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显示 2025年07月18日， 星期五 新的列表
- [1] arXiv:2507.13013 [中文pdf, pdf, html, 其他]
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标题： 流形上的Levy拉普拉斯算子和微分形式的热流标题： Levy Laplacian on manifold and heat flows of differential forms评论： 18页主题： 数学物理 (math-ph) ; 微分几何 (math.DG) ; 泛函分析 (math.FA)
勒维拉普拉斯算子是一个无限维微分算子，由于其与杨-米尔斯规范场的联系而引人注目。 文章证明了在黎曼流形上的$H^1$-路径流形上，勒维拉普拉斯的各种定义之间的等价性。 考虑了带有勒维拉普拉斯的热方程。 研究了当时间趋于无穷时，该热方程的一些解趋向于局部常数泛函的趋势。 这些解是通过紧致黎曼流形上微分形式的热流构造的。
The Levy Laplacian is an infinite-dimensional differential operator, which is interesting for its connection with the Yang-Mills gauge fields. The article proves the equivalence of various definitions of the Levy Laplacian on the manifold of $H^1$-paths on a Riemannian manifold. The heat equation with the Levy Laplacian is considered. The tendency of some solutions of this heat equation to the locally constant functionals as time tends to infinity is studied. These solutions are constructed using heat flows of differential forms on the compact Riemannian manifold.
- [2] arXiv:2507.13104 [中文pdf, pdf, html, 其他]
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标题： 从冻结中得到的椭圆长程自旋链的模族标题： Modular families of elliptic long-range spin chains from freezing评论： v1:23+5页，1图主题： 数学物理 (math-ph) ; 高能物理 - 理论 (hep-th) ; 量子代数 (math.QA) ; 精确可解与可积系统 (nlin.SI)
我们考虑通过“冻结”具有自旋的可积量子多体系统来构建具有q-变形长程相互作用的量子可积自旋链。 输入是一个自旋-Ruijsenaars系统以及其无自旋经典Ruijsenaars-Schneider系统的平衡配置。 对于一个特定的平衡选择，得到的长程自旋链具有实数谱并允许短程极限，从而提供从最近邻到长程相互作用自旋的可积插值。 我们关注椭圆情况。 我们首先定义模群在无自旋椭圆Ruijsenaars-Schneider系统上的作用，以表明对于固定的椭圆参数，它有一整个模族的经典平衡配置。 这些通常具有常数但非零动量。 然后我们使用变形量子化的框架，在任何经典平衡下冻结椭圆自旋-Ruijsenaars系统，同时保持量子可积性。 正如我们在之前的工作中所展示的，结果包括Heisenberg、Inozemtsev和Haldane-Shastry链及其xxz类似q-变形（面型），或Fukui-Kawakami的反对周期Haldane-Shastry链、Sechin-Zotov的椭圆推广，以及Matushko-Zotov的完全各向异性q-变形（顶点型）。
We consider the construction of quantum-integrable spin chains with q-deformed long-range interactions by `freezing' integrable quantum many-body systems with spins. The input is a spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars-Schneider system. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbour to long-range interacting spins. We focus on the elliptic case. We first define an action of the modular group on the spinless elliptic Ruijsenaars-Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantisation to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium whilst preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane-Shastry chains along with their xxz-like q-deformations (face-type), or the antiperiodic Haldane-Shastry chain of Fukui-Kawakami, its elliptic generalisation of Sechin-Zotov, and their completely anisotropic q-deformations due to Matushko-Zotov (vertex type).
- [3] arXiv:2507.13154 [中文pdf, pdf, html, 其他]
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标题： 拉格朗日乘数法在LCA群和无限量子自旋系统中的应用标题： The Hudson theorem in LCA groups and infinite quantum spin systems评论： 28页主题： 数学物理 (math-ph) ; 泛函分析 (math.FA) ; 量子物理 (quant-ph)
著名的Hudson定理指出，在$\mathbb{R}^d$中，高斯函数是唯一其Wigner分布处处为正的函数。 受量子信息理论的启发，D. Gross在阿贝尔群$\mathbb{Z}_d^n$上证明了一个类似的结果，当$d$为奇数时——对应于一个由$n$个qudit组成的系统——表明只有所谓的稳定化态的Wigner分布是非负的。 将这一结果扩展到有限维系统的热力学极限，自然地引导我们考虑具有紧开子群的一般$2$-正则LCA群，其中Wigner分布的非负性问题目前仍是一个开放问题。 我们通过证明如果映射$x\mapsto 2x$是保测的，则Wigner分布非负的函数恰好是二次的子特征，除了平移和乘以常数之外。 相反，如果上述映射不是保测的，Wigner分布总是会取负值。 我们详细讨论了离散群的无限和以及紧群的无限积的特殊情况，这恰好对应于无限量子自旋系统。 进一步的例子包括$n$-进系统，其中$n\geq 2$是一个任意整数（不一定是素数），以及挠群。
The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group $\mathbb{Z}_d^n$, for $d$ odd - corresponding to a system of $n$ qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general $2$-regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map $x\mapsto 2x$ is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include $n$-adic systems, where $n\geq 2$ is an arbitrary integer (not necessarily a prime), as well as solenoid groups.
