Topological phases and related topics

mardi 22 oct. 2024, 10:30 → 18:00 Europe/Paris

Salle 318 (IMB)

Invited speakers

• Ken Shiozaki (Kyoto University)
• Atsushi Ueda (Ghent University)
• Hidir-Deniz Yeral (Université de Bourgogne)

10:30 → 11:30 Higher Berry curvature of matrix product states¶

The Berry curvature is a geometric quantity that underlies parameter-dependent quantum mechanics. Recently, the concept of a higher dimensional generalization of the Berry curvature, known as the higher Berry curvature, has been proposed [1]. In $d$-spatial dimensions, the higher Berry curvature is proposed as a $(d+2)$-form that is defined for short-range entangled states. Providing a computationally feasible method for calculating the higher Berry curvature for a given short-range entangled state is challenging. We propose an explicit method to compute the higher Berry curvature for a given translationally invariant matrix product state for a 1-dimensional short-range entangled state. Our demonstration showed that summing up all the higher Berry fluxes across tetrahedra results in a nontrivial integer associated with the 3rd cohomology $H^3(X,Z)$.

[1] A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of Berry curvature, Phys. Rev. B 101, 235130 (2020).
[2] Ken Shiozaki, Niclas Heinsdorf, Shuhei Ohyama, Higher Berry curvature from matrix product states, arXiv:2305.08109.

Orateur: Ken Shiozaki (Kyoto University)

Shiozaki.pdf

11:45 → 12:30 Semisimple quantum invariants via factorization homology¶

Since their axiomatization by Atiyah in '88, topological field theories (TFTs) have developed into a bridge between topology and representation theory. One goal of my thesis is to give a construction of the 3-manifold invariants coming from an important class of three-dimensional TFTs, namely the Reshetikhin-Turaev construction and its non-semisimple generalization, via the factorization homology of surfaces. In this talk, I will explain the notion of a topological field theory and discuss some important examples in the three-dimensional case. Then I will explain how a consistent system of handlebody group representations (so-called an ansular functor) can be used to obtain a factorization homology description of the semisimple quantum invariants.

Orateur: Hidir-Deniz Yeral (Université de Bourgogne)

14:00 → 15:00 Putting chiral fermions on tensor networks as a generalized eigenvalue problem¶

Naively discretizing fermionic field theory on a lattice is known to produce unwanted fermion modes. This phenomenon is often referred to as 'fermion doubling,' which has been a significant issue when simulating chiral fermionic theory. We develop a formalism for simulating one-dimensional interacting chiral fermions on the lattice without breaking any local symmetries by defining a Fock space endowed with a semi-definite norm defined in terms of matrix product operators. This formalism can be understood as a second-quantized form of Stacey fermions [1], hence providing a possible solution for the fermion doubling problem and circumventing the Nielsen-Ninomiya theorem. The chiral states, in the analogue of edge physics, has non-orthogonal structures, and thus calls for more generic frame work called the generalized eigenvalue problem. Using this form, we find that the chiral Hamiltonian has local and hermitian features, allowing us to be simulated using tensor network methods similar to the ones used for simulating local quantum Hamiltonians. As a proof of principle, we consider a single Weyl fermion on a periodic ring with Hubbard-type nearest-neighbor interactions and construct a variational generalized DMRG code demonstrating that the ground states of the system for large system sizes can be determined efficiently [2].

[1] R. Stacey, Phys. Rev. D. 26, 468 (1982)
[2] J. Haegeman, L. Lootens, Q. Mortier, A. Stottmeister, A. Ueda, and F. Verstraete, arXiv:2405.10285 (2024)

Orateur: Atsushi Ueda (Ghent University)

Ueda.pdf