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This dissertation investigates the entanglement properties of extended quantum systems which exhibit either long-range order or quantum criticality. While we mainly focus on the von Neumann entanglement entropy, the mutual information is also studied for certain systems when a mixed state is of concern. For extended quantum systems, all the gapped local Hamiltonians, as well as a large number gapless systems, are known to follow the so called "area law", which states that the entanglement entropy is proportional to the surface area of the subsystem. However, violations of the area law, usually in a logarithmic fashion, do exist in various different systems. They are found to be associated with quantum criticality in certain one dimensional systems. For free fermions in higher dimensions, it is found that the area law is enhanced by a logarithmic factor. Besides, such logarithmic terms also appears as subleading corrections to the area law. The central goal of this dissertation is, by studying specific solvable models, to explore and discuss the connections of such logarithmic divergence of entanglement entropy to quantum criticality and long-range order, and the intrinsic relations among such terms in different dimensions, and seek generalization to interacting systems. In the first part, we study two different systems that both exhibit long-range order, namely magnetically ordered Heisenberg spin systems and Bose-Einstein condensate systems, and reveal that this logarithmic divergence (violating the area law) is not particular to quantum criticality. They are present in those long-range ordered systems as well. Therefore, caution must be taken when people try to use such divergence to detect and characterize quantum criticality. In second part, we explore the relation between logarithmic divergence in one-dimensional fermionic systems and that of free fermions in higher dimensions. We show that both logarithmic factors share the same origin - the singularity at the Fermi points or Fermi surface - via a toy model. Based on the intuition from our toy model, we make use of the tool of multi-dimensional bosonization to rigorously re-derive the entanglement entropy of free fermions in high dimensions in a simpler way. Then by the convenience of the bosonization technique, we take into account the Fermi liquid interactions, and obtain the leading scaling behavior of the entanglement entropy of Fermi liquids.
entanglement entropy, long range order, quantum critical
Date of Defense
May 11, 2012.
A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Includes bibliographical references.
Kun Yang, Professor Directing Dissertation; Phillip Bowers, University Representative; Nicholas Bonesteel, Committee Member; Jorge Piekarewicz, Committee Member; Peng Xiong, Committee Member.
Florida State University
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