Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model

We construct and analyze the phase diagram of a self-interacting matrix field in two dimensions coupled to the curvature of the non-commutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar's. The curvature term proves crucial for the diagram's structure: when turned off, the triple point collapses into the origin as matrices grow larger; when turned on, the triple point recedes from the origin proportionally to the coupling strength and the matrix size. The coupling attenuation that turns the Grosse-Wulkenhaar model into a renormalizable version of the $\phi^4_\star$-model cannot stop the triple point recession. As a result, the stripe phase escapes to infinity, removing the problems with UV/IR mixing

One-loop structure of the U(1) gauge model on the truncated Heisenberg space

We calculate divergent one-loop corrections to the propagators of the U(1) gauge theory on the truncated Heisenberg space, which is one of the extensions of the Grosse-Wulkenhaar model. The model is purely geometric, based on the Yang-Mills action; the corresponding gauge-fixed theory is BRST invariant. We quantize perturbatively and, along with the usual wave-function and mass renormalizations, we find divergent nonlocal terms of the $\Box^{-1}$ and $\Box^{-2}$ type. We discuss the meaning of these terms and possible improvements of the model.Comment: 29 page

Fazni prelazi u matričnim modelima na modifikovanom Hajzenbergovom prostoru

In this dissertation, we study a self-interacting Hermitian matrix field in two dimensions coupled to the curvature of the noncommutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar’s. We inspect the connection between the model’s curvature term, UV/IR mixing, and renormalizability. The model is numerically simulated using the Hybrid Monte Carlo method. In order to obtain the nontrivial phase structure, we first vary the scalings of the action term parameters and inspect the transition line stability under the change of matrix size. After we fix the scalings, we proceed to construct the phase diagrams and find their large matrix size limits. As a result, we establish the presence of the three phases previously found in other matrix models — the ordered, the disordered, and a purely noncommutative striped phase. The curvature term proves crucial for the diagram’s structure: when turned off, the triple point collapses into the origin as matrices grow larger; when turned on, the triple point recedes from the origin proportionally to the coupling strength and the matrix size. We use both the field eigenvalue distribution approach and a bound on the action to predict the position of the transition lines. Their simulated curvature-induced shift convincingly agrees with our analytical results. We found that the coupling attenuation that turns the Grosse-Wulkenhaar model into a renormalizable version of the λφ4 ?-model cannot stop the triple point recession. As a result, the stripe phase escapes to infinity, removing the problems with UV/IR mixing and explaining the success of the Grosse-Wulkenhaar model.U ovoj disertaciji, izučavamo samointeragujuće hermitsko matrično polje na koga deluje krivina nekomutativnog modifikovanog Hajzenbergovog prostora. U limesu beskonačnih matrica, ovaj model se svodi na renormalizabilni Grose-Vulkenharov. Cilj je da se ispita veza između člana sa krivinom, UV/IR mešanja i renormalizabilnosti modela. Numeričkoj simulaciji modela smo pristupili Hibridnim Monte Karlo metodom. Radi do- bijanja netrivijalne strukture faznog dijagrama, prvo variramo skaliranje parametara članova u dejstvu i ispitujemo stabilnost linija faznih prelaza pri promeni veličine matrica. Nakon što smo fiksirali skaliranje, konstruišemo fazne dijagrame i nalazimo njihove limese. Na ovaj način smo utvrdili pristustvo tri faze prethodno detektovane kod drugih matričnih modela — uređene, neuređene i čisto nekomutativne trakaste faze. Član sa krivinom se pokazao presudnim po strukturu dijagrama: kada je uključen, trojna tačka modela kolapsira u koordinatni početak prostora parametara s povećanjem formata matrica; kada je isključen, trojna tačka se udaljava od koordinatnog početka srazmerno parametru krivine i veličini matrice. Za predviđanje položaja linija faznih prelaza, koristili smo metod raspodela svojstvenih vrednosti polja kao i procenjivanje granica na vrednosti samog dejstva. Simulirane vrednosti ovog krivinom izazvanog pomeranja se ubedljivo slažu sa našim analitičkim rezultatima. Brizina isključivanja parametra krivine koje pretvara Grose-Vulkenharov model u re- normalizabilnu verziju λφ4 ? modela je nedovoljna da zaustavi udaljavanje trojne tačke od koordinatnog početka. Posledica toga je da trakasta faza nestaje u beskonačnosti, rešavajući problem UV/IR mešanja, čime smo objasnili uspešnost Grose-Vulkenharovog modela

