Citation
Shukla, Sujeet Kumar (2018) Tensor Network Representation of ManyBody Quantum States and Unitary Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9DF6PDB. https://resolver.caltech.edu/CaltechTHESIS:09122017161101236
Abstract
Understanding manybody quantum systems is one of the most challenging problems in contemporary condensedmatter physics. Tensor network representation of quantum states and operators are taking central stage in this pursuit and beyond. They prove to be a powerful numerical and conceptual tool, and indeed a new language altogether. This thesis investigates various aspects of these representations by focusing on two specific problems: the first half of the thesis is devoted to examining how 'stable' a tensor network representation is for twodimensional quantum states with topological order, and the second half explores the representability of various unitary loop operators with tensor networks.
In the numerical usage of the tensor networks, the tensor is varied as to find the representation of the ground states of the given Hamiltonian. In chapter two and three of this thesis we show that such a numerical program for topological phases can be 'illposed'. We show that tensor network can be an unstable representation for a topological phase: even an infinitesimal variation in the representation results in the loss of topological order, completely or partially. We diagnose this problem by identifying the exact causes of this instability, and find that it is only tensor variations in certain directions that result in instability, because they result in the condensation of bosonic quasiparticles of the phase. Such unstable variations are characterized by two properties: (1) they can replace a tensor in the tensor network without making the network collapse, and (2) their presence in the network represents the presence of a nontrivial topological charge. We prove that the general tensor representation of all stringnet models suffer with such instabilities. We propose an exact mathematical operator to project out all such unstable variations and show its efficacy for a few models by direct calculations. Such an operator can be useful in numerical programs involving such tensor representations. We also point out that such variations play a crucial role in simulating topological phase transitions and their presence can be vital in an accurate simulation.
In chapter four and five of this thesis we focus on the representability of unitary loop operators by tensor networks. Such operators not only provide an important tool in the study of dynamical process in onedimensional systems, but also in understanding and classification of symmetry protected topological phases in two dimensions. To characterize all such operators, we find a necessary and sufficient condition for any loop tensor network operator of a given length to represent a unitary operator. In particular, it is shown that all unitary operators that map local operators to local operators (localitypreserving) can always be represented by a tensor network. Localitypreserving unitary loop operators are classified by a rational index called the GNVW index defined in Ref. [1] which measures how much information 'flows' along the loop. We define RankRatio index for tensor network operators and show that it is completely equivalent to the GNVW index. Therefore, GNVW index of a unitary operator can be easily extracted from its tensor network representation. We find that, other than representing localitypreserving unitary maps, tensor networks can also represent unitary operators that map local operators to global (nonlocal) operators. These tensor network operators are found to have a longranged order similar to tensors that represent topological tensor network states in two dimensions.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  tensor network; quantum physics; topological phases  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Physics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Group:  Institute for Quantum Information and Matter  
Thesis Committee: 
 
Defense Date:  21 July 2017  
NonCaltech Author Email:  shuklasujeetk (AT) gmail.com  
Record Number:  CaltechTHESIS:09122017161101236  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:09122017161101236  
DOI:  10.7907/Z9DF6PDB  
Related URLs: 
 
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  10432  
Collection:  CaltechTHESIS  
Deposited By:  Sujeet Shukla  
Deposited On:  25 Sep 2017 22:43  
Last Modified:  08 Aug 2022 17:50 
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