Browsing by Author "Chen, Hua, committee member"

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Open Access
A quantum H*(T)-module via quasimap invariants
(Colorado State University. Libraries, 2024) Lee, Jae Hwang, author; Shoemaker, Mark, advisor; Cavalieri, Renzo, advisor; Gillespie, Maria, committee member; Peterson, Christopher, committee member; Hulpke, Alexander, committee member; Chen, Hua, committee member
For X a smooth projective variety, the quantum cohomology ring QH*(X) is a deformation of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. When X is toric with geometric quotient description V//T, the cohomology ring H*(V//T) also has the structure of a H*(T)-module. In this paper, we introduce a new deformation of the cohomology of X using quasimap invariants with a light point. This defines a quantum H*(T)-module structure on H*(X) through a modified version of the WDVV equations. We explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a semipositive toric variety.
Open Access
Expected distances on homogeneous manifolds and notes on pattern formation
(Colorado State University. Libraries, 2023) Balch, Brenden, author; Shipman, Patrick, advisor; Bradley, Mark, committee member; Shonkwiler, Clay, committee member; Peterson, Chris, committee member; Chen, Hua, committee member
Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In Chapter 1 of this dissertation, we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data. Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In Chapter 2, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces. Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE) in Chapter 3. The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results. Chapter 4 explores the measurement and characterization of order in non-equilibrium pattern forming systems. The study focuses on the use of topological measures of order, via persistent homology and the Wasserstein metric. We investigate the quantification of order with respect to ideal lattice patterns and demonstrate the effectiveness of the introduced measures of order in analyzing imperfect three-dimensional patterns and their time evolution. The paper provides valuable insights into the complex pattern formation and contributes to the understanding of order in three dimensions.
Embargo
Investigations of low-temperature reaction pathways in solid-state reactions
(Colorado State University. Libraries, 2024) Tran, Gia Thinh, author; Neilson, James R., advisor; Prieto, Amy L., committee member; Sambur, Justin B., committee member; Chen, Hua, committee member
Advances in our technology are limited by our knowledge of functional materials, and access to new, possibly better, functional materials is limited by our synthesis methods. This dissertation discusses different synthesis methods for a variety of solid state materials. At the core of this thesis are metathesis reactions i.e. double displacement reactions. Metathesis reactions allow for control over product selectivity and reaction kinetics with choice of the spectating ions. We demonstrate these characteristics with different spectating ions in metathesis and cometathesis (e.g., combining 2 halides) reactions. LaMnO3 was chosen to probe the product selectivity of anion cometathesis towards specific off-stoichiometries of LaMnO3. The metathesis reaction for BiFeO3 illustrates that prediction of thermodynamic selectivity is important, but reaction kinetics remain important as well. Kinetic studies of metathesis reactions that form YMnO3 demonstrate the importance of crystalline intermediates to modulate the reaction rates. The complexity of solid-state kinetics their kinetic regimes within a reaction can be identified through synchrotron X-ray diffraction. We attempted to synthesize LiMoO2 as precursors for the proposed phase LaMoO3. We demonstrate our considerations on the synthesis challenges and offer gained insights into alternative Mo-based systems (nitrides). Aside from metathesis reactions, we employ learned concepts to flux reactions to influence the chemical potential of N2. Synthesis of Li-Fe-O-N and Li-Mn-O-N phases was attempted under the hypothesis that alkali halide salt mixtures solubilize nitrogen and pin nitrogen's chemical potential to prevent N2 formation. Cs2SbCl6 was chosen as a single crystal target to gain clearer insights into the electronic structure. Single crystals were synthesized via hydrothermal synthesis, but preliminary conductivity measurements suggest that Cs2SbCl6 has a photoconductance below our limit detection.
