subject:(Adiabatic Limit)

University of Alberta

1. Li, Zhou. Numerical study of the crossover from free electrons to small polarons.

Degree: PhD, Department of Physics, 2012, University of Alberta

URL: https://era.library.ualberta.ca/files/w95051402

The electron-phonon interaction is one of the fundamental interactions in almost all condensed matter materials. In conventional superconductors, the electron-phonon interaction is the glue that attracts two electrons to one another to form a pair. A strong electron-phonon interaction leads to the concept of a polaron, which is an electron with lattice distortions around it. The small polaron is a polaron with spatial extent comparable to an interatomic dimension of the solid. Evidence for polarons has been identified in many experiments in superconductors and semiconductors. In this thesis we present exact calculations of the polaron. Specifically we have refined Trugman's method to solve the ground state of an electron-phonon coupled system in the whole parameter regime, and we also generalized this method to treat spin-orbit coupled systems. The most difficult regimes, which is the strong-coupling regime and the small phonon frequency limit, have been solved by these refinements. There are three representative kinds of electron-phonon interaction, the Holstein model, the Fr"ohlich model and the BLF-SSH model. In this thesis we have addressed the first and the third one. The second one, the Fr"ohlich model, is very similar to the Holstein model but the interaction is nonlocal. For the Holstein model we have observed the expected smooth crossover from free electrons to small polarons, while for the BLF-SSH model, we have studied the weak coupling regime with perturbation theory and derived a new analytical result for the one-dimensional problem.

Subjects/Keywords: Trugman's method; adiabatic limit; polaron; Holstein; cold atom; strong coupling; superconductivity; electron-phonon

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APA (6^th Edition):

Li, Z. (2012). Numerical study of the crossover from free electrons to small polarons. (Doctoral Dissertation). University of Alberta. Retrieved from https://era.library.ualberta.ca/files/w95051402

Chicago Manual of Style (16^th Edition):

Li, Zhou. “Numerical study of the crossover from free electrons to small polarons.” 2012. Doctoral Dissertation, University of Alberta. Accessed April 13, 2021. https://era.library.ualberta.ca/files/w95051402.

MLA Handbook (7^th Edition):

Li, Zhou. “Numerical study of the crossover from free electrons to small polarons.” 2012. Web. 13 Apr 2021.

Vancouver:

Li Z. Numerical study of the crossover from free electrons to small polarons. [Internet] [Doctoral dissertation]. University of Alberta; 2012. [cited 2021 Apr 13]. Available from: https://era.library.ualberta.ca/files/w95051402.

Council of Science Editors:

Li Z. Numerical study of the crossover from free electrons to small polarons. [Doctoral Dissertation]. University of Alberta; 2012. Available from: https://era.library.ualberta.ca/files/w95051402

2. Xu, Guangbo. Symplectic Vortex Equation and Adiabatic Limits .

Degree: PhD, 2013, Princeton University

URL: http://arks.princeton.edu/ark:/88435/dsp016h440s533

This thesis consists of four parts, on four separate topics in the study of the symplectic vortex equation and their adiabatic limits. In the first part, we constructed a compactication of the moduli space of twisted holomorphic maps with Lagrangian boundary condition. It generalizes the compactness theorem of Mundet-Tian in the case of closed Riemann surfaces to the case of bordered Riemann surfaces, and it is the first step in developing the open-string analogue of Mundet-Tian's program. In the second part, we studied the Morse theory of Lagrange multipliers, which is based on a joint work with Stephen Schecter. We also considered two adiabatic limits by varying a real parameter in this theory, which result in two different homology group. Via the homotopy provided by the variation of we prove that the two homology groups are isomorphic. In the third part, we considered a U(1)-gauged linear σ-model and its low-energy adiabatic limits. Via adiabatic limits, we managed to classify all affine vortices with target the complex vector space and diagonal U(1)-action, and we identify their moduli spaces, which generalizes Taubes' result. This also gives a precise meaning of the "point-like instantons" described by Witten. We also computed the associated quantum Kirwan map by compactifying the moduli space. In the fourth part, we introduce a new type of equations. It is a generalization of Witten's equation for a quasi-homogeneous polynomial W, by coupling a gauge field. The purpose of this generalization is to realize the geometric Landau-Ginzburg/Calabi-Yau correspondence predicted in string theory. This part is based on a work in progress joint with Gang Tian. Advisors/Committee Members: Tian, Gang (advisor).

