Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling

Abstract

We set up an effective field theory formulation for the renormalization flow of matrix product states (MPS) with finite bond dimension, focusing on systems exhibiting finite-entanglement scaling close to a conformally invariant critical fixed point. We show that the finite MPS bond dimension $\chi$ is equivalent to introducing a perturbation by a relevant operator to the fixed-point Hamiltonian. The fingerprint of this mechanism is encoded in the $\chi$-independent universal transfer matrix's gap ratios, which are distinct from those predicted by the unperturbed Conformal Field Theory. This phenomenon defines a renormalization group self-congruent point, where the relevant coupling constant ceases to flow due to a balance of two effects; When increasing $\chi$, the infrared scale, set by the correlation length $\xi (\chi )$, increases, while the strength of the perturbation at the lattice scale decreases. The presence of a self-congruent point does not alter the validity of the finite-entanglement scaling hypothesis, since the self-congruent point is located at a finite distance from the critical fixed point, well inside the scaling regime of the CFT. We corroborate this framework with numerical evidences from the exact solution of the Ising model and density matrix renormalization group (DMRG) simulations of an effective lattice model.

References in corpus (21)

• The density-matrix renormalization group in the age of matrix product states
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Matrix product states represent ground states faithfully
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Entanglement spectrum in one-dimensional systems
• Scaling of entanglement support for Matrix Product States
• Entanglement hamiltonians in two-dimensional conformal field theory
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Fermionic Gaussian states: an introduction to numerical approaches
• Transfer Matrices and Excitations with Matrix Product States
• Finite correlation length scaling with infinite projected entangled pair states at finite temperature
• Operator content of real-space entanglement spectra at conformal critical points
• 2D Ising Field Theory in a Magnetic Field: The Yang-Lee Singularity
• Finite-size and finite bond dimension effects of tensor network renormalization
• A scaling hypothesis for projected entangled-pair states
• Emergent conformal boundaries from finite-entanglement scaling in matrix product states
• Extracting the Speed of Light from Matrix Product States
• Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries
• Universal finite-size gap scaling of the quantum Ising chain
• Finite-Entanglement Scaling of 2D Metals
• Ising Spectroscopy I: Mesons at T < T_c

Cited by in corpus (1)

• Spiral renormalization group flow and universal entanglement spectrum of the non-Hermitian 5-state Potts model