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Real-space renormalization group methods in the age of tensor network states

Matthias Bal (UGent)
(2018)
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Abstract
This dissertation contributes to the ongoing effort of understanding the origins and applications of real-space renormalization group methods in tensor network representations of classical and quantum many-body systems. First, we construct a matrix product operator ansatz to coarse-grain real-space transfer matrices of matrix product state descriptions of one-dimensional quantum spin chains. By treating the physical spin as an impurity, we unravel the virtual entanglement degrees of freedom of matrix product states into a layered structure to reveal an inherent renormalization group scale. Secondly, we rephrase tensor network renormalization for two-dimensional classical lattice models in a manifestly nonnegative way. The resulting real-space renormalization group flow preserves positivity and hence yields an interpretation in terms of Hamiltonian flows, reconciling modern real-space tensor network renormalization methods with traditional block-spin approaches. Thirdly, we study non-local symmetries in tensor networks by expressing two-dimensional classical partition functions in terms of strange correlators of judiciously chosen product states and string-net wave functions. We exhibit and exploit the emerging non-local symmetries of the partition function at criticality and highlight parallels between topological sectors and conformal primary fields in the shared framework of matrix product operator symmetries. Additionally, we provide a complementary perspective on real-space renormalization by recognizing known tensor network renormalization methods as the approximate truncation of an exactly coarse-grained strange correlator.
Keywords
quantum many-body systems, block-spin, real-space renormalization, renormalization group theory, tensor networks, matrix product states, entanglement, quantum information, quantum mechanics

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Please use this url to cite or link to this publication:

Chicago
Bal, Matthias. 2018. “Real-space Renormalization Group Methods in the Age of Tensor Network States”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Bal, M. (2018). Real-space renormalization group methods in the age of tensor network states. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Bal M. Real-space renormalization group methods in the age of tensor network states. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2018.
MLA
Bal, Matthias. “Real-space Renormalization Group Methods in the Age of Tensor Network States.” 2018 : n. pag. Print.
@phdthesis{8550528,
  abstract     = {This dissertation contributes to the ongoing effort of understanding the origins and applications of real-space renormalization group methods in tensor network representations of classical and quantum many-body systems. First, we construct a matrix product operator ansatz to coarse-grain real-space transfer matrices of matrix product state descriptions of one-dimensional quantum spin chains. By treating the physical spin as an impurity, we unravel the virtual entanglement degrees of freedom of matrix product states into a layered structure to reveal an inherent renormalization group scale. Secondly, we rephrase tensor network renormalization for two-dimensional classical lattice models in a manifestly nonnegative way. The resulting real-space renormalization group flow preserves positivity and hence yields an interpretation in terms of Hamiltonian flows, reconciling modern real-space tensor network renormalization methods with traditional block-spin approaches. Thirdly, we study non-local symmetries in tensor networks by expressing two-dimensional classical partition functions in terms of strange correlators of judiciously chosen product states and string-net wave functions. We exhibit and exploit the emerging non-local symmetries of the partition function at criticality and highlight parallels between topological sectors and conformal primary fields in the shared framework of matrix product operator symmetries. Additionally, we provide a complementary perspective on real-space renormalization by recognizing known tensor network renormalization methods as the approximate truncation of an exactly coarse-grained strange correlator.},
  author       = {Bal, Matthias},
  keyword      = {quantum many-body systems,block-spin,real-space renormalization,renormalization group theory,tensor networks,matrix product states,entanglement,quantum information,quantum mechanics},
  language     = {eng},
  pages        = {X, 208},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {Real-space renormalization group methods in the age of tensor network states},
  year         = {2018},
}