SciPost Submission Page
A sine-square deformation approach to quantum critical points in one-dimensional systems
by Yuki Miyazaki, Shiori Tanigawa, Giacomo Marmorini, Nobuo Furukawa, Daisuke Yamamoto
Submission summary
| Authors (as registered SciPost users): | Yuki Miyazaki |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2512.14149v1 (pdf) |
| Date submitted: | Dec. 17, 2025, 4:24 a.m. |
| Submitted by: | Yuki Miyazaki |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We propose a method to determine the quantum phase boundaries of one-dimensional systems using sine-square deformation (SSD). Based on the proposition, supported by several exactly solved cases though not proven in full generality, that "if a one-dimensional system is gapless, then the expectation value of any local observable in the ground state of the Hamiltonian with SSD exhibits translational symmetry in the thermodynamic limit," we determine the quantum critical point as the location where a local observable becomes site-independent, identified through finite-size scaling analysis. As case studies, we consider two models: the antiferromagnetic Ising chain in mixed transverse and longitudinal magnetic fields with nearest-neighbor and long-range interactions. We calculate the ground state of these Hamiltonians with SSD using the density-matrix renormalization-group algorithm and evaluate the local transverse magnetization. For the nearest-neighbor model, we show that the quantum critical point can be accurately estimated by our procedure with systems of up to 84 sites, or even smaller, in good agreement with results from the literature. For the long-range model, we find that the phase boundary between the antiferromagnetic and paramagnetic phases is slightly shifted relative to the nearest-neighbor case, leading to a reduced region of antiferromagnetic order. Moreover, we propose an experimental procedure to implement the antiferromagnetic $J_1$-$J_2$ Ising couplings with SSD using Rydberg atom arrays in optical tweezers, which can be achieved within a very good approximation. We show that our protocol can control the ratio $J_2/J_1$ over a wide range, and realize a well-approximated SSD of the nearest-neighbor model. Because multiple independent scaling conditions naturally emerge, our approach enables precise determination of quantum critical points and possibly even the extraction of additional critical phenomena, such as critical exponents, from relatively small system sizes.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
The application to long-range interacting hamiltonian and to present how to apply the SSD function in realistic and slightly complex interacting geometry.
Weaknesses
Report
The authors show that the difference in the local observables between different locations in space crosses zero value at the critical point of the parameters that are size-dependent, and that the critical-point value follows a nice scaling law that converges to a true critical point at the thermodynamic limit. The authors demonstrate the method convincingly for both nearest-neighbor and long-range mixed-field Ising chains, and show excellent agreement between the extracted critical points and those obtained using conventional approaches.
One strong aspect of the work, which I think is new, is the treatment of long-range interactions, where boundary effects are notoriously severe and standard finite-size analyses become subtle. The authors’ demonstration that the SSD framework remains effective in this context broadens the applicability of the SSD framework in general and enhances its practical relevance.
Another notable strength is the detailed and concrete discussion of experimental realizability in Rydberg-atom arrays. The manuscript goes beyond a purely numerical proposal and carefully explains how the SSD Hamiltonian can be engineered through spatially dependent interactions and local fields controlled by atomic geometry and laser parameters. This part is valuable, as it bridges numerical boundary-conditioning techniques and experimentally accessible Hamiltonian engineering.
The manuscript is clearly written, well-structured, and technically sound.
I have only a few minor comments and suggestions, which can be optional:
Scope of SSD–PBC equivalence
The discussion on the (non-)universality of the equivalence between SSD and periodic boundary condition (PBC) ground states is appropriate and well-balanced. It might be helpful to more clearly delineate which parts of the present method rely on exact equivalence and which only require approximate restoration of translational invariance, especially for interacting or long-range systems.
Finite-size limitations
While the SSD clearly improves finite-size behavior, a brief quantitative discussion of residual finite-size effects (e.g., how system size requirements compare to standard OBC or PBC calculations in practice) would further clarify the practical advantages of the method.
Outlook to higher dimensions and nonlocal order
The future directions are well chosen. A short comment on potential conceptual or technical obstacles in extending the SSD approach to higher dimensions or to systems with topological order would help set realistic expectations.
Recommendation
Publish (meets expectations and criteria for this Journal)
Strengths
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The central idea, nemely, using the coincidence between the SSD ground state and the PBC ground state as a diagnostic at criticality, is well motivated and methodologically appealing.
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The proposed criteria are based on local observables that are readily accessible within DMRG, which makes the approach practically useful at moderate system sizes.
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The manuscript goes beyond locating QCPs by discussing scaling behavior and critical exponents, indicating possible extensions of the method.
Weaknesses
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The quantitative validation of the reported critical values is somewhat limited (particularly for the long-range model), and the rationale for quoting 5-6 significant digits is not fully supported by independent benchmarks. This seems primarily a matter of presentation/benchmarking rather than a fundamental limitation of the approach.
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Section 3 appears only loosely connected to the main SSD-based QCP-estimation protocol; clearer framing and/or a more explicit link to the proposed diagnostic procedure would improve the overall coherence.
Report
The authors propose a method to determine quantum critical points (QCPs) in one-dimensional systems using sine-square deformation (SSD). The method is validated by DMRG calculations for (i) the mixed-field antiferromagnetic spin-1/2 Ising chain with nearest-neighbor couplings and (ii) its long-range (power-law) counterpart. The analysis is based on the reasonable criterion that, at criticality, the SSD ground state coincides with the ground state under periodic boundary conditions. Overall, the manuscript is clearly written and the estimated critical points appear scientifically sound. I also find it interesting that the authors briefly discuss critical exponents, which suggests potential extensibility of the approach. I think the paper is essentially publishable in SciPost Physics after addressing the following minor comments, which are rather optional. 1. In Fig. 4, the size dependence of \Delta_5 and \Delta_6 is explained by stating that "these differences are attributed to the parity combinations of the reference sites." For the present model, is the intended interpretation that they exhibit a sign change when the separation between the reference sites is odd, while it shows a local minimum when the separation is even? Please clarify this point more explicitly. 2. The manuscript reports critical values with five or six significant digits in several places, and it also claims “highly accurate determination of the QCP from relatively small system sizes.” However, the quantitative accuracy is not obvious without an external benchmark (especially for the long-range model). Since at h_x=0 for the long-range model the transition is known exactly at h_z=1.00145, it would strengthen the manuscript if the authors could (if feasible) demonstrate that their SSD-based criterion reproduces this value. 3. Section 3 currently feels somewhat disconnected from the main SSD-based QCP-estimation narrative, as it focuses on a possible Rydberg implementation without applying the proposed QCP-detection protocol. Please add a brief statement clarifying the purpose of Sec. 3 and how (or why not) the Sec. 2 procedure could be applied there. If the omission is due to numerical difficulties (e.g., metastability in DMRG for near-classical and/or frustrated Ising-type models), please mention this explicitly. 4. Please correct the citation information for Ref. 52.
Recommendation
Ask for minor revision