Physics Department Theses and Dissertations

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Beyond the Exceptional Point: Exploring the Features of Non-Hermitian PT Symmetric Systems
(2022-08) Agarwal, Kaustubh Shrikant; Joglekar, Yogesh N.; Vemuri, Gautam; Ou, Zhe “Jeff”; Petrache, Horia I.; Lukens, Joseph M.
Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold. We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples. In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model. Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.
Biophysical studies of cholesterol in unsaturated phospholipid model membranes
(2013) Williams, Justin A.; Wassall, Stephen R.; Decca, Ricardo; Petrache, Horia; Zhu, Fangqiang; Todd, Brian A.
Cellular membranes contain a staggering diversity of lipids. The lipids are heterogeneously distr ibuted to create regions, or domains, whose physical properties differ from the bulk membrane and play an essential role in modulating the function of resident proteins. Many basic questions pertaining to the formation of these lateral assemblies remain. T his research employs model membranes of well - defined composition to focus on the potential role of polyunsaturated fatty acids (PUFAs) and their interaction with cholesterol (chol) in restructuring the membrane environment. Omega - 3 (n - 3) PUFAs are the main bioactive components of fish oil, whose consumption alleviates a variety of health problems by a molecular mechanism that is unclear. We hypothesize that the incorporation of PUFAs into membrane lipids and the effect they have on molecular organization may be, in part, responsible. Chol is a major constituent in the plasma membrane of mammals. It determines the arrangement and collective properties of neighboring lipids, driving the formation of domains via differential affinity for different lipids . T he m olecular organization of 1 -[ 2 H 31 ]palmitoyl -2- eicosapentaenoylphosphatidylcholine (PEPC - d 31 ) and 1 -[ 2 H 31 ]palmitoyl -2- docosahexaenoylphosphatidylcholine (PDPC -d 31 ) in membran es with sphingomyelin (SM) and chol (1:1:1 mol) was compared by solid - state 2 H NMR spectroscopy. Eicosapentaenoic (EPA) and docosahexaenoic (DHA) acids are the two major n - 3 PUFAs found in fish oil, while PEPC -d 31 and PDPC -d 31 are phospholipids containing the respective PUFAs at the sn - 2 position and a perdeuterated palmitic acid a t the sn - 1 position . Analysis of s pectra recorded as a function of temperature indicate s that in both cases, formation of PUFA - rich (less ordered) and SM - rich (more ordered) domains occurred. A surprisingly substantial proportion of PUFA was found to infil trate the more ordered domain. There was almost twice as much DHA (65%) as EPA (30%) . The implication is that n - 3 PUFA s can incorporate into lipid rafts, which are domains enriched in SM and chol in the plasma membrane, and potentially disrupt the activity of signaling proteins that reside therein. DHA, furthermore, may be the more potent component of fish oil. PUFA - chol interactions were also examined through affinity measurements. A novel method utilizing electron paramagnetic resonance (EPR) was develope d, to monitor the partitioning of a spin - labeled analog of chol , 3β - doxyl - 5α - cholestane (chlstn), between large unilamellar vesicles (LUVs) and met hyl - β - cyclodextrin (mβCD). The EPR spectra for chlstn in the two environments are distinguishable due to the substantial differences in tumbling rates , allowing the population distribution ratio to be determined by spectral simulation. Advantages of this approach include speed of implementation and a vo idance of potential artifact s associated with physical separation of LUV and mβCD . Additionally, in a check of the method, t he relative partition coefficients between lipids measured for the spin label analog agree with values obtained for chol by isothermal titration calorimetry (ITC). Results from LUV with different composition confirmed a hierarchy of decreased sterol affinity for phospholipids with increasing acyl chain unsaturation , PDPC possessing half the affinity of the corresponding monounsaturated phospholipid. Taken together, the results of these studies on model membranes demonstrate the potential for PUFA - driven alteration of the architecture of biomembranes, a mechanism through which human health may be impacted.
