Exact time-dependent solutions of interacting systems

Klobas, Katja (author), Prosen, Tomaž (mentor)

In the thesis we study exact solutions to simple interacting non-equilibrium problems. We propose two lattice models in discrete time, a model of hard-core interacting charged particles and the Rule 54 reversible cellular automata (RCA54). Both systems describe dynamics of particles (solitons) that move with fixed velocities and undergo pairwise scattering. The first problem we study are the transport properties of the model of charged particles. We start by considering the linear response regime. We show that the computation of the relevant correlation functions can be restricted to a subspace of extensive observables, in which the time evolution is simple. This enables us to obtain exact expressions of linear transport coefficients, such as the diffusion constant and Drude weight. A similar approach is applied to the evaluation of the charge profile at large times after starting from a piecewise homogeneous state, and the spatio-temporal charge-charge correlation function. We proceed by studying the full time-evolution of local observables in RCA54. We find an efficient matrix-product state (MPS) that captures the full time-evolved local density. We use the MPS to express the density profile after a special inhomogeneous quench problem, and get an exact expression of the spatio-temporal correlation function. The results show that the model exhibits ballistic transport with diffusive corrections. In the last chapter we consider space-dynamics of RCA54, i.e. dynamics of the dual model that corresponds to the exchange of roles of space and time. We obtain an MPS that encodes all the multi-time correlation functions of ultra-local observables positioned at the same spatial coordinate. We then proceed to formulate space dynamics in terms of local maps with support that is finite, but bigger than that of the local time-evolution maps. We finish by providing an equivalent circuit representation of the dynamics.

• Language: English
• Keywords: statistical physics, nonequilibrium statistical mechanics, solvable lattice models, cellular automata, solitons, exact results, matrix product states, transport
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-120396
• COBISS.SI-ID: 30100483
• Publication date in RUL: 19.09.2020

Secondary language

• Language: Slovenian
• Title: Točne časovno odvisne rešitve interagirajočih sistemov

Abstract

V doktorskem delu preučujemo točne rešitve preprostih enodimenzionalnih sistemov z interakcijo. Predstavimo dva modela na mreži, definirana v diskretnem času: model nabitih delcev s kontaktno interakcijo in reverzibilni celični avtomat, podan s pravilom 54 (RCA54). Oba sistema opisujeta dinamiko delcev (solitonov), ki se premikajo s fiksnimi hitrostmi in se v parih sipajo. Prvi obravnavani problem so transportne lastnosti modela nabitih delcev. Začnemo z režimom linearnega odziva. Pokažemo, da lahko izračun potrebnih korelacijskih funkcij omejimo na podprostor ekstenzivnih opazljivk, ki imajo preprosto časovno evolucijo. To nam omogoči, da izračunamo transportne koeficiente, kot sta npr. difuzijska konstanta in Drudejeva utež. Na podoben način izračunamo profil naboja v začetnem problemu z nehomogenim začetnim stanjem in časovno-prostorsko korelacijsko funkcijo. Nadaljujemo s časovno evolucijo lokalnih opazljivk v RCA54. Najdemo ekonomičen zapis s matrično-produktnim stanjem (MPS), ki opiše celotno časovno evolucijo lokalne gostote. MPS uporabimo za izračun gostotnega profila v nehomogenem začetnem problemu in za izpeljavo časovno-prostorske korelacijske funkcije. Rezultati pokažejo, da je transport v modelu balističen z difuzijskimi popravki. V zadnjem poglavju si ogledamo lastnosti prostorske dinamike RCA54, t.j. dinamike dualnega modela, ki ga dobimo z zamenjavo vlog prostora in časa. Najdemo MPS, ki opiše vse veččasovne korelacijske funkcije opazljivk, lokaliziranih na istem mestu. Nato nadaljujemo z zapisom prostorske dinamike v obliki komutirajočih lokalnih preslikav. Zaključimo z upodobitvijo dinamike z vezji, sestavljenimi iz lokalnih vrat.

• Keywords: statistična fizika, neravnovesna statistična mehanika, točno rešljivi modeli na mreži, celični avtomati, solitoni, točne rešitve, matrično-produktna stanja, transport