IA Scholar Query: Qed. Computing What Remains to Be Proved.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 01 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A Mathematics Pipeline to Student Success in Data Analytics through Course-Based Undergraduate Research
https://scholar.archive.org/work/2ekwbosf3ndjbevhm737yj6fla
This paper reports on Data Analytics Research (DAR), a course-based undergraduate research experience (CURE) in which undergraduate students conduct data analysis research on open realworld problems for industry, university, and community clients. We describe how DAR, offered by the Mathematical Sciences Department at Rensselaer Polytechnic Institute (RPI), is an essential part of an early low-barrier pipeline into data analytics studies and careers for diverse students. Students first take a foundational course, typically Introduction to Data Mathematics, that teaches linear algebra, data analytics, and R programming simultaneously using a project-based learning (PBL) approach. Then in DAR, students work in teams on open applied data analytics research problems provided by the clients. We describe the DAR organization which is inspired in part by agile software development practices. Students meet for coaching sessions with instructors multiple times a week and present to clients frequently. In a fully remote format during the pandemic, the students continued to be highly successful and engaged in COVID-19 research producing significant results as indicated by deployed online applications, refereed papers, and conference presentations. Formal evaluation shows that the pipeline of the single on-ramp course followed by DAR addressing real-world problems with societal benefits is highly effective at developing students' data analytics skills, advancing creative problem solvers who can work both independently and in teams, and attracting students to further studies and careers in data science.Kristin P. Bennett, John S. Erickson, Amy Svirsky, Josephine C. Seddonwork_2ekwbosf3ndjbevhm737yj6flaThu, 01 Dec 2022 00:00:00 GMTVortices in Bose-Einstein Condensates
https://scholar.archive.org/work/3xxk3e7e4zblflwlrzjjjrdw7m
This thesis surveys aspects of the very broad topic of vortex dynamics in Bose-Einstein condensates. The Gross-Pitaevskii equation (GPE) is an important and popular tool in this regard, and we discuss a number of its properties which are relevant to the study of vortex dynamics. Using the GPE, we explore the dynamics of point vortices in Bose-Einstein condensates, writing hydrodynamic relations we have not encountered elsewhere in the literature. For incompressible condensates, we pay particular attention to discussions of the structure of the Hamiltonian system, and we revisit the topic of linear stability analysis for a steadily rotating, regular polygon of vortices. Furthermore, geometric mechanics is a powerful tool in the study of fluid dynamics, and we make use of the language of geometric mechanics, in particular the Lie derivative, to gain insights into some of the consequences of describing superfluid dynamics using the GPE. We have not encountered the relations we have derived as a result of this effort elsewhere in the literature. The topics of vortex formation, general approaches to describing vortex dynamics and phonon emission are also discussed.Omofolarin Alex-Duduyemi, Henrik Jensenwork_3xxk3e7e4zblflwlrzjjjrdw7mThu, 22 Sep 2022 00:00:00 GMTA First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL
https://scholar.archive.org/work/zobvalu4sfcjzofd2qshxlfaeq
We formalize a multivariate quantifier elimination (QE) algorithm in the theorem prover Isabelle/HOL. Our algorithm is complete, in that it is able to reduce any quantified formula in the first-order logic of real arithmetic to a logically equivalent quantifier-free formula. The algorithm we formalize is a hybrid mixture of Tarski's original QE algorithm and the Ben-Or, Kozen, and Reif algorithm, and it is the first complete multivariate QE algorithm formalized in Isabelle/HOL.Katherine Kosaian, Yong Kiam Tan, André Platzerwork_zobvalu4sfcjzofd2qshxlfaeqThu, 22 Sep 2022 00:00:00 GMTEfficient Extensional Binary Tries
https://scholar.archive.org/work/pfg2pflotjbtlpnzphapf42t74
