Les Houches Lecture Notes on Tensor Networks

Abstract

Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.

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• Faster identification of optimal contraction sequences for tensor networks
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• Unified approach to Quantum and Classical Dualities
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Dualities in one-dimensional quantum lattice models: topological sectors
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Cluster state as a non-invertible symmetry protected topological phase
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Locality in Quantum Systems
• A scaling hypothesis for matrix product states
• Numerical evidence for a Haagerup conformal field theory
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Characterizing Topological Order with Matrix Product Operators
• A critical lattice model for a Haagerup conformal field theory
• Edge theories in Projected Entangled Pair State models
• Gapless edges of 2d topological orders and enriched monoidal categories
• Topological symmetry in quantum field theory
• The Bethe ansatz as a matrix product ansatz
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• Numerical Contraction of the Tensor Network generated by the Algebraic Bethe Ansatz
• The Algebraic Bethe Ansatz and Tensor Networks
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• Classifying phases protected by matrix product operator symmetries using matrix product states
• A generalization of the injectivity condition for Projected Entangled Pair States
• Going Beyond Bell's Theorem
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• S matrix from matrix product states
• Spin Topological Field Theory and Fermionic Matrix Product States
• Galois Conjugates of Topological Phases
• Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz
• Variational methods for contracting projected entangled-pair states
• Fermion masses without symmetry breaking in two spacetime dimensions
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• Matrix Product State Representations
• Topological aspects of the critical three-state Potts model
• Condensation-Driven Phase Transitions in Perturbed String Nets
• Galois conjugated tensor fusion categories and non-unitary CFT
• Kennedy-Tasaki transformation and non-invertible symmetry in lattice models beyond one dimension
• Fermionic quantum cellular automata and generalized matrix product unitaries
• Matrix product states for anyonic systems and efficient simulation of dynamics
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• Invertible bimodule categories and generalized Schur orthogonality
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• Integrability and braided tensor categories
• Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states
• Finite Semisimple Module 2-Categories
• Categorical Landau Paradigm for Gapped Phases
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Tensor-product representations for string-net condensed states
• Anomalous domain wall condensation in a modified Ising chain
• Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories
• Quantum cellular automata and categorical dualities of spin chains
• The Classification of Fusion 2-Categories
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• Generalized Landau Paradigm for quantum phases and phase transitions
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Cited by in corpus (1)

• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl