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Mathematical Results on Quantum Many-body Physics


Lemm, Marius Christopher (2017) Mathematical Results on Quantum Many-body Physics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9D21VNV.


The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics.

In this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematical analysis; quantum many-body physics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Frank, Rupert L
Thesis Committee:
  • Frank, Rupert L. (chair)
  • Makarov, Nikolai G.
  • Motrunich, Olexei I.
  • Simon, Barry M.
Defense Date:6 March 2017
Record Number:CaltechTHESIS:05252017-092503939
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter III adapted for Chapter IV adapted for Chapter V adapted for Chapter VI adapted for Chapter VII
Lemm, Marius Christopher0000-0001-6459-8046
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10207
Deposited By: Marius Lemm
Deposited On:26 May 2017 20:39
Last Modified:04 Oct 2019 00:16

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