Welcome to the home page of the online seminars on **Tensor Networks in High Energy Physics**! (www.heptnseminar.org)

This is a joint initiative of the *Gravity, Quantum Fields and Information* group at the **Albert Einstein Institute** in Potsdam (Johannes Knaute, Sukhi Singh, Michal Heller), **DESY** in Zeuthen (Karl Jansen), the **Max-Planck Institute for Quantum Optics** in Garching (Mari-Carmen Banuls), the Tensor Network initiative (Stefan Kuhn, Bianca Dittrich, Adam Lewis) at the **Perimeter Institute for Theoretical Physics** in Canada, and the **Free University of Berlin** (Jens Eisert).

Our aim is to provide an online platform for researchers working on this topic all around the globe to present their work from anywhere they like (office, home, restaurant, airport, or even the beach). We hope this seminar series can make a small contribution to cutting down costs, unnecessary travel, and carbon emissions.

**How does it work?** A link to the virtual seminar room for each talk is sent out to participating groups via our mailing list. Anyone with the link can tune-in remotely to the live stream, ask questions, and participate in the discussion.

In addition, the talks are typically recorded and posted on our YouTube channel: https://www.youtube.com/c/GravityQuantumFieldsandInformationAEI, in case you miss the live stream, and/or want to revisit the talk.

If you are interested in being added to the mailing list to receive information (including the link to the virtual seminar room) please subscribe here: https://lists.aei.mpg.de/mailman/listinfo/heptnseminar

Contact:

hep-tn@aei.mpg.de

## Upcoming Seminars

#### 18. Norbert Schuch (University of Vienna)

**When:** January 28, 2021

**Title:** *TBA*

**Abstract: **TBA

#### 19. Roman Orus (DIPC San Sebastian)

**When:** February 4, 2021

**Title:** News on tensor network simulations for quantum matter

**Abstract: **In this talk I will make an overview of recent developments at my group on the modelling and simulation of a variety of strongly correlated systems with quantum-inspired tensor networks. Specifically, I will briefly review our findings on the simulation of breathing Kagome antiferromagnets, 3d thermal bosons, 2d systems with SU(2) symmetry, and 2d/3d Kitaev models.

#### 20. Andreas Bauer (Freie Universität Berlin)

**When:** February 11, 2021

**Title:** A unified diagrammatic approach to topological fixed point models

**Abstract: **We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework established is reminiscent to that of state-sum models and lattice topological quantum field theories, but is formalized and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism directly represent discrete path integrals in Euclidean space-time. This language is more immediately related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We illustrate our formalism at hand of simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.

## Past Seminars

#### 17. Johannes Knaute (Albert Einstein Institute, Potsdam)

**When:** October 22, 2020 @ 15.30 (Berlin time)

**Title:** *From spin chains to real-time thermal field theory using tensor networks*

**Abstract: **One of the most interesting directions in theoretical high-energy and condensed-matter physics is understanding dynamical properties of collective states of quantum field theories. The most elementary tool in this quest is retarded equilibrium correlators governing the linear response theory. In this talk, I will describe how tensor networks can be used to determine retarded correlators in an ab initio way for a class of (1 + 1)-dimensional quantum field theories that arise as infrared descriptions of quantum Ising chains. I will show that, complemented with signal analysis using the Prony method, tensor network calculations for intermediate times provide a powerful way to explore the structure of singularities of the correlator in the complex frequency plane and to make predictions about the thermal response to perturbations in a class of nonintegrable interacting quantum field theories. The talk is based on arXiv:1912.08836. I then will also mention some follow-up directions emerging from this line of research.