Lookup tables (finite maps) are a ubiquitous data structure. In pure functional languages they are best represented using trees instead of hash tables. In pure functional languages within constructive logic, without a primitive integer type, they are well represented using binary tries instead of search trees. In this work, we introduce canonical binary tries, an improved binary-trie data structure that enjoys a natural extensionality property, quite useful in proofs, and supports sparseness more efficiently. We provide full proofs of correctness in Coq. We provide microbenchmark measurements of canonical binary tries versus several other data structures for finite maps, in a variety of application contexts; as well as measurement of canonical versus original tries in two big, real systems. The application context of data structures contained in theorem statements imposes unusual requirements for which canonical tries are particularly well suited.Andrew W Appel, Xavier Leroywork_pfg2pflotjbtlpnzphapf42t74Wed, 21 Sep 2022 00:00:00 GMTPion-mediated Cooper pairing of neutrons: beyond the bare vertex approximation
https://scholar.archive.org/work/d4fl3dcnjbde7cme2cxnuqzuei
In some quantum many particle systems, the fermions could form Cooper pairs by exchanging intermediate bosons. This then drives a superconducting phase transition or a superfluid transition. Such transitions should be theoretically investigated by using proper non-perturbative methods. Here we take the neutron superfluid transition as an example and study the Cooper pairing of neutrons mediated by neutral π-mesons in the low density region of a neutron matter. We perform a non-perturbative analysis of the neutron-meson coupling and compute the superfluid gap Δ, the critical density ρ_c, and the critical temperature T_c by solving the Dyson-Schwinger equation of the neutron propagator. We first carry out calculations under the widely used bare vertex approximation and then incorporate the contribution of the lowest-order vertex correction. This vertex correction is not negligible even at low densities and its importance is further enhanced as the density increases. The transition critical line on density-temperature plane obtained under the bare vertex approximation is substantially changed after including the vertex correction. These results indicate that the vertex corrections play a significant role and need to be seriously taken into account.Hao-Fu Zhu, Guo-Zhu Liuwork_d4fl3dcnjbde7cme2cxnuqzueiWed, 21 Sep 2022 00:00:00 GMTEntanglement between Valence and Sea Quarks in Hadrons of 1+1 Dimensional QCD
https://scholar.archive.org/work/7ckeca5doffihhmodtluczmqvu
The conceptual interpretation of valence- and sea-quark separation, which is a key aspect of the parton model and of an intuitive picture of hadron structure, becomes obscured by quantum effects in QCD. This suggests that there may be measures of entanglement between quark degrees of freedom that are present in QCD, but absent in the intuitive picture with a clear valence-sea (VS) separation. In this paper, we define the first rigorous measure of VS entanglement in QCD in an attempt to bring conceptual clarity to this issue, and, potentially, to find a measure of the applicability of the parton model to QCD bound states. This VS entanglement vanishes in the large-N_c limit, and it remains low when finite-N_c states resemble their large-N_c counterparts. We perform a numerical study of VS entanglement in 1+1 dimensional discrete light-cone quantized QCD, and in the process develop a method for building the color-singlet basis of 1+1d QCD that is manifestly complete and orthogonal by construction. We calculate this VS entanglement entropy for the first time and find that it is relatively low for the first few excited states of both mesons and baryons compared to all other states in the spectrum, with the VS entropy of ground state hadrons providing a minimum. We also see that for ground state mesons the entropy is well described in the 1/N_c approximation. These results suggest that low energy hadrons may be the only QCD bound states for which the large-N_c expansion, and perhaps the parton model, provide an accurate description. This work also provides the first evidence that the VS entanglement entropy of QCD in 3+1d, which would likely serve as an order parameter for the transition between quark and hadron degrees of freedom, may be perturbatively accessible through a large-N_c expansion.Peter J. Ehlerswork_7ckeca5doffihhmodtluczmqvuTue, 20 Sep 2022 00:00:00 GMTA Solution of the 4th Clay Millennium Problem about the Navier-Stokes Equations
https://scholar.archive.org/work/2gvd54husfdv3dooe4cb2gxssy
In this paper it is solved the 4 th Clay Millennium problem about the Navier-Stokes equations, in the direction of regularity (no blow-up). This is proved for the Navier-Stokes equations for the non-periodic formulation and without external forcing (homogeneous case).The proof is based on discovering a new invariant as a 2D surface density of (rotatory) momentum, derived from the well-known Helmholtz-Kelvin-Stokes velocity circulation invariant. This invariant is indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity..It is proved that not only there is no Blow-up in finite time but not even at the time T=+∞.Konstantinos E. Kyritsis, University Of Ioanninawork_2gvd54husfdv3dooe4cb2gxssyMon, 19 Sep 2022 00:00:00 GMTOn the complex structure of Yang-Mills theory