- [4] arXiv:2507.13321 [中文pdf, pdf, html, 其他]
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标题： 有限扩展费米子系统中间隙相内的自同构等价标题： Automorphic equivalence within gapped phases of infinitely extended fermion systems评论： 21页主题： 数学物理 (math-ph) ; 量子物理 (quant-ph)
我们证明了在无限扩展的格点费米子系统（以及自旋系统）的有隙相中存在自同构等价性（automorphic equivalence），这些系统具有超多项式衰减的相互作用。 作为简单的应用，我们证明了这类系统的Goldstone定理的一个版本：如果无限体积的相互作用在连续对称性下保持不变，那么任何有隙基态也将在该对称性下保持不变。
We prove automorphic equivalence within gapped phases of infinitely extended lattice fermion systems (as well as spin systems) with super-polynomially decaying interactions. As a simple application, we prove a version of Goldstone's theorem for such systems: if an infinite volume interaction is invariant under a continuous symmetry, then any gapped ground state is also invariant under that symmetry.
新提交 (展示 4 之 4 条目 )
- [5] arXiv:2507.12525 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]
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标题： 边界共形流形上的高阶结构：高阶贝里相位与边界共形场论标题： Higher Structures on Boundary Conformal Manifolds: Higher Berry Phase and Boundary Conformal Field Theory评论： 63页主题： 高能物理 - 理论 (hep-th) ; 强关联电子 (cond-mat.str-el) ; 数学物理 (math-ph)
我们在(1+1)d共形场理论(CFT)的共形边界条件空间中引入了更高贝里连接和曲率的概念，这些边界条件通过精确边缘变形相互关联，形成一个"边界共形流形"。 我们的定义建立在之前关于张量网络的研究基础上，例如矩阵乘积态(MPS)，其中三重内积或多波函数重叠在几何上起关键作用。 一方面，我们对更高贝里相位的边界共形场论(BCFT)表述为研究凝聚态系统中可逆相族提供了一种新的分析工具。 另一方面，它揭示了共形边界条件模空间中的新几何结构，超越了通过扎莫洛德奇科夫度规定义的通常黎曼结构。 当边界共形流形被解释为D膜的位置模空间时，我们的更高贝里连接与弦理论中的NS-NS$B$-场一致。 一般定义不需要这种解释，并且纯粹以场论方式表述，涉及边界条件改变(bcc)算符的相关函数。 我们还探讨了更高贝里连接与边界共形流形环空间中的泛函贝里连接之间的关系。
We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in (1+1)d conformal field theories (CFT), related to each other by exactly marginal boundary deformations, forming a "boundary conformal manifold." Our definition builds upon previous works on tensor networks, such as matrix product states (MPS), where the triple inner product or multi-wavefunction overlap plays the key geometric role. On the one hand, our boundary conformal field theory (BCFT) formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the NS-NS $B$-field in string theory. The general definition does not require such an interpretation and is formulated purely field-theoretically, in terms of correlation functions of boundary-condition-changing (bcc) operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.
- [6] arXiv:2507.12546 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]
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标题： 从高阶贝里相位的角度看共形边界条件的空间：参数化BCFT中的贝里曲率流标题： Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs评论： 11页主题： 高能物理 - 理论 (hep-th) ; 强关联电子 (cond-mat.str-el) ; 数学物理 (math-ph) ; 量子物理 (quant-ph)
在本工作中，我们研究两个主题之间的联系：边界共形场理论（BCFTs）中的共形边界条件空间与由更高贝里相位表征的体系统空间。 我们通过分析带有连续参数化共形边界条件的狄拉克费米子BCFT中的多参数谱流来探索这一联系，这些边界条件是通过将共形场理论（CFT）耦合到一族体系统引入的。 当体系统属于非平凡的更高贝里类时，相关的共形边界条件会引发普通贝里曲率的流动，在BCFT的福克空间中产生陈数泵。 这一现象是单维参数化体系统中贝里曲率流动的BCFT对应物，其中流动发生在实空间中。 基于这种对应关系，我们在BCFT框架内引入了更高贝里曲率和更高贝里不变量的概念。 我们的结果为研究共形边界态和体基态族的拓扑性质提供了新的视角：如果一族体态属于非平凡的更高贝里类，则相应的纠缠哈密顿量表现出携带贝里曲率的多参数谱流。
In this work, we study the connection between two subjects: the space of conformal boundary conditions in boundary conformal field theories (BCFTs) and the space of gapped systems characterized by higher Berry phases. We explore this connection by analyzing multi-parameter spectral flow in Dirac fermion BCFTs with continuously parametrized conformal boundary conditions, which are introduced by coupling a CFT to a family of gapped systems. When the gapped systems belong to a nontrivial higher Berry class, the associated conformal boundary conditions induce a flow of the ordinary Berry curvature, resulting in a Chern number pump in the Fock space of the BCFT. This phenomenon is the BCFT analog of Berry curvature flow in one-dimensional parametrized gapped systems, where the flow occurs in real space. Building on this correspondence, we introduce the notions of higher Berry curvature and higher Berry invariants within the BCFT framework. Our results provide a new perspective for studying the topological properties of families of conformal boundary states and gapped ground states: if a family of gapped states belongs to a nontrivial higher Berry class, then the corresponding entanglement Hamiltonians exhibit a multi-parameter spectral flow that carries Berry curvature in the Fock space.
