From Quantum Entanglement to Interactions of Elementary Excitations in Coupled Spin Chains : An Introduction to Numerical Many-Body Physics with Matrix Product States and Tensor Networks

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http://urn.fi/URN:NBN:fi:hulib-202108273578
Title: From Quantum Entanglement to Interactions of Elementary Excitations in Coupled Spin Chains : An Introduction to Numerical Many-Body Physics with Matrix Product States and Tensor Networks
Author: Niedermeier, Marcel
Contributor: University of Helsinki, Faculty of Science
Publisher: Helsingin yliopisto
Date: 2021
Language: eng
URI: http://urn.fi/URN:NBN:fi:hulib-202108273578
http://hdl.handle.net/10138/333721
Thesis level: master's thesis
Degree program: Teoreettisten ja laskennallisten menetelmien maisteriohjelma
Master's Programme in Theoretical and Computational Methods
Magisterprogrammet i teoretiska och beräkningsmetoder
Specialisation: Teoreettinen fysiikka
Theoretical Physics
Teoretisk fysik
Abstract: Matrix product states provide an efficient parametrisation of low-entanglement many-body quantum states. In this thesis, the underlying theory is developed from scratch, requiring only basic notions of quantum mechanics and quantum information theory. A full introduction to matrix product state algebra and matrix product operators is given, culminating in the derivation of the density matrix renormalisation group algorithm. The latter provides a simple variational scheme to determine the ground state of arbitrary one-dimensional many-body quantum systems with supreme precision. As an application of matrix-product state technology, the kernel polynomial method is introduced in detail as a state-of-the art numerical tool to find the spectral function or the dynamical correlator of a given quantum system. This in turn gives access to the elementary excitations of the system, such that the locations of the low-energy eigenstates can be studied directly in real space. To illustrate those theoretical tools concretely, the ground state energy, the entanglement entropy and the elementary excitations of a simple interface model of a Heisenberg ferromagnet and a Heisenberg antiferromagnet are studied. By changing the location of the model in parameter space, the dependence of the above-mentioned quantities on the transverse field and the coupling strength is investigated. Most notably, we find that the entanglement entropy characteristic to the antiferromagnetic ground state stretches across the interface into the ferromagnetic half-chain. The dependence of the physics on the value of the coupling strength is, overall, small, with exception of the appearance of a boundary mode whose eigenenergy grows with the coupling. A comparison with a localised edge field shows however that the boundary mode is a true interaction effect of the two half-chains. Various algorithmic and physics extensions of the present project are discussed, such that the code written as part of this thesis could be turned into a state-of-the-art MPS library with managable effort. In particular, an application of the kernel polynomial method to calculate finite-temperature correlators is derived in detail.
Subject: tensor networks
matrix product states
matrix product operators
spin chains
Heisenberg model
entanglement entropy
DMRG
kernel polynomial method
computational condensed matter physics


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