Title Groupe de renormalisation fonctionnel et événements rares Title (croatian) Funkcionalna renormalizacijska grupa i rijetki događaji Author Lucija Nora Farkaš Mentor Ivan Balog (mentor)Mentor Gilles Tarjus Committee member Dominique Mouhanna Committee member Pawel Marek Jakubczyk Committee member Bertrand Berche Committee member Denis Sunko (član povjerenstva)Committee member Danko Radić (član povjerenstva)Committee member Gilles Tarjus Committee member Ivan Balog (član povjerenstva)Granter University of Zagreb Defense date and country 2023-12-13, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 53 - Physics Abstract U ovoj tezi analiziramo može li derivacijski razvoj u sklopu Funkcionalne renormalizacijske Grupe opisati dugovalnu fiziku modela bez a priori poznavanja relevantnih okrupnjenih konfiguracija, cak i kad su one vrlo neuniformne. U tu svrhu istražujemo kako niski redovi derivacijskog razvoja opisuju prilazak donjoj kritičnoj dimenziji dlc skalarne ϕ4 teorije koja dijeli klasu univerzalnosti s Isingovim modelom i u kojoj se stoga očekuje da je fizika na dlc dominirana proliferacijom lokaliziranih pobuđenja (instantona). Naglašavamo da nam nije cilj pružiti još jedan teorijski opis prilaska donjoj kritičnoj dimenziji za sustave u klasi univerzalnosti Isingovog modela, budući da je to pitanje prilično dobro objašnjeno već prije nekoliko desetljeća. Naša je namjera na ovom primjeru razviti općeniti, neperturbativni (ali aproksimativni) pristup unutar FRG-a koji bi služio za probleme koji su još otvoreni a u kojima se vjeruje da neuniformne konfiguracije igraju značajnu ulogu, poput npr. niskotemperaturne faze Isingovih spinskih stakala ili donje kritične dimenzije neravnotežnog, atermalno vo.enog ("athermally driven") Isingovog modela s nasumičnim poljem (RFIM). Za ravnotežni RFIM, rigorozno je pokazano da je donja kritična dimenzija dlc = 2, no za RFIM van ravnoteže sama vrijednost dlc je i dalje upitna. FRG je vrlo pogodna metoda za takve sustave budući da se zna da i najniži stupnji aproksimacija u sklopu FRG-a mogu opisati netrivijalne neperturbativne pojave. Dodatna prednost FRG je i generalnost metode, u opreci sa specijaliziranim pristupima koji se obično moraju koristiti da bi se opisali ovakvi sustavi, poput kapljične teorije ("droplet theory") u slučaju Isingovog modela tik nad dlc gdje je nužno zatvorene domene jedne spinske orijentacije uronjene u drugu orijentaciju (t.j., kapljice) eksplicitno uvesti i tretirati. Osim toga, prostorna dimenzija d je u FRG parametar koji se može kontinuirano varirati. Pokazali smo da je konvergencija efektivnog potencijala 11 u fiksnoj tocki modela neuniformna u polju kad se dimenzija d približava dlc, s pojavom graničnog sloja ("boundary layer") oko minimuma potencijala.
Abstract (english) In this work we assess the ability of the derivativeexpansion of the FRG to quantitatively describe the long-distance physics of amodel without a priori knowledge of the relevant real-space coarse-grainedconfigurations, even when they are strongly nonuniform. To do so we investigatehow low orders of the derivative expansion describe the approach to the lower criticaldimension dlc of the Ising-like scalar φ 4 theory which is expected to becontrolled by the proliferation of localized excitations (instantons) [12–14].FRG is well suited for this task since even the lowest orders of approximationsare known to capture nontrivial nonperturbative phenomena [15, 16], and it hasan advantage of being a general approach, unlike specialized approaches onemust usually resort to capturing such behavior, like the droplet theory in thecase of the Ising model just above dlc, where enclosed domains of onespinorientation in the other (the droplets) must be introduced explicitly[12–14]. Additionally, the spatial dimension d is a parameter of the theorythat we can vary. We show that the convergence of the fixed-point effectivepotential describing the critical point of the model is nonuniform in the fieldwhen the dimension d approaches dlc, with the emergence of a boundary layeraround the minimum of the potential. < At the lowest nontrivialapproximation level (known as LPA’) this allows us to make analyticalpredictions for the value of the lower critical dimension dlc and for thebehavior of the critical temperature as d → dlc, which are both found in fairagreement with the known exact results. We check the stability of the resultsupon increasing the approximation order by studying the second order of the derivativeexpansion (∂2). The ∂2 investigation is still in progress. As of yet, less hasbeen found analytically due to the complexity coming from the nontrivial fieldrenormalization function z (ϕ) (z (ϕ) = 1 in LPA’). Numerical results above thedlc show strong indications that the boundary layer persists, tentatively withthe same behavior of the critical temperature in the dlc limit. However,predictions of the value of the lower critical dimension require matching theboundary layer solution with the rest of the field domain, which we have not achievedat the present moment. We also consider for the same Ising-like scalar φ 4model the ordered phase in the limit where d → dlc. We focus on the way theeffective potential in the FRG becomes convex as the proper spatialfluctuations, which now involve nonuniform configurations with a domain wallseparating two phases, are included in the calculation.
Keywords
renormalozacijska grupa
Keywords (english)
renormalization group
Language croatian URN:NBN urn:nbn:hr:217:081497 Promotion 2024 Project Number: KK.01.1.1.01.0004 Study programme Title: Doctoral study Type of resource Text Extent 159 str. File origin Born digital Access conditions Open access Terms of use Created on 2024-05-24 13:14:03
