## Periodic Reporting for period 3 - ERQUAF (Entanglement and Renormalisation for Quantum Fields)

Reporting period: 2020-02-01 to 2021-07-31

The goal of this project is to strengthen the synergy between quantum field theory on the one hand and the theory of entanglement methods, with in particular the framework of tensor network states, on the other hand. In high-energy physics, the widespread interest in entanglement has only been triggered recently due to the intriguing connections between entanglement and the structure of spacetime that arise in black hole physics and quantum gravity. During the past few years, direct continuum limits of various tensor network ansätze have been formulated. However, the application thereof is largely unexplored territory and holds promising potential. But also the application of (discrete) tensor networks to problems in lattice field theory has only just begin, with applications so for mostly restricted to one-dimensional field theories. Vice versa, ideas traditionally belonging to the toolbox of quantum field theory, such as path integral methods, perturbative expansions and instantons, can be applied in the context of tensor networks to obtain a more versatile toolbox for numerically probing the various aspects of many body physics and field theories alike.

This proposal formulates several advancements and developments for the theoretical and computational study of continuous quantum systems, gauge theories and exotic quantum phases, but also for establishing the intricate relation between entanglement, renormalisation and geometry in the context of the holographic principle. Ultimately, these developments will alter the way in which to approach some of the most challenging questions in physics, ranging from the simulation of cold atom systems to non-equilibrium or high-density situations in quantum chromodynamics and the standard model. As the rise of new quantum technologies is on the horizon, it is not unimaginable that these developments have an impact in this direction.

In conclusion, the overall objectives of this proposal are

• The development of a toolbox of continuous tensor networks

• The expansion of tensor network applications for field theories and gauge theories

• The extension of the toolbox of tensor network methods by introducing concepts and techniques which are commonly used in quantum field theory

This proposal formulates several advancements and developments for the theoretical and computational study of continuous quantum systems, gauge theories and exotic quantum phases, but also for establishing the intricate relation between entanglement, renormalisation and geometry in the context of the holographic principle. Ultimately, these developments will alter the way in which to approach some of the most challenging questions in physics, ranging from the simulation of cold atom systems to non-equilibrium or high-density situations in quantum chromodynamics and the standard model. As the rise of new quantum technologies is on the horizon, it is not unimaginable that these developments have an impact in this direction.

In conclusion, the overall objectives of this proposal are

• The development of a toolbox of continuous tensor networks

• The expansion of tensor network applications for field theories and gauge theories

• The extension of the toolbox of tensor network methods by introducing concepts and techniques which are commonly used in quantum field theory

On work package 1 on continuous matrix product states (cMPS), we have investigated various approaches to find cMPS ground state approximations for inhomogeneous systems. Inhomeogenous cMPS are parameterized by matrix-valued functions, which thus need to be represented using a finite number of parameters to be stored and manipulated in a computer. We have spent quite some time investigating parameterizations based on orthogonal polynomials for finite systems, but were troubled by finding a suitably matching basis for expressing e.g. the corresponding environments. Furthermore, to avoid unstable or singular behaviour near the boundaries, one needs to define and exploit the mixed canonical form for cMPS. So far, algorithms for switching between canonical forms are not completely satisfactory. Instead, we turned our attention to infinite systems with a periodic potential, which is the case most interesting for the setting of optical lattices. Here, we can use Fourier modes as a basis, and first results in this direction look promising. Furthermore, using the time dependent variational principle, the ground state algorithms is closely related to an algorithm for simulating time evolution. Using this approach, we have first experiments to simulate quenches in the optical potential.

For WP2, part of the work has focussed on the concept of a strange correlator to construct an explicit and systematic tensor network mapping between three-dimensional topological quantum field theories (TQFT) and two-dimensional conformal field theories (CFT). These theories share the underlying mathematical structure of fusion categories. Starting from the tensor network representation of string net ground states, taking unitary fusion categories as input data, we have established a mapping to corresponding critical statistical mechanics models whose critical behaviour is described by the corresponding CFT. Non-unitary extensions, as well as the highly challenging generalization to higher dimensions, are currently under investigation. Other work withing the scope of WP2 are the study of lattice field theories, where we started with an extensive study of the Gross-Neveu model. This popular toy model describes interacting fermions in (1+1) dimensions, but shares various properties with quantum chromodynamics. The results of this work will be published soon.

For WP3 on the (continuous) multiscale entanglement renormalization ansatz (MERA), work during the beginning of the project focussed on an analytical construction of a MERA for approximating the ground state of a free fermion model, based on the theory of wavelets. This first ever result of an rigorous MERA construction with provable error bounds was published in Physical Review X. Since then, we have been studying extensions of this work. Another objective within this work package has been a further characterization of the entanglement structure of the cMERA ansatz, continuing on a work published by Adrian Franco-Rubio and Guifre Vidal.

WP4 is concentrated on the use of QFT methods in the context of tensor networks. A first proposal to expand the path integral in terms of a resolution of the identity over MPS was proposed in between the time of writing the proposal and the start of the project by A.G. Green, C.A. Hooley, J. Keeling and S. Simon. We have since been collaborating to use the develop the same idea using a resolution of the identity over cMPS, as well as to build on this approach in order to know include quadratic and higher order pertubative corrections, as well as to compute instanton contributions to tensor network states. This work is well on track and should lead to one or more publications soon.

WP5 has so far received the least attention as some of the objectives formulated in this work package have been addressed in other publications arising in between the time of writing the proposal and the start of the project.

Aside from the established work packages, the field of theoretical physics is always rapidly evolving, and new interesting developments can arise and receive a great of attention in a short amount of time. Such a development is the interplay between machine learning and quantum many body physics, where one particular connection is the use of machine learning models as variational ansatz. Here, we have contributed by developing a related ansatz based on restricted Boltzmann machines, but which unlike previous proposals explicitly respects spin rotation symmetry.

For WP2, part of the work has focussed on the concept of a strange correlator to construct an explicit and systematic tensor network mapping between three-dimensional topological quantum field theories (TQFT) and two-dimensional conformal field theories (CFT). These theories share the underlying mathematical structure of fusion categories. Starting from the tensor network representation of string net ground states, taking unitary fusion categories as input data, we have established a mapping to corresponding critical statistical mechanics models whose critical behaviour is described by the corresponding CFT. Non-unitary extensions, as well as the highly challenging generalization to higher dimensions, are currently under investigation. Other work withing the scope of WP2 are the study of lattice field theories, where we started with an extensive study of the Gross-Neveu model. This popular toy model describes interacting fermions in (1+1) dimensions, but shares various properties with quantum chromodynamics. The results of this work will be published soon.

For WP3 on the (continuous) multiscale entanglement renormalization ansatz (MERA), work during the beginning of the project focussed on an analytical construction of a MERA for approximating the ground state of a free fermion model, based on the theory of wavelets. This first ever result of an rigorous MERA construction with provable error bounds was published in Physical Review X. Since then, we have been studying extensions of this work. Another objective within this work package has been a further characterization of the entanglement structure of the cMERA ansatz, continuing on a work published by Adrian Franco-Rubio and Guifre Vidal.

WP4 is concentrated on the use of QFT methods in the context of tensor networks. A first proposal to expand the path integral in terms of a resolution of the identity over MPS was proposed in between the time of writing the proposal and the start of the project by A.G. Green, C.A. Hooley, J. Keeling and S. Simon. We have since been collaborating to use the develop the same idea using a resolution of the identity over cMPS, as well as to build on this approach in order to know include quadratic and higher order pertubative corrections, as well as to compute instanton contributions to tensor network states. This work is well on track and should lead to one or more publications soon.

WP5 has so far received the least attention as some of the objectives formulated in this work package have been addressed in other publications arising in between the time of writing the proposal and the start of the project.

Aside from the established work packages, the field of theoretical physics is always rapidly evolving, and new interesting developments can arise and receive a great of attention in a short amount of time. Such a development is the interplay between machine learning and quantum many body physics, where one particular connection is the use of machine learning models as variational ansatz. Here, we have contributed by developing a related ansatz based on restricted Boltzmann machines, but which unlike previous proposals explicitly respects spin rotation symmetry.

Results developed so far:

• A rigorous MERA construction with provable error bounds for the ground states of free fermions in one and two dimensions

• 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 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.

Expected results in the near future:

• A ground state algorithm for cMPS in periodic potentials, additionally, time evolution,

• A detailed study of an asymptotically free relativistic field theory of interacting fermions using matrix product state methods

• A (c)MPS-based path integral approach, including quadratic fluctuations, perturbative corrections and instanton contributions

• A rigorous MERA construction with provable error bounds for the ground states of free fermions in one and two dimensions

• 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 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.

Expected results in the near future:

• A ground state algorithm for cMPS in periodic potentials, additionally, time evolution,

• A detailed study of an asymptotically free relativistic field theory of interacting fermions using matrix product state methods

• A (c)MPS-based path integral approach, including quadratic fluctuations, perturbative corrections and instanton contributions