附件中的文件是我写的文章，主要讨论应变对RuO2磁有序的影响，你帮我看看文章总结一下

下面是我对你这篇《基于第一性原理计算的 RuO₂ 应变诱发磁有序》的结构化总结（按“研究问题—方法—结果—机制—意义/结论”来梳理）：

1) 研究动机与核心问题

你把 RuO₂ 作为“潜在交错磁体”的候选材料，但针对“是否存在磁有序”这一点，近期实验上出现了较大争议。文章聚焦于：外延整体应变是否能触发 AFM？在临界附近的局部畸变是否能进一步诱导 SDW 成为基态？ (基于第一性原理计算的RuO2应变诱发磁有序.docx)

2) 建模与计算方法

• 整体应变参数 ε： 用来表征 RuO₂ 晶格从本征到 TiO₂ 外延匹配晶格的“连续插值”；并据此扫描磁矩随 ε 的变化，确定相变临界点。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• 局部畸变模型： 沿 (100) 扩胞成 n×1×1 超胞，保持 b=aε、c=cε 不变，仅让每个单胞的 a 按规则起伏；考虑简谐畸变与更“缺陷化”的 δ 畸变，用 Δ（Δh 或 Δδ）表征幅值、n 表征周期。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• 第一性原理细节： VASP + PBE + PAW，截断能 520 eV；单胞 9×9×13 k 网格、超胞 1×9×13；正文主要不给 SOC/DFT+U，相关对比放补充材料。(基于第一性原理计算的RuO2应变诱发磁有序.docx) (基于第一性原理计算的RuO2应变诱发磁有序.docx)

3) 关键结果（你文章的“主线结论”）

A. 仅靠整体应变：顺磁 → AFM 的临界点

Ru 原子磁矩随 ε 增大从 0 开始出现并增长，得到单胞的临界应变 εc=0.7404（无 SOC）；考虑 SOC 后 εc≈0.7340，临界点基本不被 SOC 颠覆。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

B. 局部畸变对“是否出现磁有序”的边界影响较小，但决定“磁性细节”

在简谐畸变下，Δh 从 8‰/14‰/20‰ 增大时，εc 只“缓慢下降”（从 0.7404 到约 0.72/0.69/0.66），说明磁有序出现与否主要由整体 ε 决定，局部畸变更多在细节层面调制磁矩与能量竞争。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

C. 临界附近 + 足够局部畸变：SDW 可成为比 AFM 更低能的基态

你定义 ΔE = E_SDW − E_AFM 来判定谁更稳定。在 ε≈εc 时，ΔE 随 Δh 呈非单调，并在一定幅度（例如 Δh≥12‰）时可变为负值，意味着 SDW 成为更低能态；且你指出外延薄膜里 10‰ 量级局部畸变是常见的，因此这种机制在实验上是“有机会发生”的。(基于第一性原理计算的RuO2应变诱发磁有序.docx) (基于第一性原理计算的RuO2应变诱发磁有序.docx)

4) 机制解释与“可检验推论”

A. 为什么 SDW 可能赢过 AFM：能量主要来自“波节（节点）区域”

你提出假设：SDW 在波节处本征地趋向“低磁矩/近无磁”，当局部压缩进一步压低该区域磁矩时，SDW 反而能在波节处获得相对 AFM 的能量优势，从而推动 ΔE 变负；并给出可检验判断：** ΔE 的主要贡献来自波节附近**。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

B. SDW 的“最有利波长”来自畸变幅度/周期竞争：约 7–10 nm

你计算得到 ΔE per cell 随周期 n 先降后升，并观察到“Δh 越小，最稳定 SDW 所需 n 越大”，最终提炼出更易稳定 SDW 的特征波长范围 约 7–10 nm（对应 16a–22a）。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

C. 低能激发来源的推论：AFM 与 SDW 近简并导致 GHz 尺度能量差

你进一步提出：临界附近且存在局部畸变时，AFM 与 SDW 的能量差仅 10–100 μeV（~10 GHz），可解释实验上观察到的“低能激发”与传统 AFM THz 自旋波尺度的差异。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

5) 文章结论与意义（你最后落点）

总体上，你给出一个统一图像：整体应变决定“是否进入 AFM”；临界附近叠加局部畸变决定“是否可能出现 SDW 作为潜在基态”，从而为 RuO₂ 薄膜磁性实验结论分歧提供解释，并提出通过应力/畸变维度调控自旋输运的新思路。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

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如果你愿意，我也可以在这个总结基础上，帮你再做一版更“投稿友好”的一页式摘要（背景→方法→三条结果→一句话意义），或者把你文中“关键图（Fig.1–6）各自想表达的结论”用一句话逐张对齐，方便你检查叙事是否紧凑。

你导师担心“太繁琐”其实是对的：APS/PRB 的风格里，标题要自洽、简洁、信息量高，避免塞太多限定语/术语。(APS Journals)
但你这篇的关键新意之一，确实是“临界整体应变附近，局部畸变可以把 SDW 拉成更稳定的态”，而不仅是“应变诱导 AFM”。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

怎么办：用“最少的词”把局部畸变放进去（不啰嗦）

你可以把“local distortion”作为一个并列驱动因素或一个限定机制写进标题，而不是把模型细节（harmonic/δ、n、Δ）写进去。PRB 读者看到 local distortions 就够了，细节放摘要/正文。

我建议你优先考虑这两种写法（都符合“短而信息密度高”）：

写法 A：并列驱动（最稳妥，也最不繁琐）

• Epitaxial strain and local distortions induce magnetic order in RuO₂
• Global strain and local distortions induce magnetic order in RuO₂

这一类标题等价于告诉读者：你讲的是“整体应变 + 局部畸变”共同决定磁性相图/基态选择（与你文章主线一致）。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

写法 B：突出你的“第二步结果”（局部畸变稳定 SDW）

• Local distortions near critical strain stabilize a spin-density-wave state in RuO₂
• Local distortions stabilize spin-density waves near the strain-driven magnetic transition in RuO₂

如果你更希望读者一眼看到“SDW”这条新意，就用这类。

关于 “from first-principles calculations”

在 PRB 里这句通常可省略，因为方法在摘要/计算细节里会写清楚；去掉它能显著减短标题、也更像 PRB 常见风格。(APS Journals)
如果你想保留“第一性原理”标签，建议改成更短的前缀式：

• First-principles study of strain- and distortion-induced magnetic order in RuO₂

和导师沟通的“折中方案”

你可以把上面写法 A 当作折中：只多加 local distortions 两个词，但把你论文的核心叙事（整体应变决定是否磁有序、局部畸变决定临界附近的 SDW 竞争）准确放进标题。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
同时你也可以让导师放心：模型细节不进标题，而是在摘要第一段第二句就点明“near critical strain, local distortions stabilize SDW (10–100 μeV; 7–10 nm)”——这样标题不繁琐，但卖点不会丢。

如果你愿意，我可以按你现在 PRB 的写法习惯（大小写、连字符、是否用 epitaxial/global）帮你把候选标题再精修到 3 个“最像 PRB 文章标题”的最终版本。

可以，而且从你这篇的建模方式看，“局部应变”其实比“局部畸变”更贴切——你是在固定整体应变 ε 的前提下，让每个单胞的晶格常数 a 在空间上起伏（Δh / Δδ 控制幅值、n 控制周期），这本质上就是“非均匀/局域应变场（local strain inhomogeneity / fluctuations）”的一个模型化表达。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

但我建议你在英文标题里不要直接写成 bare 的 local strain（太短时容易让人误解成“又施加了一种新的宏观应变分量”），更 PRB 友好的写法通常是：

• local strain inhomogeneity（局部应变不均匀）
• local strain fluctuations（局部应变涨落）
• inhomogeneous strain（非均匀应变）

这样既简洁，又能准确对应你“整体应变 + 局部起伏”这条主线。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

我给你 4 个不繁琐、但能点出“局部应变”的标题备选（按信息密度从低到高）

A. 最像你原题、改动最小（导师一般最容易接受）

1. Strain- and local-strain-induced magnetic order in RuO₂
   （短；“local strain”稍微口语，但还能接受）

B. 更推荐：把“局部应变”的物理含义说清（仍然不长） 2) Epitaxial strain and local strain inhomogeneity induce magnetic order in RuO₂
3) Magnetic order in RuO₂ driven by epitaxial strain and local strain fluctuations

