Tensor network methods have become workhorses in the study of strongly correlated systems in condensed matter physics and increasingly also in the context of molecular systems. While their variational character provides results of remarkable accuracy, the capture quantum states featuring low entanglement well, a set of states often referred to as the 'physical corner'. While this physical describes ground states of local Hamiltonian problems from the condensed matter context well, the same cannot be said for problems involving molecules in quantum chemistry or systems undergoing time evolution. In this talk I will provide an overview over tensor network methods augmented by fermionic mode transformations in the context of molecular problems , condensed-matter simulations , and time evolution . If time allows, I might briefly mention two new applications of tensor networks in being a design principle for building quantum devices  and in machine learning [5,6].  Fermionic orbital optimisation in tensor network states (C. Krumnow, L. Veis, Ö. Legeza, J. Eisert, Phys. Rev. Lett. 117, 210402 (2016))  Towards overcoming the entanglement barrier when simulating long-time evolution (C. Krumnow, J. Eisert, Ö. Legeza, arXiv:1904.11999)  Dimension reduction with mode transformations: Simulating two-dimensional fermionic condensed matter systems (C. Krumnow, L. Veis, J. Eisert, Ö. Legeza, arXiv:1906.00205)  Simulating topological tensor networks with Majorana qubits (C. Wille, R. Egger, J. Eisert, A. Altland, Phys. Rev. B 99, 115117 (2019))  Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning, I. Glasser, R. Sweke, N. Pancotti, J. Eisert, J. I. Cirac, arXiv:1907.03741, NeurIPS (2019))  Tensor network approaches for learning non-linear dynamical laws (A. Goessmann, M. Goette, I. Roth, G. Kutyniok, J. Eisert, R. Sweke, Submitted to ICML2020 (2020))
Nonlinear resonances are ubiquitous in nature. They play a central role in determining the long time (in)stability of dynamical systems ranging from dissociating molecules to escaping asteroids. In Hamiltonian systems with three or more degrees of freedom the network of resonances results in a connected chaotic layer called as the Arnold web. Understanding transport, local and global, on the Arnold web is essential for identifying novel dynamical regimes. In the quantum setting additional transport pathways between "remote" regions on the web open up due to dynamical tunnelling, hence potentially altering the classical dynamical stabilities. This "tug-of-war" between classical and quantum transport pathways has interesting consequences for coherent control and thermalisation is small isolated systems. Thus, the classical notions of Kolmogorov-Arnold-Moser (KAM) eternal stability, the long time Nekhoroshev stability, and the Chirikov instability maybe tempered by quantum mechanics in intriguing ways. In this talk I will discuss examples of dynamics on the web and highlight the important and universal features on the web. Moreover, I will show the analogy between seemingly disparate dynamical systems that would benefit from classical-quantum correspondence study of the dynamics on the Arnold web.
Chaotic transport in phase space connects very different states of the system. Transport times are often restricted due to the presence of partial barriers, which allow for just a small flux across them. For systems with two degrees of freedom this is well understood, however, in higher-dimensional systems, like the 3-body problem or chemical reactions, there are still many fundamental questions: Which invariant objects in phase space give rise to a partial barrier, which is the most restrictive one, and is there still a so-called turnstile mechanism that acts like a revolving door? In the talk I will first explain the low-dimensional case, where partial barriers are obtained from periodic orbits. Subsequently, I will present the answers for the higher-dimensional case.
Landau-Fermi liquid theory is commonly associated with the existence of coherent quasiparticles, i.e., of a simple pole in the single-particle Green’s function at the chemical potential with residue Z<1. On the contrary, its quantum impurity counterpart, the Nozières local Fermi liquid theory. remains valid also when the impurity density of states displays a pseudo gap instead of the standard Abrikosov-Suhl Kondo resonance. I will show that this asymmetry is only apparent, namely that Landau’s Fermi-liquid-like expressions of thermodynamic and dynamic quantities can be recovered even in the absence of coherent quasiparticles. Specifically, after a brief historical overview of Landau-Fermi liquid theory and Nozières's local one, I will revisit the standard microscopic derivation of the former, and show that it holds under a more general hypothesis that includes, as a particular case, the conventional Fermi liquids with well-defined quasiparticles.
In standard text books on electrodynamics, Maxwell's laws are often illustrated using field lines -- a notion that lacks relativistic covariance and obscures the superposition principle (see e.g. R. Feynman in: The Feynman Lectures on Physics, Vol II, Chapter 1-5). In this colloquium, an improved variant of the field line picture is presented: the discrete approximation of the electromagnetic field by chains (B and D are modeled as 1-chains, E and H as 2-chains). The language of chains offers a simple, direct and intuitive approach to Maxwell's electrodynamics, without compromising physical or mathematical correctness. Fundamental aspects such as parity invariance or the transcription of the theory to curved space-time are especially transparent in this approach. The talk illustrates the chain picture at a number of examples including vortex motion in superconductors, spin-orbit coupling, equivalence between problems of magneto- and electrostatics, emission of an electromagnetic signal by the discharge of a capacitor, quantum Hall-Ohm law. The colloquium is an outgrowth of the speaker's 30 years of experience teaching electrodynamics to physics students.
In 2019, Google researchers carried out an experiment where they achieved "quantum supremacy". This is defined as when a quantum computer performs *any* task in a way that a classical computer takes impractical amounts of time to simulate its outcome. This is a huge milestone for the field, but several milestones lie ahead. In this talk, I will describe a potential algorithmic road to "quantum advantage", ie. the moment when a quantum computer carries out a *practical* task on a system that is unreachable by current classical computers. To reach quantum advantage presumably many years of hardware development and algorithmic development lie ahead of us. The current era of quantum computing, coined near-term intermediate-scale quantum computing (NISQ) by John Preskill presents many challenges and opportunities. I will discuss this algorithmic road in the context of algorithms for the simulation of molecules and materials, as well as will briefly mention quantum machine learning applications.
I discuss the higher-order topological field theory and response of topological crystalline insulators with no other symmetries. The geometry and topology of the system is organised in terms of the elasticity tetrads which are ground state degrees of freedom labelling the lattice geometry, topological responses and higher-form conservation laws on sub-dimensional manifolds of the bulk system. In a crystalline insulator in arbitrary dimensions, they classify the topological responses in a transparent fashion in terms of a lattice embedding and higher-order global symmetries. In the continuum limit of the lattice, the elasticity tetrads can be used to derive the higher-order or embedded topological responses to global U(1) symmetries, such as electromagnetic gauge fields with explicit formulas for the quasi-topological invariants in terms Green's functions. I will discuss the examples of topological polarisation, quantum Hall and axionic charge density wave and multipole insulators in detail. The proposed framework further bridges the recently appreciated connections between topological field theory, higher form symmetries and gauge fields, fractonic excitations, elasticity and topological defects with restricted mobility in crystalline insulators.