**YouTube link:** https://youtu.be/yMnKtPndwZ4

#### 16. Jutho Haegeman (University of Ghent)

**When:** October 8, 2020 @ 15.30 (Berlin time)

**Title:** *Lattice regularization and entanglement structure of the Gross Neveu model*

**Abstract: **We construct a Hamiltonian lattice regularisation of the N-flavour Gross-Neveu model that manifestly respects the full O(2N) symmetry, preventing the appearance of any unwanted marginal perturbations to the quantum field theory. In the context of this lattice model, the dynamical mass generation is intimately related to the Coleman-Mermin-Wagner and Lieb-Schultz-Mattis theorem. In particular, the model can be interpreted as lying at the first-order phase transition line between a trivial and symmetry-protected topological (SPT) phase, which explains the degeneracy of the elementary kink excitations. We show that our Hamiltonian model can be solved analytically in the large N limit, producing the correct expression for the mass gap. Furthermore, we perform extensive numerical matrix product state simulations for N=2, thereby recovering the emergent Lorentz symmetry and the proper non-perturbative mass gap scaling in the continuum limit. Finally, our simulations also reveal how the continuum limit manifests itself in the entanglement spectrum. As expected from conformal field theory we find two conformal towers, one tower spanned by the linear representations of O(4), corresponding to the trivial phase, and the other by the projective i.e. spinor representations, corresponding to the SPT phase.

**YouTube link:** https://www.youtube.com/watch?v=XgYJf286n1c

#### 15. Laurens Lootens (University of Ghent)

**When:** September 24, 2020 @ 15.30 (Berlin time)

**Title:** *Matrix product operator symmetries and intertwiners for topological and critical lattice models*

**Abstract: **Projected entangled pair state (PEPS) descriptions of (2+1)d topologically ordered states are characterized by non-local matrix product operator (MPO) symmetries on the entanglement degrees of freedom of the PEPS tensors. Various explicit tensor network representations and their symmetries have been known for a long time, examples being G-injective and string-net PEPS. In this talk, based on our recent work (arXiv:2008.11187), I will synthesize and generalize these results by showing that the consistency conditions for MPO symmetries amount to the defining equations of a bimodule category, thereby showing that the classification of these PEPS and MPO symmetries amounts to the classification of the data of a bimodule category. Different PEPS representations of the same state can be related by an MPO intertwiner, which can be thought of as a generalized gauge transformation on the virtual level, providing an important step towards a general fundamental theorem of PEPS. Additionally, these new PEPS representations allow us to construct tensor network representations of domain walls between different topological phases. Finally, all these results have an immediate application to critical lattice models described by CFT, and I will illustrate this by giving an example of orbifolding or simple current extension on the lattice.

**YouTube link: **https://www.youtube.com/watch?v=NyvxFzE7cFo

#### 14. Erez Zohar

**When:** July 23, 2020 @ 15.30 (Berlin time)

**Title:** *Absorbing fermionic statistics by lattice gauge fields and eliminating fermions*

**Abstract: **Fermionic statistics include the parity superselection rule, having to do with the global Z_2 (parity) symmetry of fermionic Hamiltonians. In lattice theories, this symmetry can be gauged and made local, with "parity Gauss laws" relating the local fermionic parity on each site with the divergence of parity gauge fields around it. These local constraints allow one to transfer the statistics from the fermionic matter to the gauge field. Lattice gauge theories, whose gauge group contains a normal Z_2 subgroup, offer this possibility without introducing any extra ingredients. I will discuss a transformation we have recently derived, which maps lattice gauge theories with fermionic matter to equivalent theories with hardcore bosonic matter. In the resulting models, the interaction of the matter with the gauge fields is slightly changed, but in a local way. Unlike in the Jordan-Wigner procedure, this mapping gives rise to no nonlocality and can be performed in arbitrary space dimensions. Furthermore, it can be extended to models without any gauge field, by the introduction of an auxiliary, non-dynamical Z_2 field which will absorb the statistics.

**YouTube link: **https://www.youtube.com/watch?v=I1I5JDDR1pg

#### 13. Philipp Hauke

**When:** July 10, 2020 @ 15.30 (Berlin time)

**Title:** *Quantum simulating lattice gauge theories – How to make a quantum simulator obey Gauss’s law*

**Abstract: **The difficulty of tackling the out-of-equilibrium dynamics of gauge theories on classical computers is spurring a worldwide effort to solve these problems on dedicated quantum simulator devices. In this talk, I will discuss recent progress towards quantum simulation of gauge theories. A particular focus will lie on a central issue in this context: How to ensure that the quantum simulator fulfills the local symmetry that defines the gauge theory? In other words, how can one ensure that the ultracold atoms or superconducting qubits in a quantum simulator behave as electrons, positrons, and electric field, and mimic Gauss’s law of electrodynamics? I will present our theoretical effort to quantify – and mitigate – the influence of microscopic violations of gauge symmetry [1] as well as the first quantum simulation experiment that measured the fulfillment of Gauss’s law [2]. Through these discussions, I will aim at outlining a roadmap towards mature and practically relevant quantum simulation of gauge theories.

