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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/2411.03954v2 (pdf) |
| Date submitted: | 2025-02-26 08:39 |
| 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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Author comments upon resubmission
List of changes
- Page 3, removed the sentence “This has been the subject of several studies.”
- Beginning of Section 2: We have made the distinction of thermodynamic limit and
infinite degrees of freedom
- Page 6, added sentence on further details on the scaling hypothesis and reference.
- Page 5, thus affecting ``thus affecting the ratio of the gap ratios'' to ``thus affecting the ratio of the gaps''
- Beginning of Section 2.1: Added a paragraph on introduction on the transfer matrix and motivation of studying MPS TMs.
- Fig 2 description: symmetry-broken to non-conserving
- Figure 3(b): x-axis label changed from $L/\xi_0 \chi^\kappa$ to $L/\xi(\chi)$, label for dashed green line changed from $L^{15/4}$ to $(L/\xi)^{15/4}$
- Below Eq.~(13) and describing Fig.~3(a) correction of quantity explained from ``this relation is shown'' (i.e. $\delta E_0$) to ``$E_0 - \epsilon \, L$'' and in the sentence referencing Fig.~3(a), a clarification and definition of $\epsilon_0$
- Page 12. This result aligns with our numerical observations shown in Fig.~3(b) [added:] when replacing the renormalization scale \(L\) with \(L/\xi(\chi)\), being the relevant length scale in the crossover regime from finite-size to finite-entanglement scaling.
- Below Eq.~(34) add spontaneous:
However, in the spontaneous symmetry-breaking phase (SSB), the quasi-particle excitations are domain walls. Add also sentence and reference to algebraic Bethe ansatz with its analytical solutions.
- On page 13, changed ``transitionally'' into ``translationally''.
- Typo in the caption of Fig 6, $1-1/\xi$ instead of $1-1/L$.
- Page 21, added clarification below Eq. (40) on the confinement of the theory
- Page 17, Eqs. (34) and (35), added expression in terms of the perturbation $\delta$.
- Page 17, define acronym SSB as ``spontaneous symmetry-breaking phase''
- Page 18, adding further clarification for why the SSB phase is selected by iDMRG.
- On page 22, added subclause specifying that the relevant operator is rendered marginal in combination with its scale dependent coupling (twice)
- Right before A.11, corrected $\ket{\psi}$ to be $\ket{\psi_0}$, and $e$ to $e_0$.
- Below (B.1) updated the self-duality mapping to the correct one from the original reference.
- Below Eq.~(34): moved the sentence from before Eq.~(35) about the two elementary excitations needed to create any correlation
- Several minor orthographic corrections