Detecting scaling in phase transitions on the truncated Heisenberg algebra

We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. The model reduces to the renormalizable Grosse-Wulkenhaar model in an infinite matrix size limit and exhibits a purely non-commutative non-uniformly ordered phase. Particular attention is given to scaling of model’s parameters. We additionally provide the infinite matrix size limit for the disordered to ordered phase transition line

Friction and Stiffness Dependent Dynamics of Accumulation Landslides with Delayed Failure

We propose a new model for landslide dynamics under the assumption of a delay failure mechanism. Delay failure is simulated as a delayed interaction between adjacent blocks, which mimics the relationship between the accumulation and feeder part of the accumulation slope. The conducted research consisted of three phases. Firstly, the real observed movements of the landslide were examined to exclude the existence or the statistically significant presence of background noise. Secondly, we propose a new mechanical model of an accumulation landslide dynamics, with introduced delay failure, and with variable friction law. Results obtained indicate the onset of a transition from an equilibrium state to an oscillatory regime if delayed failure is assumed for different cases of slope stiffness and state of homogeneity/heterogeneity of the slope. At the end, we examine the influence of different frictional properties (along the sliding surface) on the conditions for the onset of instability. Results obtained indicate that the increase of friction parameters leads to stabilization of sliding for homogeneous geological environment. Moreover, increase of certain friction parameters leads to the occurrence of irregular aperiodic behavior, which could be ascribed to the regime of fast irregular sliding along the slope

Supplementary material for: Prekrat, D., Todorović-Vasović, K. N., Vasović, N.,& Kostić, S.. (2024). Complex global dynamics of conditionally stable slopes: effect of initial conditions. in Frontiers in Earth Science Frontiers Media., 12 - 2024 https://doi.org/10.3389/feart.2024.1374942

In the present paper, we investigate the effect of the initial conditions on the dynamics of the spring-block landslide model. The time evolution of the studied model, which is governed by a system of stochastic delay differential equations, is analyzed in the mean-field approximation, which qualitatively exhibits the same dynamics as the initial model. The results of the numerical analysis show that changing the initial conditions has different effects in different parts of the parameter space of the model. Namely, moving away from the fixed-point initial conditions has a stabilizing effect on the dynamics when the noise, the friction parameters a (higher values) and c as well as the spring stiffness k are taken into account. The stabilization manifests itself in a complete suppression of the unstable dynamics or a partial limitation of the effect of some friction parameters. On the other hand, the destabilizing effect of changing the initial conditions occurs for the lower values of the friction parameters a and for b. The main feature of destabilization is the complete suppression of the sliding regime or a larger parameter range with a transient oscillatory regime. Our approach underlines the importance of analyzing the influence of initial conditions on landslide dynamics.Supplementary material for: [https://doi.org/10.3389/feart.2024.1374942]Related to published version: [https://farfar.pharmacy.bg.ac.rs/handle/123456789/5561

Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics

In the present paper, we propose a new model for landslide dynamics, in the form of the spring-block mechanical model, with included delayed interaction and the effect of the background seismic noise. The introduction of the random noise in the model of landslide dynamics is confirmed by the surrogate data testing of the recorded ambient noise within the existing landslide in Serbia. The performed research classified the analyzed recordings as linear stationary stochastic processes with Gaussian inputs. The proposed mechanical model is described in the form of a nonlinear dynamical system: a set of stochastic delay-differential equations. The solution of such a system is enabled by the introduction of mean-field approximation, which resulted in a mean-field approximated model whose dynamics are qualitatively the same as the dynamics of the starting stochastic system. The dynamics of the approximated model are analyzed numerically, with rather unexpected results, implying the positive effect of background noise on landslide dynamics. Particularly, the increase of the noise intensity requires higher values of spring stiffness and displacement delay for the occurrence of bifurcation. This confirms the positive stabilizing effect of the increase in noise intensity on the dynamics of the analyzed landslide model. Present research confirms the significant role of noise in landslides near the bifurcation point (e.g., creeping landslides)

