Hubig, Claudius (2017): Symmetryprotected tensor networks. Dissertation, LMU München: Fakultät für Physik 

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Abstract
The simulation and numerical study of large, strongly correlated quantum systems containing Fermions or using realtime evolution in finite dimensions is still an essentially unsolved problem, primarily due to the exponential growth of the Hilbert state space with system size and the occurrence of the socalled sign problem in Monte Carlo studies. In this area, the use of tensornetwork methods, for onedimensional systems chief among them the density matrix renormalisation group (DMRG) and matrixproduct states (MPS), has grown in importance in recent years. This thesis first recapitulates the use of nonabelian symmetries such as SU(2)Spin in arbitrary tensor networks with an extensive review of the published literature including detailed algorithms and implementation hints. Implementing such symmetries can lead to a considerably more efficient representation of states in the tensor network. This part is intended to be suitable as an implementationoriented introduction to tensor networks in general and the implementation of nonabelian symmetries in particular. Second, it introduces a series of technical improvements for the MPS methods. These improvements include a faster convergence scheme for MPSDMRG, a systematic approach to the construction of matrixproduct operators and an improved Krylov time evolution method as well as the combination of several wellknown techniques into a single tensor network toolkit, SYTEN. The effectiveness of these improvements is demonstrated in numerical examples. Third, the toolkit is applied to the study of two models of current research interest: A onedimensional spin chain in a staggered external magnetic field is studied and confinement of the elementary spinon excitations, as predicted by analytical arguments, found numerically using realtime evolution and evaluation of the dynamical structure factor. Additionally, the Hubbard model in two dimensions is studied extensively at various system sizes, geometries, interaction strengths U and filling factors n using up to 30'000 SU(2)Spinsymmetric states equivalent to approx. 100'000 states in other MPSDMRG implementations. Hints of a possible phase coexistence in the region 0.85 < n < 0.95 are found at intermediate interaction strengths U = 4 and U = 6 as well as a consistently striped ground state in the region n ≈ 0.875.
Dokumententyp:  Dissertation (Dissertation, LMU München) 

Keywords:  tensor networks, density matrix renormalization group, nonabelian symmetries, matrixproduct states 
Themengebiete:  500 Naturwissenschaften und Mathematik
500 Naturwissenschaften und Mathematik > 530 Physik 
Fakultäten:  Fakultät für Physik 
Sprache der Hochschulschrift:  Englisch 
Datum der mündlichen Prüfung:  30. Oktober 2017 
1. Berichterstatter/in:  Schollwöck, Ulrich 
MD5 Prüfsumme der PDFDatei:  1418bbdd1a1d6c723fe16207fac8424d 
Signatur der gedruckten Ausgabe:  0001/UMC 25063 
ID Code:  21348 
Eingestellt am:  03. Nov. 2017 14:08 
Letzte Änderungen:  03. Nov. 2017 14:08 