arXiv subjects

Juraj Tekel

Publications and source records attributed to Juraj Tekel.

At least 19 recordsLinked to original sources

arXiv2026

Numerical Study of Scalar Field Theory on the Fuzzy Onion

We study the behaviour of the scalar field theory on the fuzzy onion model -- a three-dimensional matrix model consisting of concentric fuzzy spheres of gradually increasing radii. We use a numerical method of Hamiltonian Monte Carlo simulations to study the phase structure of this theory. We identify the field phases, investigate and describe a phenomenon of dynamical phase transitions and attempt to reconstruct phase transition lines. We compare the results with the well-studied phase structure of the fuzzy sphere. Finally, we identify two boundaries on the phase transitions of the theory, a uniform phase boundary and a critical boundary between the disordered and non-uniform phase.

hep-th↗

arXiv2025

Pinpointing Triple Point of Noncommutative Matrix Model with Curvature

We study a Hermitian matrix model with a quartic potential, modified by a curvature term $\mathrm{tr}(R\Phi^2)$, where $R$ is a fixed external matrix. Inspired by the truncated Heisenberg algebra formulation of the Grosse--Wulkenhaar model, this term breaks unitary invariance and, through perturbative expansion, induces an effective multitrace matrix model. We analyze the resulting action both analytically and numerically, including Hamiltonian Monte Carlo simulations, focusing on two features closely tied to renormalizability: the shift of the triple point and the suppression of the noncommutative striped phase. Our findings show that the curvature term drives the phase structure toward renormalizable behavior by removing the striped phase in the large-$N$ limit, while also unexpectedly revealing a possible novel multi-cut phase observed at the level of finite matrix size.

hep-th↗

arXiv2025

Hydrogen Atom in a Fuzzy Spherical Cavity

The fuzzy onion model formed by connecting a set of concentric fuzzy spheres of increasing radius is motivated by studies of quantum space but can also be used to study standard physics. The main feature of the model is that functions in three-dimensional space -- like scalar fields or wavefunctions -- are expressed in terms of Hermitian matrices of a certain structure. Relevant equations are then matrix equations, and some problems, such as searching for the energy spectrum for fixed quantum numbers $(l,m)$, can be expressed as an eigenvalue problem. We show how this simple approach can reproduce the results of other studies analyzing the hydrogen atom in a spherical cavity. We also test the effect of the short-distance quantum structure of the space on these solutions -- not looking for the phenomenological consequences, as the scale of quantum space is many orders below the order of the Bohr radius, but to understand the effect of quantum space in general. We observe a set of solutions without a classical counterpart which have been suggested also in a former theoretical study.

hep-th↗

arXiv2024

Cubic asymmetric multitrace matrix model

We analyze multitrace random matrix models with the help of the saddle point approximation and we introduce a multitrace term of type $-c_1c_3$ to the action. We obtain the numerical phase diagram of the model, with a stable asymmetric phase and the triple point. Furthermore, we examine response functions in this model.

hep-th↗

arXiv2024

The Fuzzy Onion: An Initial Study

In our previous contribution, we introduced a matrix formulation of a three-dimensional quantum space named the fuzzy onion. The novel part of the construction is the radial derivative term, which has been defined to recover the correct continuum limit. Here, we describe a numerical simulation of the scalar field theory in this space and test some physical properties of the model with emphasis on the interaction between neighbouring layers.

hep-th↗

arXiv2024

Fuzzy Onion as a Matrix Model

We propose a matrix model realisation of a three-dimensional quantum space. It has an onion-like structure composed of concentric fuzzy spheres of increasing radius. The angular part of the Laplace operator is inherited from that of the fuzzy sphere. The radial part is constructed using operators that relate matrices of various sizes using the matrix harmonic expansion. As an example of this approach, we produce a numerical simulation of a scalar quantum field theory, the classical heat transfer, study the quantum mechanical hydrogen atom, and consider some analytical aspects of the scalar field theory on this space.

hep-th↗

arXiv2023

Phase transitions in a $Φ^4$ matrix model on a curved noncommutative space

In this contribution, we summarize our recent studies of the phase structure of the Grosse-Wulkenhaar model and its connection to renormalizability. Its action contains a special term that couples the field to the curvature of the noncommutative background space. We first analyze the numerically obtained phase diagram of the model and its three phases: the ordered, the disordered, and the noncommutative stripe phase. Afterward, we discuss the analytical derivation of the effective action and the ordered-to-stripe transition line, and how the obtained expression successfully explains the curvature-induced shift of the triple point compared to the model without curvature. This shift also causes the removal of the stripe phase and makes the model renormalizable.

hep-th↗

arXiv2022

On quarkonium masses in 3D non-commutative space

We modify the calculation of quarkonium masses using the radial WKB and Pekeris-type approximations for the case of three-dimensional, rotationally invariant non-commutative space. We obtain corrections to the charmonium ($\text{c}\bar{\text{c}}$), bottomonium ($\text{b}\bar{\text{b}}$) and bottom-charmed meson ($\text{c}\bar{\text{b}}$) masses due to the discrete short distance structure of the space introduced by the space non-commutativity. For the fundamental length at the Planck scale we obtain relative correction at the order of $10^{-39}$, and taking into account the current experimental data, we obtain the upper bound at the order of ${10^{-18}\textrm{ m}}$ for the fundamental length scale of space.