Characterization and Measurement of Hybrid Gas Journal Bearings
(2012-08-28) Lawrence, Tom Marquis; Kemple, Marvin D.; Decca, Ricardo; Joglekar, Yogesh; Petrache, Horia; Akay, Hassan; Krousgrill, Charles Morton
This thesis concentrates on the study of hybrid gas journal bearings (bearings with externally pressurized mass addition). It differs from most work in that it goes back to “basics” to explore the hydrodynamic phenomena in the bearing gap. The thesis compares geometrically identical bearings with 2 configurations of external pressurization, porous liners where mass-addition compensation is varied by varying the liner’s permeability, and bushings with 2 rows of 6 feedholes where the mass-addition compensation is varied by the feedhole diameter. Experimentally, prototype bearings with mass-addition compensation that spans 2 orders of magnitude with differing clearances are built and their aerostatic properties and mass addition characteristics are thoroughly tested. The fundamental equations for compressible, laminar, Poiseuille flow are used to suggest how the mass flow “compensation” should be mathematically modeled. This is back-checked against the experimental mass flow measurements and is used to determine a mass-addition compensation parameter (called Kmeas) for each prototype bushing. In so doing, the methodology of modeling and measuring the mass addition in a hybrid gas bearing is re-examined and an innovative, practical, and simple method is found that makes it possible to make an “apples-to-apples” comparison between different configurations of external pressurization. This mass addition model is used in conjunction with the Reynolds equation to perform theory-based numerical analysis of virtual hybrid gas journal bearings (CFD experiments). The first CFD experiments performed use virtual bearings modeled to be identical to the experimental prototypes and replicate the experimental work. The results are compared and the CFD model is validated. The ontological significance of appropriate dimensionless similitude parameters is re-examined and a, previously lacking, complete set of similitude factors is found for hybrid bearings. A new practical method is developed to study in unprecedented detail the aerostatic component of the hybrid bearings. It is used to definitively compare the feedhole bearings to the porous liner bearings. The hydrostatic bearing efficiency (HBE) is defined and it is determined that the maximum achievable hydrostatic bearing efficiency (MAHBE) is determined solely by the bearing’s mass addition configuration. The MAHBE of the porous liner bearings is determined to be over 5 times that of the feedhole bearings. The method also presents a means to tune the Kmeas to the clearance to achieve the MAHBE as well as giving a complete mapping of the hitherto misunderstood complex shapes of aerostatic load versus radial deflection curves. This method also rediscovers the obscure phenomenon of static instability which is called in this thesis the “near surface effect” and appears to be the first work to present a practical method to predict the range of static instability and quantify its resultant stiffness fall-off. It determines that porous liner type bearings are not subject to the phenomenon which appears for feedhole type bearings when the clearance exceeds a critical value relative to its mass-addition compensation. The standing pressure waves of hydrostatic and hybrid bearings with the 2 configurations of external pressurization as well as a geometrically identical hydrodynamic bearing are studied in detail under the methodology of the “CFD microscope”. This method is used to characterize and identify the development, growth, and movement of the pressure wave extrema with increased hydrodynamic action (either increasing speed or increasing eccentricity). This method is also used to determine the “cause” of the “near surface effect”. A gedanken experiment is performed based on these results which indicates that a bearing with a “stronger aerostatic strength” component should be more stable than one with a low aerostatic strength component. Numerical instability “speed limits” are found that are also related to the hydrostatic strength of the bearing. The local conditions in the standing waves are characterized in terms of their local Mach number, Knudsen number, Reynolds number, and Taylor Number. It is concluded that low eccentricity bearing whirl can be attributed to the off load-line orientation of the bearing load force caused by the overlay of the hydrodynamic bearing standing wave onto the hydrostatic bearing wave of the hybrid bearing, whereas it is hypothesized that aperiodic and random self-excited vibration which occurs at high eccentricity, as reported in the literature, is probably due to shock waves, turbulence, near surface effect, and slip at local areas of the standing wave.
Developing an approach to improve beta-phase properties in ferroelectric pvd-hfp thin films
(2020-05) Dale, Ashley S.; Cheng, Ruihua; Petrache, Horia; Wassall, Stephen
Improved fabrication of poly(vinylindenefluoride)-hexafluoropropylene (PVDF-HFP) thin films is of particular interest due to the high electric coercivity found in the beta-phase structure of the thin film. We show that it is possible to obtain high-quality, beta-phase dominant PVDF-HFP thin films using a direct approach to Langmuir-Blodgett deposition without the use of annealing or additives. To improve sample quality, an automated Langmuir-Blodgett thin film deposition system was developed; a custom dipping trough was fabricated, a sample dipping mechanism was designed and constructed, and the system was automated using custom LabVIEW software. Samples were fabricated in the form of ferroelectric capacitors on substrates of glass and silicon, and implement a unique step design with a bottom electrode of copper with an aluminum wetting layer and a top electrode of gold with an aluminum wetting layer. Samples were then characterized using a custom ferroelectric measurement program implemented in LabVIEW with a Keithley picoammeter/voltage supply to confirm electric coercivity properties. Further characterization using scanning electron microscopy and atomic force microscopy confirmed the improvement in thin film fabrication over previous methods.
