# Thorsten B Wahl

# Thorsten B Wahl

Postdoctoral Research Assistant

I received my PhD in the group of J. Ignacio Cirac, Max Planck Institute of Quantum Optics, Garching, Germany. My research is focused on the application of tensor network states to open problems in Condensed Matter and High Energy Physics.

Tensor network states are non-perturbative ansatz wave functions for lattice systems (such as spin lattices or particles hopping on a lattice). At first sight, describing the physical properties of such model systems appears exponentially hard, as the dimension of the Hilbert space grows exponentially with the number of lattice sites. However, the local nature of realistic interactions heavily constrains the complexity of actual physical systems: Realistic ground states occupy only a polynomially large submanifold of the exponentially large Hilbert space. For systems with an energy excitation gap, this submanifold is characterised by states which fulfil the area law of entanglement. That is, the entanglement between a block of sites and the rest grows only like the boundary of that region - which is highly non-generic. Tensor network states also fulfil the area law, which suggests that they constitute an efficient representation of the sought-after physical submanifold. This has been shown rigorously in one dimension [Phys. Rev. B 73, 094423 (2006)], and there is strong evidence for the same to be true in higher dimensions [Phys. Rev. B 91, 045138 (2015)].

While lattice systems are always approximations of actual physical systems, they often capture the essential physical properties of such systems. Prominent examples are spin chains describing magnetic insulators in one dimension, the two-dimensional Hubbard model, which is believed to describe high-temperature superconductivity, and Lattice Gauge Theories in High Energy Physics. Note, that tensor network states have also been generalised to continuum systems [Phys. Rev. Lett. 104, 190405 (2010)].

I currently work on the application of tensor network states to the phenomenon of many-body localisation, chiral topological systems and Lattice Gauge Theories, as explained in the other sections.

Non-interacting metallic systems in one and two dimensions (without spin-orbit coupling or spontaneously broken time reversal symmetry) are fragile to arbitrarily weak disorder, i.e., disorder induces localisation of all single-particle orbitals - a phenomenon known as Anderson localisation. If one includes interactions, one might anticipate that they destroy localisation as they lead to mixing of all single-particle orbitals. However, it turns out that in one dimension localisation survives for sufficiently strong disorder, which has been dubbed many-body localisation [Ann. Phys. 321, 1126 (2006); Phys. Rev. Lett. 95, 206603 (2005)]. Many-body localised systems do not thermalise, i.e., they retain a memory of their initial state for arbitrarily long times. All eigenstates of such systems fulfil the area law of entanglement and as a result can be efficiently approximated by tensor network states [Phys. Rev. Lett. 114, 170505 (2015)].

Building on a previous approach to represent all eigenstates by a single tensor network [Phys. Rev. B 94, 041116 (2016)], we devised an ansatz for such a tensor network whose approximation error decreases inversely polynomially with computational time [Phys. Rev. X 7, 021018 (2017)]. We work on applying and extending our scheme in order to describe important many-body localisation effects, such as the emergence of conserved, effective spin degrees of freedom [Phys. Rev. B, 184201 (2018); arXiv:1811.00442], symmetry and localisation protection of all eigenstates [Phys. Rev. B 98, 054204 (2018); arXiv:1808.05656] and the short-time phenomenon of many-body localization in two dimensions [Nat. Phys. 15 (2019)], where exact diagonalisation is unfeasible.

There used to be the wide-spread expectation that tensor network states were not able to exactly represent ground states of chiral topological systems (such as integer or fractional quantum Hall systems). Such a shortcoming would also have cast doubts on their ability to efficiently approximate such systems. However, we and another independent research group [Phys. Rev. Lett. 111, 236805 (2013); Phys. Rev. B 92, 205307 (2015)] demonstrated that tensor network states can have chiral topological properties if they are critical, i.e., have algebraically decaying correlations. In the non-interacting limit, this turns out to be a necessary condition. With interactions, the same seems to be true based on all known examples [Phys. Rev. Lett. 114, 106803 (2015); Phys. Rev. B 91, 224431 (2015); ...], although there is no rigorous proof. We are currently working on gaining more physical intuition as to why free fermionic chiral tensor networks have to be critical [Phys. Rev. B 90, 115133 (2014)] with the aspiration of extending such arguments to the interacting case.

Lattice Gauge Theories are discretisations of Quantum Field Theories and thus constitute a tool to describe such theories numerically. The continuum limit is obtained by taking the lattice spacing to zero and accounting for remnant effects such as the fermion doubling problem. However, numerical simulations have previously been constrained to quantum Monte Carlo methods, which suffer from the sign problem and thus become unfeasible for finite fermion density.

Tensor network states can be readily extended to fermionic systems and do not suffer from the sign problem. To describe Lattice Gauge Theories, gauge fields can easily be included into the links between tensors. This scheme enabled the numerical simulation of one-dimensional field theories with arbitrary fermion density [Phys. Rev. A 90, 042305 (2014)]. We put forth tensor network ansatzes for U(1) and SU(2) Lattice Gauge Theories in two dimensions [Ann. Phys. 363, 385 (2015); Ann. Phys 374, 84 (2016)] and analysed their physical properties. We are planning to employ our approach for the variational description of Lattice Gauge Theories.