Title
Phase transitions in matrix models on the truncated Heisenberg space: doctoral dissertation
Creator
Prekrat, Dragan, 1982-
CONOR:
81012489
Copyright date
2023
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Autorstvo 3.0 Srbija (CC BY 3.0)
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Language
Serbian
Cobiss-ID
Theses Type
Doktorska disertacija
description
Datum odbrane: 03.03.2023.
Other responsibilities
Academic Expertise
Prirodno-matematičke nauke
Academic Title
-
University
Univerzitet u Beogradu
Faculty
Fizički fakultet
Alternative title
Fazni prelazi u matričnim modelima na modifikovanom Hajzenbergovom prostoru
Publisher
[D. Prekrat]
Format
103 str.
Abstract (en)
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.
Abstract (sr)
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.
Authors Key words
Noncommutative geometry • Grosse-Wulkenhaar model • Matrix models
Phase transitions • Renormalizability • Monte Carlo simulations
Authors Key words
Nekomutativna geometrija • Grose-Vulkenharov model • Matrični modeli
Fazni prelazi • Renormalizabilnost • Monte Karlo simulacije
Classification
530.122.3(043.3)
544.015.4:530.23(043.3)
Type
Tekst
Abstract (en)
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.
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