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 λφ 4 with mass µ 2 and lattice spacing a, we demonstrate a double data collapse for the correlation length δξ(µ, λ, D) = ξ (α − αc)(δ/a) −1/ν with D the bond dimension, δ the gap between eigenvalues of the transfer matrix, and αc = µ 2 R /λ the parameter which fixes the critical quantum field theory.

Tensor Renormalization Group for interacting quantum fields

“…We will start here the analysis of its numerical performance using as benchmark a λφ 4 theory. This theory has been studied with tensor networks techniques which imply the discretization of the field variables in [26,35,36,40,43] and with continuous tensor networks techniques in [52,53].…”

Lattice regularisation and entanglement structure of the Gross-Neveu model

“…dimensions [21,26,28,34,43], most of the tensor network effort has so far been invested in QFTs in (1+1) dimensions, and in particular the λφ 4 model [8,9,17,18,20,22,24] and the Schwinger model, i.e. (1+1)-dimensional quantum electrodynamics, (as well as nonabelian generalizations thereof) [19, 23, 25, 27, 29-31, 33, 35-39, 41, 44].…”

“…We refer to refs. [17,96] and use these techniques to extrapolate reported mass values to the infinite bond dimension limit.…”

“…In recent years, the use of tensor network methods has been proposed as an alternative [6], with the promise that these are able to access dynamical information and do not suffer from sign problems in the case of fermionic densities or far-from-equilibrium situations [7]. One can apply tensor renormalisation group techniques as an alternative to Monte Carlo sampling to the path integral in discretised spacetime [8][9][10][11][12][13][14][15][16][17][18]. Alternatively, one can target the wave functional using a tensor network ansatz and apply variational techniques using the field theory hamiltonian (where only the spatial dimensions are discretised) .…”

Entanglement compression in scale space: From the multiscale entanglement renormalization ansatz to matrix product operators

“…The MPS case underlies the very successful density matrix renormalization group algorithm [2,3], and its success for simulating critical systems stems from the fact that the finite bond dimension χ approximation plays the role of an infrared (IR) cutoff by effectively adding a small relevant perturbation to the critical system with an amplitude monotonically decreasing as a function of χ . This allows us to construct a theory of finite entanglement scaling for uniform (infinite) MPSs in a vein similar to finite length scaling methods used in exact diagonalization or Monte Carlo [4][5][6][7][8]. Alternatively, matrix product operators (MPOs) have been proven to provide a faithful approximation of thermal states of quantum spin chain Hamiltonians at any finite temperature, with a bond dimension scaling as a power of the inverse temperature, thereby providing an avenue for finite temperature scaling [9][10][11][12].…”