The Effect of Acyl Chain Unsaturation on Phospholipid Bilayer
(2010-02-26T17:51:02Z) Soni, Smita Pravin; Wassall, Stephen R.; Petrache, Horia; Kemple, Marvin D.; Rader, Andrew J.
Each biological cell is surrounded by a membrane that consists of many different kinds of lipids. The lipids are mainly composed of phospholipids, which form a fluid bilayer that serves as the platform for the function of membrane bound proteins regulating cellular activity. In the research described in this thesis we employed solid state 2H NMR, complemented by DSC (differential scanning calorimetry) and MD (molecular dynamics) simulations, to study the effect of PUFA (polyunsaturated fatty acids) and TFA (trans fatty acids) on molecular organization in protein-free model membranes of controlled composition. These two classes of unsaturated fatty acid incorporate into membrane lipids and have, respectively, a beneficial and harmful impact on health. The aim is to gain insight into the molecular origin of this behavior. DHA (docosahexaenoic acid), which with 6 "natural" cis double bonds is the most highly unsaturated PUFA found in fish oils, and EA (elaidic acid), which with only a single "unnatural" trans double bond is the simplest manmade TFA often found in commercially produced food, were the focus. 2H NMR spectra for [2H31]-N-palmitoylsphingomyelin ([2H31]16:0SM) in SM/16:0-22:6PE (1-palmitoyl-2-docosahexaenoylphosphatidylethanolamine)/cholesterol (1:1:1 mol) mixed membranes were recorded. This system served as our PUFA-containing model. The spectra are consistent with lateral separation into nano-sized (< 20 nm) domains that are SM-rich/cholesterol-rich (raft), characterized by higher chain order, and DHA-rich/cholesterol-poor (non-raft), characterized by lower chain order. The aversion cholesterol has for DHA, as opposed to the affinity cholesterol has for predominantly saturated SM, excludes the sterol from DHA-containing PE-rich domains and DHA from SM-rich/cholesterol-rich domains. It is the formation of highly disordered membrane domains that we hypothesize is responsible, in part, for the diverse health benefits associated with dietary consumption of DHA. 2H NMR spectra for 1-elaidoyl-2-[2H35]stearoylphosphatidylcholine (t18:1-[2H35]18:0PC) and 1-oleoyl-2-[2H35]stearoylphosphatidylcholine (c18:1-[2H35]18:0PC) were recorded to compare membranes with respect to a trans vs. cis ("natural") double bond. The spectra indicate that while a trans double bond produces a smaller deviation from linear conformation than a cis double bond, membrane order is decreased by a comparable amount because the energy barrier to rotation about the C-C single bonds either side of a trans or cis double bond is reduced. Although EA adopts a conformation somewhat resembling a saturated fatty acid, the TFA is almost as disordered as its cis counterpart oleic acid (OA). We speculate that EA could be mistaken for a saturated fatty acid and infiltrate lipid rafts to disrupt the high order therein that is necessary for the function of signaling proteins.
Exceptional Points and their Consequences in Open, Minimal Quantum Systems
(2022-08) Muldoon, Jacob E.; Joglekar, Yogesh; Decca, Ricardo; Cheng, Rui; Vemuri, Gautam; Cincio, Lukasz
Open quantum systems have become a rapidly developing sector for research. Such systems present novel physical phenomena, such as topological chirality, enhanced sensitivity, and unidirectional invisibility resulting from both their non-equilibrium dynamics and the presence of exceptional points. We begin by introducing the core features of open systems governed by non-Hermitian Hamiltonians, providing the PT -dimer as an illustrative example. Proceeding, we introduce the Lindblad master equation which provides a working description of decoherence in quantum systems, and investigate its properties through the Decohering Dimer and periodic potentials. We then detail our preferred experimental apparatus governed by the Lindbladian. Finally, we introduce the Liouvillian, its relation to non-Hermitian Hamiltonians and Lindbladians, and through it investigate multiple properties of open quantum systems.
