In this work I present a constructive method for finding critical points of the Ginzburg-Landau energy functional using the method of Sobolev gradients. I give a description of the construction of the Sobolev gradient and obtain convergence results for continuous steepest descent with this gradient. I study the Ginzburg-Landau functional with magnetic field and the Ginzburg-Landau functional without magnetic field. I then present the numerical results I obtained by using steepest descent with the discretized Sobolev gradient.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22233号 / 理博第4547号 / 新制||理||1653(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 河合 俊哉, 教授 大槻 知忠, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Research in superconductivity is heading towards overcoming the limitations imposed by extreme conditions, and promising candidates in this pursuit are superconductors made from hydrides. Carbonaceous Sulfur Hydride (CSH) was reported in Nature 586, 373-377 (2020) as a room-temperature superconductor in the pressure range of 140-267 GPa; however, there is controversy in the literature regarding these results. Here, we use density functional theory to confirm the hypothesis of Nature 596, E9-E10 (2021) that a metallic path is the reason for the sharp drop in resistance interpreted in Nature 586, 373-377 (2020) as indicative of a weak type 2 superconductor. We find that the metallic behavior of CSH is dominated by sulfur p-orbitals, and not by metallization of hydrogen. If CSH would be a superconductor, the predicted Ginzburg Landau parameter would be 1356.9, reflecting an unusually strong type 2 superconductor and thus contradicting the interpretation of Nature 586, 373-377 (2020). The fact that we find no metallic states below 220 GPa casts doubts on the onset of superconductivity at 140 GPa reported in Nature 586, 373-377 (2020). Additionally, the small fraction of active hydrogen density of states at the Fermi level shows that CSH is not a high-temperature superconductor.

京都大学 / 0048 / 新制・課程博士 / 博士(エネルギー科学) / 甲第17913号 / エネ博第285号 / 新制||エネ||59(附属図書館) / 30733 / 京都大学大学院エネルギー科学研究科エネルギー基礎科学専攻 / (主査)教授 岸本 泰明, 教授 中村 祐司, 教授 前川 孝 / 学位規則第4条第1項該当 / Doctor of Energy Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20885号 / 理博第4337号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 拓郎, 教授 中島 啓, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

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This thesis investigates the unique properties of 2D devices such as twisted semiconducting bilayers and a-RuCl₃ heterostructures employing scanning tunneling microscopy (STM) and spectroscopy (STS) probes. The research presented here sheds light on the vast opportunities that 2D materials provide in condensed matter systems as well as future device applications. Among 2D materials, transition metal dichalcogenide (TMD) heterobilayers provide a promising platform to study many quantum phenomena such as excitonic states due to their tunability of band gap. In addition, TMDs are excellent candidates to achieve localized states and carrier confinement, crucial for single photon emitters used in quantum computation and information. We begin this thesis with a brief overview of STM/STS and utilizing these techniques on 2D materials in the first and second chapters.The third chapter of this work investigates the twisted bilayer of WSe₂ and MoSe₂ in the H-stacking configuration using STM/STS which was previously challenging to measure. The spectroscopic results obtained from the heterobilayer indicate that a combination of structural rippling and electronic coupling generates unexpectedly large \moire potentials, in the range of several hundred meV. Our analysis reveals that the \moire structure and internal strain, rather than interlayer coupling, are the main factors of the moire potential. Large moire potentials lead to deeply trapped carriers such as electron-hole pairs, so-called excitons. Our findings open new routes toward investigating excitonic states in twisted TMDs. In the next chapter, we investigate the ultralocalized states of twisted WSe₂/MoSe₂ nanobubbles. Mechanical and electrical nanostructurings are expected to modify the band properties of transition metal dichalcogenides at the nanoscale. To visualize this effect, we use STM and near-field photoluminescence to examine the electronic and optical properties of nanobubbles in the semiconducting heterostructures. Our findings reveal a significant change in the local bandgap at the nanobubble, with a continuous evolution towards the edge of the bubble. Moreover, at the edge of the nanobubble, we show the formation of in gap bound states. A continuous redshift of the interlayer exciton on entering the bubble is also detected by the nano-PL. Using self-consistent Schrodinger-Poisson simulations, we further show that strong doping in the bubble region leading to band bending is responsible for achieving ultralocalized states. Overall, this work demonstrates the potential of 2D TMDs for developing well-controlled optical emitters for quantum technologies and photonics. We next turn to the effect of the electric field in band gap tuning of WSe₂/WS₂ heterobilayer. The tunability of band gap is a crucial element in device engineering to achieve quantum emitters. The electrostatic gate generates doping and an electric field giving access to continuous tunability, higher doping level, and integration capability to nanoelectronic devices. We employ scanning tunneling microscopy (STM) and spectroscopy (STS) to probe the band properties of twisted heterobilayer with high energy and spatial resolution. We observe continuous band gap tuning up to several hundreds of meV change by sweeping the back gate. We introduced a capacitance model to take into account the finite tip size leading to an enhanced electric field. The result of our calculation captures well the band gap change observed by STS measurements. Our study offers a new route toward creating highly tunable semiconductors for carrier confinement in quantum technology. In the next chapters, we focus on a-RuCl₃ heterointerfaces. We first explore the nanobubble of graphene/a-RuCl3 to create sharp p-n junctions. The ability to create sharp lateral p-n junctions is a critical requirement for the observation of numerous quantum phenomena. To accomplish this, we used a charge-transfer based heterostructure consisting of graphene and a-RuCl₃ to create nanoscale lateral p-n junctions in the vicinity of nanobubbles. Our approach relied on a combination of scanning tunneling microscopy (STM) and spectroscopy (STS), as well as scattering-type scanning near-field optical microscopy (s-SNOM), which allowed us to examine both the electronic and optical responses of these nanobubble p-n junctions. Our results showed a massive doping variation across the nanobubble with a band offset of 0.6 eV. Further, we observe the formation of an abrupt junction along nanobubble boundaries with an exceptionally sharp lateral width (<3 nm). This is one order of magnitude smaller length scale than previous lithographic methods. Our work paves the way toward device engineering via interfacial charge transfer in graphene and other low-density 2D materials. In chapter 7, we describe the use of low-temperature scanning tunneling microscopy (STM) measurements to observe the \moire pattern in graphene/a-RuCl3 heterostructure to validate the InterMatch method. This method is effective in predicting the charge transfer, strain, and stability of an interface. The InterMatch method was applied to moire patterns of graphene/a-RuCl3 to predict the stable interface structure. STM topographs show three regions with distinct moire wavelengths due to atomic reconstructions. Using the InterMatch method, we perform a comprehensive mapping of the space of superlattice configurations and we identify the energetically favorable superlattices that occur in a small range of twist angles. This range is consistent with the STM results. Moreover, the spectra on these regions exhibit strong resonances with the spacing between resonances following the expectation from Landau levels on a Dirac spectrum due to strain and doping. The results of our scanning tunneling microscopy (STM) measurements confirm that the InterMatch method is effective in predicting the charge transfer and stability of interfaces between materials. We next investigate WSe₂/a-RuCl₃ heterostructure through a multi-faceted approach. Our exploration encompassed diverse techniques such as STM, and optical measurements. We detect a significant charge transfer between the two layers by STM measurements, leading to a shift in the Fermi level towards the valence band of WSe₂. Our findings are supported by optical measurements and DFT calculations, which confirm the p-doped WSe₂ observed through STM. The results of this work highlight a-RuCl₃ potential for contact engineering of TMDs and unlocking their functionalities for the next generation optoelectronic devices. In the last chapter of this thesis, I provide a brief conclusion as well as a few future directions and insights for investigating 2D materials.

Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. We verify this conjecture for a wide class of singularities on the level of Frobenius algebras, generalizing work of Krawitz [10]. We also review the relevant parts of the constructions.

Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubic--quintic Ginzburg--Landau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non--integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse--type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this dissertation, we develop a theoretical framework for these novel classes of solutions. In the first part, we use a traveling wave reduction or a so--called spatial approximation to comprehensively investigate the bifurcations of plane wave and periodic solutions of the CGLE. The primary tools used here are Singularity Theory and Hopf bifurcation theory respectively. Generalized and degenerate Hopf bifurcations have also been considered to track the emergence of global structure such as homoclinic orbits. However, these results appear difficult to correlate to the numerical bifurcation sequences of the dissipative solitons. In the second part of this dissertation, we shift gears to focus on the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. For this part, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the two alternative starting formulations for the Lagrangian are recent and not well explored. Also, after extensive discussions with David Kaup, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler--Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well--known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors constructed via the method of multiple scales yield solitons whose amplitudes are non--stationary or time dependent. In particular, pulsating, snake (and, less easily, creeping) dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves --- their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Results will be presented for the pulsating and snake soliton cases. Chaotic evolution of the trial function parameters in chaotic regimes identified using dynamical systems analysis would yield chaotic solitary waves. The method also holds promise for detailed modeling of chaotic solitons as well. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.

We study the fluctuation effects in the seminal cyclic predator-prey model in population dynamics due to Robert May and Warren Leonard both in the zero-dimensional and two-dimensional spatial version. We compute the mean time to extinction of a stable set of coexisting populations driven by large fluctuations. We see that the contribution of large fluctuations to extinction can be captured by a quasi-stationary approximation and the Wentzel–Kramers–Brillouin (WKB) eikonal ansatz. We see that near the Hopf bifurcation, extinctions are fast owing to the flat non-Gaussian distribution whereas away from the bifurcation, extinctions are dominated by large fluctuations of the fat tails of the distribution. We compare our results to Gillespie simulations and a single-species theoretical calculation. In addition, we study the spatio-temporal pattern formation of the stochastic May--Leonard model through the Doi-Peliti coherent state path integral formalism to obtain a coarse-grained Langevin description, i.e. the Complex Ginzburg Landau equation with stochastic noise in one complex field. We see that when one restricts the internal reaction noise to small amplitudes, one can obtain a simple form for the stochastic noise correlations that modify the Complex Ginzburg Landau equation. Finally, we study the effect of coupling a spatially extended May--Leonard model in two dimensions with symmetric predation rates to one with asymmetric rates that is prone to reach extinction. We show that the symmetric region induces otherwise unstable coexistence spiral patterns in the asymmetric May--Leonard lattice. We obtain the stability criterion for this pattern induction as we vary the strength of the extinction inducing asymmetry. This research was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-17-1-0156. / Doctor of Philosophy / In the field of ecology, the cyclic predator-prey patterns in a food web are relevant yet independent to the hierarchical archetype. We study the paradigmatic cyclic May--Leonard model of three species, both analytically and numerically. First, we employ well--established techniques in large-deviation theory to study the extinction of populations induced by large but rare fluctuations. In the zero--dimensional version of the model, we compare the mean time to extinction computed from the theory to numerical simulations. Secondly, we study the stochastic spatial version of the May--Leonard model and show that for values close to the Hopf bifurcation, in the limit of small fluctuations, we can map the coarse-grained description of the model to the Complex Ginsburg Landau Equation, with stochastic noise corrections. Finally, we explore the induction of ecodiversity through spatio-temporal spirals in the asymmetric version of the May--Leonard model, which is otherwise inclined to reach an extinction state. This is accomplished by coupling to a symmetric May-Leonard counterpart on a two-dimensional lattice. The coupled system creates conditions for spiral formation in the asymmetric subsystem, thus precluding extinction.