20.500.12556/RUL-109343 Transport v integrabilni klasični spinski prostorsko-časovni mreži Transport in an integrable classical spin space-time lattice V delu študiramo transport v Landu-Lifshitzovem modelu na mreži v eni prostorski dimenziji, ki predstavlja klasičen analog kvantne spinske verige. Model motiviramo s kvantnomehansko obravnavo izmenjalne interakcije. Vpeljemo koncept integrabilnosti v klasični mehaniki ter zapišemo enačbe gibanja modela z uporabo Laxovih operatorjev in pogoja ničelne ukrivljenosti. Integrabilnost modela sledi iz involutivne lastnosti para matrik monodromij, ki jo dokažemo z uporabo R-matrike. Predstavimo Trotterjev razcep Liouvillove enačbe, ki nam omogoči učinkovito simulacijo dinamike modela. Naiven razcep modela se izkaže za neintegrabilnega, kar potrdimo z izračunom Lyapunovega spektra. Uvedemo novo integrabilno posplošitev Landau-Lifshitzovega modela na diskretno krajevno-časovno mrežo ter poiščemo dvodelčni hamiltonian, ki generira želeno dinamiko ter novo R-matriko modela. Pokažemo, da je takšen model sam svoj dual. V nemagnetiziranem stanju se model izkaže za superdifuzivnega, z analizo korelacijskih funkcij pokažemo, da spada v Kardar-Parisi-Zhangov univerzalnostni razred. V magnetiziranem stanju prevlada balistično obnašanje, opazimo bogato odvisnost korelacijske funkcije magnetizacije od celotne magnetizacije. In the present work we study the transport properties of the lattice Landau-Lifshitz model in one spatial dimensions, which represents a classical analogue of a quantum spin chain. The model is motivated by a quantum mechanical treatment of the exchange interaction. We introduce the concept of integrability in classical mechanics and write down the model’s equations of motion using Lax operators and the zero-curvature condition. The integrability of the models follows from an involutive property of a pair of monodromy matrices which we prove using an R-matrix. We present the Trotter decomposition of the Liouville equation, which allows for an efficient simulation of the model dynamics. The naive decomposition of the model turns out to be nonintegrable, which we verify by computing the Lyapunov spec- trum. We introduce a novel integrable generalization of the Landau-Lifshitz model on the discrete time lattice and compute the two-body Hamiltonian that generates the requisite dynamics and find a novel R-matrix. We show that the model is self-dual. The model turns out to be superdiffusive in a non-magnetized state, by analyzing its correlation functions we further show it belongs into the Kardar-Parisi-Zhang universality class. In a magnetized state ballistic trasnport predominates, a rich dependance of the correlation function of magnetization upon net magnetization is observed. Landau-Lifshitzov model na mreži transport integrabilnost samodualnost Laxov operator R-matrika Trotterjev razcep Liouvillova enačba integrabilna trotterizacija Lyapunov spekter superdifuzivnost Kardar-Parisi-Zhangov univerzalnostni razred lattice Landau-Lifshitz model transport integrability self-duality Lax operator R-matrix Trotter decomposition Liouville equation integrable trotterization Lyapunov spectrum superdiffusivity Kardar-Parisi-Zhang universality class true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2019-08-30 14:59:49 2019-08-30 14:59:53 2023-08-30 11:16:06 0000-00-00 00:00:00 2019 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 97870 3351396 836.pdf 836.pdf 1 154DA609D5EC107BB3E5B925BA9525B2 6d0c512da4bae8719696300c0d5428b7efb568019b6161cd942f3ed05e7bc6fb d668b1bd-a1b6-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=120875 Fakulteta za matematiko in fiziko 0 0 0