# Theory Seminar

 Date Seminars 2017   (Go to Seminars 2016, Seminars 2015, Seminars 2014, and earlier therein) 09.01.2017 Bipartite charge fluctuations in Z_2 topological insulators and superconductors Invited speaker: Loïc Herviou (Ecole Polytechnique, ENS) Bipartite charge fluctuations (BCF) have been introduced to provide an experimental indication of many-body entanglement.They are a very efficient and useful tool to characterize phase transitions in a large variety of charge-conserving models in one and two dimensions In this seminar, we study the BCF in generic one- and two-dimensional Z_2 (topological) models such as the Kitaev chain, spin-orbit insulators, the graphene and the Haldane model, where the charge we observe is no longer conserved. In one-dimension, we demonstrate that at phase transitions characterized by a linear dispersion, the BCF probe the change in a winding number that allows to pinpoint the transition and corresponds to the topological invariant for standard models. Additionally, we prove that a sub-dominant logarithmic contribution is still present at the exact critical point. Its quantized coefficient is universal and a characteristic of the critical model. In two dimensions, a similar structure appears. While the area term no longer reveal directly the phase transition, a subdominant logarithmic term is still present. Similarly to the entanglement entropy, it depends on the exact shape of the considered region, with contributions of the corner of the regions only. 25.01.2017 Approaching non-Abelian Lattice Gauge Theories with Tensor Networks Stefan Kühn In recent years the Tensor Network approach to lattice gauge theories has proven itself as promising alternative to the conventional Monte Carlo methods widely used. In contrast to Monte Carlo simulations, numerical methods based on Tensor networks do not suffer from the sign problem, thus allowing to address problems and parameter regimes which are inaccessible with Monte Carlo. However, even for for the simplest non-Abelian gauge models with dynamical fermions, the computational effort typically grows quickly. Hence, current Tensor Network simulations for non-Abelian models are rather limited. In this talk I will address the case of a SU(2) lattice gauge theory. I will show how, starting from a basis of color neutral states, the gauge field for systems on finite lattices with open boundary conditions can be integrated out, thus greatly reducing the degrees of freedom. While this formulation is completely general, it trivially allows to truncate the maximum color-electric flux in the system, thus making it particularly suitable for a Tensor Network approach. As a proof of principle I will present numerical results for the low lying spectrum obtained with Matrix Product States for a family of truncated SU(2) models. 01.02.2017 Decay of correlations in systems of fermions with long-range interactions at non-zero temperature Senaida Hernandez We study correlations in fermionic systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anti-commuting operators based on long-range Lieb-Robinson type bounds. Our result shows that correlations between such operators in fermionic long-range systems of spatial dimension $D$ with at most two-site interactions decaying algebraically with the distance with an exponent $\alpha \geq 2\,D$, decay at least algebraically with an exponent arbitrarily close to $\alpha$. Our bound is asymptotically tight, which we demonstrate by numerically analyzing density-density correlations in a 1D quadratic (free, exactly solvable) model, the Kitaev chain with long-range interactions. Away from the quantum critical point correlations in this model are found to decay asymptotically as slowly as our bound permits. 08.02.2017 An introduction to Variational Monte-Carlo Ivan Glasser Variational Monte-Carlo methods are used to determine the energy of a wave function and to optimize its parameters in order to approximate the ground state of a many-body quantum system. In this talk I will give a general introduction to Variational Monte-Carlo. Starting from the early days of Monte-Carlo integration I will explain how these methods can be used to compute energies of wave functions. I will then give an overview of modern methods to optimize the energy of a wave function with many parameters. 15.02.2017 Simulating non-Equilibrium systems with Matrix Product States Julian Roos Understanding out of equilibrium remains a challenge for classical and quantum systems. There is no general non-equilibrium statistical mechanics framework to resort to, if one is interested in the statistical properties of observables in far from equilibrium situations. The theory of large deviations can fill this gap in some cases and Tensor Networks are one possibility to explore this problem from a numerical side. Matrix Product States can capture the properties of non-equilibrium stationary states of many classical and quantum models. On Wednesday I will discuss how people have used these techniques for the simplest problem of particle hopping on a 1D lattice. 21.02.2017 at 14:00 Dissipation induced topological states: A recipe Invited speaker: Moshe Goldstein (Tel-Aviv University) It has recently been realized that driven-dissipative dynamics, which usually tends to destroy subtle quantum interference and correlation effects, could actually be used as a resource. By proper engineering of the reservoirs and their couplings, one may drive a system towards a desired quantum-correlated steady state, even in the absence of internal Hamiltonian dynamics. An intriguing class of quantum phases is characterized by topology, including the quantum Hall effect and topological insulators and superconductors. Which of these noninteracting topological states can be achieved as the result of purely dissipative Lindblad-type dynamics? Recent studies have only provided partial answers to this question. In this talk I will present a general recipe for the creation, classification, and detection of states of the integer quantum Hall and 2D topological insulator type as the outcomes of coupling a system to reservoirs, and show how the recipe can be realized with ultracold atoms and other quantum simulators. The mixed states so created can be made arbitrarily close to pure states. I will discuss ways to extend this construction to other topological phases, including non-Gaussian ones, such as fractional quantum Hall state. 02.03.2017 at 11:30 Majorana quasi-particles from angular momentum conservation Invited speaker: Fernardo Iemini (International Centre for Theoretical Physics - Trieste, Italy) We show how angular momentum conservation can stabilise a quasi-topological phase of matter supporting Majorana qausi-particles as edge modes. Differently from typical scenarios, where such quasi-particles require the presence of superconductivity, we investigate orbital SU (2) × SU (2) Hubbard models in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional models. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated with numerical simulations. Our results pave the way toward the observation of Majorana edge modes with Alkaline-earth-like fermions in optical lattices, where the basic ingredients for our recipe - spin-orbit coupling and strong inter-orbital interactions - have been observed over the last two years. 15.03.2017 Efficient representation of fully many-body localized systems using tensor networks Invited speaker: Thorsten Wahl (Oxford University) Many-body localization (MBL) is currently an intensely studied topic and characterized by the fact that certain strongly disordered systems fail to thermalize. For sufficiently strong disorder in one dimension, all eigenstates of MBL systems fulfill the area law of entanglement. This makes tensor network states ideally suited to represent such fully many-body localized systems. Building on the ansatz proposed in Phys. Rev. B 94, 041116(R) (2016), I will present a tensor network that is able to capture the full set of eigenstates of such MBL systems efficiently: For a given system size, local observables can be approximated with an error that decreases as an inverse polynomial of the computational cost, which is an exponential improvement over the previous ansatz. If the system size is increased, the computational cost needs to grow only linearly with the system size in order to keep the accuracy fixed. The technique turns out to be highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the dynamical phase transition. Finally, the power of the technique is demonstrated on systems of 72 sites, where clear signatures of the phase transition can be seen.