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ENtanglement renormalization and GAuGE Symmetry

Final Report Summary - ENGAGES (ENtanglement renormalization and GAuGE Symmetry)

The project ENGAGES has allowed Dr L. Tagliacozzo to better understand the relations between the novel developments in the theory of quantum many-body systems and lattice gauge theories (LGT).

One of the open problems in LGT is indeed related to the existence of phases that cannot be characterised by local operators. Indeed different phases in LGT are related to the same symmetries, something that inevitably questions the classification of phases introduced by Landau.

In the context of Abelian LGT, this problem can be overcome by using dual non-local order parameters and attributing different dual symmetries to the phases of the Abelian LGT. In the context of non-Abelian gauge theories, these dual non-local order parameters have been proposed and characterised to a great extent based on the idea of Abelian dominance and the related Abelian projections. However, these ideas are still largely debated due mainly to the privileged role of some Abelian subgroup of the full non-Abelian symmetries that many researchers today still find hard to justify on general grounds.

In the context of topological models in two dimensions, on the other hand, these years have seen a lot of work that relate the properties of bulk phases to the properties of a theory defined on the boundaries. This has been typically done in the context of resonating valence bound states and in the context of simple topological models with discrete groups.

Dr L. Tagliacozzo in these two years has been able to rederive important results about the existence of LGT for continuous group in spin systems. With the new derivation, L. Tagliacozzo has been able to set the framework to apply the new ideas of phase classifications to the case of both Abelian and non-Abelian LGT with continous group. In particular, Dr Tagliacozzo has explicitly worked out the theory for the:

1) two-dimensional (2D) U(1) lattice gauge theories built with spin one half constituents (also called Abelian gauge magnet);
2) 2D SU(2) lattice gauge theory built with four dimensional (two spin one half) constituents.

In this way, Dr Tagliacozzo has been able to:

1) propose the first 2D implementation of these model on optical lattices that once realised will give access to both ground state and out of equilibrium experimental studies of Abelian and non- Abelian LGT;
2) a tensor network ansatz for gauge invariant states of the above models that used in numerical simulations will allow to obtain a full characterisation of ground state properties (phases) and short-time out of equilibrium dynamics.

The socio-economic impact of the project is huge, since with its expertise about lattice gauge theories L. Tagliacozzo has contributed positioning ICFO at the head of a huge international quest for designing alternative simulations of LGT, beyond the Monte Carlo tools used today, that despite their enormous success present shortcomings difficult to overcome when dealing with out-of equilibrium dynamics and the presence of a finite density of fermions.

L. Tagliacozzo has helped in giving a huge strategic advantage to the group at ICFO when compared for example with the group of P. Zoller (Innsbruck) or the joint collaboration of Prof. Reznik (Tel Aviv) and P. Cirac (Garching), since at ICFO, the group is now able to both design quantum simulations (as the other two competing groups) and perform classical simulations (unlike the other two competing groups) of the Abelian and non-Abelian gauge magnets.

The scientific impact of the project is also huge with eleven papers being published during the two years of the project two of them completely dedicated to the Abelian and non-Abelian gauge magnets.

As a whole the main deliverables of the project have all been achieved. The three main categories are:

1) development of a theory of entanglement in LGT: this has been done in papers of both section A and B of the pubblication list;
2) simulations of Abelian and non-Abelian LGT has been discussed in papers;
3) developments related to out-of equilibrium dynamics have been discussed in papers;
4) numerical toolbox for both Abelian and non-Abelian LGT will appear soon in a forthcoming papers.

The project has also led D. Tagliacozzo to contribute to ongoing effort at ICFO about design of interesting quantum simulations protocols.

From the point of view of increasing the collaborations between ICFO and leading scientists, L. Tagliacozzo has been promoting a series of talk given by experts in the field of tensor networks, among others:

- talk by G. Vidal on the branching MERA;
- talk by B. Pirvu on excitations in systems with periodic boundary conditions (a work in collaboration with F. Verstraete);
- talk by J. Molina about the relations between entanglement renormalisation and AdS/CFT;
- talk by V. Stojevic about continuous tensor networks for field theories;
- talk by E. Tonni about the scaling of entanglement negativity in critical systems.

Regarding the aspects of transfer of knowledge, now ICFO has a group of young PhD students that master the tensor network techniques that L. Tagliacozzo has been teaching them. Among others, T. Koffel (who has moved back to ENS Paris), P. Hauke (who has now moved to Innsbruck) and A. Zamora (ICFO). With all of these students, L. Tagliacozzo has published important papers on peer reviewed journals.

This is a comprehensive summary overview of results, conclusions and the socio-economic impacts of the project. The publishable report shall be formatted to be printed as a stand alone paper document. This report should address a wide audience, including the general public.

Most of the advances made during the project are summarised on L. Tagliacozzo webpage: where updated contact details for L. Tagliacozzo will always been made available.