C. 如果你希望标题直接“卖”你最亮的结果（临界附近 + 局部效应 → SDW） 4) Local strain fluctuations near critical epitaxial strain stabilize a spin-density wave in RuO₂
（这句和你文中“临界附近局部畸变可诱导 SDW 成为更低能态”的结论是对齐的。(基于第一性原理计算的RuO2应变诱发磁有序.docx)）

为了避免审稿人纠结术语：正文里一句话“兜底”

如果你把标题写“local strain …”，我建议你在模型首次出现的位置用一句话把术语钉死，比如：

“We model local strain fluctuations (i.e., local lattice-constant modulations) at fixed global strain ε …”

这和你现在对“整体应变 ε + 局部起伏 Δ”的定义完全一致，也能避免“你到底是畸变还是应变？”这种语义争论。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

如果你告诉我你更想突出哪条主结果（“应变诱导 AFM”还是“临界区 SDW”），我可以把上面 4 个再精修成最像 PRB 标题的 2–3 个最终版本（大小写、连字符、是否用 epitaxial/global 等都一起统一）。

1) 在本文里把“局部畸变”换成“局部应变”，可以吗？

可以，但我建议不要直接一刀切把所有“局部畸变”都替换成“局部应变”，而是用更精确、也更不容易被审稿人挑语义的表述：

• 更推荐：“局部应变不均匀 / 局部应变涨落（非均匀应变）”
• 英文更推荐：local strain inhomogeneity / local strain fluctuations / inhomogeneous strain

原因是：你模型里“局部畸变”本质上是在固定整体应变 ε 的前提下，让局部晶格常数 $a_i$ 发生空间起伏（由 Δ 和 n 控制），这在物理上确实等价于“非均匀局部应变场”。但如果只写“局部应变（local strain）”，有些读者会误以为你又施加了一个额外的宏观应变分量。你文中其实也已经把它解释成“不改变整体应变 ε 的局部畸变”这一点了。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

最稳妥的写法：首次出现用

局部畸变（非均匀/局部应变涨落）
后文为了简洁，可以统一简称“局部应变不均匀”或“局部应变涨落”。

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2) 你给的这段中文（摘要部分）英文翻译（我做了少量“PRB风格”润色）

English (translated and lightly polished):
RuO$_2$, proposed as a potential altermagnet, is expected to serve as a new-generation spintronic material enabling precise control of spin transport; however, the very existence of magnetic order in RuO$_2$ has recently been widely questioned. Here, based on first-principles calculations and taking epitaxial RuO$_2$ films grown on TiO$_2$ substrates as an example, we systematically investigate the magnetic ordering of RuO$_2$ thin films under epitaxial strain. We find that a sufficiently large global strain can drive RuO$_2$ films from a nonmagnetic state to an antiferromagnetic (AFM) state. Near the strain-driven magnetic critical point, sufficiently strong local strain inhomogeneity (i.e., inhomogeneous strain modeled as local lattice-constant modulations) can stabilize a spin-density-wave (SDW) state over the AFM state, with an energy difference on the order of 10–100 $\mu$eV and a characteristic wavelength of about 7–10 nm. The strain-induced magnetic-ordering mechanism provides a unified microscopic picture for the experimental controversy regarding altermagnetism in RuO$_2$ thin films, and highlights the key roles of global strain and local strain inhomogeneity in tuning RuO$_2$ magnetism and spin transport. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

我做了哪些“有意识的修改”（已尽量少改，只为更像 PRB）

1. 把“局部畸变(非均匀应变)”翻成 local strain inhomogeneity，并用括号一句话钉死你模型含义（局部晶格常数调制），避免审稿人纠结“distortion vs strain”。
2. “顺磁态”我这里写成 nonmagnetic state：因为 DFT 自洽里对应的是“无磁矩解”，严格来说比“paramagnetic”更不容易被挑刺；如果你全文想保持“paramagnetic”，也可以把第一处改回 paramagnetic (PM)，但最好统一全稿术语。

如果你接下来把文章其它部分也发我（或直接按章节），我可以继续按同样标准翻译，并顺手把“局部畸变/局部应变”相关术语在全文做一致性处理。

下面是你给的这段 INTRODUCTION 中文的英文翻译（按 PRB 常见写法做了轻微润色，同时尽量保持你原意与信息结构不变）。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

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English translation (lightly polished for PRB style)

In recent years, research on altermagnetism has rapidly gained momentum. Altermagnets exhibit no net magnetization in real space; however, owing to their reduced crystal symmetry, they can host a pronounced spin splitting in momentum space, thereby combining key features of both ferromagnets and antiferromagnets. As a result, altermagnets have attracted considerable attention in spintronics, where microscopic control of magnetic order is central. Meanwhile, the high-frequency response, stray-field-free advantage, and lattice-anisotropy–driven transport properties of altermagnets have opened new opportunities for expanding the application landscape of magnetic materials. A number of phenomena have been reported in altermagnets, including the anomalous Hall effect [41–44], terahertz (THz) emission, and spin–charge interconversion [45–47], which provide new ideas for developing next-generation spintronic devices and for improving technologies such as MRAM, GMR, and TMR [48–50]. In this context, the prediction of a large spin-splitting energy and a high Néel temperature in RuO$_2$ [1], together with theoretical groundwork on antiferromagnetism in RuO$_2$ [2–4] and rapid experimental progress on the altermagnetic spin-splitting effect (ASSE) in RuO$_2$ [5], have jointly propelled RuO$_2$ to become a research hotspot in this field.

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我做了哪些“必要的小修改”（你说有修改要告诉你）

1. **“因此受到了…重视”**这句我改成了更 PRB 的表达：用 have attracted considerable attention in spintronics… 来承接原因与结果，语气更自然。
2. “也为磁性材料的拓展了新的应用领域” 原句略缺主语/宾语衔接，我翻译时补全为 opened new opportunities for expanding the application landscape of magnetic materials，不改变意思，只是语法更完整。
3. “受到了自旋电子学方向这类关注微观调控材料磁性方向学科的重视” 这句中文很长，英文里我拆解为从句：in spintronics, where microscopic control of magnetic order is central，读起来更顺。

如果你希望更“硬核/更物理论文腔”（比如强调 spin-symmetry breaking、momentum-dependent spin splitting without SOC 等），我也可以给你一个更偏理论物理表达的版本。

English translation (lightly polished for PRB style)

Despite the great promise of RuO$_2$ for advancing spintronic platforms, its magnetic ordering has long remained highly controversial. Early studies suggested that RuO$_2$ hosts relatively weak antiferromagnetic order, and resonant x-ray scattering was used to identify antiferromagnetic signatures in both bulk and thin-film samples, yielding a magnetic correlation length of about 5 nm in films [2,6]. On the other hand, a substantial body of experiments supports RuO$_2$ being closer to a Pauli-paramagnetic metal: $\mu$SR results indicate the absence of observable internal dynamic magnetic fields and long-range magnetic order [7,8]; some ARPES measurements also show a band structure more consistent with a nonmagnetic state [9,10]; and quantum oscillation (QO) experiments pose an even stronger challenge to the existence of antiferromagnetic order [11]. Meanwhile, several works have emphasized that the magnetic state of RuO$_2$ is intrinsically fragile—for instance, the relatively small Fermi velocity near the Fermi surface makes its magnetism highly susceptible to external perturbations [13,14]. Given that RuO$_2$ features a finite Coulomb repulsion together with a relatively wide bandwidth, placing it in a strongly covalent, intermediate-coupling regime, such complex magnetic behavior is not unexpected. Accordingly, it has been proposed that extrinsic factors—such as oxygen vacancies, epitaxial strain, and symmetry breaking at surfaces or interfaces—may play an important role in generating magnetism in thin films [13,15,16], where these effects are often pronounced. Indeed, most experiments reporting antiferromagnetic order are based on RuO$_2$ thin films, and spin-transport experiments supporting altermagnetism in RuO$_2$ have also predominantly been carried out in film geometries [17,18]. Moreover, it has been reported that tuning the electron count via electron- or hole-type impurities can modify the Néel-vector orientation and enhance the experimental signatures of antiferromagnetism [19,20], further highlighting the critical impact of extrinsic effects on RuO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

