Verallgemeinerte Symmetrien in Quantentensornetzwerken

1) Global symmetries play a fundamental role in quantum many-body physics (QMBP). A key question is how a global symmetry is encoded at the level of the local constituents, that is, the microscopic understanding of this feature. The solution to this question has led, for example, to the classification of symmetry protected topological (SPT) orders, outside Landau's paradigm of symmetry breaking, its application to quantum computation and its generalization to higher dimensions. The current state-of-the-art approach focuses on matrix product states (MPSs), which give a microscopic description of ground states of one-dimensional local gapped Hamiltonians which are symmetric under groups, while more general tensor network states (TNS) and symmetries of important systems are currently not well understood.

2)The aim of this project is to develop new analytical tools on tensor networks to locally characterize exotic symmetric phases of matter. This proposal tackles major problems to gain a general understanding of symmetric phases beyond standard SPT phases. The final goal is to capture and algebraically formalize certain infinite-dimensional classes of tensor networks. First, we will focus on global symmetries representing quantum groups on regular MPSs and secondly, we will study regular symmetries on field TNS (FTNS). On top of the results we are aiming at, as a byproduct, we would obtain a classification of their corresponding phases of matter, with an entanglement characterization and a construction of order parameters and we would study their higher dimensional analogous phases.

3)The approach of this proposal is to formulate the problem from an algebraic point of view and use its representation theory. The tools used are tensor networks describing both the states, as MPSs and FTNS, and also the global symmetries as matrix product operators (MPO). TNS provide a unique access to the entanglement space, whose properties determine the quantum phase. Our methods will focus on characterizing the properties of TNS at the entanglement level focusing on how the action of the symmetry, in particular the MPO tensor, is translated locally.

4)The methods of this project correspond to a novel proposal to tackle questions which it is not possible to answer by the case-by-case approach of previous studies. The novelty lies on the generalized perspective of the problem given by constructing the algebraic framework covering MPO representations of quantum groups and the class FTNS. This approach will enable us to characterize symmetries locally, providing us with the preferred knowledge to answer unsolved questions and propose new cutting-edge ideas.

5)The applicant, Dr. Jose Garre Rubio, is a theoretical physicist specialized in analytical tools for TN describing topological phases. The host, Prof. German Sierra, is a pioneer of conformal field theories and entanglement theory and a leading expert in mathematical methods in QMBP.