A scaling hypothesis for matrix product states

Abstract

We revisit the question of describing 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. Critical exponents and central charge are determined by optimizing the exponents such as 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 $\lambda {\varphi }^4$ with mass ${\mu }^2$ and lattice spacing $a$, we demonstrate a double data collapse for the correlation length $\delta \xi (\mu ,\lambda ,D)=\tilde{\xi }\left((\alpha -{\alpha }_c)(\delta /a{)}^{-1/\nu }\right)$ with $D$ the bond dimension, $\delta$ the gap between eigenvalues of the transfer matrix, and ${\alpha }_c={\mu }_R^2/\lambda$ the parameter which fixes the critical quantum field theory.

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Cited by in corpus (41)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Tensor lattice field theory with applications to the renormalization group and quantum computing
• Variational methods for contracting projected entangled-pair states
• Computing the renormalization group flow of two-dimensional ${\varphi }^4$ theory with tensor networks
• A scaling hypothesis for projected entangled-pair states
• Kekulé spiral order in magic-angle graphene: a density matrix renormalization group study
• Quantum computing critical exponents
• Tensor Networks Can Resolve Fermi Surfaces
• Emergent conformal boundaries from finite-entanglement scaling in matrix product states
• The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study
• Relativistic continuous matrix product states for quantum fields without cutoff
• Extracting the Speed of Light from Matrix Product States
• Tensor Network Based Finite-Size Scaling for Two-Dimensional Classical Models
• Collective Monte Carlo updates through tensor network renormalization
• Gaussian Continuous Tensor Network States for Simple Bosonic Field Theories
• Correlation Functions From Tensor Network Influence Functionals: The Case of the Spin-Boson Model
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• Variational method in relativistic quantum field theory without cutoff
• Extended Quantum Spin Liquid with Spinon-like Excitations in an Anisotropic Kitaev-Gamma Model
• Sketching phase diagrams using low-depth variational quantum algorithms
• Superconductivity in a Topological Lattice Model with Strong Repulsion
• Lattice regularisation and entanglement structure of the Gross-Neveu model
• Even-odd effects in the $J_1-J_2$ SU($N$) Heisenberg spin chain
• On quenches to the critical point of the three states Potts model -- Matrix Product State simulations and CFT
• Continuous Wigner-Mott transition at $\nu =1/5$
• Finite-Entanglement Scaling of 2D Metals
• Symmetric projected entangled-pair states analysis of a phase transition in coupled spin-1/2 ladders
• The Chiral Gross-Neveu model on the lattice via a Landau-forbidden phase transition
• Tensor Renormalization Group for interacting quantum fields
• Chiral Spin Liquid and Quantum Phase Transition in the Triangular Lattice Hofstadter-Hubbard Model
• Finite-size scaling on the torus with periodic projected entangled-pair states
• Entanglement compression in scale space: from the multiscale entanglement renormalization ansatz to matrix product operators
• Finite Correlation Length Scaling of Disorder Parameter at Quantum Criticality
• Non-Landau quantum phase transition in modulated SU(N) Heisenberg spin chains
• Bridging Rokhsar-Kivelson Type and Generic Quantum Phase Transitions via Thermofield Double States
• Quasi-Fermi liquid behavior in a one-dimensional system of interacting spinless fermions
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• Les Houches Lecture Notes on Tensor Networks
• Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling
• Emergent Intermediate Phase in the $J_1$-$J_2$ XY model from Tensor Network Approaches
• Quantum Supercritical Crossover with Dynamical Singularity