Open Access
Quantum magnetism in the rare-earth pyrosilicates
(Colorado State University. Libraries, 2021) Hester, Gavin L., author; Ross, Kate, advisor; Chen, Hua, committee member; Gelfand, Martin, committee member; Shores, Matthew, committee member
In recent years, both physicists and non-physicists have shown immense interest in the burgeoning field of quantum computing and the possible applications a quantum computer could be used for [1]. However, current quantum computers suffer from issues of decoherence: where the quantum state used for computation is broken by external noise. A new possible avenue for quantum computation would be to use systems that are intrinsically protected from some level of noise, such as topologically protected states. Topological states are inherently protected from small perturbations due to their topological nature. However, to exploit this feature of topologically protected systems more experimental realizations are needed to better understand the underlying mechanisms. This has motivated a surge in interest of condensed matter systems with topologically protected states, such as the quantum spin liquid or fractional quantum Hall systems. A current focus in the subfield of quantum magnetism has focused on using the anisotropic exchange properties of the rare-earth (La - Lu) ions to find quantum spin liquid states, such as the Kitaev spin liquid that is predicted for systems exhibiting a honeycomb lattice. The Kitaev model is an exactly solvable model with a quantum spin liquid ground state, allowing for precise comparison between experiment and theory. Currently, no system has been rigorously proven to be a Kitaev spin liquid but developing our understanding of the underlying physical mechanisms in these systems may allow for the "engineering" of systems that are likely to be Kitaev spin liquids. The desire to understand the underlying mechanisms for quantum spin liquids and other quantum ground states led to the study of the three-honeycomb rare-earth pyrosilicate compounds discussed in this dissertation. The first compound, Yb2Si2O7, is a quantum dimer magnet system with the first evidence for a rare-earth based triplon Bose-Einstein condensate. Inelastic neutronscattering, specific heat, and ultrasound velocity measurements showed a characteristic (for triplon Bose-Einstein condensates) dome in the field-temperature phase diagram and provided evidence for predominantly isotropic exchange, something that is not typically expected for rare-earth systems. Following this work on Yb2Si2O7, our focused turned to two of the Er3+ rare-earth pyrosilicates. The first of these Er3+-based pyrosilicates measured was D-Er2Si2O7. Previous work on D-Er2Si2O7 discovered a highly anisotropic g-tensor, an antiferromagnetic ground state, and modeled some of the magnetic field induced transitions via Monte-Carlo simulations [2]. Our work followed up on this with AC susceptibility, powder inelastic neutron scattering, and powder neutron diffraction measurements to further investigate the ground state of this quantum magnet. Through this we discovered that the system enters an antiferromagnetic state with the spins almost aligned along the previously determined local Ising-axis [2]. The inelastic neutron scattering spectrum show a gapped excitation at zero field - consistent with Ising-like exchange. Transverse field AC susceptibility shows a change in the susceptibility at 2.65 T. These signatures indicate that D-Er2Si2O7 exhibits predominantly Ising-like exchange and that a transition can be induced by a field applied transverse to the Ising axis. This allows for the possibility of D-Er2Si2O7 bein g a new experimental realization of the Transverse Field Ising Model (TFIM). The TFIM is a simple, anisotropic exchange, theoretically tractable model exhibiting quantum criticality with few experimental examples, making new experimental examples of this model highly desired. These intriguing results on D-Er2Si2O7 and Yb2Si2O7 led to an interest in the polymorph formed at lower synthesis temperatures, C-Er2Si2O7, which happens to be isostructural to Yb2Si2O7. Measurements of the neutron diffraction, specific heat, and magnetization/susceptibility in this system allowed for us to determine that C-Er2Si2O7 magnetically orders at 2.3 K into an antiferromagnetic Néel state. While this is the expected ground state for an isotropically exchange coupled honeycomb system, C-Er2Si2O7 does not form a "perfect" honeycomb lattice and it is interesting that C-Er2Si2O7 magnetically orders while Yb2Si2O7 does not. Understanding the ground state for C-Er2Si2O7 will allow for bettering our understanding of Yb2Si2O7 and rare-earth quantum magnet ground states by comparing the properties of the two systems. Overall, the work on these three compounds required numerous experimental techniques, models, and theoretical understanding. It is my hope that the preliminary understanding for these three pyrosilicates will motivate future work within the rare-earth pyrosilicate family and provide a family of rare-earth quantum magnets that can be studied to improve our understanding of novel quantum states.