- [7] arXiv:2507.12627 (交叉列表自 math.AP) [中文pdf, pdf, html, 其他]
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标题： 非线性Hartree方程的量子散射态的半经典极限标题： Semi-classical limit of quantum scattering states for the nonlinear Hartree equation评论： 40页主题： 偏微分方程分析 (math.AP) ; 数学物理 (math-ph)
本文研究了半经典条件下量子粒子的长时间动力学。 首先，我们证明对于具有短程相互作用势的非线性Hartree方程，小数据解满足色散界限并且它们发生散射，其中小性条件和界限与表示约化普朗克常数的小参数$\hbar\in(0,1]$无关。 然后，取半经典极限$\hbar\to0$，我们证明此类量子散射态的Wigner变换弱-*收敛到Vlasov方程对应的经典散射态。 作为直接结果，我们在不假设初始数据正则性的前提下，建立了Vlasov方程的小数据散射。 我们的分析基于自由薛定谔流的新统一色散估计，该估计简单但至关重要，可用于包含奇异相互作用势，如逆幂律势$\frac{1}{|x|^a}$其中$1<a<\frac{5}{3}$。
This article concerns the long-time dynamics of quantum particles in the semi-classical regime. First, we show that for the nonlinear Hartree equation with short-range interaction potential, small-data solutions obey dispersion bounds and they scatter, where the smallness conditions and the bounds are independent of the small parameter $\hbar\in(0,1]$ representing the reduced Planck constant. Then, taking the semi-classical limit $\hbar\to0$, we prove that the Wigner transforms of such quantum scattering states converge weakly-* to the corresponding classical scattering states for the Vlasov equation. As a direct consequence, we establish small-data scattering for the Vlasov equation without assuming regularity on initial data. Our analysis is based on a new uniform dispersion estimate for the free Schr\"odinger flow, which is simple but crucial to include singular interaction potentials such as inverse power-law potential $\frac{1}{|x|^a}$ with $1<a<\frac{5}{3}$.
- [8] arXiv:2507.12662 (交叉列表自 cond-mat.stat-mech) [中文pdf, pdf, html, 其他]
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标题： 在阿基米德晶格上行走：来自布洛赫能带理论的见解标题： Walking on Archimedean Lattices: Insights from Bloch Band Theory评论： 27页主题： 统计力学 (cond-mat.stat-mech) ; 中尺度与纳米尺度物理 (cond-mat.mes-hall) ; 强关联电子 (cond-mat.str-el) ; 数学物理 (math-ph)
返回的格点行走是起始于给定格点并在$n$步后返回同一格点的移动序列。 确定给定长度$n$的返回行走总数是一个典型的图论问题，与统计物理和凝聚态物理中的格点模型有关。 我们通过与布洛赫能带理论建立联系，推导出十一种二维阿基米德格点上的返回行走数的解析表达式。 我们通过一种替代方法对结果进行验证，该方法依赖于计算大型图的邻接矩阵的矩，我们详细解释了这些图的构造。 作为凝聚态物理的应用，我们使用公式计算阿基米德格点上紧束缚模型的状态密度。 虽然阿基米德格点提供了足够丰富的结构，并在此处为了具体性而选择，但我们的技术可以直接推广到其他二维或更高维的欧几里得格点。
Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands. We benchmark our results through an alternative method that relies on computing the moments of adjacency matrices of large graphs, whose construction we explain explicitly. As a condensed matter physics application, we use our formulas to compute the density of states of tight-binding models on the Archimedean lattices. While the Archimedean lattices provide a sufficiently rich structure and are chosen here for concreteness, our techniques can be generalized straightforwardly to other two- or higher-dimensional Euclidean lattices.