[1] Jad C. Halimeh, Philipp Hauke, *Reliability of lattice gauge theories*, arXiv:2001.00024 [cond-mat.quant-gas] (2020), *Staircase prethermalization and constrained dynamics in lattice gauge theories*, arXiv:2004.07248 [cond-mat.quant-gas] (2020), *Origin of staircase prethermalization in lattice gauge theories*, arXiv:2004.07254 [cond-mat.str-el] (2020), Jad C. Halimeh, Robert Ott, Ian P. McCulloch, Bing Yang, Philipp Hauke, *Robustness of gauge-invariant dynamics in ultracold-atom gauge theories, *arXiv:2005.10249 [cond-mat.quant-gas].

[2] Bing Yang, Hui Sun, Robert Ott, Han-Yi Wang, Torsten V. Zache, Jad C. Halimeh, Zhen-Sheng Yuan, Philipp Hauke, Jian-Wei Pan, *Observation of gauge invariance in a 71-site quantum simulator*, arXiv:2003.08945 [cond-mat.quant-gas] (2020).

**YouTube link: **https://www.youtube.com/watch?v=z4bQZz-rNLs

#### 12. Ananda Roy

**When:** June 25, 2020 @ 15.30 (Berlin time)

**Title:** Simulating Quantum Field Theories with Quantum Circuits

**Abstract: **Investigation of strongly interacting quantum field theories (QFTs) remains one of the outstanding challenges of modern physics. Quantum simulation has the potential to be a crucial technique towards solving this problem. By harnessing the power of quantum information processing, quantum simulation can potentially perform tasks deemed intractable by the classical information processing paradigm. In this talk, I will show that mesoscopic quantum electronic circuit lattices, built with superconducting capacitors and Josephson junctions, can simulate certain bosonic QFTs in 1+1 space-time dimensions. In contrast to conventional spin-chain lattice-regularizations, quantum circuits faithfully capture the non-perturbative properties of these QFTs and are experimentally-realizable with modern-day, superconducting circuit technology. I will begin with the free, compactified boson conformal field theory and analyze its entanglement Hamiltonian for different boundary conditions using analytical and numerical (density matrix renormalization group) techniques. Subsequently, I will describe a quantum circuit lattice for an integrable deformation of the free compactified boson QFT: the quantum sineGordon model. I will present analytical and numerical computations for the various thermodynamic properties of this model.

**YouTube Link: **https://youtu.be/5uqua3QbpSI

#### 11. Karel Van Acoleyen

**When:** June 18, 2020 @ 15.30 (Berlin time)

**Title:** *Entanglement compression in scale space: from MERA to MPOs*

**Abstract: **The multiscale entanglement renormalisation ansatz (MERA) provides a constructive algorithm for realising wavefunctions that are inherently scale invariant. Unlike conformally invariant partition functions however, the finite bond dimension χ of the MERA provides a cut-off in the fields that can be realised. In this talk I will present our recent work (arXiv:1912.10572) in which we demonstrate that this cut-off is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension χ. This is achieved by constructing an explicit mapping between the isometries of a MERA and the local tensors of the MPO.

**YouTube Link: **https://www.youtube.com/watch?v=4oED7twJQg4

#### 10. Luca Tagliocozzo

**When:** May 22, 2020 @ 15.30 (Berlin time)

**Title: ***Signatures of universality out of equilibrium*

**Abstract: **I will discuss the results contained in https://arxiv.org/abs/1909.07381 on how the entanglement spectrum of a region made by adjacent constituents in a one dimensional quantum system becomes universal after quenches at the critical point or across it.

**YouTube Link: **https://www.youtube.com/watch?v=OYxNBM5GeUQ

#### 9. Javier Molina-Vilaplana

**When:** May 08, 2020 @ 15.30 (Berlin time)

**Title:** *Non-Gaussian Entanglement Renormalization for Quantum Fields*

**Abstract: **The multiscale entanglement renormalization ansatz (MERA), which was originally proposed as a variational method to obtain the ground state of spin chains systems, consists of a real space renormalization group technique that, iteratively, removes the quantum correlations between small adjacent regions of space at each length scale. A continuous version of MERA (cMERA) was proposed for free field theories. Motivated, among others, by the conjecture that cMERA is a realization of the AdS/CFT correspondence, a rigorous and (non)perturbative formalism for interacting theories turns out to be essential to advance in this program.

