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Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)

Numerical Study of the Coulomb Problem in the Fuzzy Onion Model

Samuel Kováčik, Juraj Tekel, Patrik Rusnák

Published Dec 9, 2025• 0 citations
Publisher

Abstract

The fuzzy onion model is a three-dimensional quantum space model expressed in terms of Hermitian matrices of a certain form. It is a convenient model as it is manageable and can be easily put on a computer. It can also be related to various physical contexts, capturing the essence of the quantum structure of space. In this work, we briefly summarise the construction of the model and then show how it can be used for the effects of quantum space on spherically symmetrical problems, the Coulomb one, for example. A possible connection to the earlier stages of the universe is discussed.

DOI
10.22323/1.490.0313

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The fuzzy onion model is a three-dimensional quantum space model expressed in terms of Hermitian matrices of a certain form. It is a convenient model as it is manageable and can be easily put on a computer. It can also be related to various physical contexts, capturing the essence of the quantum structure of space. In this work, we briefly summarise the construction of the model and then show how it can be used for the effects of quantum space on spherically symmetrical problems, the Coulomb one, for example. A possible connection to the earlier stages of the universe is discussed.
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Continuous gauge theories, because of their bosonic degrees of freedom, have an infinite-dimensional local Hilbert space. Encoding these degrees of freedom on qubit-based hardware demands some sort of “qubitization” scheme, where one approximates the behavior of a theory while using only finitely many degrees of freedom. We propose a novel qubitization strategy for gauge theories, called “fuzzy gauge theory,” building on the success of the fuzzy σ -model in earlier work. We provide arguments that the fuzzy gauge theory lies in the same universality class as regular gauge theory, in which case its use would obviate the need of any further limit besides the usual spatial continuum limit. Furthermore, we demonstrate that these models are relatively resource-efficient for quantum simulations. Published by the American Physical Society 2024
The three-dimensional cubic conformal field theory governs the critical behavior of Heisenberg magnets with cubic anisotropy. Studying this theory nonperturbatively is challenging because its most easily accessible observables are numerically very close to those of the more symmetric O ( 3 ) model. In this work, we overcome this difficulty using the fuzzy sphere regularization method. By adding a cubic-invariant two-body interaction to the quantum rotor Hamiltonian used for the O ( 3 ) model, we break the continuous rotational symmetry by construction and unambiguously isolate the cubic critical point. Using exact diagonalization and the density matrix renormalization group, we calculate the scaling dimensions of several key operators, including the leading scalar singlets, and resolve the splitting of the O ( 3 ) rank-two traceless symmetric tensor into the E g and T 2 g representations of the cubic group. Our results are consistent with existing Monte Carlo, conformal perturbation theory, and ɛ expansion benchmarks, demonstrating the power of the fuzzy sphere in resolving closely spaced universality classes.
A set of generalized coherent states for the one-dimensional Coulomb problem in coordinate representation is constructed. At first, we obtain a mapping for proper transformation of the one-dimensional Coulomb problem into a nonrotating four-dimensional isotropic harmonic oscillator in the hyperspherical space, and the generalized coherent states for the one-dimensional Coulomb problem is then obtained in exact closed form. This exactly soluble model can provide an adequate means for a quantum coherency description of the Coulomb problem in one dimension, sample for coherent aspects of the exciton model in one-dimension example in high-temperature superconductivity, semiconductors, and polymers. Also, it can be useful for investigating the coherent scattering of the Coulomb particles in one dimension.
The concept of fuzzy qubit is proposed as an approach to emulating quantum interference using fuzzy logic. This makes it possible to implement certain quantum computing algorithms, especially quantum associative memory, on hardware dedicated to fuzzy logic. The fuzzy qubit model is briefly reviewed and the fuzzy implementation of basic 1 and 2 qubit gates, such as Hadamard or C-not gates, is presented. The possibility of emulating quantum algorithms is discussed. Then quantum associative memory is presented and its implementation using the fuzzy qubit model is discussed. The advantages over the previous associative memory models, both quantum and classical, are discussed.