Tensor Network States and Algorithms in the presence of Abelian and non-Abelian Symmetries

Sukhbinder Singh (2011). Tensor Network States and Algorithms in the presence of Abelian and non-Abelian Symmetries PhD Thesis, School of Mathematics & Physics, The University of Queensland.

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Author Sukhbinder Singh
Thesis Title Tensor Network States and Algorithms in the presence of Abelian and non-Abelian Symmetries
School, Centre or Institute School of Mathematics & Physics
Institution The University of Queensland
Publication date 2011-05
Thesis type PhD Thesis
Supervisor Prof. Guifre Vidal
Total pages 189
Total colour pages 74
Total black and white pages 115
Subjects 01 Mathematical Sciences
Abstract/Summary Understanding and classifying phases of matter is a vast and important area of research in modern physics. Of special interest are phases at low temperatures where quantum effects are dominant. Theoretical progress is thwarted by a general lack of analytical solutions for quantum many-body systems. Moreover, perturbation theory is often inadequate in the strongly interacting regime. As a result, numerical approaches have become an indispensable tool to address such problems. In recent times, numerical approaches based on tensor networks have caught widespread attention. Tensor network algorithms draw on insights from Quantum Information theory to take advantage of special entanglement properties of low energy quantum many-body states of lattice models. Examples of popular tensor networks include Matrix Product States, Tree tensor Network, Multi-scale Entanglement Renormalization Ansatz and Projected Entangled Pair States. The main impediment of these methods comes from the fact that they can only represent states with a limited amount of entanglement. On the other hand, exploitation of symmetries, a powerful asset for numerical methods, has remained largely unexplored for a broad class of tensor networks algorithms. In this thesis we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically protect the symmetry and exploit it for computational gain in tensor network simulations. Our formalism is generic. It can readily be adapted to specic tensor network representations and to a wide spectrum of physical symmetries. The latter includes conservation of total particle number (U(1) symmetry) and of total angular momentum (SU(2) symmetry), and also more exotic symmetries (anyonic systems). The generality of the formalism is due to the fact that the symmetry constraints are imposed at the level of individual tensors, in a way that is independent of the details of the tensor network. As a result, we are led to a framework of symmetric tensors. Such tensors are then used as building blocks for tensor network representations of quantum-many states in the presence of symmetry. For a long time several physical problems of immense interest have remained elusive to numerical methods mostly owing to extremely high simulation costs. These include systems of frustrated magnets and interacting fermions that are relevant in the context of quantum magnetism and high temperature superconductivity. With symmetry now as a potent ally, tensor network algorithms may nally be used to draw positive insights about such systems.
Keyword tensor networks, symmetry, U(1), SU(2), MERA, MPS, spin networks, anyons
Additional Notes 33, 38-49, 53-56, 61-82, 85, 87, 88, 89, 91, 92, 116, 122, 124, 126, 128, 131, 134, 138, 140, 141, 143, 145, 149, 150, 151, 155, 156, 158, 160, 165, 166, 167, 168, 170, 171, 174, 176, 177, 178

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Created: Fri, 20 May 2011, 11:50:12 EST by Mr Sukhbinder Singh on behalf of Library - Information Access Service