Contributor info: Prof. Masaki Oshikawa

Jan T. Schneider, Atsushi Ueda, Yifan Liu, Andreas M. Läuchli, Masaki Oshikawa, Luca Tagliacozzo

SciPost Phys. 18, 142 (2025) · published 29 April 2025 | Toggle abstract · pdf

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 (CFT). 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 evidence from the exact solution of the Ising model and density matrix renormalization group (DMRG) simulations of an effective lattice model.

Yifan Liu, Haruki Shimizu, Atsushi Ueda, Masaki Oshikawa

SciPost Phys. 17, 099 (2024) · published 2 October 2024 | Toggle abstract · pdf

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

Linhao Li, Masaki Oshikawa, Yunqin Zheng

SciPost Phys. 17, 013 (2024) · published 15 July 2024 | Toggle abstract · pdf

Symmetry protected topological (SPT) phases are one of the simplest, yet nontrivial, gapped systems that go beyond the Landau paradigm. In this work, we study an extension of the notion of SPT for gapless systems, namely, gapless symmetry protected topological states. We construct several simple gapless-SPT models using the decorated defect construction, which allow analytical understanding of non-trivial topological features including the symmetry charge under twisted boundary conditions, and boundary (quasi)-degeneracy under open boundary conditions. We also comment on the stability of the gapless-SPT models under symmetric perturbations, and apply small-scale exact diagonalization when direct analytic understanding is not available.