- [9] arXiv:2507.13093 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]
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标题： 自旋-自旋关联函数在二维伊辛自由费米子量子场论中，由低快速度阈值的费米子数测量加权的渐近行为标题： Asymptotics of spin-spin correlators weighted by fermion number measurements with low rapidity threshold in the 2D Ising free-fermion QFT评论： 33页主题： 高能物理 - 理论 (hep-th) ; 高能物理 - 现象学 (hep-ph) ; 数学物理 (math-ph)
在本工作中，我们研究了在低快速度阈值$Y$之上的大量费米子的平均数量，这是在欧几里得距离$r$处自旋-自旋两点关联函数的因子展开的基础，在二维伊辛量子场论的自由质量费米子点处。 尽管处于壳外自由性，自旋算符仍然远离高斯型，它们在渐近态中以复杂的关联产生粒子。 我们展示了如何将数量可观测量仍纳入可积的双曲正弦-戈登/皮亚诺勒三阶框架，并由两个变量的线性微分方程控制$(r,Y)$。 我们展示了微分方程以及在$r\rightarrow 0$，$e^{Y}r={\cal O}(1)$缩放极限中出现的两个关键缩放函数的信息，如何结合以完全确定可观测量在$r$小值下的渐近行为，在$\lambda$扩展形式中。 另一方面，缩放函数通过直接求和指数形式因子展开进行分析，推广了传统的伊辛连接计算。 我们仔细展示了奇异点如何在物理值极限$\lambda \pi \rightarrow 1$中抵消，以及如何对在这个值处坍缩的幂修正进行重求和。 特别是，我们展示了对于物理的$\lambda$值，标度函数与伊辛 CFT 中的积分四点函数相关，并且在标度极限下继续控制数量可观测量的渐近行为，直到${\cal O}(r^3)$。
In the work, we study the averaged number of massive fermions above a low rapidity threshold $Y$, underlying the form-factor expansions of the spin-spin two-point correlators at an Euclidean distance $r$, in the 2D Ising QFT at the free massive fermion point. Despite the on-shell freeness, the spin operators are still far away from being Gaussian, and create particles in the asymptotic states with complicated correlations. We show how the number observables can still be incorporated into the integrable Sinh-Gordon/Painleve-III framework and controlled by linear differential equations with two variables $(r,Y)$. We show how the differential equations and the information of two crucial scaling functions arising in the $r\rightarrow 0$, $e^{Y}r={\cal O}(1)$ scaling limit, can be combined to fully determine the small-$r$ asymptotics of the observables, in the $\lambda$-extended form. The scaling functions, on the other hand, are analyzed by summing the exponential form-factor expansions directly, generalizing the traditional Ising connecting computations. We show carefully, how the singularities cancel in the physical value limit $\lambda \pi \rightarrow 1$ and how the power-corrections that collapse at this value can be resummed. In particular, we show for the physical $\lambda$-value, the scaling functions are related to an integrated four-point function in the Ising CFT and continue to control the asymptotics of the number-observables in the scaling limit up to ${\cal O}(r^3)$.
- [10] arXiv:2507.13134 (交叉列表自 math.AG) [中文pdf, pdf, html, 其他]
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标题： 代数几何中的时间-能量原理标题： The Time-Energy Principle in Algebraic Geometry评论： 16页主题： 代数几何 (math.AG) ; 数学物理 (math-ph)
我们考虑量子力学中的时间-能量不确定性原理，并在其堆栈的上下文中提供其代数几何解释。
We consider the time-energy uncertainty principle from Quantum Mechanics and provide its Algebro-Geometric interpretation within the context of stacks.
- [11] arXiv:2507.13219 (交叉列表自 math.AG) [中文pdf, pdf, 其他]
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标题： 顶点函数对于弓簇及其镜像对称标题： Vertex functions for bow varieties and their Mirror Symmetry评论： 56页。欢迎提出评论！主题： 代数几何 (math.AG) ; 数学物理 (math-ph) ; 量子代数 (math.QA) ; 表示理论 (math.RT)
在本文中，我们研究有限类型$A$箱形变体的顶点函数。 顶点函数是$K$理论中$I$函数的类似物，3d 镜像对称预测，一个变体及其 3d 镜像对偶的顶点函数所满足的$q$差分方程在变量替换后是相同的，该变量替换交换了各个参数的角色。 因此，顶点函数由椭圆函数矩阵相关联，该矩阵预期为 M. Aganagic 和 A. Okounkov 的椭圆稳定包络。 我们证明了所有这些陈述。 我们证明的策略是将其简化为余切丛的完全旗流形的情况，对于这种情况，$q$差分方程可以明确地与 Macdonald 差分方程相联系。 这种简化的关键要素，具有独立兴趣，涉及将部分旗流形的余切丛的顶点函数与“更细”的旗流形的顶点函数相关联。 我们的公式涉及将某些凯勒参数（也称为诺维科夫参数）特化到顶点函数的奇点。 在$\hbar \to \infty$极限下，这一陈述预计会退化为关于旗流形$I$函数的类似结果。
In this paper, we study the vertex functions of finite type $A$ bow varieties. Vertex functions are $K$-theoretic analogs of $I$-functions, and 3d mirror symmetry predicts that the $q$-difference equations satisfied by the vertex functions of a variety and its 3d mirror dual are the same after a change of variable swapping the roles of the various parameters. Thus the vertex functions are related by a matrix of elliptic functions, which is expected to be the elliptic stable envelope of M. Aganagic and A. Okounkov. We prove all of these statements. The strategy of our proof is to reduce to the case of cotangent bundles of complete flag varieties, for which the $q$-difference equations can be explicitly identified with Macdonald difference equations. A key ingredient in this reduction, of independent interest, involves relating vertex functions of the cotangent bundle of a partial flag variety with those of a ``finer" flag variety. Our formula involves specializing certain K\"ahler parameters (also called Novikov parameters) to singularities of the vertex functions. In the $\hbar \to \infty$ limit, this statement is expected to degenerate to an analogous result about $I$-functions of flag varieties.
- [12] arXiv:2507.13269 (交叉列表自 math.PR) [中文pdf, pdf, 其他]
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标题： 二维热核关于$\sqrt{8/3}$-Liouville Brownian 运动的界限标题： Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion评论： 133页，8图主题： 概率 (math.PR) ; 数学物理 (math-ph) ; 复变量 (math.CV)
利乌维尔布朗运动（LBM）是在利乌维尔量子引力（LQG）表面上的典型扩散过程。 在本工作中，我们建立了当$\gamma=\sqrt{8/3}$时，LBM 的热核关于$\sqrt{8/3}$-LQG 度量的上下界，这些界在指数中的多项对数因子范围内是精确的。
Liouville Brownian motion (LBM) is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $\gamma=\sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential.
