Localization Tensor Revisited: Geometric-Probabilistic Foundations and a Structure-Factor Criterion under Periodic Boundaries

• URL: http://arxiv.org/abs/2602.14779v1
• Date: Mon, 16 Feb 2026 14:29:57 GMT
• Title: Localization Tensor Revisited: Geometric-Probabilistic Foundations and a Structure-Factor Criterion under Periodic Boundaries
• Abstract summary: We revisit the localization tensor (LT) from geometric and probabilistic perspectives.<n>We identify the diagonal part, Cpp, as a "localization function" C(p), whose small-momentum behavior determines the LT in the thermodynamic limit.<n>We show that the finite-momentum behavior of C(p), together with an inverse participation ratio (IPR)-based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization.
• Score: 2.5694382720775306
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density-density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frechet mean) on the circle, and (ii) a metric-free one based on the mutual information I, which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, Cpp, as a "localization function" C(p), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C(p), together with an inverse participation ratio (IPR)-based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su-Schrieffer-Heeger and Aubry-Andre models, with and without interactions, and suggest that structure factor-based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.

Related papers

• Characterizing Heterogeneous Rates in Finite Mixture Estimation via Partial Optimal Transport [88.55403215843967]
  Voronoi-based partial optimal transport (VPOT) for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure.<n>Results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components.
  arXiv  Detail & Related papers  (2026-09-15T04:32:53Z)
• Entanglement Quantification via Symmetric Extensions: A Resource Theory Hierarchy [3.662037602213926]
  We introduce a hierarchy of entanglement measures Ek based on k-symmetric PPT extensions.<n>Each Ek, defined via a minimal eigenvalue shift, is faithful, convex, and monotone under free operations.<n>The hierarchy strictly refines PPT-robustness at k = 1, detects bound entanglement at k = 2, and converges exactly to the separability measure as k -> infinity.
  arXiv  Detail & Related papers  (2026-07-18T20:39:35Z)
• Inducing Comparability of Factorised Probability Distributions [4.622328826769001]
  We propose an extension scheme for any two given models and establish the formal foundation: Unmatched components are completed using conditionally uniform extensions.<n>This preserves the probabilistic semantics while enabling the application of well-defined distributional discrepancy measures.
  arXiv  Detail & Related papers  (2026-06-23T08:33:06Z)
• A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning [52.452902154360565]
  We introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning.<n>MCBP explicitly models the intrinsic curvature of the data manifold and computes pointwise mean curvature without requiring principled manifold parametrization.<n>Experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance.
  arXiv  Detail & Related papers  (2026-05-05T20:19:09Z)
• Intrinsic Pointer Basis and Irreversible Classicality from Coherence Contraction [0.0]
  We analyze the irreversible emergence of classical behavior from a reduced quantum description via a canonical intrinsic decomposition of the density operator.<n>We prove that the quadratic coherence functional is a Lyapunov functional under pure-dephasing or interaction-picture evolution.<n>This yields a canonical state-dependent operational classicality criterion via the normalized cohesion index.
  arXiv  Detail & Related papers  (2026-04-25T13:41:22Z)
• Conformal Robust Set Estimation [41.99844472131922]
  We propose a robust conformal method based on a non-conformity score defined as the half-mass radius around a point.<n>We show that the resulting conformal regions are marginally valid for any sample size and converge in probability to a robust population central set.
  arXiv  Detail & Related papers  (2026-04-20T15:59:23Z)
• Geometry-Aware Probabilistic Circuits via Voronoi Tessellations [12.790945282316763]
  Probabilistic circuits (PCs) enable exact and tractable inference but employ data independent mixture weights that limit their ability to capture local geometry of the data manifold.<n>We propose Voronoi tessellations (VT) as a natural way to incorporate geometric structure directly into the sum nodes of a PC.<n>We formalize this incompatibility and develop two complementary solutions: (1) an approximate inference framework that provides guaranteed lower and upper bounds for inference, and (2) a structural condition for VT under which exact tractable inference is recovered.
  arXiv  Detail & Related papers  (2026-03-12T13:56:35Z)
• On the Structural Non-Preservation of Epistemic Behaviour under Policy Transformation [51.56484100374058]
  We formalise such information-conditioned interaction patterns as behavioural dependency.<n>This induces a probe-relative notion of $$-behavioural equivalence and a within-policy behavioural distance.<n>Results identify structural conditions under which probe-conditioned behavioural separation is not preserved under common policy transformations.
  arXiv  Detail & Related papers  (2026-02-24T22:55:21Z)
• Latent Object Permanence: Topological Phase Transitions, Free-Energy Principles, and Renormalization Group Flows in Deep Transformer Manifolds [0.5729426778193398]