In this seminar, a non-Gaussian cMERA tensor network for interacting quantum field theories (icMERA) is presented. This consists of a continuous tensor network circuit in which the generator of the entanglement renormalization of the wavefunction is nonperturbatively extended with nonquadratic variational terms. The icMERA circuit nonperturbatively implements a set of scale dependent nonlinear transformations on the fields of the theory, which suppose a generalization of the scale dependent linear transformations induced by the Gaussian cMERA circuit.

Here we present these transformations for the case of self-interacting scalar and fermionic field theories. We show how the icMERA tensor network can be fully optimized for the self interacting scalar theory in (1+1) dimensions. This allows us to evaluate, nonperturbatively, the connected parts of the two- and four-point correlation functions.

Our results show that icMERA wavefunctionals encode proper non-Gaussian correlations of the theory, thus providing a new variational tool to study phenomena related with strongly interacting field theories.

Based on: https://arxiv.org/abs/2003.08438

**YouTube Link:** https://youtu.be/p0hega-v7cw

#### 8. Philippe Corboz

**When:** April 24, 2020 @ 15.30 (Berlin time)

**Title: ***Simulation of strongly correlated systems with infinite projected entangled-pair states (iPEPS)*

**Abstract: **An infinite projected entangled pair state (iPEPS) is a variational tensor network ansatz to represent 2D ground states in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension of the tensors. Thanks to several methodological advances in recent years iPEPS has become a very powerful tool for the study of 2D strongly correlated systems, in particular models where quantum Monte Carlo fails due to the negative sign problem. In this talk I will first give an introduction to the iPEPS ansatz and algorithms for ground state simulations, and in the second part I will highlight some of the recent advances with iPEPS, including simulations at finite temperature, the computation of excitations, and the study of critical phenomena.

**YouTube Link: **https://www.youtube.com/watch?v=qgQc2dTVqFk

#### 7. Jonathan Sorce

**When:** April 15, 2020 @ 16.00 (Berlin time)

**Title: ***Status update on tensor networks and quantum gravity*

**Abstract: **In recent years, the "tensor networks" used to describe condensed matter systems have been discovered to share many qualitative features in common with holographic theories of quantum gravity. Studying these features in tensor networks has helped us build intuition for how the corresponding features should work in quantum gravity; most importantly, tensor networks helped us develop a tractable framework for understanding holographic quantum error correction. A little over a year ago (1812.01171), my collaborators and I proved that every geometric state in a holographic theory of quantum gravity can be represented as a tensor network, explaining the success of the "toy model." I will explain this proof, comment on its implications, and discuss recent work from other authors on the tensor network/holography correspondence. The aim of the talk will be to communicate the lessons from tensor networks that should inform our general approach to quantum gravity research; it should be intelligible to an audience with little to no tensor-network experience.

**YouTube Link: **https://www.youtube.com/watch?v=SmIV6hl0NFc

#### 6. Germán Sierra

**When:** March 27, 2020 @ 15.30 (Berlin time)

**Title:** *Tensor Networks with infinite bond dimension*

**Abstract:** In the last few years Tensor Networks (TN) have become a standard technique to study the properties of many body systems. Its origins can be traced back to the AKLT model in 1987, and the Density Matrix Renormalization Group in 1992. They were the first examples of Matrix Product States that were actively investigated by the cond-mat community in the 90's. By the turn of the century, the quantum information community understood the success of the MPS in terms of quantum entanglement. This led to new TNs like PEPS and MERA. In all these TNs the dimension of the auxiliary space, that mediates the entanglement between the physical degrees of freedom, is finite. This feature limits the application of MPS to critical systems in 1D whose low energy states violate the area law of the entanglement entropy. In this talk I will introduce TNs whose bond dimension is infinite, that allows us to overcome this problem and describe critical spin chains and Fractional Quantum Hall systems.