- [13] arXiv:2507.13342 (交叉列表自 cond-mat.stat-mech) [中文pdf, pdf, 其他]
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标题： 扩展在有限连通性模型中产生自旋玻璃序：来自LDPC码理论的严格且直观的方法标题： Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes评论： 27+28页，16+12图主题： 统计力学 (cond-mat.stat-mech) ; 无序系统与神经网络 (cond-mat.dis-nn) ; 数学物理 (math-ph)
具有许多由大势垒分隔的局部最小值的复杂自由能景观被认为是在各种物理系统中导致玻璃行为的原因。 这是与自旋玻璃中的副本对称性破缺（RSB）相关的一种启发式图景，但RSB仅在某些具有全连接性的平均场模型中被严格验证过。 在本工作中，我们为有限连通性、非欧几里得扩展图上的局部相互作用模型族提供了有限温度自旋玻璃序的严格证明。 为此，我们完全绕过了RSB形式主义，而是利用了这些模型与某些低密度奇偶校验（LDPC）码之间的数学等价性。 我们使用了码扩展性，这是LDPC码的一个属性，它保证了在基态周围存在广泛的能量势垒。 结合一些额外的温和假设，这使我们能够将低温吉布斯态显式地分解为不相交的组件，每个组件都包含一个与景观局部最小值相关的渐近长寿命状态。 每个组件最多携带总权重的指数级小部分，且几乎所有组件都不包含基态——我们将这些一起定义为自旋玻璃序。 该证明是基本的，并以相同的方式处理各种扩展图拓扑，包括现有方法如空腔法失效的短环图。 我们的结果严格适用于足够大的p值的稀释p自旋玻璃，尽管尚未证明，但我们预计我们的假设也适用于更广泛的码族。 受此启发，我们数值研究了两个简单的模型，在随机正则图和双曲空间的正则镶嵌上进行研究。 我们表明，这两个模型随着温度的变化经历了两次转变，分别对应弱遍历性破缺和自旋玻璃序的出现。
Complex free-energy landscapes with many local minima separated by large barriers are believed to underlie glassy behavior across diverse physical systems. This is the heuristic picture associated with replica symmetry breaking (RSB) in spin glasses, but RSB has only been rigorously verified for certain mean-field models with all-to-all connectivity. In this work, we give a rigorous proof of finite temperature spin glass order for a family of models with local interactions on finite-connectivity, non-Euclidean expander graphs. To this end, we bypass the RSB formalism entirely, and instead exploit the mathematical equivalence of such models to certain low-density parity check (LDPC) codes. We use code expansion, a property of LDPC codes which guarantees extensive energy barriers around ground states. Together with mild additional assumptions, this allows us to construct an explicit decomposition of the low-temperature Gibbs state into disjoint components, each hosting an asymptotically long-lived state associated with a local minimum of the landscape. Each component carries at most an exponentially small fraction of the total weight, and almost all components do not contain ground states -- which we take together to define spin-glass order. The proof is elementary, and treats various expanding graph topologies on the same footing, including those with short loops where existing approaches such as the cavity method fail. Our results apply rigorously to diluted p-spin glasses for sufficiently large p, and while unproven, we also expect our assumptions to hold in a broader family of codes. Motivated by this, we numerically study two simple models, on random regular graphs and a regular tesselation of hyperbolic space. We show that both models undergo two transitions as a function of temperature, corresponding to the onset of weak ergodicity breaking and spin glass order, respectively.
交叉提交 (展示 9 之 9 条目 )
- [14] arXiv:2407.17918 (替换) [中文pdf, pdf, html, 其他]
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标题： 有界域中具有非零散度的电场的矢量层析成像标题： Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains评论： 在脑电源成像中的应用主题： 数学物理 (math-ph)
在向量断层扫描（VT）中，目标是使用线积分数据来重建未知的多维向量场。 在二维VT的情况下，通常需要两种类型的线积分数据。 这些数据对应于向量场沿积分线的平行和垂直投影的积分。 VT方法是非侵入性的、非干扰性的，并且比传统的点测量提供更多的场信息；它们通常用于重建无散度（或无源）的速度和流场。 在本文中，我们表明VT也可以用于重建具有非零散度的场。 特别是，我们研究了由偶极子源在有界域中产生的电场，这例如出现在脑电图（EEG）源成像中。 据我们所知，VT以前未被用于重建此类场。 我们详细解释了理论背景，电场逆问题的推导以及线积分的数值近似。 我们表明，借助从横向测量和向量拉普拉斯算子构造的两个稀疏性约束，可以利用纵向测量重建具有非零散度的场。 与EEG源成像相比，我们注意到VT不需要对源进行数学建模。 通过数值模拟，我们展示了可以使用VT正确估计电场的模式，并且可以从场的重构幅度中准确确定源活动的位置。
In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along integration lines. VT methods are non-invasive, non-intrusive and offer more information on the field than classical point measurements; they are typically used to reconstruct divergence-free (or source-free) velocity and flow fields. In this paper, we show that VT can also be used for the reconstruction of fields with non-zero divergence. In particular, we study electric fields generated by dipole sources in bounded domains which arise, for example, in electroencephalography (EEG) source imaging. To the best of our knowledge, VT has not previously been used to reconstruct such fields. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergence can be reconstructed from the longitudinal measurements with the help of two sparsity constraints that are constructed from the transverse measurements and the vector Laplace operator. As a comparison to EEG source imaging, we note that VT does not require mathematical modelling of the sources. By numerical simulations, we show that the pattern of the electric field can be correctly estimated using VT and the location of the source activity can be determined accurately from the reconstructed magnitudes of the field.
