Nonequilibrium integrable quantum dynamical systems

Zadnik, Lenart (author), Prosen, Tomaž (mentor)

We present two directions of research in statistical mechanics of nonequilibrium onedimensional quantum systems. One is related to isolated integrable models, the other one to exactly solvable dissipatively driven spin chains. The device enabling their analysis is identified as the quantum group symmetry. In the framework of isolated systems we focus on the concept of integrable Floquet driven systems. In particular, we show how to build such systems out of the basic constituents of integrability structure. This allows substantiated and conclusive statements,for instance, about the transport phenomena. Despite their inherent relation, the presented integrable periodically driven models originate in two different ideas. On the one hand we have Trotterisations of integrable spin chains, concieved in an effort to better understand the dynamics of spin models. Then, there are systems originating in the attempts to solve quantum field theory in an algebraically closed form, for example, the quantum Hirota equation. The bulk of our consideration of isolated quantum systems consists of the construction of extensive conservation laws that either (i) constitute the generalised Gibbs ensemble and the hydrodynamic description of thermalisation after a quantum quench, or (ii) prevent the decay of current autocorrelations and thus characterise ideal transport in the system. The exact way in which these charges determine the dynamics stems from the symmetries of the quantum group representations. The second part of the exposition is dedicated to open quantum systems. Again we distinguish two formally related settings. The first one is that of a boundary driven quantum cellular automaton, our particular example being the integrable Trotterisation of the Heisenberg magnet. Starting from a repeated interaction protocol, in which the system under scrutiny is repeatedly coupled to the environment, we introduce the Kraus map as a general form of a dissipative time evolution of the density matrix. We then solve for its unique nonequilibrium steady state, using integrability structure of the model. The other setting is that of a dissipatively boundary driven spin chain in the continuous time. Here we present the recently developed formalism of inhomogeneous Lax structure. Using it we demonstrate the solvability of the XXZ and XYZ spin chains, acted upon by the Lindblad operators that polarise the boundary spins in arbitrary directions. The ansatz for the steady state is particularly interesting, since it exhibits a previously unknown integrability structure, differing from site to site in the spin chain. This structure can independently produce nontrivial conservation laws in an isolated spin chain with arbitrary boundary fields.

• Language: English
• Keywords: Integrable quantum cellular automata, Trotterisation of the Heisenberg magnet, quantum Hirota model, quasilocal charges, Drude weight, Mazur bound, inhomogeneous Lax operators
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-111487
• COBISS.SI-ID: 3374436
• Publication date in RUL: 02.10.2019

Secondary language

• Language: Slovenian
• Title: Neravnovesni integrabilni kvantni dinamični sistemi

Abstract

Predstavljeni sta dve področji statistične mehanike neravnovesnih kvantnih sistemov v eni razsežnosti. Prvo obravnava izolirane integrabilne modele, drugo pa točno rešljive, disipativno gnane spinske verige. Analizo teh sistemov omogoča simetrija kvantnih grup. V okviru izoliranih sistemov se osredotočimo na pojem integrabilnih Floquetovo gnanih sistemov. Pokažemo, kako jih tvoriti iz osnovnih gradnikov integrabilne strukture, ki omogoča točne zaključke, na primer o transportnih pojavih. Kljub njuni sorodnosti, detajlno predstavljena intagrabilna periodično gnana modela izvirata iz dveh različnih idej. Na eni strani imamo Trotterizacije integrabilnih spinskih verig, s pomočjo katerih želimo razumeti dinamiko spinskih modelov. Po drugi strani nas zanimajo modeli, katerih cilj je algebraičen poskus reševanja kvantnih teorij polja, na primer kvantni model Hirote. Naša obravnava izoliranih kvantnih sistemov temelji na izgradnji ekstenzivnih ohranjenih količin, ki bodisi (i) tvorijo posplošen Gibbsov ansambel in hidrodinamski opis termalizacije v kvantnem začetnem problemu bodisi (ii) preprečujejo pojemanje avtokorelacij toka in tako poosebljajo balistični transport. Vpliv teh količin na dinamiko zavisi od simetrij upodobitev kvantnih grup. V drugem delu se posvečamo odprtim kvantnim sistemom. Spet ločimo dva sorodna scenarija. Prvi je robno gnan kvantni celični avtomat, kjer kot primer uporabimo integrabilno Trotterizacijo Heisenbergovega magneta. Iz protokola, v katerem je obravnavan sistem ponavljajoče se sklopljen z okolico, izpeljemo Krausovo preslikavo kot splošno obliko disipativnega časovnega razvoja gostotne matrike. S pomočjo integrabilne strukture modela nato poiščemo enolično rešitev za neravnovesno stacionarno stanje take dinamike. Drugi scenarij je spinska veriga v zveznem času, disipativno gnana na robovih. Tu predstavimo pred kratkim razvit formalizem nehomogene Laxove strukture. Z njegovo pomočjo pokažemo rešljivost modelov XXZ in XYZ, v katerih disipativni Lindbladovi operatorji polarizirajo robne spine v poljubnih smereh. Nastavek za stacionarno stanje je posebej zanimiv prav zaradi prej neznane prostorsko nehomogene integrabilne strukture,ki omogoča izgradnjo netrivialnih ohranitvenih zakonov v izolirani spinski verigi s poljubnimi robnimi magnetnimi polji.

• Keywords: Integrabilni kvantni celični avtomati, Trotterizacija Heisenbergovega magneta, kvantni model Hirote, kvazilokalni ohranitveni zakoni, Drudejeva utež, Mazurjeva meja, nehomogeni Laxovi operatorji