我做的改动（都不改变原意，只是更像 PRB 写法）

• 把长句拆成更清晰的并列结构（尤其是“另一方面…μSR/ARPES/QO”那段），读起来更顺。
• “缺乏可观测的动态磁场与磁有序”翻成 absence of observable internal dynamic magnetic fields and long-range magnetic order：更贴近实验结论的常用表述。
• “非本征因素可能是薄膜中磁性的重要成因”用 extrinsic factors … play an important role in generating magnetism in thin films：语气更学术、更中性。

下面是这段的英文翻译（按 PRB 常见的 introduction 语气做了轻微润色，但不改你的逻辑与信息点）： (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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English translation (lightly polished)

Building on these considerations, a growing number of studies have begun to focus on how strain influences RuO$_2$, and how strain-induced lattice distortions modify charge transport and magnetic ordering to enable device-level tuning [21–23]. This research direction is particularly well motivated because the fragile Fermi-surface structure of RuO$_2$ renders its magnetic state unusually sensitive to the lattice constants. Previous work has shown that doping can markedly affect its magnetism by altering the lattice [53]. It has also been reported that RuO$_2$ thin films may exhibit superconductivity under specific strain conditions [54–56]. Even in studies of chemical catalysis, the strong impact of RuO$_2$ lattice parameters on the activation energy has been repeatedly emphasized [62]. Motivated by these insights, experiments have started to explore fine control of substrate–film coupling via the film thickness [51,58]. Application-oriented studies have further proposed using epitaxial strain to control the details of spin transport in RuO$_2$, thereby expanding the tuning “knobs” for spintronic devices [52,59]. Overall, the influence of strain on RuO$_2$ magnetism has attracted systematic attention at the theoretical, experimental, and device/application levels.

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我做了哪些“有意识的小修改”（不改含义，只让英文更地道）

1. 把“电输运”译成 charge transport（比 electrical transport 更常用于凝聚态/自旋输运语境）。
2. “实现器件层面的调制”译成 device-level tuning，更 PRB。
3. 最后一句把“三个层面”写成 theoretical / experimental / device/application，读起来更顺。

如果你希望全文统一用 electrical transport（而不是 charge transport），我也可以把后续段落都按同一术语风格统一。

English translation (lightly polished for PRB style)

Using the widely studied system of RuO$_2$ thin films epitaxially grown on TiO$_2$ substrates as an example, we approximate the substrate effect as a uniform (global) strain applied to the RuO$_2$ lattice, and on this basis systematically investigate how local lattice distortions affect the magnetism of RuO$_2$. We show that, even without invoking the still-debated DFT+$U$ treatment, RuO$_2$ can undergo a transition from a paramagnetic state to an antiferromagnetic (AFM) state once the lattice distortion becomes sufficiently strong. Furthermore, when the local distortion (i.e., strain inhomogeneity) is sufficiently pronounced, a spin-density-wave (SDW) state may become energetically more favorable than the AFM state, with an energy difference reaching the 10–100 $\mu$eV scale. These results provide a consistent microscopic picture for the rich extrinsic magnetic ordering observed in RuO$_2$ thin films, and help clarify the mechanisms by which strain influences the magnetism and spin-transport behavior of RuO$_2$ films. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

我做的修改（不改含义，只做更清晰/更像 PRB 的表达）

1. 把“基底影响近似为整体应变”译成 uniform (global) strain，强调是“宏观均匀应变”的近似。
2. “局部畸变(应变非均匀性)”我保留 local distortion，并用括号明确它等价于 strain inhomogeneity，避免术语歧义。

下面是你这段 COMPUTATIONAL METHOD 的英文翻译（按 PRB 常见写法做了轻微润色，但不改信息点）。(基于第一性原理计算的RuO2应变诱发磁有序.docx) (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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English translation (polished for PRB style)

Our first-principles calculations were performed using the VASP package within the framework of density functional theory (DFT). The Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional was adopted, and the interaction between valence electrons and ionic cores was described using the corresponding pseudopotentials (PAW datasets) [28–33]. The valence configurations were Ru_pv (4p$^6$4d$^6$5s$^1$), O (2s$^2$2p$^4$), and Ti (3p$^6$3d$^3$4s$^1$). The plane-wave energy cutoff was set to 520 eV. A smearing width of 0.15 eV was used with the ISMEAR = 1 scheme. The lattice constants of RuO$_2$ and TiO$_2$ employed in this work are listed in Table I. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

For calculations of the rutile primitive cell, the Brillouin zone was sampled using a Γ-centered 9×9×13 $k$-point mesh; a denser 18×18×26 mesh was used for density-of-states (DOS) calculations. For supercells expanded along the $a$ axis, a Γ-centered 1×9×13 $k$-point mesh was adopted, and the DOS was computed using a 2×18×26 mesh. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

For structural convergence, the ionic relaxation criterion for primitive-cell calculations was set to a Hellmann–Feynman force < 0.001 eV/Å. For extensive scans involving a large number of supercells, to control the computational cost while maintaining accuracy, we required the forces on O atoms to be within 0.05 eV/Å in supercell calculations. The rationale for this threshold, together with comparisons to more stringent relaxation settings and the methodology used, is provided in Supplementary Material S3 [63]. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Data post-processing and visualization were mainly carried out using VESTA and VASPKIT [34,60]. The main-text results are primarily based on calculations without spin–orbit coupling (SOC) and without DFT+$U$; comparisons and additional analyses involving SOC and DFT+$U$ are presented in Supplementary Materials S1 and S2 [63]. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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我对英文做的“必要小改动”（方便你核对）

1. 把“赝势”写成了 pseudopotentials (PAW datasets)：VASP 语境里更标准（你文献 [30–33] 也对应 PAW）。
2. 我没有把 ISMEAR=1 强行解释成 Methfessel–Paxton order 1 放进正文（避免啰嗦），但如果你想更规范，可以加一句：first-order Methfessel–Paxton smearing (ISMEAR=1)。
3. 你这里 Ti 的势在中文写的是 Ti(...)，如果你实际用的是 Ti_pv（你文稿里英文版本似乎写过 Ti_pv），建议全文统一势的命名，避免审稿人挑一致性。

如果你继续发下一段（比如整体应变 ε 的定义 + Fig.1 那段），我可以把公式前后的衔接也按 PRB 习惯一起顺一下。

English translation (lightly polished for PRB style)

RuO$_2$ is a strain-sensitive material. The fragility of its Fermi-surface structure makes its magnetic state susceptible to strong modulation by lattice strain. Given the good lattice match between TiO$_2$ and RuO$_2$, their similar rutile structures facilitate the epitaxial growth of single-crystalline RuO$_2$, providing a favorable platform for spin-transport studies. Therefore, it is natural to discuss strain-induced magnetic order in RuO$_2$ by taking the substrate-induced lattice strain from TiO$_2$ as a starting point. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

To quantify the global strain effect of the TiO$_2$ substrate on epitaxial RuO$_2$ films, we introduce a global strain parameter $\varepsilon$, which describes the degree to which the RuO$_2$ lattice is continuously deformed from its intrinsic lattice toward that of TiO$_2$ [Fig. 1(a)]. When $\varepsilon \to 0$, the lattice returns to the intrinsic RuO$_2$ values; when $\varepsilon \to 1$, it approaches the TiO$_2$ lattice. (基于第一性原理计算的RuO2应变诱发磁有序.docx)
The lattice constants are defined as

$$a_{\varepsilon}=\varepsilon a_{\mathrm{TiO_2}}+(1-\varepsilon)a_{\mathrm{RuO_2}},\qquad c_{\varepsilon}=\varepsilon c_{\mathrm{TiO_2}}+(1-\varepsilon)c_{\mathrm{RuO_2}}.$$

Figure 1(b) shows the evolution of the Ru magnetic moment as a function of $\varepsilon$. With increasing $\varepsilon$, the system can undergo a transition from a paramagnetic (nonmagnetic) state to an antiferromagnetic (AFM) state, as evidenced by the Ru moment emerging from zero and gradually increasing. A finer scan yields a critical strain $\varepsilon_c=0.7404$ (without SOC) for the RuO$_2$ primitive cell. Upon including SOC, the critical point shifts slightly to $\varepsilon_c=0.7340$, indicating that the above conclusion is not qualitatively altered by SOC; the main difference lies in the moment magnitude. For $\varepsilon>\varepsilon_c$, SOC reduces the moment by about 10% compared with the no-SOC results. Therefore, in the following we primarily discuss the results without SOC, while the similarities and differences between SOC and no-SOC cases are provided in Supplementary Material S2 [63]. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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我做的必要修改（不改含义，只是修正明显笔误/更顺的英文表达）

