Periodic Reporting for period 4 - ERQUAF (Entanglement and Renormalisation for Quantum Fields) Reporting period: 2021-08-01 to 2023-01-31 Summary of the context and overall objectives of the project The aim of this project is to explore the relationship between quantum field theory and the theory of entanglement methods, specifically tensor network states. Entanglement has recently gained interest in high-energy physics due to its connections with black hole physics and quantum gravity. While some progress has been made in formulating tensor network ansätze for continuous quantum systems, their application is mostly unexplored. Similarly, ideas from quantum field theory, such as path integral methods and perturbative expansions, can be used in the context of tensor networks to study many-body physics and field theories. The project proposes to develop a toolbox of continuous tensor networks, expand tensor network applications to field theories and gauge theories, and introduce concepts and techniques from quantum field theory to the tensor network toolbox. The advancements made in this project could potentially have an impact on simulating cold atom systems, non-equilibrium situations in quantum chromodynamics and the standard model, and upcoming quantum technologies. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far 1. We have developed an algorithm to find continuous matrix product state (cMPS) approximations of the ground state of finite-size inhomogeneous systems. We encountered difficulties with early attempts but made a breakthrough by switching to a finite element approach, which also allowed us to efficiently compute the environments. The results were published in Physical Review Letters. We also developed an open-source toolbox that includes these developments and others, such as a novel optimization algorithm based on Riemannian optimization. Our team has also turned our attention to infinite systems with a periodic potential and used Fourier modes as a basis, with promising initial results. We have incorporated all these developments into the toolbox, which we will continue to develop.2. we been developing tensor network descriptions of quantum field theories (QFTs), specifically topological and conformal field theories, and exploring questions related to the entanglement structure of QFTs. Our work started with the development of a strange correlator to construct a systematic tensor network mapping between three-dimensional topological quantum field theories and two-dimensional conformal field theories. We also proposed and investigated the first quantum lattice model that realizes a low energy theory that could be described by a putative Haagerup conformal field theory.We also studied the Gross-Neveu model and its chiral extension using matrix product state techniques. Our lattice construction realized all the well-known features of the QFT and focused on the highly nontrivial entanglement spectrum for the Gross-Neveu model. For the chiral Gross-Neveu model, we recovered the continuous chiral symmetry as an emerging symmetry on a line of Landau-forbidden phase transitions. Additionally, we published other projects in which we studied anomalies using an entanglement and tensor network perspective.3. we focused on using the multiscale entanglement renormalization ansatz (MERA) to approximate ground states of quantum systems. We developed a MERA for approximating the ground state of a free fermion model using the theory of wavelets, which was published in Physical Review X. Our work also provided a branching MERA construction for the ground state of hopping free fermions in two dimensions. We also studied the ability of Projected Entangled-Pair States (PEPS) to approximate states exhibiting Fermi surfaces and found that topological aspects related to the description of chiral phases of matter were important in this study. Our findings have been published in Physical Review Letters, and we are currently preparing follow-up work for publication.4. We focused on enhancing the tensor network toolbox through collaborations with field theory and differential geometry. They proposed expanding the path integral through a resolution of identity over matrix product states (MPS) and expanded this approach to continuous matrix product states (cMPS) and instanton contributions to tensor network states. Other completed projects included low-parameter families of tensor network states, finite entanglement scaling behavior, geometry of variational methods, an algorithm for simulating real-time scattering processes, and optimization algorithms based on Riemannian optimization.5. we explored the connections between entanglement, renormalisation, and holography, with a focus on tensor network representations of critical quantum states, holography in the AdS/CFT correspondence, and quantum error correcting codes. Although the interest in this topic has decreased since the proposal was written, we obtained a novel interpretation of MERA and entanglement compression in scale space, which can be related to the approximation of thermal states as matrix product operators by the lattice equivalent of a conformal mapping.In addition to the established work packages, we also contributed to the rapidly evolving field of theoretical physics by exploring the interplay between machine learning and quantum many-body physics. Specifically, we developed a variational ansatz based on restricted Boltzmann machines that explicitly respects spin rotation symmetry, which has potential applications in machine learning models. We also improved sampling strategies to facilitate the use of variational Monte Carlo for optimizing tensor network states, particularly PEPS. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) List of results:• A ground state algorithm for cMPS in inhomogeneous potentials, and, more generally, a cMPS open-source package for ground states and excitations• An explicit tensor network mapping between topological quantum states in (2+1) dimensions and critical lattice models in (2+0) dimensions using the strange correlator concept.• A detailed study of an asymptotically free relativistic field theory of interacting fermions (Gross Neveu model) using matrix product state methods.• A rigorous MERA construction with provable error bounds for the ground states of free fermions in one and two dimensions.• A investigation and confirmation that PEPS can efficiently approximate states with a Fermi surface (in a scaling regime).• A tensor network encapsulation of perturbative expansions, which allows to promote the perturbative coefficients to variational parameters and results in a low-parameter family of tensor networks with an insightful structure.• A scaling hypothesis for the finite entanglement scaling behaviour of the MPS approximation of critical states• A novel algorithm for simulating real-time scattering processes of interacting quasiparticles in quantum spin chains• A set of novel optimization algorithms for (isometric) tensor networks based on ideas from Riemannian optimisation• A novel interpretation of the MERA bond dimension in terms of the MPO representation of a thermal state• A neural network ansatz for quantum many body systems with arbitrary global symmetries, including non-abelian and even anyonic (categorical) symmetries.