Title Efekti konačne veličine u XY lancu Title (english) Finite-Size Effects in the XY Chain Author Vanja Marić Mentor Fabio Franchini (mentor)Mentor Denis Sunko (mentor)Committee member Denis Sunko (predsjednik povjerenstva)Committee member Fabio Franchini (član povjerenstva)Committee member Mario Basletić (član povjerenstva)Committee member Hrvoje Buljan (član povjerenstva)Committee member Damir Pajić (član povjerenstva)Granter University of Zagreb Defense date and country 2017-07-14, Croatia Scientific / art field, NATURAL SCIENCES Abstract XY lanac u poprečnom magnetskom polju je prototipni egzaktno rješiv model sa zanimljivim i poželjnim svojstvima: njegov dvodimenzionalni fazni dijagram, karakteriziran parametrom anizotropije i snagom vanjskog magnetskog polja, sadrži dva različita kvantna prijelaza na temperaturi apsolutne nule, a model se uvijek može preslikati u sistem slobodnih (bez spina) fermiona. Jedan od faznih prijelaza je klasa univerzalnosti slobodnih fermiona (c = 1 CFT), a drugi je klasa univerzalnosti 1D kvantnog Isingovog modela (c = 1=2 CFT). XY lanac je zapravo generalizacija Isingovog lanca. Dok se obje linije faznih prijelaza mogu opisati konformalnom teorijom polja (CFT), bikritična točka u kojoj se te linije susreću je nekonformalna, jer ima kvadratičan spektar. Cilj diplomskog rada je uočiti nekonformalnu prirodu bikritične točke u XY lancu u numeričkom eksperimentu kroz veličine dostupne također u modelima koji nisu egzaktno rješivi, kao što su entropija zapetljanosti i svojstvene vrijednosti reducirane matrice gustoće . Na taj način želimo u budućnosti predložiti numeričke provjere je li multikritična točka u proizvoljnom modelu konformalna ili nije. Najprije uvodimo XY lanac i koristeći standarde analitičke tehnike pronalazimo njegovo mikroskopsko rješenje. Kao i Isingov model XY lanac pokazuje fazu slomljene Z2 simetrije, u kojoj postoje dva degenerirana osnovna stanja u termodinamičkom limesu velikog sistema. Primijetivši manjak rezultata o egzaktnoj degeneraciji osnovnog stanja, istražujemo problem koristeći numeričke i analitičke metode. Analitička metoda, koja uključuje kompleksnu analizu i Fourierov red, dala je odgovor za Isingov model. Koristeći tu metodu za općenitiji XY lanac pronalazimo ovisnost degeneracije o broju spinova u slučaju iščezavajućeg magnetskog polja i pokazujemo odsutnost degeneracije kada je parametar anizotropije veći od 1 i broj spinova paran. Vraćajući se glavnom cilju rada, izvodimo svojstvene vrijednosti reducirane matrice gustoće i entropiju zapetljanosti u XY lancu te konstruiramo numerički algoritam za njihovo računanje. Uspijevamo primijetiti nekonformalnu prirodu bikritične točke uspoređivanjem entropije zapetljanosti i najveće svojstvene vrijednosti reducirane matrice gustoće s predviđanjima konformalne teorije polja u blizini i daleko od bikritične točke.
Abstract (english) The XY chain in a transverse field is a prototypical exactly solvable model with interesting and desirable properties: its two-dimensional phase diagram, characterized by an anisotropy parameter and the strength of an external magnetic field, hosts two different quantum phase transitions at zero temperature, while the model can always be mapped into a system of free (spin-less) fermions. One of the phase transitions is the universality of free fermions (c = 1 CFT), while the other is that of the 1D quantum Ising model (c = 1=2 CFT). As a matter of fact, the XY chain is a generalization of the Ising chain. While both lines of phase transitions can be described by Conformal Field Theory, the bi-critical point at which these lines meet is non-conformal, since it has a quadratic spectrum. The aim of this thesis is to observe the non-conformal nature of the bi-critical point in the XY chain in a numerical experiment trough quantities accessible also in non-exactly solvable models, such as the entanglement entropy and the eigenvalues of the reduced density matrix. In this way, we want to propose in the future numerical tests whether a multi-critical point in an arbitrary model is conformal or not. First we introduce the XY chain and using standard analytical methods find its microscopic solution. Like the Ising model, the XY chain has a phase of broken Z2 symmetry, where the model shows two degenerate ground states in the thermodynamic limit of infinite system length. Having noticed a scarcity of results on the exact energy degeneracy between two putative ground states, we examine the problem using numerical and analytical methods. The analytical method, employing complex analysis and Fourier series, has given a definite answer for the Ising model. Using this method for the more general XY chain we find the dependence of the exact degeneracy on the number of spins in the special case of zero magnetic field and show the absence of an exact degeneracy when the anisotropy is greater than 1 and the number of spins is even. Coming back to the main aim of the thesis, we derive the reduced density matrix eigenvalues and the entanglement entropy in the XY chain and construct a numerical algorithm for calculating them. We accomplish to discriminate the non-conformal nature of the bi-critical point by comparing the entanglement entropy and the largest reduced density matrix eigenvalue with the conformal field theory predictions near and far from the bi-critical point.
Keywords
XY lanac u poprečnom magnetskom polju
zapetljanost
kvantni fazni prijelazi
degenerirana osnovna stanja
nekonformalna točka
Keywords (english)
XY chain in a transverse magnetic field
entanglement
quantum phase transitions
ground state degeneracy
non-conformal point
Language croatian URN:NBN urn:nbn:hr:217:052835 Study programme Title: Physics; specializations in: Research Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2018-03-15 16:58:27