1. 你原文里有一句“当 $\varepsilon>c$ 时”，这里我按语境修正为“当 $\varepsilon>\varepsilon_c$ 时”（应是笔误）。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
2. “顺磁态”在 DFT 语境容易被理解为“无磁矩解”，我在英文里用了 paramagnetic (nonmagnetic) 这种写法来避免审稿人抠字眼；如果你全文统一用 paramagnetic (PM) 我也可以帮你统一。

English translation (lightly polished for PRB style)

Substrate-induced lattice strain is not unique to RuO$_2$ films epitaxially grown on TiO$_2$; it is also commonly encountered for other substrates such as MgF$_2$ and Al$_2$O$_3$ [64]. Considering the intrinsic magnetic instability of RuO$_2$ thin films, typical local structural imperfections—including nonideal growth associated with the substrate–film heterojunction, epitaxial defects, interfacial effects, or local relaxations—may all affect the magnetic ordering of RuO$_2$ (and the impact could be even more pronounced in nonepitaxial cases). Therefore, describing the realistic film system solely by a global strain parameter $\varepsilon$ is insufficient: global strain captures the overall lattice deformation but misses local details, while local magnetic features in thin films may have a substantial influence on the system. Accordingly, in addition to the global strain $\varepsilon$, we introduce local distortions (strain inhomogeneity) to examine their roles in magnetic ordering.

Specifically, we construct $n\times 1\times 1$ supercells by expanding RuO$_2$ along the (100) direction as a model for local distortions (inhomogeneous strain), where $n$ denotes the modulation period measured in the number of primitive cells. Within the supercell, we keep $b=a_{\varepsilon}$ and $c=c_{\varepsilon}$ fixed, and apply a prescribed modulation only to the lattice constant $a$ of each primitive cell as a function of the cell index $i$ $(i=1,2,\ldots,n)$ [Fig. 2(a)]. We consider two types of local-distortion (inhomogeneous-strain) models: harmonic and $\delta$-type distortions. For the harmonic case, the distortion has period $n$ and forms a spatially periodic modulation within the supercell, with $\Delta_h$ denoting the amplitude of the local distortion (strain inhomogeneity). The periodicity of the harmonic function ensures that the global strain $\varepsilon$ remains unchanged, making it an idealized model that is convenient for understanding generic effects of smooth fluctuations. For the $\delta$-type case, the distortion also has period $n$, but the lattice constant $a$ varies in a $\delta$-function-like manner within the supercell: $\Delta_{\delta}$ characterizes the relative amplitude of the most stretched cell, while the remaining cells provide compensating distortions so that the global strain $\varepsilon$ is conserved. This model is useful for mimicking situations with strongly localized distortions (strain inhomogeneity) in thin films, closer to defect-like scenarios. The lattice-constant constructions for these two models are given by: (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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I made a couple of small edits (to improve clarity / fix likely typos)

1. 你原文里 “i(i=1,2,…, n_i=1,2,…,n)” 我按语境理解为 $i=1,2,\ldots,n$（应是笔误/排版混入了 $n$）。
2. 中文里反复出现“局部畸变(非均匀应变)”很多次，英文里我统一为 local distortions (strain inhomogeneity)，避免重复但不改变含义。

如果你把后面的“公式如下”那两条公式贴出来，我可以把这一段最后一句补全成带公式的最终版本，并把符号（$\Delta_h,\Delta_\delta,n,i$）在全文统一到最标准的 PRB 记号。

English translation (lightly polished for PRB style)

Because local distortions affect the lattice constants differently at different positions, the Ru magnetic moments become position dependent [Fig. 2(a)]. From preliminary calculations, we find that the Ru moment is maximized near the center of the tensile region and minimized near the center of the compressive region. This indicates that tensile strain favors magnetic ordering whereas compressive strain suppresses it, consistent with the primitive-cell results. In the following, we characterize this amplitude-modulated antiferromagnetism using the maximum and minimum local moments. We set $n=9$ and systematically examine the effects of $\varepsilon$ and $\Delta$ on magnetic ordering.

As shown in Fig. 2(b), for the harmonic distortion, increasing $\Delta_h$ from 8‰ and 14‰ to 20‰ leads to a slow reduction of the global strain required for the tensile maximum to develop magnetic order (defined by $\mu_{\max} > 0.01\,\mu_B$): the corresponding critical strain decreases from 0.7404 to approximately 0.72, 0.69, and 0.66. By contrast, the moment at the maximum compression remains nearly zero for $\varepsilon < \varepsilon_c$, but starts to increase rapidly once $\varepsilon > \varepsilon_c$. This implies that the overall magnetic response is governed primarily by the global strain $\varepsilon$: even in the presence of local distortions, the system generally needs to approach the vicinity of $\varepsilon_c$ to establish global magnetic order. The role of local distortion (strain inhomogeneity) is mainly to produce spatially nonuniform onset—magnetism emerges in tensile regions at smaller $\varepsilon$ than in compressive regions, i.e., the two do not develop magnetic order simultaneously. Consequently, for systems with local distortions it is more appropriate to speak of a strain window for the onset of magnetic order rather than a single sharply defined critical strain, and this window is ultimately constrained by (and anchored to) $\varepsilon_c$. Representative spatial distributions of the atomic moments are provided in Supplementary Material S4 [63]. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

When the $\delta$-type distortion is adopted, the critical strains for magnetism at the maximum tensile site are approximately 0.73, 0.70, and 0.67 for $\Delta_{\delta}=8‰, 14‰,$ and $20‰$, respectively, while the moment at the maximum compression site remains nearly zero for $\varepsilon<\varepsilon_c$ and rises rapidly for $\varepsilon>\varepsilon_c$. These results are very close to those obtained for harmonic distortions [Fig. 2(c)]. Although the spatial profile of the $\delta$-type distortion differs substantially from a harmonic fluctuation, it yields a similar critical strain at the tensile maximum for the same amplitude $\Delta$, and the overall magnetism remains strongly dependent on the global strain. This indicates that whether magnetic order emerges is determined mainly by the global strain $\varepsilon$, whereas local distortions (strain inhomogeneity) primarily modify the moment distribution rather than the phase boundary itself. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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我对原文做的“必要小改动/更正”（不改变意思）

• 把你那句重复的比较（“要比压缩…要比压缩…”）在英文里合并成一句：tensile 区域更早出现磁性，compressive 区域更晚，避免重复但含义一致。
• “顺磁态”我仍按 DFT 语境写成更稳妥的 magnetic order emerges / moment nearly zero 的表述（避免审稿人抠“DFT 里的 paramagnetic 是否严格成立”）。

如果你希望全文把“局部畸变”统一替换为“局部应变不均匀/涨落”，我也可以在后续翻译里帮你把术语统一成 local strain inhomogeneity / fluctuations，并在第一次出现处加一句定义来彻底消除歧义。

明白了——接下来我会严格保留你括号里的备选/解释用语（例如“局部畸变(非均匀应变)”我就翻成 local distortion (inhomogeneous strain)，不再替你提前“二选一”或省略括号内容），方便你之后统一替换风格。

下面把你介意的那两段我按“括号保留版”重新给一遍（含必要的英文润色，但不删括号信息）：

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① 局部畸变(非均匀应变)模型段（括号保留版）

Substrate-induced lattice strain occurs not only in epitaxial RuO$_2$ films grown on TiO$_2$ substrates, but is also common for substrates such as MgF$_2$ and Al$_2$O$_3$ [64]. Considering the magnetic instability of RuO$_2$ thin films, common local distortions (inhomogeneous strain)—such as non-ideal growth caused by the substrate–film heterojunction, epitaxial defects in the film, interfacial effects, or local relaxations—may all influence the magnetic ordering of RuO$_2$ (and the effect may be even more pronounced in non-epitaxial cases). Therefore, discussing realistic films solely in terms of the global strain $\varepsilon$ is insufficient, because global strain highlights overall properties rather than local details, while local magnetic details in thin films may have a significant impact on the system. Hence, in addition to the global strain $\varepsilon$, we introduce local distortions (inhomogeneous strain) to investigate their roles in magnetic ordering.