- [15] arXiv:2112.09705 (替换) [中文pdf, pdf, html, 其他]
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标题： 精确的Schur指标闭式表达式标题： The exact Schur index in closed form评论： 65页；v2：小的澄清，参考文献添加；v3：更正了拼写错误；v4：更正了拼写错误主题： 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph)
四维N=2超共形场理论的超共形指标的Schur极限包含了该理论受保护谱的丰富物理信息。 对于拉格朗日模型，此指标的极限可以通过多变量椭圆函数的围道积分来计算。 然而，令人惊讶的是，到目前为止，它尚未能以封闭的解析形式进行精确求值。 在本文中，我们提出了一种基本的方法，通过利用被积函数的椭圆性，来解决这些积分中的大部分。 我们的结果表现为一个有限和（乘积）的形式，这些和是广泛研究过的带味道的Eisenstein级数。 特别是，我们推导出所有类型a1的S类理论的完全带味道的Schur指标的简洁公式，提出了所有具有规范群SU(N)的N=4超杨-Mills理论的无味道Schur指标的猜想，并给出了各种低秩规范理论的指标的显式表达式。 我们还讨论了对非拉格朗日理论的应用、模性质和缺陷指标。
The Schur limit of the superconformal index of a four-dimensional N = 2 superconformal field theory encodes rich physical information about the protected spectrum of the theory. For a Lagrangian model, this limit of the index can be computed by a contour integral of a multivariate elliptic function. However, surprisingly, so far it has eluded exact evaluation in closed, analytical form. In this paper we propose an elementary approach to bring to heel a large class of these integrals by exploiting the ellipticity of their integrand. Our results take the form of a finite sum of (products of) the well-studied flavored Eisenstein series. In particular, we derive a compact formula for the fully flavored Schur index of all theories of class S of type a1, we put forward a conjecture for the unflavored Schur indices of all N=4 super Yang-Mills theories with gauge group SU(N), and we present closed-form expressions for the index of various other gauge theories of low ranks. We also discuss applications to non-Lagrangian theories, modular properties, and defect indices.
- [16] arXiv:2402.05105 (替换) [中文pdf, pdf, html, 其他]
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标题： 关于无扭率 torsional Newton--Cartan 几何中的规范变换标题： On gauge transformations in twistless torsional Newton--Cartan geometry评论： 15+3页（正文，参考文献）。v2：扩展的结果，扩展的讨论，新增参考文献主题： 广义相对论与量子宇宙学 (gr-qc) ; 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph)
扭曲无关的扭转纽顿-卡坦（TTNC）几何有两种变体，类型I和类型II，它们的区别在于其规范变换。 在TTNC几何中存在一个特定的局部伽利略不变函数，在现有文献中被赋予了不同的名称，我们将其称为“局部伽利略不变势”。 我们证明，在两种类型的TTNC几何中，总存在一个局部规范变换，将局部伽利略不变势变为零。 对于类型I TTNC几何，这是由于对应于规范参数的方程形式为哈密顿-雅可比方程。 在类型II TTNC几何的情况下，我们执行次级空间微分同胚。 在这两种情况下，我们的论证严格证明了在几何场仅具有有限阶可微性的情况下，各自规范变换的存在性。 这改进了文献中典型的“规范固定”论证，这些论证需要解析性。 我们考虑了我们结果的两个应用。 首先，它推广了标准纽顿-卡坦几何中的一个经典结果。 其次，它允许以两种新方式（局部地）参数化TTNC几何：一种是通过空间度量和一个单位类时向量场，另一种是通过类空向量的分布和一个正定余度量。
Twistless torsional Newton--Cartan (TTNC) geometry exists in two variants, type I and type II, which differ by their gauge transformations. In TTNC geometry there exists a specific locally Galilei-invariant function, called by different names in existing literature, that we dub the `locally Galilei-invariant potential'. We show that in both types of TTNC geometry, there always exists a local gauge transformation that transforms the locally Galilei-invariant potential to zero. For type I TTNC geometry, we achieve this due to the corresponding equation for the gauge parameter taking the form of a Hamilton--Jacobi equation. In the case of type II TTNC geometry, we perform subleading spatial diffeomorphisms. In both cases, our arguments rigorously establish the existence of the respective gauge transformation also in case of only finite-degree differentiability of the geometric fields. This improves upon typical arguments for `gauge fixing' in the literature, which need analyticity. We consider two applications of our result. First, it generalises a classical result in standard Newton--Cartan geometry. Second, it allows to (locally) parametrise TTNC geometry in two new ways: either in terms of just the space metric and a unit timelike vector field, or in terms of the distribution of spacelike vectors and a positive-definite cometric.
