Jake fluktuacije u kritičnim sustavima

master's thesis

• Title (english): Strong fluctuations in critical systems
• Author: Lucija Nora Farkaš
• Mentor: Ivan Balog (mentor)
• Mentor: Marinko Jablan (mentor)
• Committee member: Emil Tafra (predsjednik povjerenstva)
• Committee member: Marinko Jablan (član povjerenstva)
• Committee member: Ivan Balog (član povjerenstva)
• Committee member: Matko Milin (član povjerenstva)
• Committee member: Ivica Smolić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2019-06-14, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

Razmatramo dugovalnu aproksimaciju Isingovog modela koja odgovara ϕ4 teoriji polja sa skalarnim poljem ϕ. Ovaj model služi nam kao predložak sustava s kontinuiranim faznim prijelazom. Fluktuacije dobivaju na značaju smanjenjem prostorne dimenzije. Dimenzija ispod koje prijelaz nestaje naziva se donjom kritičnom dimenzijom. Donja kritična dimenzija odgovara dl = 1, gdje je egzaktno rješenje Isingovog modela poznato. Ipak, ovdje primjenjujemo metode neperturbativne renormalizacijske grupe (NPRG) kako bismo našli aproksimativno rješenje. Krajnji cilj je u budućnosti razviti nov pristup kritičnim statističkim sustavima u donjoj kritičnoj dimenziji preko NPRG koji bi bio primjenjiv i na sustave koje ne možemo egzaktno riješiti. Koristeći određene neperturbativne aproksimacije nalazimo opis prijelaza u blizini i na donjoj kritičnoj dimenziji. Nalazimo da se tako određena donja kritična dimenzija dobro slaže s dl = 1 te da je opća slika prijelaza konzistentna s egzaktnim rezultatom. Novi je rezultat ovoga rada da efektivni potencijal U pokazuje neanalitičnost na konačnom polju, ali na način da se novodobiveni rezultati u potpunosti slažu s fizikalnim ponašanjem sustava.

Abstract (english)

We consider the long wave approximation of the Ising model which corresponds to the ϕ4 field theory with a scalar field ϕ. This model is used as a paradigm of a system with a continuous phase transition. Fluctuations are gaining importance with the decrease of the spatial dimension. The dimension at which the transition disappears is called the lower critical dimension. The lower critical dimension for the present system is dl = 1, where the exact solution is known. Still, here we attempt to apply the methods of the nonperturbative renormalisation group (NPRG) to obtain an approximative solution. The ultimate goal is to, in future, develop a novel approach to handling critical statistical systems in the lower critical dimension via NPRG that could be extended to systems without exact solutions. Using certain nonperturbative approximation schemes, we find a description of the system in the vicinity and at the lower critical dimension. We find the lower critical dimension to be in good agreement with the exact dl = 1. A new result that we find is that the effective potential U shows nonanalyticity at a finite field, but in a way that is consistent with the expected physical behavior of the system.

• Keywords: fluktuacije; kritično ponašanje; fazni prijelaz; Isingov model; donja kritična dimenzija; renormalizacijska grupa
• Language: croatian
• URN:NBN: urn:nbn:hr:217:140879
• Study programme: Title: Physics; specializations in: Research; Course: Research; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra fizike (magistar/magistra fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-04 10:19:03