**YouTube Link: **https://www.youtube.com/watch?v=iV8r-wBbU60

#### 5. Adam G. M. Lewis

**When:** February 28, 2020 @ 15.30 (Berlin time)

**Title:** *Fermionic Hartle-Hawking Vacua From a Staggered Lattice Scheme*

**Abstract:** I will discuss work in collaboration with Guifré Vidal towards simulation of quantum fields in curved spacetimes. We eventually mean to simulate strongly interacting fields, out of equilibrium, coupled to spacetime curvature in various ways. This study concerns the more modest goal of computing renormalized, quadratic expectation values of free Dirac fields installed upon fixed, two dimensional Lorentzian spacetimes. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation values of quadratic operators in the continuum. As illustrative applications, we show how to recover the Unruh effect in flat spacetime, how to compute renormalized expectation values in the Hawking-Hartle vacuum of a 2-dimensional "Schwarzschild" black hole, and how to do the same in the Bunch-Davies vacuum of dS2.

**YouTube Link:** https://youtu.be/Vu538q3G0wY

#### 4. Simone Montangero

**When:** February 14, 2020 @ 15.30 (Berlin time)

**Title:** *Tensor network methods applied to high energy physics problems*

**Abstract:** We briefly introduce tensor network methods, a classical numerical approach that promises to become a powerful tool to support future quantum simulations and computations, providing guidance, benchmarking and verification of the quantum computation and simulation results. We review some of the latest achievements we obtained: the gauge-invariant formulation of tensor networks and their application to abelian and non-abelian, one- and two-dimensional lattice gauge theories in regimes where Monte Carlo methods efficiency is hindered by the sign problem. Finally, we present the application of tensor network machine learning techniques to the event classification of LHCb simulated data.

**YouTube link: **https://youtu.be/vrZHkyDvYhI

#### 3. Frank Verstraete

**When:** January 24, 2020 @ 15.30 (Berlin time)

**Title:** *Quantum symmetries in tensor networks*

**Abstract:** Tensor networks and more specifically matrix product operators provide a natural framework for describing nonlocal symmetries in lattice spin systems. It will be argued that those matrix product operators form representations of tensor fusion categories, and that they lead to simple lattice representations of topological and conformal field theories. We will construct algebraic equations defining the topological / conformal sectors, and construct explicitly all excitations using the operator-state correspondence.

**YouTube link: **https://www.youtube.com/watch?v=IHe5YYsEK7k.

#### 2. Bartlomiej Czech

**When:** December 6, 2019 @ 15.30 (Berlin time)

**Title:** *What does the Chern-Simons formulation of AdS3 gravity tell us about complexity?*

**Abstract:** I will explain how to realize the wavefunction of a CFT2 ground state as a network of Wilson lines in the Chern-Simons formulation of AdS3 gravity. The position and shape of the network encode the scale at which the wavefunction is defined. The structure of the network is that of a Matrix Product State (MPS) whose constituent tensors effect the Operator Product Expansion. A general argument suggests identifying the "density of complexity" of this MPS network with the extrinsic curvature of the bulk cutoff surface, which by the Gauss-Bonnet theorem agrees with the Complexity = Volume proposal. The viewpoint I offer departs from the circuit paradigm of complexity and dispenses with reference states. Instead, recognizing that field theory states are functionals which send observables to their expectation values, I propose to think of state complexity as the algorithmic complexity of constructing such functionals.

**YouTube link: **https://www.youtube.com/watch?v=2p-mo-LdZxw

#### 1. Ignacio Cirac (Opening seminar of the series)

**When:** November 8, 2019 @ 15.00 (Berlin time)

**Title:** *Tensor Networks and Lattice Gauge Theories*

**Abstract:** Certain Quantum Many-body states can be efficiently described in terms of tensor networks. Those include Matrix Product States (MPS), Projected Entangled-Pair States (PEPS), or the Multi-scale Entanglement Renormalization Ansatz. They play an important role in quantum computing, error correction, or the description of topological order in condensed matter physics, and are widely used in computational physics. In the last years, it has also been realized their suitability to describe Lattice Gauge Theories, at least in the context of MPS in low dimensions. In this talk, I will review some of the basic ideas about tensor networks and their applications to lattice gauge theories, and explain current efforts to extend them to higher dimensions using PEPS.

**YouTube link: **https://youtu.be/hdb82b1kazw