Specifically, we build an $n\times 1\times 1$ supercell by expanding RuO$_2$ along the (100) direction as a local distortion (inhomogeneous strain) model, where $n$ is the period (counted in the number of unit cells). In the supercell, we keep $b=a_{\varepsilon}$ and $c=c_{\varepsilon}$ fixed, and apply a prescribed, unit-cell-index-dependent distortion (inhomogeneous strain) only to the lattice constant $a$ for each unit cell indexed by $i$ $(i=1,2,\ldots,n)$ [Fig. 2(a)]. We consider two types of local distortion (inhomogeneous strain) models: harmonic distortion (strain) and $\delta$-type distortion (strain). For the harmonic distortion (strain), the lattice distortion (inhomogeneous strain) has a period $n$ and produces a spatially periodic modulation within the supercell, where $\Delta_h$ denotes the amplitude of the local distortion (strain inhomogeneity). The periodicity of the harmonic function ensures that the global strain $\varepsilon$ is not changed, making it a more idealized model that facilitates understanding the generic effects of continuous fluctuations. The $\delta$-type distortion (strain) also has a period $n$, but the lattice constant $a$ varies in a $\delta$-function-like manner within the supercell: $\Delta_{\delta}$ characterizes the relative amplitude of the most stretched unit cell in the supercell; the remaining unit cells provide compensating distortions so that the global strain $\varepsilon$ remains unchanged. This model can be used to mimic situations with local strong distortion (inhomogeneous strain) in thin films. The lattice-constant construction formulas for these two types of local distortion (inhomogeneous strain) are given by: (基于第一性原理计算的RuO2应变诱发磁有序.docx)

注：你原文里 “i(i=1,2,…, n_i=1,2,…,n)” 这处我按语境在英文里写成了 $i=1,2,\ldots,n$，否则英文会显得像符号写错（你若想保留原貌也可以，我可以照排版原样放进去）。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

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② 磁矩分布与“应变磁有序区间”段（括号保留版）

Considering that local distortions (inhomogeneous strain) affect the lattice constants differently at different positions, the Ru magnetic moments vary with position [Fig. 2(a)]. Based on preliminary calculations, we note that the Ru moment is largest at the center of the tensile region and smallest at the center of the compressive region, implying that tension favors magnetic ordering whereas compression suppresses magnetic ordering, consistent with the primitive-cell results. In the following, we use the maximum and minimum moments to characterize the magnetism of this amplitude-modulated antiferromagnet. We temporarily set $n=9$ to systematically examine the effects of $\varepsilon$ and $\Delta$ on magnetic ordering. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

As shown in Fig. 2(b), under harmonic distortion (strain), as $\Delta_h$ increases from 8‰ and 14‰ to 20‰, the global strain required for magnetic ordering at the maximum tensile position (defined by $\mu_{\max}>0.01\,\mu_B$) decreases slowly from 0.7404 to approximately 0.72, 0.69, and 0.66. Meanwhile, the moment at the maximum compressive position is nearly zero for $\varepsilon<\varepsilon_c$, but begins to increase rapidly for $\varepsilon>\varepsilon_c$. This indicates that $\varepsilon$ plays the dominant role in controlling the overall magnetism: for systems with different local distortions (strain inhomogeneity), the global strain still needs to approach the vicinity of $\varepsilon_c$ to establish overall magnetic ordering. The presence of local distortions (inhomogeneous strain) mainly causes different regions to develop magnetic order to different extents; specifically, the tensile region develops magnetic ordering (with $\mu_{\max}>0.01\,\mu_B$) at a smaller strain than the compressive region, i.e., the onset is not simultaneous. This suggests that for systems with local distortions (inhomogeneous strain), it is less appropriate to define a single critical strain; instead, similar to how amorphous melting occurs over a temperature interval rather than at a fixed melting point, there exists a strain window for magnetic ordering in systems with local distortions (inhomogeneous strain), and this window is ultimately modulated by $\varepsilon_c$. (Part of the atomic-moment distributions is provided in Supplementary Material S4 [63].) (基于第一性原理计算的RuO2应变诱发磁有序.docx)

When the $\delta$-type distortion (strain) is adopted, for $\Delta_{\delta}=8‰,14‰,$ and $20‰$, the critical strains for magnetism at the maximum tensile position are approximately 0.73, 0.70, and 0.67, respectively. The moment at the maximum compressive position remains nearly zero for $\varepsilon<\varepsilon_c$ and starts to increase rapidly for $\varepsilon>\varepsilon_c$, in close agreement with the harmonic-distortion results [Fig. 2(c)]. Although the spatial distribution of the $\delta$-type distortion (strain) differs substantially from that of the harmonic distortion (strain), it yields similar critical strains at the maximum tensile position for the same amplitude parameter. Moreover, the overall magnetism still exhibits a strong dependence on the global strain, indicating that the emergence of magnetic order is mainly determined by the global strain $\varepsilon$, while local distortions (strain inhomogeneity) primarily modify the moment distribution rather than the phase boundary itself. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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如果你同意这个处理方式，我后面翻译会按一个固定规则走：中文里括号怎么放，我英文里就怎么保留（必要时只做语法顺句、不做术语替你取舍）。你把下一段继续贴我就行。

下面是你这段的英文翻译（保持你原句结构与术语，且不删你文中的括号/符号表达）： (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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English translation

Based on the systematic calculations of the above global distortion/strain and local distortion, our main results can be summarized as follows:

(1) The magnetic structure of RuO$_2$ is highly sensitive to lattice distortion; when the global strain reaches a certain level, the system undergoes a transition from a paramagnetic state to an antiferromagnetic (AFM) state. This conclusion remains essentially unchanged after including SOC.

(2) For a given global strain $\varepsilon$, the local distortion $\Delta$ can indeed modulate the magnetism to some extent; however, the phase boundary of magnetic ordering is still mainly controlled by $\varepsilon$.

Although it is difficult to realize a global strain identical to our model in experiments, the global strain level achievable in epitaxial thin films is typically on the order of the parameters considered here [51,64] (in this work, at the critical strain, $a$ changes by 21‰ and $c$ changes by 35‰ compared with the unstrained case; in the above references, the measured strain within a few nanometers from the interface is about 10‰–40‰). Meanwhile, the specific value of $\varepsilon$ depends on extrinsic factors such as the substrate and film thickness, which provides a natural route to rationalize the diversity of current experimental characterizations of RuO$_2$ thin-film magnetism.

More importantly, across the paramagnetic–ferromagnetic transition, the band structure of RuO$_2$ does not exhibit an abrupt change [Fig. 3(a)]. In terms of the Fermi surface, only the red portion shows a pronounced modification, while the blue portion changes only slightly [Fig. 3(b)]. This is consistent with existing ARPES and QO measurements. Therefore, the magnetic-ordering mechanism driven by global strain + local distortions can, to some extent, alleviate concerns raised by band-structure experiments regarding magnetic ordering in RuO$_2$.

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我需要提醒你的一处可能笔误/术语不一致（供你后续改稿时确认）

你这里写的是“顺磁-铁磁转变 / paramagnetic–ferromagnetic transition”，但在前文你的主线是“顺磁 → 反铁磁(AFM)”。从上下文逻辑看，这里很可能应为 paramagnetic–AFM transition（或更宽泛写 paramagnetic–magnetically ordered transition）。你可以回头对照 Fig. 3 对应的计算态到底是 FM 还是 AFM，再决定这里用哪个更准确。

下面是这段的英文翻译（我保留了你原文中的括号内容，如 (IC-SDW)、(IEC)、$(\pi,0,\pi/2)$ 等；行文做了少量 PRB 风格的顺句处理，但不替你“二选一”改术语）： (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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English translation

As studies on antiferromagnetic order in RuO$_2$ have progressed, related work has reported an antiferromagnetic incommensurate spin-density-wave (IC-SDW) state in RuO$_2$–Py thin-film systems, and discussed in depth the low-energy excitations arising from the coupling between the RuO$_2$ spin-density wave and the spin waves in the surface Py layer [17]. This low-energy excitation differs from the RuO$_2$ spin-wave spectrum predicted in earlier theoretical calculations [24]. In particular, the GHz-range low-energy mode appears to be at odds with the THz excitations typically expected for antiferromagnets, revealing the rich antiferromagnetic behavior of RuO$_2$. Related experiments have also suggested the presence of certain strain–spin fluctuations in RuO$_2$ [57]. Considering the instability of the RuO$_2$ Fermi surface—where the bands are relatively flat near $(\pi,0,\pi/2)$, implying a low Fermi velocity and a tendency toward Fermi-surface nesting, hence the potential for forming a spin-density wave [3]—extrinsic factors such as lattice distortion, temperature effects, interlayer exchange coupling (IEC), and strong correlations may all reshape its magnetic-instability landscape and affect the tendency toward an SDW state [25–27,61]. This implies that, under specific external conditions, RuO$_2$ may host an SDW state and exhibit magnetic dynamical responses distinct from those of the AFM state.