hep-ph↗

arXiv2022

Eigenvalue-flipping Algorithm for Matrix Monte Carlo

Many physical systems can be described in terms of matrix models that we often cannot solve analytically. Fortunately, they can be studied numerically in a straightforward way. Many commonly used algorithms follow the Monte Carlo method, which is efficient for small matrix sizes but cannot guarantee ergodicity when working with large ones. In this paper, we propose an improvement of the algorithm that, for a large class of matrix models, allows to tunnel between various vacua in a proficient way, where sign change of eigenvalues is proposed externally. We test the method on two models: the pure potential matrix model and the scalar field theory on the fuzzy sphere.

hep-lat↗

arXiv2022

String modes, propagators and loops on fuzzy spaces

We present a systematic organization of functions and operators on the fuzzy 2-sphere in terms of string modes, which are optimally localized in position and momentum space. This allows to separate the semi-classical and the deep quantum regime of non-commutative quantum field theory and exhibits its nonlocal nature. This organization greatly simplifies the computation of loop contributions, avoiding oscillatory integrals and providing the effective action directly in position space. UV/IR mixing is understood as nonlocality arising from long string modes in the loops. The method is suited for any quantized symplectic space.

hep-th↗

arXiv2022

Beyond second-moment approximation in fuzzy-field-theory-like matrix models

We investigate the phase structure of a special class of multi-trace hermitian matrix models, which are candidates for the description of scalar field theory on fuzzy spaces. We include up to the fourth moment of the eigenvalue distribution into the multi-trace part of the probability distribution, which stems from the kinetic term of the field theory action. We show that by considering different multi-trace behavior in the large moment and in the small moment regimes of the model, it is possible to obtain a matrix model, which describes the numerically observed phase structure of fuzzy field theories. Including the existence of uniform order phase, triple point, and an approximately straight transition line between the uniform and non-uniform order phases.

hep-th↗

arXiv2020

Multitrace matrix models of fuzzy field theories

We review analytical approaches to scalar field theory on fuzzy spaces. We briefly outline the matrix description of these theories and describe various approximations to the relevant matrix model. We discuss the challenge of obtaining a consistent approximation that includes the higher moments of the theory.

hep-th↗

arXiv2020

Fuzzy field theories and related matrix models

We review the description of scalar field theories on fuzzy spaces by Hermitian random matrix models. After reminding the reader of the relevant aspects of the random matrix theory and construction of the fuzzy spaces, we summarize the most important results for the scalar fields on such spaces. We then introduce the multi-trace matrix models relevant for the analytical description of scalar field theories on fuzzy spaces and show to what extent they do, and to what extent they do not, describe the know phase structure of $ϕ^4$ theory on the fuzzy sphere.

hep-th↗

arXiv2020

Second moment fuzzy-field-theory-like matrix models

We solve a multitrace matrix model approximating the real quartic scalar field theory on the fuzzy sphere and obtain its phase diagram. We generalize this method to models with modified kinetic terms and demonstrate its use by investigating models related to the removal of the UV/IR mixing. We show that for the fuzzy sphere a modification of the kinetic part of the action by higher derivative term can change the phase diagram of the theory such that the triple point moves further from the origin.

hep-th↗

arXiv2018

Asymmetric hermitian matrix models and fuzzy field theory

We analyze two types of hermitian matrix models with asymmetric solutions. One type breaks the symmetry explicitly with an asymmetric quartic potential. We give the phase diagram of this model with two different phase transitions between the one cut and two cut solutions. The second type, describing real scalar field theory on fuzzy spaces, breaks the symmetry spontaneously with multitrace terms. We present two methods to study this model, one direct and one using a connection with the first type of models. We analyze the model for the fuzzy sphere and obtain a phase diagram with the location of the triple point in a good agreement with the most recent numerical simulations.

hep-th↗

arXiv2018

Matrix Models of Fuzzy Field Theories

We briefly review the connection between the fuzzy field theories and matrix models and describe the main features of the models that appear. We summarize the different approaches to their analysis, some of the recent results and the challenges to be addressed in the future.

hep-th↗

arXiv2016

Phase diagram of scalar field theory on fuzzy sphere and multitrace matrix models

We study the phase diagram of the scalar field theory on the fuzzy sphere described as a particular multitrace matrix model. We consider perturbative and nonperturbative terms in the kinetic term effective action and describe consequences for the asymmetric regime, the free energy and the location of the triple point within the approximation.

hep-th↗

arXiv2016

Matrix model approximations of fuzzy scalar field theories and their phase diagrams

We present an analysis of two different approximations to the scalar field theory on the fuzzy sphere, a nonperturbative and a perturbative one, which are both multitrace matrix models. We show that the former reproduces a phase diagram with correct features in a qualitative agreement with the previous numerical studies and that the latter gives a phase diagram with features not expected in the phase diagram of the field theory.

hep-th↗