Scaling Hypothesis for Matrix Product States

2019/07/19 by Bram Vanhecke, Jutho Haegeman, Karel Van Acoleyen +2
Mathematics · Physics and Astronomy · #Ansatz #Central charge #Critical exponent #Eigenvalues and eigenvectors #Gauge theory #Ising model #Lambda #Lattice (music) #Lattice field theory #Mass gap #Mathematical analysis #Mathematical physics #Mathematics #Opinion Dynamics and Social Influence #Phase transition #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum entanglement #Quantum field theory #Quantum many-body systems #Quantum mechanics #Scaling #Scaling dimension #Transfer matrix #cond-mat.stat-mech #hep-lat #hep-th #quant-ph

paper · pdf · doi:10.1103/physrevlett.123.250604

published as Phys. Rev. Lett. 123, 250604 (2019)

arxiv created 2019/07/19 · openalex publication_date 2019/12/18 · arxiv updated 2019/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of \ensuremathλ\ensuremathφ⁴ with mass parameter \ensuremathμ² and lattice spacing a, we demonstrate a double data collapse for the correlation length \ensuremathδ\ensuremathξ(\ensuremathμ,\ensuremathλ,D)=\stackrel\texttildelow\ensuremathξ((\ensuremathα\ensuremath-\ensuremathα_c)(\ensuremathδ/a)^\ensuremath-1/\ensuremathν) with D the bond dimension, \ensuremathδ the gap between eigenvalues of the transfer matrix, and \ensuremathα_c=\ensuremathμ_R²/\ensuremathλ the parameter which fixes the critical quantum field theory.

Citations

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