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Fluctuations of the spectral form factor in the kicked Ising chain
Flack, Ana (Author), Prosen, Tomaž (Mentor) More about this mentor... This link opens in a new window, Bertini, Bruno (Co-mentor)

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Abstract
The main goal of this thesis is to study the fluctuations of the kicked Ising spin chain and then compare the results to the predictions of random matrix theory (RMT), which is believed to be an indicator of quantum chaos. This is achieved by applying a recently discovered method based on the duality between the propagation in time and in space. Before the results are presented, some relevant concepts from RMT will be introduced. In addition to the general introduction of this theory, the connection to quantum chaos is also discussed. The thesis continues with the definition of the kicked Ising model and the space-time duality. By investigating the model at the self-dual point, where transfer matrix is unitary in both space and time, we calculate the first power of the trace of the Floquet propagator and it further confirms the previously obtained result for the averaged spectral form factor (SFF). Then the same steps are used to estimate the variance of the SFF. These results are further tested by two numerical methods, the Monte-Carlo simulations and the power method, which is applied to the dual quantum propagator. The predictions for all higher-order moments are also presented at the end. Our results show that, contrary to the expectations, the fluctuations of the self-dual kicked Ising model do not agree with the results obtained in the scope of RMT. These findings are interesting due to the fact that it is believed that non-integrable chaotic quantum many-body systems generally agree with random matrix theory predictions.

Language:English
Keywords:Quantum chaos, Random matrix theory, Spectral form factor, Kicked Ising spin chain, Many-body quantum physics.
Work type:Master's thesis/paper (mb22)
Tipology:2.09 - Master's Thesis
Organization:FMF - Faculty of Mathematics and Physics
Year:2020
COBISS.SI-ID:20060675 Link is opened in a new window
Views:118
Downloads:61
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Secondary language

Language:Slovenian
Title:Fluktuacije spektralnega oblikovnega faktorja v brcani Isingovi verigi
Abstract:
Glavni cilj te magistrske naloge je raziskati fluktuacije spektralnega oblikovnega faktorja brcanega Isingovega modela, ki je v samo-dualni točki in jih primerjati z napovedmi iz teorije naključnih matrik, ki velja za pokazatelja kaosa v kvantnih sistemih. Za izračun fluktuacij je ključna nova metoda, ki sloni na dualnosti med propagacijo sistema v času in kraju. Preden so predstavljeni rezultati, so opisani ključni pojmi iz teorije naključnih matrik. Posebna pozornost je posvečena tudi povezavi med omenjeno teorijo in kvantnim kaosom. V nadaljevanju je definiran obravnavani brcan Isingov model in nato je podrobneje opisana samo-dualnost, ki je značilna zanj. Del naloge, ki se posveča rezultatom se začne z izračunom sledi Floquetovega operatorja, ki dodatno potrdi ujemanje med povprečjem spektralnega oblikovnega faktorja in rezultatom, ki velja za naključne matrike. Nato je z enako metodo izračunana še variacija. Ti rezultati so preverjeni z uporabo dveh numeričnih metod, Monte-Carlo simulacijami in potenčno metodo, ki je uporabljena za iskanje lastnih vektorjev dualnega propagatorja. Na koncu je predstavljena tudi napoved za višje momente oblikovnega faktorja. Rezultati so presenetljivi, saj kažejo, da se fluktuacije ne ujemajo z napovedmi iz teorije naključnih matrik. Izkaže se, da so kljub ujemanju povprečja višji momenti večji od napovedi. To odstopanje je zanimivo, saj v splošnem velja, da imajo kaotični mnogodelčni sistemi spektralno statistiko, ki je enaka tisti, ki jo opazimo pri naključnih matrikah.

Keywords:Kvantni kaos, teorija naključnih matrik, spektralni oblikovni faktor, brcana Isingova spinska veriga, mnogodelčna kvantna mehanika

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