In fact, lattice–SDW coupling can play a decisive role in SDW formation in some materials [38], but this is in tension with the underlying logic of standard DFT total-energy calculations. Without explicitly accounting for lattice–SDW coupling, the AFM state is typically energetically favored. As can be seen from Fig. 4(b), in the absence of local strain, the excess energy of the SDW state relative to the AFM state, $\Delta E$, increases rapidly to the meV scale once magnetic order is established. Physically, at the SDW nodes the Néel vector reverses and the local moment is strongly suppressed, which leads to an exchange-energy penalty and an additional energetic cost associated with the essentially nonmagnetic Ru atoms at the nodes, making it difficult for the SDW to emerge as a low-energy self-consistent solution. However, RuO$_2$ thin films inherently involve global strain and local distortions, so it is reasonable to evaluate the SDW state by introducing an a priori periodic local distortion, which is also similar to the strategy used in total-energy calculations of the SDW in Cr [37].

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我做的“轻微修改”（不改含义，只为英文更顺）

• 把“GHz 低能激发与 THz 激发矛盾”翻成更中性的 appears to be at odds with，语气更像 PRB 引言。
• “这意味着……”那句我拆成了 This implies that… 的收束句，让段落逻辑更清晰，但信息点不变。

如果你下一段里有“局部畸变(非均匀应变)”这种括号，我也会继续逐字保留括号来翻译。

English translation (with your wording/logic preserved)

Specifically, we first construct a “global strain + local distortion” model to investigate how local distortions affect the magnetism of RuO$_2$ thin films. Since the period of the local distortion is half of the SDW period, we need to build a locally distorted supercell containing two local-distortion periods in order to compute the SDW state. Taking the sinusoidal distortion model with $n=8$ as an example [Fig. 4(a)], we construct a $16\times 1\times 1$ locally distorted supercell. The lattice constants are defined as follows:

$$\textbf{Harmonic Distortion:}\quad a_i = a_{\varepsilon}\Bigl[1-\Delta_h \cos\Bigl(\frac{(2i-1)\theta}{2n}\Bigr)\Bigr],\qquad b=a_{\varepsilon},\qquad c=c_{\varepsilon}.$$

Here, $i$ is the unit-cell index, and $\Delta_h$ is the amplitude parameter of the local distortion.

We define $\Delta E = E_{\mathrm{SDW}}-E_{\mathrm{AFM}}$, and evaluate the relative stability of the SDW and AFM states by computing $\Delta E$ for different $\Delta_h$ values [Fig. 4(c)]. When $\varepsilon>\varepsilon_c$, $\Delta E$ increases rapidly as $\Delta_h$ decreases; to achieve $\Delta E<0$, the required local-distortion strength becomes more stringent with increasing $\varepsilon$, consistent with previous literature [36]. When $\varepsilon<\varepsilon_c$, global magnetic order is more difficult to establish, making the emergence of the SDW state more challenging. Notably, when $\varepsilon \approx \varepsilon_c$, $\Delta E$ exhibits a nonmonotonic dependence on $\Delta_h$, first increasing and then decreasing, and the SDW becomes the lower-energy state for $\Delta_h \ge 12‰$. Considering that local distortions on the order of $10‰$ are fairly common in epitaxial RuO$_2$ thin films, this suggests that the SDW state is more likely to become a potential ground state near the critical $\varepsilon$ when accompanied by a certain degree of local distortion. When $\varepsilon$ is far away from the critical regime, the system tends to remain in the paramagnetic state or adopt the AFM state. Overall, local distortion is an important driving factor for SDW formation.

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我对原文做的“仅排版/语法层面”的处理（不替你选词）

• 你给的公式在中文里括号/分式排版不够清晰，我在英文里只做了数学排版补全（把应有的括号和分母写清楚），符号含义不变。
• 其他内容我没有替你把术语“定死”，也没有擅自删改你对物理趋势的表述。

English translation (keeping your parentheses/optional phrasing intact)

On this basis, we propose a hypothesis to explain why the SDW state may become the ground state: as the local-distortion amplitude $\Delta_h$ increases, the SDW state intrinsically satisfies that the magnetic moment approaches zero at the node region (波节), and when the compressive region further suppresses the local moment, the SDW can instead gain an energetic advantage over the AFM state around the nodes, making $\Delta E$ more likely to become negative. This leads to a testable conclusion: the dominant contribution to $\Delta E$ should come from the vicinity of the nodes (波节). Considering the nonlocal nature of AFM, the moment at the nodes in the AFM state should not immediately drop to zero, but should decrease gradually with increasing $\Delta_h$; in contrast, in regions where the moment is maximal, the local moments of AFM and SDW should tend to become identical, and thus the net contribution to $\Delta E$ should be small.

We perform calculations for different $n$ at $\varepsilon=\varepsilon_c$. As shown in Fig. 5(a), when $\Delta_h \ge 5‰$, $\Delta E$ decreases gradually with increasing $\Delta_h$. We further compare the spatial distributions of Ru local moments in AFM and SDW [Fig. 5(b)]: the maximum moments in both magnetic structures increase with $\Delta_h$ and gradually converge, consistent with the prediction that the tensile region moments converge and contribute little to $\Delta E$. Meanwhile, the minimum moment of the AFM state decreases as $\Delta_h$ increases, indicating that the moment in the node region is continuously suppressed but does not become zero. Therefore, the node region is the primary source of the moment difference between AFM and SDW, which supports the statement that “$\Delta E$ is mainly contributed by the nodes (波节).”

We can also obtain a second testable implication: as the period $n$ increases, a “nearly nonmagnetic” region is more likely to form around the nodes (波节), and the energetic advantage of SDW relative to AFM should gradually emerge. To this end, at $\varepsilon=\varepsilon_c$ we calculate $\Delta E$ per cell (不同周期$\Delta E$ per cell易于讨论) for different $\Delta_h$, as well as the evolution of AFM/SDW moments with $n$. The results show that $\Delta E$ per cell first decreases and then increases with $n$ [Fig. 5(c)], while the AFM moment at the nodes decreases rapidly as $n$ increases [Fig. 5(d)]. This physical picture can be understood as a competition between two energy contributions: as $n$ increases from small values, the “nearly nonmagnetic” region near the nodes expands and the wavelength becomes longer, reducing the exchange-energy cost and making it easier for the SDW to become the ground state; however, when $n$ increases further, the nearly nonmagnetic region gradually exceeds the constraint imposed by the nonlocality of AFM, so that the AFM node moment also approaches zero and becomes closer to the SDW, causing $\Delta E$ per cell to increase again and potentially approach zero. Meanwhile, a longer period also weakens the coupling between neighboring magnetically ordered regions, making the moment magnitudes of the two states more similar.

We further observe that the smaller $\Delta_h$ is, the larger $n$ is required to reach the minimum of $\Delta E$ per cell; i.e., the smaller the distortion amplitude, the longer the wavelength required for the most stable SDW. When $\Delta_h=8‰, 14‰, 20‰$, the minima of $\Delta E$ per cell occur at $n=8, 9, 11$, respectively, and these $\Delta_h$ values are reasonably consistent with experimental situations [51] (晶格常数涨落为10‰–30‰). This implies that the characteristic SDW wavelength is $16a$–$22a$, i.e., about 7–10 nm.

下面是你这段的 PRB 风格英文翻译（并按你的要求保留你文中括号/占位提示，比如“(插入引用文献)”、以及“波节(node)”这种中英并列）： (基于第一性原理计算的RuO2应变诱发磁有序.docx)

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We further analyze the density of states (DOS) within the local-distortion model. Taking the harmonic distortion with $\varepsilon=\varepsilon_c$, $n=8$, and $\Delta_h=24\permil$ as an example, we compare the Ru-projected DOS at different positions [Fig. 6(a)]. Ru sites away from the node region (波节) exhibit a clear spin-splitting feature, and additional calculations indicate that the DOS evolution is continuous (smooth).