- [17] arXiv:2407.07970 (替换) [中文pdf, pdf, html, 其他]
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标题： 阿贝尔-普里姆映射在存在额外对合情况下的逆标题： Inversion of the Abel--Prym map in presence of an additional involution评论： 17页主题： 代数几何 (math.AG) ; 数学物理 (math-ph)
与黎曼曲面对称幂到其雅可比簇的阿贝尔映射不同，阿贝尔-普里姆映射通常不能通过与雅可比逆问题相关的传统技术及其主要组成部分，即黎曼消去定理来逆转。这是因为相应的黎曼消去定理的对应物给出的点数是普里姆流形维度的两倍。然而，如果黎曼曲面有一个与定义普里姆流形的对合交换的第二个对合，并且满足某种附加条件，则可以定义一个雅可比逆问题的类似物，并用普里姆θ函数来表示。我们提出了这些条件，并将满足它们的对合对称为第一类对合对。我们给出了对合对成为第一类对合对的必要条件，并给出了一系列具有此类对合对的曲线的例子，主要是Hitchin系统的谱曲线，以及Kovalewski系统的谱曲线。
Unlike Abel map of the symmetric power of a Riemann surface onto its Jacobian, the Abel--Prym map generically can not be reversed by means of conventional technique related to the Jacobi inversion problem, and of its main ingredient, namely the Riemann vanishing theorem. It happens because the corresponding analog of the Riemann vanishing theorem gives twice as many points as the dimension of the Prym variety. However, if the Riemann surface has a second involution commuting with the one defining the Prym variety and satisfying a certain additional condition, an analog of the Jacobi inversion can be defined, and expressed in terms of the Prym theta function. We formulate these conditions and refer to the pairs of involutions satisfying them as to pairs of the first type. We formulate necessary conditions for the pair of involutions to be a pair of the first type, and give a series of examples of curves with such pairs of involutions, mainly spectral curves of Hitchin systems, and also a spectral curve of the Kovalewski system.
- [18] arXiv:2407.13327 (替换) [中文pdf, pdf, 其他]
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标题： 关于相干态的熵和复杂性标题： On entropy and complexity of coherent states评论： 14页。重写以包含$SL(d+1,\C)$的最一般情况。K\"ahler势的变形方程对于$d>1$数值求解。即将发表在《几何与物理杂志》上。主题： 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph) ; 辛几何 (math.SG)
熵和复杂度的亲缘关系通过群$SL(d+1,\C)$的相干态示例被指出。 两者都是从对应于 Fubini-Study 度量的球面基础几何的 Kähler 位势获得的。 熵被证明等于用对偶辛变量表示的 Kähler 位势，即环状流形的 Guillemin 位势。 关联两个状态的复杂度的对数被证明等于 Calabi 的 diastasis 函数。 通过考虑其变形，指示了 Fubini-Study 度量的最优性。
Consanguinity of entropy and complexity is pointed out through the example of coherent states of the group $SL(d+1,\C)$. Both are obtained from the K\"ahler potential of the underlying geometry of the sphere corresponding to the Fubini-Study metric. Entropy is shown to be equal to the K\"ahler potential written in terms of dual symplectic variables as the Guillemin potential for toric manifolds. The logarithm of complexity relating two states is shown to be equal to Calabi's diastasis function. Optimality of the Fubini-Study metric is indicated by considering its deformation.
- [19] arXiv:2412.06533 (替换) [中文pdf, pdf, html, 其他]
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标题： 常磁场下磁 Dirichlet Laplacian 特征值的数值优化标题： Numerical Optimization of Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field评论： 29页，13图，2表；更新至发表版本期刊参考： J. Math. Phys. 66 (7): 072104 (2025)主题： 优化与控制 (math.OC) ; 数学物理 (math-ph) ; 偏微分方程分析 (math.AP) ; 谱理论 (math.SP)
我们为磁 Dirichlet Laplacian 在不同强度的恒定磁场下的前七个特征值提供了数值最小化器。 通过适应 Antunes 和 Freitas 的方法，我们在最小化过程中使用梯度下降法，并结合基本解方法进行特征值计算。 值得注意的是，当磁通量超过目标特征值的索引时，最小化器始终是一个圆盘。
We present numerical minimizers for the first seven eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field in a wide range of field strengths. Adapting an approach by Antunes and Freitas, we use gradient descent for the minimization procedure together with the Method of Fundamental solutions for eigenvalue computation. Remarkably, we observe that when the magnetic flux exceeds the index of the target eigenvalue, the minimizer is always a disk.
- [20] arXiv:2502.09510 (替换) [中文pdf, pdf, html, 其他]
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标题： 具有阶数为n的赫米特函数的Gabor系统，以及过采样大于n+1但不是框架的情况标题： Gabor systems with Hermite functions of order n and oversampling greater than n+1 which are not frames评论： 版本2包含扩展的引言和一个包含开放问题和新例子的章节。16页，7图。将发表于采样理论、信号处理和数据分析主题： 泛函分析 (math.FA) ; 数学物理 (math-ph)
我们证明了对于格点上的Gabor系统使用Hermite函数的足够密度条件在一般情况下并不充分。 这是从一个关于Zak变换零点如何确定整数过采样Gabor系统的框架性质的结果中得出的。
We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak transform determine the frame property of integer over-sampled Gabor systems.