Subjects/Keywords: adiabatic limit; compactness; vortex equation

…cohomology (see [38]); Gaio-Salamon considered certain adiabatic limit of the… …dimension 4 via “dimensional reduction”; the adiabatic limits in gauged σ-model also reduces to an… …x5B;39, Lemma 5.1] 1. The limit Hol(A, 0) := lim →0 γ exists. 2. Any τ ∈ T… …bounded with respect to the chosen metric. Because of the existence of nontrivial limit holonomy…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Xu, G. (2013). Symplectic Vortex Equation and Adiabatic Limits . (Doctoral Dissertation). Princeton University. Retrieved from http://arks.princeton.edu/ark:/88435/dsp016h440s533

Chicago Manual of Style (16^th Edition):

Xu, Guangbo. “Symplectic Vortex Equation and Adiabatic Limits .” 2013. Doctoral Dissertation, Princeton University. Accessed April 13, 2021. http://arks.princeton.edu/ark:/88435/dsp016h440s533.

MLA Handbook (7^th Edition):

Xu, Guangbo. “Symplectic Vortex Equation and Adiabatic Limits .” 2013. Web. 13 Apr 2021.

Vancouver:

Xu G. Symplectic Vortex Equation and Adiabatic Limits . [Internet] [Doctoral dissertation]. Princeton University; 2013. [cited 2021 Apr 13]. Available from: http://arks.princeton.edu/ark:/88435/dsp016h440s533.

Council of Science Editors:

Xu G. Symplectic Vortex Equation and Adiabatic Limits . [Doctoral Dissertation]. Princeton University; 2013. Available from: http://arks.princeton.edu/ark:/88435/dsp016h440s533

ETH Zürich

3. Antony, Charel. Gradient Trajectories Near Real And Complex A2-singularities.

Degree: 2018, ETH Zürich

URL: http://hdl.handle.net/20.500.11850/284182

In this thesis, the existence and uniqueness of gradient trajectories near an A₂-singularity are analysed. The A₂-singularity is called a birth-death critical point in the real case. The birth-death critical point appears in a one-parameter family of functions. Such a family of functions has precisely two Morse critical points of index difference one, on the birth side. The result of the real case states that these two critical points are joined by a unique gradient trajectory up to time-shift. Here the gradient flow is defined with respect to any family of Riemannian metrics. This can be viewed as a converse to Smale's cancellation theorem. We also look at the complex analogue of the result in Picard – Lefschetz theory. This analogue considers a holomorphic one-parameter family with an A₂-singularity. Such a family has two critical Morse critical points near the singularity for every small non-zero parameter. We prove that the two Lagrangian vanishing cycles associated to these critical points intersect transversally in exactly one point in all regular fibres along a straight line. The result is obtained by analysing the gradient trajectories of the real part of these functions. Both proofs start with a normal form in local coordinates for such families of functions. The gradient equations in these coordinates can be rescaled into a fast-slow system of non-linear differential equation. Existence will rely on an adiabatic limit analysis whereas uniqueness follows from a Conley index pair construction. The latter construction will also show that connecting gradient trajectories cannot leave the local charts. Even though the proof of these two results follow from similar lines of argument, the real case cannot be reduced to the complex case and vice versa. Advisors/Committee Members: Salamon, Dietmar, Frauenfelder, Urs, Biran, Paul.

Subjects/Keywords: Birth-death; Critical point; Gradient flow; A_2 singularity; vanishing cycles; Whitney Lemma; Adiabatic Limit; Conley Index Pair; Existence and uniqueness of solutions; info:eu-repo/classification/ddc/510; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Antony, C. (2018). Gradient Trajectories Near Real And Complex A2-singularities. (Doctoral Dissertation). ETH Zürich. Retrieved from http://hdl.handle.net/20.500.11850/284182

Chicago Manual of Style (16^th Edition):

Antony, Charel. “Gradient Trajectories Near Real And Complex A2-singularities.” 2018. Doctoral Dissertation, ETH Zürich. Accessed April 13, 2021. http://hdl.handle.net/20.500.11850/284182.

MLA Handbook (7^th Edition):

Antony, Charel. “Gradient Trajectories Near Real And Complex A2-singularities.” 2018. Web. 13 Apr 2021.

Vancouver:

Antony C. Gradient Trajectories Near Real And Complex A2-singularities. [Internet] [Doctoral dissertation]. ETH Zürich; 2018. [cited 2021 Apr 13]. Available from: http://hdl.handle.net/20.500.11850/284182.

Council of Science Editors:

Antony C. Gradient Trajectories Near Real And Complex A2-singularities. [Doctoral Dissertation]. ETH Zürich; 2018. Available from: http://hdl.handle.net/20.500.11850/284182

Open Access Theses and Dissertations