Hubig, Claudius (2017): Symmetryprotected tensor networks. Dissertation, LMU München: Faculty of Physics 

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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.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  tensor networks, density matrix renormalization group, nonabelian symmetries, matrixproduct states 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 530 Physics 
Faculties:  Faculty of Physics 
Language:  English 
Date of oral examination:  30. October 2017 
1. Referee:  Schollwöck, Ulrich 
MD5 Checksum of the PDFfile:  1418bbdd1a1d6c723fe16207fac8424d 
Signature of the printed copy:  0001/UMC 25063 
ID Code:  21348 
Deposited On:  03. Nov 2017 14:08 
Last Modified:  03. Nov 2017 14:08 