https://scholar.archive.org/work/fbxr2qvup5hkjm24wrd6jpzkky
We consider the coupled set of spectral Dyson-Schwinger equations in Yang-Mills theory for ghost and gluon propagators, which gives us access to the ghost and gluon spectral functions. The set-up is used for a systematic analytic evaluation of the constraints on generalised spectral representations in Yang-Mills theory that are most relevant for informed spectral reconstructions. We also provide numerical results for the coupled set of spectral functions for a large range of potential mass gaps of the gluon, and discuss the limitations and extensions of the present work.Jan Horak, Jan M. Pawlowski, Nicolas Winkwork_fbxr2qvup5hkjm24wrd6jpzkkyFri, 16 Sep 2022 00:00:00 GMTBrownian Axion-like particles
https://scholar.archive.org/work/su7zg42fh5gvzeb5fhut52stoy
We study the non-equilibrium dynamics of a pseudoscalar axion-like particle (ALP) weakly coupled to degrees of freedom in thermal equilibrium by obtaining its reduced density matrix. Its time evolution is determined by the in-in effective action which we obtain to leading order in the (ALP) coupling but to all orders in the couplings of the bath to other fields within or beyond the standard model. The effective equation of motion for the (ALP) is a Langevin equation with noise and friction kernels obeying the fluctuation dissipation relation. A "misaligned" initial condition yields damped coherent oscillations, however, the (ALP) population increases towards thermalization with the bath. As a result, the energy density features a mixture of a cold component from misalignment and a hot component from thermalization with proportions that vary in time (cold) e^-Γ t+(hot) (1-e^-Γ t), providing a scenario wherein the "warmth" of the dark matter evolves in time from colder to hotter. As a specific example we consider the (ALP)-photon coupling g a E⃗·B⃗ to lowest order, valid from recombination onwards. For T ≫ m_a the long-wavelength relaxation rate is substantially enhanced Γ_T = g^2 m^2_a T/16π. The ultraviolet divergences of the (ALP) self-energy require higher order derivative terms in the effective action. We find that at high temperature, the finite temperature effective mass of the (ALP) is m^2_a(T) = m^2_a(0)[ 1-(T/T_c)^4], with T_c ∝√(m_a(0)/g), suggesting the possibility of an inverted phase transition, which when combined with higher derivatives may possibly indicate exotic new phases. We discuss possible cosmological consequences on structure formation and the effective number of relativistic species.Shuyang Cao, Daniel Boyanovskywork_su7zg42fh5gvzeb5fhut52stoyFri, 16 Sep 2022 00:00:00 GMTMultiloop Feynman integrals for precision calculations in Quantum Chromodynamics
https://scholar.archive.org/work/mdfmhlvpejehhmdobtnwucpqfa
In this thesis we present analytic results for the calculation of multiloop Feynman integrals contributing to virtual corrections of various 2 → 2 and 2 → 3 scattering processes in Quantum Chromodynamics. The first part consists of an overview of the current methods and tools for the analytic computation of loop integrals and the manipulation of the special functions that appear in their results. Using Integration-By-Parts identities, one can reduce all multiloop Feynman integrals to a so-called finite basis of master integrals. We construct a computational framework for the analytic calculation of these master integrals, based on the Simplified Differential Equations approach, in conjunction with the ideas of working with a specific basis of master integrals that satisfies so-called canonical differential equations. The method of expansion-by-regions is employed for the determination of the necessary boundary terms. In the second part we present results for specific one-, two-and three-loop master integrals. We first consider planar three-loop master integrals relevant to 2 → 2 scattering with one external leg off-shell. These results contribute to the scattering amplitudes for a vector boson decaying to 3-jets or gg → H + jet in gluon fusion at Next-to-Next-to-Next-to-Leading-Order (N3LO). Furthermore we present one-and two-loop results for five-point scattering with on off-shell leg that are relevant to NNLO corrections to scattering processes such as W + 2 jets production at the LHC. Finally we study several one-loop five-point master integrals involving massless propagators and up to three off-shell legs and one massive propagator with up to two off-shell legs.Nikolaos Syrrakos, National Technological University Of Athenswork_mdfmhlvpejehhmdobtnwucpqfaThu, 15 Sep 2022 00:00:00 GMTAt What Level Should One Cluster Standard Errors in Paired and Small-Strata Experiments?