- [21] arXiv:2505.01290 (替换) [中文pdf, pdf, html, 其他]
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标题： $η$正则化和泛函测度标题： $η$ regularisation and the functional measure评论： 32页主题： 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph) ; 量子物理 (quant-ph)
在本文中，我们重新审视Fujikawa的手征异常的路径积分表述，并开发了一个广义框架，以系统地定义正则化的函数测度。 这一构造将$\eta$正则化方案扩展到算子语言，使谱不对称性和测度变换之间的联系完全明确。 在从正则化测度恢复Fujikawa的手征异常表达式之前，我们探讨了与异常相关的未定义谱和背后更深层次的数论结构，通过平滑渐近的视角对其进行解释。 我们的方法统一了两种互补的观点：Fujikawa的分析正则化和Atiyah-Singer指标定理给出的拓扑表征。 我们进一步研究了测度在正则化尺度变化下的变换，并推导出一个函数$\iota_E(\Lambda)$，该函数编码这种依赖性，展示了其Mellin矩如何支配发散性的出现。 最后，我们评论了正则化测度、$\eta$正则化和广义Schwinger固有时形式之间的概念关系，特别关注二维Schwinger模型。
In this paper, we revisit Fujikawa's path integral formulation of the chiral anomaly and develop a generalised framework for systematically defining a regularised functional measure. This construction extends the $\eta$ regularisation scheme to operator language, making the connection between spectral asymmetry and measure transformation fully explicit. Before recovering Fujikawa's expression for the chiral anomaly from the regularised measure, we explore the deeper number-theoretic structure underlying the ill-defined spectral sum associated with the anomaly, interpreting it through the lens of smoothed asymptotics. Our approach unifies two complementary perspectives: the analytic regularisation of Fujikawa and the topological characterisation given by the Atiyah-Singer index theorem. We further investigate how the measure transforms under changes to the regularisation scale and derive a function $\iota_E(\Lambda)$ that encodes this dependence, showing how its Mellin moments govern the appearance of divergences. Finally, we comment on the conceptual relationship between the regularised measure, $\eta$ regularisation, and the generalised Schwinger proper-time formalism, with a particular focus on the two-dimensional Schwinger model.
- [22] arXiv:2506.16164 (替换) [中文pdf, pdf, 其他]
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标题： 卡罗尔镜像标题： The Carrollian Kaleidoscope评论： 特邀评论，222页，包含多个图表；v2：224页，添加了评论和参考文献主题： 高能物理 - 理论 (hep-th) ; 强关联电子 (cond-mat.str-el) ; 广义相对论与量子宇宙学 (gr-qc) ; 数学物理 (math-ph) ; 核理论 (nucl-th)
卡罗尔群是在光速趋于零的庞加莱群极限中出现的，并最初被当作一种数学奇观而被忽视。然而，最近的发展证明了这一点并非如此。卡罗尔对称性和共形卡罗尔对称性现在无处不在，从凝聚态物理到量子引力，出现在各种物理现象中。本综述旨在为读者提供进入这一快速发展的领域的入门途径。在介绍了相关对称性的基础知识后，我们详细阐述了卡罗尔场论和卡罗尔共形场论（CCFT）的构建。然后，我们关注应用。其中最流行的应用是通过一个一维余维的对偶CCFT来构建渐近平坦时空（AFS）的全息理论。我们回顾了AFS$_3$/CCFT$_2$的早期工作，然后深入分析了4D AFS的对偶构建。另外两组重要的应用是在流体力学和凝聚态物理中，我们对此进行了详细讨论。卡罗尔流体力学首先作为相对论流体力学的$c\to 0$极限引入，然后从基于对称性的方法中重建。我们讨论了与超相对论流动的关系以及与夸克-胶子等离子体的联系，并给出了Bjorken流和Gubser流模型的具体例子。在凝聚态应用中，我们涵盖了与分形子、平带以及Luttinger液体模型中的相分离的联系。最后，我们简要概述了其他感兴趣的主题，包括弦理论和黑洞视界。
The Carroll group arises in the vanishing speed of light limit of the Poincar\'{e} group and was initially discarded as just a mathematical curiosity. However, recent developments have proved otherwise. Carroll and conformal Carroll symmetries are now ubiquitous, appearing in diverse physical phenomena starting from condensed matter physics to quantum gravity. This review aims to provide the reader a gateway into this fast-developing field. After an introduction and setting the stage with basics of the symmetry in question, we detail the construction of Carrollian and Carrollian Conformal field theories (CCFT). We then focus on applications. By far the most popular of these applications is in the context of the construction of holography in asymptotically flat spacetimes (AFS) in terms of a co-dimension one dual CCFT. We review the early work on AFS$_3$ /CCFT$_2$ before delving into an in-depth analysis for the construction of the dual to 4D AFS. Two other important sets of applications are in hydrodynamics and in condensed matter physics, which we discuss in detail. Carroll hydrodynamics is introduced as the $c\to 0$ limit of relativistic hydrodynamics first and then reconstructed from a symmetry based approach. Relations to ultrarelativistic flows and connections to the quark-gluon plasma are discussed with concrete examples of the Bjorken and Gubser flow models. In condensed matter applications, we cover connections to fractons, flat bands, and phase separation in Luttinger liquid models. To conclude, we give very brief outlines of other topics of interest including string theory and black hole horizons.