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Spektralne lastnosti modela $t$-$J$ in večdelčna lokalizacija
Šuntajs, Jan (Author), Bonča, Janez (Mentor) More about this mentor... This link opens in a new window, Vidmar, Lev (Co-mentor)

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Abstract
V magistrskem delu preučujemo prehod med ergodično in večdelčno lokalizirano (ang.~\emph{many-body localized}, v nadaljevanju MBL) fazo v modelu $t$-$J$ ob prisotnosti spinskega oziroma potencialnega nereda. Pri tem se osredotočimo na primera dopiranja z eno vrzeljo in tretjinskega dopiranja z vrzelmi, prehod med fazama pa zasledujemo kot funkcijo velikosti ustreznega tipa nereda v sistemu. Naša numerična analiza temelji na polni diagonalizaciji modelskih hamiltonk, pri presoji ergodičnosti oziroma večdelčne lokaliziranosti pa uporabljamo tri različne indikatorje, katerih lastnosti se v ergodičnem oziroma MBL primeru bistveno razlikujejo. Izračuna povprečnega razmerja razmikov med sosednjimi energijskimi nivoji $\langle \tilde{r}\rangle$ in \emph{spektralnega oblikovnega faktorja} (ang.~\emph{spectral form factor}, v nadaljevanju SFF) temeljita na analizi statističnih lastnosti energijskih spektrov modelskih hamiltonk, medtem ko je tretji indikator izračun prepletenostne entropije vseh lastnih stanj sistema. Izračunane vrednosti $\langle\tilde{r}\rangle$ in SFF se v ergodični fazi ujemajo z vrednostmi, ki jih dobimo z izračunom omenjenih količin v t.i. gaussovskem ortogonalnem ansamblu (v nadaljevanju GOE) naključnih matrik. Na drugi strani dobimo v primeru nastopa MBL rezultate, značilne za sisteme, v katerih so energijski nivoji medsebojno neodvisno porazdeljeni v skladu s Poissonovo verjetnostno porazdelitvijo. V primeru izračuna prepletenostne entropije je za ergodične sisteme značilno volumsko skaliranje prepletenostne entropije visoko vzbujenih lastnih stanj v spektru. V MBL sistemih so vsa lastna stanja v spektru šibko prepletena in je zanje značilno površinsko skaliranje prepletenostne entropije. V modelu $t$-$J$ za primer tretjinskega dopiranja z vrzelmi vsi naši indikatorji nakazujejo, da povečevanje nereda v sistemu vodi do prehoda med ergodično in MBL fazo tako v primeru spinskega kot potencialnega nereda. Enako velja za primer dopiranja z eno vrzeljo ob prisotnosti spinskega nereda, medtem ko se zdi primer ene vrzeli in potencialnega nereda drugačen od preostalih. Naši indikatorji namreč nakazujejo, da povečevanje nereda v tem primeru ne vodi do prehoda v MBL fazo. Kolikor nam je znano, je naše delo eden izmed prvih primerov uporabe izračuna SFF pri presoji ergodičnosti oziroma večdelčne lokaliziranosti preučevanih sistemov. Zaradi enostavne implementacije je v literaturi pogosta uporaba izračuna $\langle\tilde{r}\rangle$, pri katerem upoštevamo le korelacije med najbližjimi nivoji v energijskem spektru. Pri izračunu SFF na drugi strani upoštevamo korelacije med vsemi energijskimi nivoji v spektru, s čimer za ceno zahtevnejše implementacije dobimo precej podrobnejši vpogled v lastnosti sistema, denimo v obnašanje sistema na različnih časovnih skalah. Kot pokažemo, je naša presoja ergodičnosti oziroma večdelčne lokaliziranosti na podlagi izračunov SFF konsistentna s presojo na podlagi izračunov $\langle\tilde{r}\rangle$.

Language:Slovenian
Keywords:model $t$-$J$, nered, večdelčna lokalizacija, statistika energijskega spektra, spektralni oblikovni faktor, prepletenostna entropija
Work type:Master's thesis/paper (mb22)
Tipology:2.09 - Master's Thesis
Organization:FMF - Faculty of Mathematics and Physics
Year:2018
COBISS.SI-ID:3240804 This link opens in a new window
Views:599
Downloads:491
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Secondary language

Language:English
Title:Spectral properties of the $t$-$J$ model and many-body localization
Abstract:
In this final thesis we investigate the transition between the ergodic and many-body localized (MBL) phase in the $t$-$J$ model in the presence of spin or potential disorder. We concentrate on the two distinct cases, namely the case of one-hole doping and the case of one-third doping, where the transition between phases is investigated as a function of the magnitude of the appropriate type of disorder. Our numerical approach is based on full numerical diagonalization of the model Hamiltonians, where three different indicators are used in order to test the systems for their ergodic or MBL properties. The calculations of the mean ratio of the adjacent level-spacings $\langle \tilde{r}\rangle$ and of the spectral form-factor (SFF) are based on the analysis of the statistical properties of the investigated Hamiltonians' energy spectra. The third indicator we use is the calculation of the entanglement entropy of all system's eigenstates. In the ergodic phase, the calculated values of $\langle\tilde{r}\rangle$ and SFF match the ones obtained in the so-called gaussian orthogonal ensemble (GOE) of random matrices. The results are completely different in the MBL phase where they match those for systems in which the energy levels are distributed independently according to the Poisson probability distribution. The entanglement entropy of the highly excited eigenstates in the ergodic regime obeys the so-called volume-law scaling while in the MBL phase all the states in the spectrum are weakly entangled and thus obey the area law scaling of the entanglement entropy. All of our indicators imply that an increase in system's disorder leads towards a transition between an ergodic and the MBL phase in the case of one-third doping for both spin and potential disorder. The same holds true in the case of one-hole doping in the presence of spin disorder while the case of only one hole and potential disorder is somewhat different from the remaining ones. As our indicators show, an increase in disorder does not lead towards a transition in the MBL phase in this case. As far as we know, we are among the very first to use the calculation of SFF as an idicator of system's ergodicity or the presence of MBL. Due to its rather simple implementation, the calculation of $\langle\tilde{r}\rangle$ is commonly used in order to test for ergodicity or the presence of MBL in a given system. In calculating $\langle\tilde{r}\rangle$, we only consider the correlations between the nearest levels in an energy spectrum. The calculation of SFF, on the other hand, accounts for correlations between all the energy levels in the spectrum. It thus allows for a more comprehensive insight in the system's properties, such as its behaviour at different time scales, at the expense of a more involved numerical implementation. In terms of determining the systems' ergodicity or the presence of MBL, the results of our SFF calculations match the predictions of our $\langle\tilde{r}\rangle$ calculations.

Keywords:$t$-$J$ model, disorder, many-body localization, energy spectrum statistics, spectral form factor, entanglement entropy

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