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Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling
by Jan T. Schneider, Atsushi Ueda, Yifan Liu, Andreas M. Läuchli, Masaki Oshikawa, Luca Tagliacozzo
Submission summary
| Authors (as registered SciPost users): | Yifan Liu · Andreas Läuchli · Jan Thorben Schneider · Atsushi Ueda |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2411.03954v3 (pdf) |
| Date accepted: | 2025-04-22 |
| Date submitted: | 2025-04-08 12:27 |
| Submitted by: | Schneider, Jan Thorben |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
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.
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List of changes
Below is a list of changes from v2 to v3 to the preprint with arXiv identifier \texttt{arXiv:2411.03954}.
- Page 8, added 'numerically' to 'observed numerically' in the first paragraph of 2.1.
- Page 9, expanded the explanation on the symmetry-perserving simulation and added reference to later Section.
- Page 9, added clarification that the TM spectrum in the Ising models is real, but not generically.
- Page 11, expanded definition of the Casimir energy contribution and explained factors.
- Page 12, added a paragraph on the effect of renormalization on the symmetry-preserving and non-preserving simulations of the Ising model.
- Page 16, above Eq. (25), changed $m$ to $l$ when counting the number of fermionic modes.
- Page 17, 18, we have taken out previous Eq. (33), the single particle gap. We have also slightly restructured the chain of arguments setting $m=+1$ in our lattice model definition. Lastly, we added a reference to the reply to the report of Referee 2, which includes an attached file showing the plot that verifies that the variational ground state energy error is approximately symmetric w.r.t. the coupling $\delta$
- Page 18, corrected the wrong placement of the factor $1/2$ in previous Eqs. (34) and (35)
- Page 18, moved the paragraph explaining why the iDMRG algorithm always chooses the SSB phase as opposed to the PM phase to this spot. Previously, it was located just before Sec. 5.2.
- Page 20, corrected label Fig. 8(c) to Fig 8.(a).
- Several minor orthographic corrections
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)