https://scholar.archive.org/work/yj5ztsmptvhgndsqhpso7jakpu
In clustered and paired experiments, to estimate treatment effects, researchers often regress their outcome on the treatment and pair fixed effects, clustering standard errors at the unit-of-randomization level. We show that even if the treatment has no effect, a 5%-level t-test based on this regression will wrongly conclude that the treatment has an effect up to 16.5% of the time, an error rate much larger than the researcher's 5% target. To achieve their targeted error rate, researchers should instead cluster standard errors at the pair level. Using simulations, we show that similar results apply to clustered experiments with small strata.Clément de Chaisemartin, Jaime Ramirez-Cuellarwork_yj5ztsmptvhgndsqhpso7jakpuThu, 15 Sep 2022 00:00:00 GMTMulti-Channel Luttinger Liquids at the Edge of Quantum Hall Systems
https://scholar.archive.org/work/gc6i5ikltjg7hm4m2eu2wyeihe
We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory.Vieri Mastropietro, Marcello Portawork_gc6i5ikltjg7hm4m2eu2wyeiheThu, 15 Sep 2022 00:00:00 GMTOn Schrödingerist Quantum Thermodynamics
https://scholar.archive.org/work/uaehy3tqzzhcjozow6umskqcce
From the point of view of Schr\"odingerism, a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions, rather than classical particle configurations or Copenhagenist "found" values. Recapitulating the historical sequence, we consider here several models of magnets that classically can exhibit a phase transition to a low-temperature magnetized state. We formulate wavefunction analogues including a "Schr\"odingerist QUantum Ising Model" (SQUIM), a "Schr\"odingerist Curie-Weiss Model" (SCWM), and others. We show that the SQUIM with free boundary conditions and distinguishable "spins" has no finite-temperature phase transition, which we attribute to entropy swamping energy. The SCWM likewise, even assuming exchange symmetry in the wavefunction. But a variant model with "wavefunction energy" (introduced in prior communications about Schr\"odingerism and the Measurement Problem) does have a phase transition to a magnetised state. Our principle technique involves transforming the problem to one in probability theory, then applying results from Large Deviations, particularly the G\"artner-Ellis Theorem. Finally, we discuss Gibbs vs. Boltzmann/Einstein entropy in the choice of the quantum thermodynamic ensemble, as well as open problems. PhySH: quantum theory, quantum statistical mechanics, large deviation & rare event statistics.Leonardo De Carlo, W. David Wickwork_uaehy3tqzzhcjozow6umskqcceWed, 14 Sep 2022 00:00:00 GMTThe Universe at the MeV era: neutrino evolution and cosmological observables
https://scholar.archive.org/work/ayycaeivabeyvookillwsvv5lu
Neutrino physics in the early Universe is key to our understanding of later cosmological stages, such as primordial nucleosynthesis (BBN) or the formation of large-scale structures. The coming decade promises new experimental results to explore and constrain cosmological models even more precisely - which requires robust theoretical predictions. This PhD thesis presents a study of the evolution of neutrinos in the first seconds after the Big Bang, more precisely when the temperature of the Universe is of the order of one mega-electronvolt. This evolution is obtained numerically by solving kinetic equations for which we propose a new derivation. A first application is the calculation of the so-called "standard" decoupling in order to calculate the cosmological parameter quantifying the energy density of the primordial relativistic species, N_eff, to a precision of a few ten-thousandths. This study has highlighted the possibility of effectively describing the phenomenon of flavour oscillations, taking advantage of the large separation of time scales involved. Such an approximation is then adapted and validated in the case of non-zero asymmetries between neutrinos and antineutrinos. Finally, we study semi-analytically the consequences of incomplete neutrino decoupling on BBN, in order to understand how the primordial abundances of helium and deuterium are affected by this physics.Julien Frousteywork_ayycaeivabeyvookillwsvv5luWed, 14 Sep 2022 00:00:00 GMTSnowmass Theory Frontier: Astrophysics and Cosmology
https://scholar.archive.org/work/xxfbp56zevetxffcazrya73sfy
