1D self-similar fractals with centro-symmetric Jacobians: asymptotics and modular data

• URL: http://arxiv.org/abs/2401.15515v1
• Date: Sat, 27 Jan 2024 22:07:58 GMT
• Title: 1D self-similar fractals with centro-symmetric Jacobians: asymptotics and modular data
• Authors: Radhakrishnan Balu
• Abstract summary: We establishs of growing one dimensional self-similar fractal graphs, they are networks that allow multiple edges weighted between nodes. An additional structure is endowed with the repeating units of centro-symmetric Jacobians in the adjacency of a linear graph creating a self-similar fractal.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We establish asymptotics of growing one dimensional self-similar fractal graphs, they are networks that allow multiple weighted edges between nodes, in terms of quantum central limit theorems for algebraic probability spaces in pure state. An additional structure is endowed with the repeating units of centro-symmetric Jacobians in the adjacency of a linear graph creating a self-similar fractal. The family of fractals induced by centro-symmetric Jacobians formulated as orthogonal polynomials that satisfy three term recurrence relations support such limits. The construction proceeds with the interacting fock spaces, T-algebras endowed with a quantum probability space, corresponding to the Jacobi coefficients of the recurrence relations and when some elements of the centro-symmetric matrix are constrained in a specific way we obtain, as the same Jacobian structure is repeated, the central limits. The generic formulation of Leonard pairs that form bases of conformal blocks and probablistic laplacians used in physics provide choice of centro-symmetric Jacobians widening the applicability of the result. We establish that the T-algebras of these 1D fractals, as they form a special class of distance-regular graphs, are thin and the induced association schemes are self-duals that lead to anyon systems with modular invariance.

Related papers

• Fractals and spontaneous symmetry breaking with type-B Goldstone modes: a perspective from entanglement [0.0]
  The one-dimensional spin-$s$ $rm SU(2)$ ferromagnetic Heisenberg model, as a paradigmatic example for spontaneous symmetry breaking ( SSB) with type-B Goldstone modes (GMs), is expected to exhibit an abstract fractal underlying the ground state subspace. This intrinsic abstract fractal is here revealed from a systematic investigation into the entanglement entropy for a linear combination of factorized (unentangled) ground states on a fractal decomposable into a set of the Cantor sets. Our argument may be extended to any quantum many-body systems undergoing SSB with type-B GM
  arXiv  Detail & Related papers  (2024-07-25T10:24:11Z)
• Equivalence of dynamics of disordered quantum ensembles and semi-infinite lattices [44.99833362998488]
  We develop a formalism for mapping the exact dynamics of an ensemble of disordered quantum systems onto the dynamics of a single particle propagating along a semi-infinite lattice. This mapping provides a geometric interpretation on the loss of coherence when averaging over the ensemble and allows computation of the exact dynamics of the entire disordered ensemble in a single simulation.
  arXiv  Detail & Related papers  (2024-06-25T18:13:38Z)
• Wigner's Theorem for stabilizer states and quantum designs [0.6374763930914523]
  We describe the symmetry group of the stabilizer polytope for any number $n$ of systems and any prime local dimension $d$. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. We extend an observation of Heinrich and Gross and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments.
  arXiv  Detail & Related papers  (2024-05-27T18:00:13Z)
• Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
  We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
  arXiv  Detail & Related papers  (2024-02-26T06:23:39Z)
• Digitizing lattice gauge theories in the magnetic basis: reducing the breaking of the fundamental commutation relations [0.0]
  We present a digitization scheme for the lattice $mathrmSU(2)$ gauge theory Hamiltonian in the $mathitmagnetic$ $mathitbasis$, where the gauge links are unitary and diagonal. The digitization is obtained from a particular partitioning of the $mathrmSU(2)$ group manifold, with the canonical momenta constructed by an approximation of the Lie derivatives on this partitioning.
  arXiv  Detail & Related papers  (2023-11-20T17:11:20Z)
• A Unified Framework for Discovering Discrete Symmetries [17.687122467264487]
  We consider the problem of learning a function respecting a symmetry from among a class of symmetries. We develop a unified framework that enables symmetry discovery across a broad range of subgroups.
  arXiv  Detail & Related papers  (2023-09-06T10:41:30Z)
• Geometric Neural Diffusion Processes [55.891428654434634]
  We extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We show that with these conditions, the generative functional model admits the same symmetry.
  arXiv  Detail & Related papers  (2023-07-11T16:51:38Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation [64.49871502193477]
  We propose a novel semi-supervised subspace clustering method, which is able to simultaneously augment the initial supervisory information and construct a discriminative affinity matrix. Comprehensive experimental results on six commonly-used benchmark datasets demonstrate the superiority of our method over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-05-21T01:47:17Z)
• One-dimensional symmetric phases protected by frieze symmetries [0.0]
  We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions.
  arXiv  Detail & Related papers  (2022-02-25T18:41:26Z)
• Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds [70.32518963244466]
  A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant.
  arXiv  Detail & Related papers  (2021-06-10T19:54:19Z)

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.