# Tensor network methods, area laws and topological order in condensed matter

Quantum matter exhibits a remarkable wealth of phenomena, originating from basic local laws of interactions.Tensor network states constitute a powerful machinery of numerically solving such systems, as well as analytically characterizing their properties. Notions of topological order or the classification of phases can be elegantly expressed in terms of such tensor networks. At the heart of the insight why tensor network states approximate ground states of locally interacting models well is the area law for the entanglement entropy, defining what is often referred to as the "physical corner of Hilbert space". We are concerned with various aspects of tensor network states, topological order, and scaling laws for entanglement entropies in quantum matter systems, both in the practical-numerical and mathematical-conceptual reading.

**Selected group publications**

- A tensor network annealing algorithm for two-dimensional thermal states

Physical Review Letters 122, 070502 (2019) - Holography and criticality in matchgate tensor networks

Science Advances 5, eaaw0092 (2019) - Single-shot holographic compression from the area law

Physical Review Letters 122, 190501 (2019) - Fermionic orbital optimisation in tensor network states

Physical Review Letters 117, 210402 (2016) - Fermionic topological quantum states as tensor networks

Physical Review B 95, 245127, 574 (2017) - Approximating local observables on projected entangled pair states

Physical Review A 95, 060102 (2017) - A positive tensor network approach for simulating open quantum many-body systems

Physical Review Letters 116, 237201 (2016) - Diagnosing topological edge states via entanglement monogamy

Physical Review Letters 116, 130501 (2016) - Solving frustration-free spin systems

Physical Review Letters 105, 060504 (2010) - Contraction of fermionic operator circuits and the simulation of strongly correlated fermions

Physical Review A 80, 042333 (2009) - Many-body localisation implies that eigenvectors are matrix-product states

Physical Review Letters 114, 170505 (2015) - Matrix product operators and states: NP-hardness and undecidability

Physical Review Letters 113, 160503 (2014) - Wick's theorem for matrix product states

Physical Review Letters 110, 040401 (2013) - Unifying variational methods for simulating quantum many-body systems

Physical Review Letters 100, 130501 (2008) - Entropy, entanglement, and area: analytical results for harmonic lattice systems

Physical Review Letters 94, 060503 (2005) - Statistics dependence of the entanglement entropy

Physical Review Letters 98, 220603 (2007) - Exploring local quantum many-body relaxation by atoms in optical superlattices

Physical Review Letters 101, 063001 (2008) - Unitary circuits for strongly correlated fermions

Physical Review A 81, 050303(R) (2010) - Topological insulators with arbitrarily tunable entanglement

Physical Review B 89, 195120 (2014) - Search for localized Wannier functions of topological band structures via compressed sensing

Physical Review B 90, 115110 (2014) - Tensor network methods with graph enhancement

Physical Review B 84, 125103 (2011) - Real-space renormalization yields finite correlations

Physical Review 105, 010502 (2010)

**Group reviews**

- Area laws for the entanglement entropy

Reviews of Modern Physics 82, 277 (2010) - Entanglement and tensor network states

Modeling and Simulation 3, 520 (2013) - Quantum many-body systems out of equilibrium

Nature Physics 11, 124 (2015)