We summarize progress made in theoretical astrophysics and cosmology over the past decade and areas of interest for the coming decade. This Report is prepared as the TF09 "Astrophysics and Cosmology" topical group summary for the Theory Frontier as part of the Snowmass 2021 process.Daniel Green, Joshua T. Ruderman, Benjamin R. Safdi, Jessie Shelton, Ana Achúcarro, Peter Adshead, Yashar Akrami, Masha Baryakhtar, Daniel Baumann, Asher Berlin, Nikita Blinov, Kimberly K. Boddy, Malte Buschmann, Giovanni Cabass, Robert Caldwell, Emanuele Castorina, Thomas Y. Chen, Xingang Chen, William Coulton, Djuna Croon, Yanou Cui, David Curtin, Francis-Yan Cyr-Racine, Christopher Dessert, Keith R. Dienes, Patrick Draper, Peizhi Du, Sebastian A. R. Ellis, Rouven Essig, Raphael Flauger, Chee Sheng Fong, Joshua W. Foster, Jacopo Fumagalli, Keisuke Harigaya, Shunsaku Horiuchi, Mikhail M. Ivanov, Yonatan Kahn, Simon Knapen, Rebecca K. Leane, Hayden Lee, Erik W. Lentz, Matthew Lewandowski, Mariangela Lisanti, Andrew J. Long, Marilena Loverde, Azadeh Maleknejad, Liam McAllister, Samuel D. McDermott, Robert McGehee, P. Daniel Meerburg, Joel Meyers, Azadeh Moradinezhad Dizgah, Moritz Münchmeyer, Nadav Joseph Outmezguine, Enrico Pajer, Gonzalo A. Palma, Aditya Parikh, Jong-Chul Park, Annika H. G. Peter, Guilherme L. Pimentel, Sébastien Renaux-Petel, Nicholas L. Rodd, Bibhushan Shakya, Gary Shiu, Eva Silverstein, Marko Simonovic, Rajeev Singh, Charlotte Sleight, Volodymyr Takhistov, Philip Tanedo, Massimo Taronna, Brooks Thomas, Natalia Toro, Yu-Dai Tsai, Edoardo Vitagliano, Mark Vogelsberger, Benjamin Wallisch, Benjamin D. Wandelt, Risa H. Wechsler, Christoph Weniger, W. L. Kimmy Wu, Weishuang Linda Xu, Masaki Yamada, Hai-Bo Yu, Zhengkang Zhang, Yi-Ming Zhong, Kathryn Zurekwork_xxfbp56zevetxffcazrya73sfyWed, 14 Sep 2022 00:00:00 GMTLevel-strategyproof Belief Aggregation Mechanisms
https://scholar.archive.org/work/div2emgu75ah7f2eiuf4w5zsre
In the problem of aggregating experts' probabilistic predictions over an ordered set of outcomes, we introduce the axiom of level-strategyproofness (level-SP) and prove that it is a natural notion with several applications. Moreover, it is a robust concept as it implies incentive compatibility in a rich domain of single-peakedness over the space of cumulative distribution functions (CDFs). This contrasts with the literature which assumes single-peaked preferences over the space of probability distributions. Our main results are: (1) a reduction of our problem to the aggregation of CDFs; (2) the axiomatic characterization of level-SP probability aggregation functions with and without the addition of other axioms; (3) impossibility results which provide bounds for our characterization; (4) the axiomatic characterization of two new and practical level-SP methods: the proportional-cumulative method and the middlemost-cumulative method; and (5) the application of proportional-cumulative to extend approval voting, majority rule, and majority judgment methods to situations where voters/experts are uncertain about how to grade the candidates/alternatives to be ranked.[We are grateful to Thomas Boyer-Kassem, Roger Cooke, Aris Filos-Ratsikas, Hervé Moulin, Clemens Puppe and some anonymous EC2021 referees for their helpful comments and suggestions.]Rida Laraki, Estelle Varlootwork_div2emgu75ah7f2eiuf4w5zsreTue, 13 Sep 2022 00:00:00 GMTHow the Higgs potential got its shape
https://scholar.archive.org/work/b356glqgkvgwfewtqjonldnaea
String-localized quantum field theory allows renormalizable couplings involving massive vector bosons, without invoking negative-norm states and compensating ghosts. We analyze the most general coupling of a massive vector boson to a scalar field, and find that the scalar field necessarily comes with a quartic potential which has the precise shape of the shifted Higgs potential. In other words: the shape of the Higgs potential has not to be assumed, but arises as a consistency condition. We derive this result as a consequence of the "Principle of String-Independence" (PSI) for renormalized perturbation theory with string-localized interactions: While the renormalizable interaction density may be localized along an auxiliary "string", the S-matrix (and also the local observables of the theory) must not depend on it. Along the way, we develop a model-independent scheme how the PSI is implemented in string-localized perturbation theory, and how it can be used as a tool to "renormalize the non-renormalizable".Jens Mund, Karl-Henning Rehren, Bert Schroerwork_b356glqgkvgwfewtqjonldnaeaTue, 13 Sep 2022 00:00:00 GMTAnomalous Ward identities for on-shell amplitudes at the conformal fixed point
https://scholar.archive.org/work/itnu2wdelrenpj6eclh42hbvuy
Conformal symmetry underlies many massless quantum field theories, but little is known about the consequences of this powerful symmetry for on-shell scattering amplitudes. We study this problem in a dimensionally-regularized ϕ^3 model at the conformal fixed point. We show that the on-shell renormalised amplitudes satisfy anomalous conformal Ward identities. Each external on-shell state contributes two terms to the anomaly, both proportional to the corresponding momentum. The first term is proportional to the elementary field anomalous dimension, and thus involves only lower-loop information. We argue that the second term is governed by collinear regions of loop momentum, and that it can be represented as the convolution of a universal function and lower-order amplitudes. The computation of the conformal anomaly is then simpler than that of the amplitude at the same perturbative order, giving our anomalous conformal Ward identities a strong predictive power in perturbation theory. This result is also of practical importance for dimensionally-regularised amplitudes away from the conformal fixed point. Indeed the knowledge of the on-shell amplitude at the conformal fixed point, together with lower loop information, fixes entirely the amplitude away from the fixed point. While our analysis is done in a cubic scalar toy model, we expect analogous results to hold for amplitudes in other quantum field theories with classical conformal symmetry, such as Yang-Mills and Yukawa theories.Dmitry Chicherin, Johannes Henn, Simone Zoiawork_itnu2wdelrenpj6eclh42hbvuyMon, 12 Sep 2022 00:00:00 GMT"Most tantumising state of affairs": Mathematical and non-mathematical in quantum-like understanding of thinking
https://scholar.archive.org/work/uo4v64hkovahdjaazeirr5iwua
This article addresses the effectiveness of the predictive modeling of cognition and behavior based on quantum principles and some of the reasons for this effectiveness. It also aims, however, to explore the limitations of mathematical modeling so based, quantum-like (Q-L) modeling, and all mathematical modeling, including classical-like (C-L), in considering human cognition and behavior. It will discuss certain alternative approaches to both, essentially philosophical in nature, although sometimes found in literary works, approaches that, while not quantitative, may help compensate for limitations of mathematical modeling there. Most Q-L and C-L approaches beyond physics are realist, insofar as they offer representations of human thinking by the formalism of quantum or classical physical theories. The position adopted in this article is based on the non-realist assumption that such a representation may not be possible, which is not the same as that it is impossible. I designate interpretations that do not make this assumption reality-without-realism, RWR, interpretations, and in considering mental processes as ideality-without-idealism, IWI, interpretations.Arkady Plotnitskywork_uo4v64hkovahdjaazeirr5iwuaMon, 12 Sep 2022 00:00:00 GMTGeometric phase and its applications: topological phases, quantum walks and non-inertial quantum systems
https://scholar.archive.org/work/5mmfqmks3rctthjmbjmw26x6pu
Geometric phase plays a fundamental role in quantum theory and accounts for wide phenomena ranging from the Aharanov-Bohm effect, the integer and fractional quantum hall effects, and topological phases of matter, including topological insulators, to name a few. In this thesis, we have proposed a fresh perspective of geodesics and null phase curves, which are key ingredients in understanding the geometric phase. We have also looked at a number of applications of geometric phases in topological phases, quantum walks, and non-inertial quantum systems. The shortest curve between any two points on a given surface is a (minimal) geodesic. They are also the curves along which a system does not acquire any geometric phase. In the same context, we can generalize geodesics to define a larger class of curves, known as null phase curves (NPCs), along which also the acquired geometric phase is zero; however, they need not be the shortest curves between the two points. We have proposed a geometrical decomposition of geodesics and null phase curves on the Bloch sphere, which is crucial in improving our understanding of the geometry of the state space and the intrinsic symmetries of geodesics and NPCs. We have also investigated the persistence of topological phases in quantum walks in the presence of an external (lossy) environment. We show that the topological order in one and two-dimensional quantum walks persist against moderate losses. Further, we use the geometric phase to detect the non-inertial modifications to the field correlators perceived by a circularly rotating two-level atom placed inside a cavity.Vikash Mittalwork_5mmfqmks3rctthjmbjmw26x6puSun, 11 Sep 2022 00:00:00 GMT