These results suggest that, within an idealized model with periodic distortions, the SDW state can emerge as a potential ground state of RuO$_2$. In realistic thin films, however, local distortions are disordered fluctuations, and the real-space profile of the SDW is expected to adjust accordingly. Since the local distortions that induce an SDW do not possess a unique wavelength, we construct a class of local-distortion models in which the position of the compressed region can be tuned, thereby enabling control over the pinning position of the SDW node (波节) (see Supplementary Material S5 [63] for details). Our calculations show that the SDW node is pinned to the center of the maximally compressed region and follows its motion [Fig. 6(b)]. This, on the one hand, demonstrates that the node position responds to local distortions; on the other hand, it implies that the “SDW wavelength” in RuO$_2$ is not a strictly unique constant, but is closely tied to the characteristic length scale of the local distortion (and its spatial distribution).

In practice, local distortions extending over several unit cells—caused by defects, interfacial steps, or local relaxations—are rather common, and their influence is often confined to nm-scale regions (插入引用文献). To capture this type of “strong distortion with a finite spatial extent,” we introduce a more realistic $\delta$-type local-distortion model to mimic the realistic situation. The local lattice distortion is defined as

$$\textbf{$\delta$-type Distortion:}\quad a_i= \begin{cases} a_{\varepsilon}(1-\Delta_{\delta}), & 1\le i\le n_0\ \text{or}\ n+1\le i\le n+n_0,\\[4pt] a_{\varepsilon}\!\left(1+\dfrac{n_0}{\,n-n_0\,}\Delta_{\delta}\right), & \text{otherwise}, \end{cases} \qquad b=a_{\varepsilon},\qquad c=c_{\varepsilon}.$$

Here $n_0$ denotes the number of compressed unit cells within each magnetically ordered region, while $n-n_0$ corresponds to the stretched unit cells near the node (波节, node). This construction ensures $\langle a_i\rangle=a_{\varepsilon}$ over one period, and is closer to the thin-film reality of “local strong distortion + finite interaction range.”

In the calculations, we set $n=10$ and $\varepsilon=\varepsilon_c$, construct a $20\times 1\times 1$ locally distorted supercell, and tune the “width” of the compressed region by varying $n_0$, in order to examine its impact on the relative stability of the SDW and AFM states [Fig. 6(c)]. The results show that for $n_0=1$, $\Delta E$ increases with increasing $\Delta_{\delta}$; for $n_0=2,3,4$, $\Delta E$ also increases with $\Delta_{\delta}$, while the slope of $\Delta E(\Delta_{\delta})$ decreases as $n_0$ becomes larger. This indicates that stabilizing the SDW relative to the AFM requires a sufficiently extended locally compressed region. It also implies that the existence (or absence) of an SDW is intimately linked to the local-compression conditions in the film, consistent with the trends obtained from the harmonic-distortion analysis.

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如果你愿意，我也可以把这一段再进一步“PRB 化”到更硬的语气（比如把 potential ground state 换成更谨慎的 energetically competitive / metastable，以及把“连续”具体化为“no discontinuity across the SDW stabilization regime”等），但我先按你原意最大程度直译+顺句了。

PRB-style translation

Combining the above results, we arrive at two conclusions and one inference:

(1) Local distortions near the critical global strain can stabilize an SDW ground state in RuO$_2$ thin films. In this regime, the energetic competition between AFM and SDW is governed primarily by the local environment around the nodes (波节). Enhanced compression at the nodes amplifies the difference between the two states in this region, thereby increasing the energetic advantage of the SDW over the AFM. As the spatial extent of the “compressed / nonmagnetic tendency” region around the nodes increases, $\Delta E$ can exhibit a nonmonotonic behavior, i.e., it first decreases and then rises again.

(2) The characteristic SDW wavelength in RuO$_2$ thin films is set by the distortion strength (characteristic length scale). A distortion length of approximately 7–10 nm is most favorable for stabilizing the SDW. If the distortion length is shorter, exchange interactions substantially increase the energy cost associated with the Néel-vector reversal (at the node), making the SDW less competitive. If the distortion length is longer, the energy difference between AFM and SDW decreases and the two states approach near degeneracy.

(3) Inference: the low-energy excitations observed in RuO$_2$ thin films may originate from the near-degenerate competition between the AFM and SDW states. Close to the critical global strain and in the presence of local distortions, the AFM–SDW energy difference is only 10–100 $\mu$eV ($\sim$10 GHz), which is far below the THz-scale magnon energies typical of conventional antiferromagnets.

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CONCLUSIONS

In this work, we investigate magnetic ordering in epitaxial RuO$_2$ thin films by analyzing the effects of global strain and local distortions on non-intrinsic magnetic order. When the substrate-induced global strain on RuO$_2$ is sufficiently strong, RuO$_2$ undergoes a transition from a paramagnetic state to an AFM state. Near the paramagnetic–AFM critical strain, introducing sufficiently strong local distortions can render an SDW state energetically competitive and potentially stabilize it as the ground state. The energy difference between the SDW and AFM states depends on both the characteristic length scale and the amplitude of the local distortion. In particular, when the distorted region extends over several lattice constants and the distortion amplitude reaches the $\sim 10\permil$ level, the SDW can become the ground state.

Given the pronounced sensitivity of epitaxial RuO$_2$ thin-film magnetism to both global strain and local distortions, the strain-induced magnetic ordering mechanism revealed here has two implications. On the one hand, it provides a natural framework to rationalize the disparate experimental conclusions reported for epitaxial RuO$_2$ films under different morphologies, epitaxial conditions, and interfacial environments, and may help interpret anomalous magnetic responses in these systems. On the other hand, it suggests additional routes for controlling spin transport in epitaxial RuO$_2$ films, offering computational support for strain-based tuning and expanding the stress/strain control “knobs” available for spintronic devices. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

PRB-style English translation (with your parentheses/notation preserved)

DFT+$U$. DFT+$U$ has been examined in a large number of studies, but only a few works have discussed DFT+$U$ under strain [51], reaching conclusions close to ours. We also investigated the DFT+$U$ behavior as a function of $\varepsilon$. Figure S1(a) summarizes primitive-cell calculations for different $U$ and $\varepsilon$: regardless of the value of $\varepsilon$, when $U<1$ eV and $\varepsilon<\varepsilon_c$, the system does not undergo a transition from a nonmagnetic state to a magnetic state, indicating that these two parameters influence the magnetism in an essentially independent manner. (基于第一性原理计算的RuO2应变诱发磁有序.docx)
Figure S2(b) shows, for $\varepsilon=\varepsilon_c$, $n=16$, and $\Delta_h=20\permil$, the evolution of $\Delta E$ as well as the magnetic moments of the AFM and SDW states as $U$ increases. One finds that $U$ suppresses the emergence of the SDW to some extent. Meanwhile, the variation of the minimum AFM moment also suggests an enhanced nonlocal character of the magnetism induced by $U$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

SOC. Upon including spin–orbit coupling (SOC), the magnetic moment in RuO$_2$ is reduced, while $\varepsilon_c$ changes only weakly. This indicates that SOC does not play a major role in determining whether magnetism appears, but it does reduce the magnitude of the moment. For example, when the lattice constant $a$ is tensile/compressive distorted according to

$$a=a_{\varepsilon}(1+\Delta),\qquad b=a_{\varepsilon},\qquad c=c_{\varepsilon},$$

where $\Delta$ is the tensile ratio of $a$, our calculations at $\varepsilon=\varepsilon_c$ show that the moment obtained with SOC is reduced by nearly 20% compared with the no-SOC case [Fig. 8(a)]. This implies that, in the local-distortion model, the moments differ quantitatively once SOC is included, and the energetic penalty associated with nonmagnetic regions becomes lower. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

In SDW calculations, the SDW in RuO$_2$ originates from an energetic advantage at the node region (波节). However, SOC calculations for the locally distorted RuO$_2$ model indicate that SOC enhances the nonlocality of the magnetic response [Fig. 8(b)], such that a larger $n$ (wavelength) is required to realize a potential SDW ground state. Concretely, in AFM calculations the maximum moment is about 20% smaller than in the no-SOC case, whereas the node-region moment becomes about 50% larger. As a result, the SDW stabilization conditions become more stringent and harder to access computationally when SOC is included, although SOC does not overturn our previous discussion of AFM versus SDW in RuO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

