Transport in one-dimensional quantum systems
In this thesis we present research on one-dimensional quantum systems performed using a combination of numerical and analytic techniques, with a particular focus on spin transport. We first review the numeric and analytic methods used in our works and present some of the useful tricks we have employ...
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| Format: | Doctoral or Postdoctoral Thesis |
| Language: | English |
| Published: |
2020
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| Online Access: | https://repozitorij.uni-lj.si/IzpisGradiva.php?id=120461 https://repozitorij.uni-lj.si/Dokument.php?id=135256&dn= https://repozitorij.uni-lj.si/Dokument.php?id=135259&dn= https://repozitorij.uni-lj.si/Dokument.php?id=135258&dn= https://repozitorij.uni-lj.si/Dokument.php?id=135260&dn= https://plus.si.cobiss.net/opac7/bib/31334403?lang=sl |
| Summary: | In this thesis we present research on one-dimensional quantum systems performed using a combination of numerical and analytic techniques, with a particular focus on spin transport. We first review the numeric and analytic methods used in our works and present some of the useful tricks we have employed throughout the work. We begin by studying integrable systems, more precisely we study the paradigmatic Heisenberg model. Using tensor network algorithms we study both low and high temperature physics in the model and observe a lack of ballistic transport in the gapped regime. Additionally, at the SU(2) symmetric point we find transport to be superdiffusive with the spin-spin correlation function showing Kardar-Parisi-Zhang scaling. We then move on to discrete time integrable models by studying the integrable Trotterisation of the Heisenberg model.This allows for easier numeric simulations, which enables us to observe Kardar-Parisi-Zhang physics in another clean quantum system. Additionally, we then focus on the ballistic regime of the model, where we employ the Mazur lower bound for the Drude weight using quasilocal conserved charges, and find that it exhibits fractal behaviour. Our focus then drifts away from integrability into more generic systems. We begin by studying a free model with a single perturbed link. In the first case we study a non-interacting perturbation, which allows us to complement our numerical results with analytics, as the model remains free. We show that in the presence of a single link perturbation transport remains ballistic. Numerically we can expand this to the case where the perturbation is interacting and the complete model is no longer integrable. Lastly, we study the effects of breaking the integrability globally by imposing either a random or quasiperiodic field. We observe that the quasiperiodic model's behaviour is qualitatively different from that of the random field model when interactions are turned on. By studying the single particle resonances of the model we find a Fibonacci structure inherent to the quasiperiodic model. Breaking the quasiperiodicity even slightly destroys this structure and suggests a possible avenue to transport engineering in interacting quantum systems. V disertaciji predstavimo raziskave na temo transporta v eno-dimenzionalnih kvantnih sistemih, ki smo jih opravili s kombinacijo numeričnih in analitičnih metod. Pri tem se osredotočamo predvsem na spinski transport v izbranih sistemih. Najprej bomo predstavili numerične in analitične metode, ki smo jih uporabili pri naših raziskavah, ter predstavili nekatere uporabne trike, ki so nam pri tem pomagali. Začnemo s študijami integrabilnih sistemov, natančneje s paradigmatičnim Heisenbergovim modelom. Z uporabo numeričnih metod, ki temeljijo na matrično produktnih nastavkih, lahko opazujemo transport pri nizkih in visokih temperaturah. Opazimo, da v režimu s spektralno vrzeljo transport ni balističen, kljub temu da je naš model integrabilen. Še bolj nenavadno pa je dejstvo, da je v SU(2) simetrični točki model superdifuziven, spinsko-spinska korelacijska funkcija pa se obnaša v skladu s Kardar-Parisi-Zhang fiziko. Nadaljujemo z modeli v diskretnem času, kjer se osredotočimo na integrabilno Trotterizacijo Heisenbergovega modela. S tem si olajšamo numerične simulacije, kar nam omogoča doseganje bolj natančnih rezultatov, s katerimi lahko potrdimo obstoj Kardar-Parisi-Zhang fizike v čistih kvantnih sistemih. V tem modelu se dodatno osredotočimo tudi na balistični režim, kjer s pomočjo Mazurjeve neenakosti določimo spodnjo mejo za Drudejevo utež, ki ima fraktalno obliko. To storimo s pomočjo kvazilokalnih ohranitvenih zakonov, ki jih za ta model tudi izpeljemo. Nato preusmerimo pozornost stran od integrabilnosti v bolj generične sisteme. Kot prvi korak v to smer raziskujemo transport v neinteragirajočem XX modelu, kjer uvedemo perturbacijo na eni sami povezavi. Pokažemo, da v takem primeru transport ostane balističen. V primeru, ko perturbacija ni interagirajoča, sistem ostane rešljiv, v tem primeru podpremo numerične simulacije z analitičnimi izračuni. Numerično lahko to razširimo tudi na primere z interagirajočimi perturbacijami, ki pa niso več analitično rešljivi. Na koncu si ogledamo še primer, kjer integrabilnost zlomimo globalno. To storimo na dva način, v prvem primeru s popolnoma naključnim poljem na vsakem mestu, v drugem pa s kvaziperiodičnim poljem. Opazimo, da je obnašanje kvaziperiodičnega sistema znatno drugačno od popolnoma naključnega, ko sistemu dodamo interakcije. Z izračuni eno-delčnih resonanc modela opazimo, da se v kvaziperiodičnem modelu pojavi zanimiva Fibonaccijeva struktura. Zanimivo je, da tudi najmanjša perturbacija v kvaziperiodičnem polju zlomi to strukturo, kar odpira pot k transportnemu inženiringu v interagirajočih kvantnih sistemih. |
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