Induced magnetoelectric coupling at a ferroelectric-ferromagnetic interface
(2013-11-08) Carvell, Jeffrey David; Ruihua, Cheng; Joglekar, Yogesh; Decca, Ricardo; Petrache, Horia; Hu, Jiangping
Preparation and characterization of multiferroic materials in which ferroelectricity and ferromagnetism coexist would be a milestone for functionalized materials and devices. First, electric properties of polyvinylidene (PVDF) films fabricated using the Langmuir-Schaefer method have been studied. Films of different thickness were deposited on silicon substrates and analyzed using several techniques. X-ray diffraction (XRD) data showed that PVDF films crystallize at an annealing temperature above 130 °C. Polarization versus electric field (PE) ferroelectric measurements were done for samples prepared with electrodes. PE measurements show that the coercivity of the films increases as the maximum applied electric field increases. The coercivity dependence on the frequency of the applied electric field can be fitted as . The results also show that the coercivity decreases with increasing the thickness of PVDF film due to the pinning effect. Next, we have demonstrated that those PVDF properties can be controlled by applying an external magnetic field. Samples were created in a layered heterostructure, starting with a Fe thin film, PVDF above that, and followed by another thin film of Fe. Extended X-ray absorption fine structure (EXAFS) spectroscopy was used to study the interface between PVDF polymer films and ferromagnetic iron thin films. Conventional EXAFS was applied to identify the structure of a Fe film sandwiched between two PVDF layers. An electric signal was then applied to the polymer to study the effects polarizing the polymer has on the Fe atoms at the interface. This shows that the Fe atoms diffuse into the PVDF layer at the interface between the two layers. Polarizing the film causes further diffusion of Fe atoms into the polymer. We also found that as the applied magnetic field is changed, the switching of electric polarization for the PVDF displayed a dependence on the external magnetic field. We also noticed that both the coercivity and polarization for the PVDF polymer display hysteretic features as the applied magnetic field is changed. We also found that the thickness of both the iron layers and the PVDF layer has an effect on the magnetoelectric coupling in our samples. The same strain applied to a thicker PVDF layer becomes tougher to flip the polarization compared to a thinner PVDF layer. As the iron film thickness increases, the strain also increases, and the polarization of the PVDF polymer is more easily flipped. We also found that the magnetoelectric sensitivity increases as both the PVDF and iron layers increase in thickness. We have shown that it is possible to control the ferroelectric properties of a PVDF film by tuning the magnetic field in a heterostructure. Our experiments show a coupling between the electric polarization and applied magnetic field in multiferroic heterostructures much larger than any previously reported values. Previous reports have used inorganic materials for the ferroelectric layer. Organic polymers have an electric dipole originating at the molecular level due to atoms with different electronegativity that are free to rotate. To flip the polarization, the chains must rotate and the position of the atoms must change. This increases the force felt locally by those chains. Using this polymer, we are able to increase the magnetoelectric coupling.
