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Exact solutions of many-body quantum chaotic systems
Kos, Pavel (Author), Prosen, Tomaž (Mentor) More about this mentor... This link opens in a new window

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Abstract
Chaotic quantum many-body systems exhibit complex dynamics, which presents an obstacle in their analytical and numerical understanding. Therefore, many aspects of these generic many-body systems are still poorly understood. In this thesis, we present exact solutions for chaotic many-body quantum systems. We focus on kicked Ising spin chains and quantum circuits with an additional property -- dual-unitarity. This property asserts that quantum evolution remains unitary upon switching space and time and it is essential for obtaining exact solutions. We start by focusing on quantum chaos. One of the key goals of quantum chaos is to explain why spectral fluctuations of simple local models match the spectral fluctuations of random matrices. This was resolved in a few-body context using a semiclassical limit but remained enigmatic in many-body contexts without classical limits. After defining the setting, we prove that the spectral fluctuations match random matrix theory in three different cases by computing the Fourier transform of spectral density's two-point function: spectral form factor. This property validates these models as chaotic. Next, we compute the correlation functions in dual-unitary systems and classify them according to different degrees of ergodicity. We further characterise these chaotic systems by examining their dynamical complexity, captured by the local operator entanglement entropy. We proceed beyond previous discussions by treating correlation functions in perturbed dual-unitary models, thus establishing the stability of these models. The quest to obtain rigorous results leads us to a minimal setting of the operator evolution, relevant for noisy (time-depended) systems. In this setting, we can calculate the correlation functions for perturbed dual-unitary models as well as for a new class of chaotic models corresponding to diagrams with no merges. Finally, we use these models to probe the spectral fluctuations in noisy systems, by which we connect the generalised spectral form factor with the correlation functions.

Language:English
Keywords:quantum chaos, quantum many-body systems, exact solutions, non-equilibrium statistical mechanics, Floquet systems, quantum circuits, spectral form factor, kicked Ising spin chains
Work type:Doctoral dissertation (mb31)
Organization:FMF - Faculty of Mathematics and Physics
Year:2021
Views:120
Downloads:55
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Secondary language

Language:Slovenian
Title:Točne rešitve mnogodelčnih kvantnih kaotičnih sistemov
Abstract:
Dinamika kaotičnih mnogodelčnih kvantnih sistemov je kompleksna, kar predstavlja oviro za analitično in numerično razumevanje teh sistemov. To je glavni razlog, da je veliko vidikov dinamike v teh sistemih še vedno slabo razumljenih. V tem doktorskem delu predstavimo nekaj točnih rešitev zgoraj omenjenih sistemov. Osredotočimo se na brcane Isingove spinske verige in kvantna vezja z dodatno lastnostjo: dualno unitarnostjo. Ta lastnost zahteva unitarnost kvantne evolucije pri zamenjanih vlogah prostora in časa in je ključna za izpeljavo točnih rešitev. Najprej se osredotočimo na kvantni kaos. Eden glavnih ciljev teorije kvantnega kaosa je razložiti, zakaj se spektralne fluktuacije preprostih lokalnih sistemov ujemajo z napovedjo teorije naključnih matrik. Omenjeno ujemanje je bilo pojasnjeno v nekajdelčnih sistemih s pomočjo semiklasične limite, medtem ko je v mnogodelčnih sistemih ostalo nepojasnjeno. Po vzpostavitvi in definiranju problemov dokažemo ujemanje spektralnih fluktuacij v treh različnih vrstah sistemov in s tem pokažemo, da so ti sistemi kaotični. To dosežemo z izračunom Fourierove transformiranke dvotočkovnih korelacijskih funkcij spektralnih črt -- spektralnega oblikovnega faktorja. Nato se posvetimo izračunu dvotočkovnih korelacijskih funkcij v dualno-unitarnih modelih, ki jih razvrstimo glede na ergodičnost teh korelacij. Te sisteme nadaljnje karakteriziramo glede na njihovo dinamično kompleksnost, ki jo opiše prepletenostna entropija lokalnih operatorjev. V naslednjem koraku razširimo našo obravnavo korelacijskih funkcij, na primer zmotenih (perturbiranih) dualno-unitarnih modelov, s čimer pokažemo njihovo stabilnost. Želja po točnih rešitvah nas pripelje do minimalnega modela operatorske evolucije, ki opiše šumeče časovno odvisne sisteme. V tem kontekstu lahko točno izračunamo korelacijske funkcije za zmotene dualno-unitarne modele in za novo družino kaotičnih modelov (``brez združitev''). Na koncu uporabimo te modele za študijo spektralnih fluktuacij v časovno odvisnih šumečih sistemih, s čimer povežemo posplošen spektralni oblikovni faktor s korelacijskimi funkcijami.

Keywords:kvantni kaos, kvantni mnogodelčni sistemi, točne rešitve, neravnovesna statistična fizika, kvantna vezja, spektralni oblikovni faktor, brcana Isingova spinska veriga

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