  We study the emergence of multi-step reasoning in deep Transformer language models through a geometric and statistical-physics lens.<n>We formalize the forward pass as a discrete coarse-graining map and relate the appearance of stable "concept basins" to fixed points of this renormalization-like dynamics.<n>The resulting low-entropy regime is characterized by a spectral tail collapse and by the formation of transient, reusable object-like structures in representation space.
  arXiv  Detail & Related papers  (2026-01-16T23:11:02Z)
• From Tail Universality to Bernstein-von Mises: A Unified Statistical Theory of Semi-Implicit Variational Inference [0.12183405753834557]
  Semi-implicit variational inference (SIVI) constructs approximate posteriors of the form $q() = int k(| z) r(dz)$<n>This paper develops a unified "approximation-optimization-statistics'' theory for such families.
  arXiv  Detail & Related papers  (2025-12-05T19:26:25Z)
• Continuum Model of Isospectrally Patterned Lattices [0.0]
  We derive a continuum analogue of the IPL which allows us to determine analytically its eigenvalue spectrum and eigenstates.<n>This continuum model breaks chiral symmetry but still shows a pairing of partner states with positive and negative energies except for the ground state.
  arXiv  Detail & Related papers  (2025-10-06T06:18:04Z)
• Graph-based Clustering Revisited: A Relaxation of Kernel $k$-Means Perspective [73.18641268511318]
  We propose a graph-based clustering algorithm that only relaxes the orthonormal constraint to derive clustering results.<n>To ensure a doubly constraint into a gradient, we transform the non-negative constraint into a class probability parameter.
  arXiv  Detail & Related papers  (2025-09-23T09:14:39Z)
• CP$^2$: Leveraging Geometry for Conformal Prediction via Canonicalization [51.716834831684004]
  We study the problem of conformal prediction (CP) under geometric data shifts.<n>We propose integrating geometric information--such as geometric pose--into the conformal procedure to reinstate its guarantees.
  arXiv  Detail & Related papers  (2025-06-19T10:12:02Z)
• A New Theoretical Perspective on Data Heterogeneity in Federated Optimization [39.75009345804017]
  In federated learning (FL), data heterogeneity is the main reason that existing theoretical analyses are pessimistic about the convergence rate. In particular, for many FL algorithms, the convergence rate grows dramatically when the number of local updates becomes large. This paper aims to bridge this gap between theoretical understanding and practical performance by providing a theoretical analysis from a new perspective.
  arXiv  Detail & Related papers  (2024-07-22T11:52:58Z)
• Grounding Continuous Representations in Geometry: Equivariant Neural Fields [26.567143650213225]
  We propose a novel CNF architecture which uses a geometry-informed cross-attention to condition the NeF on a geometric variable.<n>We show that this approach induces a steerability property by which both field and latent are grounded in geometry.<n>We validate these main properties in a range of tasks including classification, segmentation, forecasting, reconstruction and generative modelling.
  arXiv  Detail & Related papers  (2024-06-09T12:16:30Z)
• Beyond the Edge of Stability via Two-step Gradient Updates [49.03389279816152]
  Gradient Descent (GD) is a powerful workhorse of modern machine learning. GD's ability to find local minimisers is only guaranteed for losses with Lipschitz gradients. This work focuses on simple, yet representative, learning problems via analysis of two-step gradient updates.
  arXiv  Detail & Related papers  (2022-06-08T21:32:50Z)
• Semi-Discrete Normalizing Flows through Differentiable Tessellation [31.474420819149724]
  We propose a tessellation-based approach that learns quantization boundaries on a continuous space, complete with exact likelihood evaluations. This is done through constructing normalizing flows on convex polytopes parameterized through a differentiable Voronoi tessellation. We show improvements over existing methods across a range of structured data modalities, and find that we can achieve a significant gain from just adding Voronoi mixtures to a baseline model.
  arXiv  Detail & Related papers  (2022-03-14T03:06:31Z)
• Entropic trust region for densest crystallographic symmetry group packings [0.8399688944263843]
  Molecular crystal structure prediction seeks the most stable periodic structure given the chemical composition of a molecule and pressure-temperature conditions. Modern CSP solvers use global optimization methods to search for structures with minimal free energy within a complex energy landscape induced by intermolecular potentials. We propose a class of periodic packings restricted to crystallographic symmetry groups (CSG) and design a search method for the densest CSG packings in an information-geometric framework.
  arXiv  Detail & Related papers  (2022-02-24T08:39:07Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Federated Functional Gradient Boosting [75.06942944563572]
  We study functional minimization in Federated Learning. For both FFGB.C and FFGB.L, the radii of convergence shrink to zero as the feature distributions become more homogeneous.
  arXiv  Detail & Related papers  (2021-03-11T21:49:19Z)
• Localisation in quasiperiodic chains: a theory based on convergence of local propagators [68.8204255655161]
  We present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators. Analysing the convergence of these continued fractions, localisation or its absence can be determined, yielding in turn the critical points and mobility edges. Results are exemplified by analysing the theory for three quasiperiodic models covering a range of behaviour.
  arXiv  Detail & Related papers  (2021-02-18T16:19:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.