Injection Current Modulated Parity-Time Symmetry in Coupled Semiconductor Lasers
(2021-08) Thomas, Luke; Vemuri, Gautam; Gavrin, Andrew; Petrache, Horia; Wassall, Stephen
This research investigates the characteristics of Parity Time symmetry breaking in two optically coupled, time delayed semiconductor lasers. A theoretical model is used to describe the controllable parameters in the experiment and intensity output of the coupled lasers. The PT parameters we control are the spatial separation between the two lasers, the frequency detuning, and the coupling strength. We find that the experimental data agrees with the predictions from the theoretical model confirming the intensity behaviors of the lasers, and the monotonic change in PT-threshold as a function of coupling scaled by the time delay
Insights on PUFA-containing lipid membranes probed by MD simulations
(2017-04-14) Leng, Xiaoling; Wassall, Stephen
The cell membrane serves as a barrier between the interior and exterior of a living cell. Its main structural component is the lipid bilayer, which is composed of various kinds of lipids that segregate into domains. These lipid domains, distinguished in composition and physical properties from the bulk lipids that surround them, are believed to modulate the function of resident proteins by providing an appropriate lipid environment. Polyunsaturated fatty acids (PUFA) are a type of fatty acid that contain multiple C=C double bonds. They have a lot of health benefits, which may originate in part due to their incorporation into lipids in the plasma membrane. Hypotheses that PUFA-containing lipids laterally separate into domains and/or modulate the structure of existing domains have been raised to explain the fundamental role played by PUFA. In our research, we use molecular dynamics (MD) simulations to simulate model membranes composed of PUFA-containing phospholipids and to investigate their interaction with cholesterol and vitamin E that are influential membrane constituents. The presumptive function for vitamin E in membranes is to protect PUFA against oxidation. Although the chemistry of the process is well established, the role played by the molecular structure that we address with atomistic molecular dynamics (MD) simulations remains controversial. We compared the behavior of vitamin E in lipid bilayers composed of 1-stearoyl-2-docosahexaenoylphosphatidylcholine (SDPC, 18:0-22-6PC) and 1-stearoyl-2-oleoylphosphatidylcholine (SOPC, 18:0-18:1PC) via all-atom MD simulations at 37° C. SDPC represents a PUFA-containing lipid, and SOPC serves as monounsaturated control. From the calculation of van der Waals energy of interaction between vitamin E and fatty acid (FA) chains, we found higher probability that the PUFA chains surround the chromanol head group on vitamin E. This is further demonstrated by probability density maps of acyl chains around vitamin E molecules. Also, an ability to more easily penetrate deep into the PUFA containing bilayer of vitamin E is detected by faster flip-flop rate of vitamin E observed in the SDPC bilayers. These results showed that the high disorder of polyunsaturated docosahexaenoic acid (DHA) chains allows vitamin E to easily tunnel down into the bilayer and often brings the PUFA chains up to the surface of the bilayer, improving the likelihood that the reactive (hydroxyl) group on vitamin E would encounter a lipid peroxyl radical and terminate the oxidation process. Thus, the simulations indicate that the molecular structure of vitamin E supports its role as an antioxidant in a PUFA-containing membrane. A subsequent study on the partitioning of vitamin E into PUFA-containing lipids was done by analyzing the binding energy of vitamin E in the corresponding lipid bilayer. The binding energy is obtained from the potential of mean force (PMF) profile of vitamin E alone the membrane normal direction (z), which is calculated from umbrella sampling MD simulations. We found the binding in SDPC is smaller in SOPC, indicating that vitamin E does not prefer PUFA-containing phospholipids. The flip-flop rate was also estimated from the PMF profile, confirming that vitamin E flip-flops across the SDPC bilayer more easily than the SOPC bilayer. From the simulations it was noted that the membrane deforms as vitamin E is pulled out, which suggests interactions between the phospholipids contribute to the binding energy of the vitamin E. In a final study, a comparison was made between the effect on membrane organization of the three types of long chain omega-3 (n-3) PUFA found in fish oils: eicosapentaenoic acid (EPA, 20:5), DHA (22:6) and docosapentaenoic acid (DPA, 22:5). MD simulations were run on lipid bilayers composed of 1-stearoyl-2-eicosapentaenoylphosphatidylcholine (EPA-PC, 18:0-20:5PC), 1-stearoyl-2-docosapentaenoylphosphatidylcholine (DPA-PC, 18:0-22:5PC), SDPC (DHA-PC, 18:0-22:6PC) and, as a monounsaturated control, SOPC (OA-PC, 18:0-18:1PC) in the absence and presence of cholesterol. By analyzing the physical properties such as membrane order and thickness, we found all three n-3 PUFAs disorder the membrane. The disordering is greatest with EPA and least with DPA. Unique among the n-3 PUFA-containing membranes, there is region of high order in the upper portion of the DPA chain. The PUFA-containing lipids were found to less favorably interact with cholesterol compared to the OA-containing lipid, which is caused by their disorder. We speculate that differences between DPA, DHA and EPA might potentially modulate their effect on lipid domain formation.
An investigation of parity and time-reversal symmetry breaking in tight-binding lattices
(2014) Scott, Derek Douglas; Joglekar, Yogesh; Decca, Ricardo; Petrache, Horia; Tarasov, Vitaly; Csathy, Gabor
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.

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