *-quantum hypergroups via a new quantum Goursat lemma.
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Extracting conformal data from finite-size tensor-network flow in critical two-dimensional classical models
A framework identifies self-consistent finite-size windows in tensor-network flows to extract conformal data from critical 2D Ising and clock models across multiple renormalization schemes.