Towards removal of striped phase in matrix model description of fuzzy field theories

The UV/IR-mixing phenomenon of non-commutative field theories is manifested by the existence of a non-local, striped phase in the scalar field theory, where the field does not oscillate around the same value in the whole space. We will consider modifications of the standard "kinetic term plus potential" actions for Hermitian matrix models which describe theories free of the UV/IR-mixing and discuss the expected receding of the striped phase in the phase diagram. We will present results for the case of the modified theory on the fuzzy sphere and for the truncated Heisenberg algebra formulation of the Grosse-Wulkenhaar model on the plane.Corfu Summer Institute 2022 "22nd Hellenic School and Workshops on Elementary Particle Physics and Gravity", August 28th to October 1st, 2022, Corfu, Greec

Fazni prelazi u matričnim modelima na modifikovanom Hajzenbergovom prostoru

In this dissertation, we study a self-interacting Hermitian matrix field in two dimensions coupled to the curvature of the noncommutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar’s. We inspect the connection between the model’s curvature term, UV/IR mixing, and renormalizability. The model is numerically simulated using the Hybrid Monte Carlo method. In order to obtain the nontrivial phase structure, we first vary the scalings of the action term parameters and inspect the transition line stability under the change of matrix size. After we fix the scalings, we proceed to construct the phase diagrams and find their large matrix size limits. As a result, we establish the presence of the three phases previously found in other matrix models — the ordered, the disordered, and a purely noncommutative striped phase. The curvature term proves crucial for the diagram’s structure: when turned off, the triple point collapses into the origin as matrices grow larger; when turned on, the triple point recedes from the origin proportionally to the coupling strength and the matrix size. We use both the field eigenvalue distribution approach and a bound on the action to predict the position of the transition lines. Their simulated curvature-induced shift convincingly agrees with our analytical results. We found that the coupling attenuation that turns the Grosse-Wulkenhaar model into a renormalizable version of the λφ4 ?-model cannot stop the triple point recession. As a result, the stripe phase escapes to infinity, removing the problems with UV/IR mixing and explaining the success of the Grosse-Wulkenhaar model.U ovoj disertaciji, izučavamo samointeragujuće hermitsko matrično polje na koga deluje krivina nekomutativnog modifikovanog Hajzenbergovog prostora. U limesu beskonačnih matrica, ovaj model se svodi na renormalizabilni Grose-Vulkenharov. Cilj je da se ispita veza između člana sa krivinom, UV/IR mešanja i renormalizabilnosti modela. Numeričkoj simulaciji modela smo pristupili Hibridnim Monte Karlo metodom. Radi do- bijanja netrivijalne strukture faznog dijagrama, prvo variramo skaliranje parametara članova u dejstvu i ispitujemo stabilnost linija faznih prelaza pri promeni veličine matrica. Nakon što smo fiksirali skaliranje, konstruišemo fazne dijagrame i nalazimo njihove limese. Na ovaj način smo utvrdili pristustvo tri faze prethodno detektovane kod drugih matričnih modela — uređene, neuređene i čisto nekomutativne trakaste faze. Član sa krivinom se pokazao presudnim po strukturu dijagrama: kada je uključen, trojna tačka modela kolapsira u koordinatni početak prostora parametara s povećanjem formata matrica; kada je isključen, trojna tačka se udaljava od koordinatnog početka srazmerno parametru krivine i veličini matrice. Za predviđanje položaja linija faznih prelaza, koristili smo metod raspodela svojstvenih vrednosti polja kao i procenjivanje granica na vrednosti samog dejstva. Simulirane vrednosti ovog krivinom izazvanog pomeranja se ubedljivo slažu sa našim analitičkim rezultatima. Brizina isključivanja parametra krivine koje pretvara Grose-Vulkenharov model u re- normalizabilnu verziju λφ4 ? modela je nedovoljna da zaustavi udaljavanje trojne tačke od koordinatnog početka. Posledica toga je da trakasta faza nestaje u beskonačnosti, rešavajući problem UV/IR mešanja, čime smo objasnili uspešnost Grose-Vulkenharovog modela