We further perform calculations using the $\delta$-distortion, taking $n_0=n/2$ (when $n_0$ is a half-integer, only the extra half-lattice is modified). Figure 8(c) shows $\Delta E$ versus $n$ for different $\Delta_{\delta}$: even for relatively large $\Delta_{\delta}$, the minimum has not yet appeared, further supporting the conjecture that SOC enhances nonlocality. Figure 8(d) presents $\Delta E$ as a function of $\Delta_{\delta}$ for different $n$, showing trends similar to the no-SOC results, indicating that the above arguments remain valid with SOC included. (基于第一性原理计算的RuO2应变诱发磁有序.docx)
Although SOC increases the critical local distortion required for the SDW to become the ground state, this is essentially because the increased nonlocality makes the wavelength in the adopted local-distortion model not long enough to stabilize the SDW. This also suggests that the SDW wavelength obtained in the no-SOC calculations is underestimated (偏小). (基于第一性原理计算的RuO2应变诱发磁有序.docx)

小提醒（不改你原意，只是排版一致性）： 你这段里写的是 Fig. S1(a)、Fig. S2(b)，但你当前稿件里补充材料对应位置似乎标成 Fig. 7(a)、Fig. 5(b)（见你文档片段）。后续定稿时建议把补充材料图号统一一下，避免 PRB 审稿人读图时困惑。

下面是你给出的这段内容的 PRB 风格英文翻译（我顺手把 Hellman 改为更规范的 Hellmann–Feynman 拼写；另外把一些“we can see”改成更学术的表述——这些算轻微润色，我会在末尾提醒你可能需要统一的地方）。

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(1) DFT+U

DFT+U has been investigated in a substantial number of previous studies; however, only a limited number of works have addressed the DFT+U behavior under strain conditions [51], and they reported results broadly consistent with ours. Here we further examine the role of DFT+U at different values of the global strain parameter ε. Figure S1(a) summarizes unit-cell calculations as functions of U and ε. For all ε considered, when $U<1\ \text{eV}$ and $\varepsilon<\varepsilon_c$, no transition from a nonmagnetic to a magnetic state is obtained, indicating that U and ε affect the magnetism in largely independent ways. Figure S2(b) shows the U dependence of $\Delta E$ as well as the magnetic moments of the AFM and SDW states at $\varepsilon=\varepsilon_c$, $n=16$, and $\Delta_h=20\permil$. These results suggest that increasing U tends to suppress the stabilization of the SDW state. Meanwhile, the evolution of the minimum AFM moment also indicates an enhancement of the nonlocal character of magnetism induced by U.

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(2) SOC

Upon including spin–orbit coupling (SOC), the magnetic moments of RuO$_2$ are reduced, whereas the critical strain $\varepsilon_c$ remains nearly unchanged. This implies that SOC does not play a decisive role in whether magnetism emerges, but it does reduce the magnitude of the ordered moments. For example, when the lattice constant $a$ is tensilely/compressively distorted according to

$$a=a_\varepsilon(1+\Delta),\quad b=a_\varepsilon,\quad c=c_\varepsilon,$$

the moments at $\varepsilon=\varepsilon_c$ decrease by nearly 20% once SOC is included [Fig. 8(a)]. This difference implies that, within the local-distortion model, the magnetic-moment distribution is modified and the energetic penalty associated with locally nonmagnetic configurations is reduced.

In our SDW analysis, the SDW stabilization originates from an energetic advantage near the nodes. However, calculations with SOC for the locally distorted models indicate that SOC enhances the magnetic nonlocality [Fig. 8(b)], such that a larger period $n$ (i.e., a longer wavelength) is required to realize a putative SDW ground state. Concretely, in the AFM calculations the maximum Ru moment is reduced by ~20% relative to the no-SOC case, while the moment at the SDW node increases by ~50%. As a result, the conditions for stabilizing the SDW become more stringent and the computations become more demanding. Nevertheless, SOC does not overturn our qualitative conclusions regarding the competition between AFM and SDW in RuO$_2$.

We further performed SOC calculations using the δ-type distortion model with $n_0=n/2$ (for half-integer $n_0$, only the extra half cell is modified accordingly). Figure 8(c) shows $\Delta E$ as a function of $n$ for different $\Delta_\delta$. Even for relatively large $\Delta_\delta$, the minimum of $\Delta E$ is not reached within our accessible range of $n$, which provides additional support for the SOC-enhanced nonlocality picture. Figure 8(d) shows $\Delta E$ as a function of $\Delta_\delta$ for different $n$, exhibiting trends similar to the no-SOC results; therefore, the above discussion remains valid when SOC is included. Overall, SOC increases the local-distortion threshold required for the SDW to become the ground state, essentially because the enhanced nonlocality demands longer distortion wavelengths than those adopted in our practical supercell models. This also suggests that the SDW wavelengths obtained without SOC are likely underestimated.

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(3) Hellmann–Feynman force / ionic relaxation strategy

The Hellmann–Feynman forces quantify the residual forces acting on atoms. For large supercells, using $\text{EDIFFG}=-0.05$ already corresponds to a relatively tight force criterion. Given the large number of supercell calculations in this work, to reduce the number of ionic steps we adopt the following strategy: each primitive cell is first fully relaxed (EDIFFG = −0.001) and then assembled into the supercell. This procedure reduces the maximum force on O atoms from ~0.30 eV/Å to below ~0.04 eV/Å in most cases, so that additional ionic relaxation is unnecessary for the majority of supercell calculations. Because the magnetism of RuO$_2$ is relatively fragile, we further benchmarked a tighter setting with EDIFFG = −0.005 to assess the influence of ionic relaxation. The corresponding results at $n=16$ and $\varepsilon=\varepsilon_c$ are summarized in Graph 2, showing that both the magnetic moments and $\Delta E$ remain essentially unchanged. Therefore, even for magnetically fragile RuO$_2$, employing the looser criterion EDIFFG = −0.05 is still reliable within our numerical resolution.

By contrast, if one computes $\Delta E$ in the supercell without applying the above “pre-relaxed unit cell” procedure and without imposing a force threshold (EDIFFG) for the supercell relaxation, an error of about $1\text{–}5\ \mu\text{eV}$ per cell may arise, rendering the results unreliable.

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(4) Local moment distributions (for selected cases)

We provide representative distributions of the AFM local moments for both the sinusoidal distortion and the δ-type distortion at $n=9$, $\varepsilon=0.74, 0.70$, and $\Delta=20\permil, 14\permil, 8\permil$ [Fig. S4(a,b)]. We also present the moment distributions at $n=16$ and $\varepsilon=0.74$ for different $\Delta_h$ values [Fig. S4(c,d)].

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(5) Implementation for shifting the center of the compressed region

We consider $\varepsilon=\varepsilon_c$, $n=10$, and $\Delta_h=1.0\%$, using a $20\times 1\times 1$ locally distorted supercell. We introduce a tunable parameter $n_1$ and define the harmonic distortion as

$$\text{Harmonic distortion: }\quad a_i= \begin{cases} a_\varepsilon\left(1-\Delta_h\cos\left(\dfrac{(2i-1)i}{2n_1}\right)\right), & i<n_1 \\ a_\varepsilon\left(1-\Delta_h\cos\left(\dfrac{(2i-1)\theta}{2(n-n_1)}\right)\right), & n_1\le i<2n \end{cases} ,\quad b=a_\varepsilon,\quad c=c_\varepsilon .$$

This construction enables controlled translation of the compressed-lattice region by tuning $n_1$; specifically, $n_1$ corresponds to the index of the center of the compressed region.

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我对这段翻译做的“轻微修改/建议”（你可以选择是否采纳）

1. 图号/补充材料编号可能需要统一：你这里写了 Fig. S1(a)、Fig. S2(b)，但后面又出现 Fig. 7a/5b（你正文/补充材料不同版本可能混着了）。建议你最终提交前统一为 PRB 常见写法：** Fig. S1(a), Fig. S1(b)** 这种连续编号。
2. Graph 2：PRB 更常见用 Fig. Sx，不太用 Graph。你可以考虑改成 Fig. S3 或 Table S1（如果主要是数据表）。
3. 公式里 $\cos$ 的自变量：你给的最后一段公式里有一些符号（比如 $(2i-1)i/2n_1$ 和 $\theta$）看起来可能是你草稿里没来得及整理的版本；我按原样翻译保留了，但你最终最好把角度变量写成一致形式（例如 $\theta=\pi/n$ 或者明确 $\theta$ 的定义），这样更 PRB。

如果你把下一段也贴出来，我可以继续按同样风格翻译，并顺手帮你把补充材料段落的叙述逻辑调成 PRB 更常见的“现象—结论—解释”的结构。