Paul A. McClarty, Max Planck Institute for the Physics of Complex Systems Topological Magnons

I give an overview of the insights we and other people have had into the band structure of magnons and discuss in some detail three main topics from our work: (i) the robustness of topological edge states in the presence of magnon interactions (ii) visualization of spin-momentum locking in magnon systems (iii) the non-Hermitian topology of spontaneous magnon decay.

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter τ . It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity τ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure τ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

[1] Bruno Mera. Localization anisotropy and complex geometry in two-dimensional insulators. Phys. Rev. B, 101:115128, Mar 2020.

In recent years there has been growing interest in open quantum systems described by non-Hermitian Hamiltonians in various fields. In this talk I present results on the quantum evolution of Gaussian wave packets generated by a non-Hermitian Hamiltonian in the semiclassical limit of small $\hbar$. This yields a generalisation of the Ehrenfest theorem for the dynamics of observable expectation values. The resulting equations of motion for dynamical variables are coupled to an equation of motion for the phase-space metric — a phenomenon having no analogue in Hermitian theories. The insight that can be gained by this classical description will be demonstrated for a number of example systems.

Derivation of macroscopic statistical laws, such as Fourier's, Ohm's or Fick's laws, from reversible microscopic equations of motion is one of the central fundamental problems of statistical physics. In recent years we have witnessed a remarkable progress in understanding the dynamics and nonequilibrium statistical physics of integrable systems. This encourages us to attempt to understand the aforementioned connection at least in specific classes of nontrivial integrable systems with strong interactions. In my talk I will introduce a family of reversible cellular automata, which model systems of interacting particles, and for which we can prove the existence of diffusion and exactly solve several interesting paradigms of statistical physics, e.g.: nonequilibrium steady states of the system between two stochastic reservoirs, the problem of relaxation to the nonequilibrium steady state, or even the problem of explicit time evolution of macroscopic states, for instance, the solution of inhomogeneous quench problems and the calculation of dynamical structure factor in highly entropic equilibrium states.

Laughlin state is an $N$-particle wave function, describing the fractional quantum Hall effect (FQHE). We define and construct Laughlin states on genus-$g$ Riemann surface, prove topological degeneracy and discuss adiabatic transport on the corresponding moduli spaces. Mathematically, the problems around Laughlin states involve subjects as asymptotics of Bergman kernels for higher powers of line bundle on a surface, large-$N$ asymptotics of Coulomb gas-type integrals, vector bundles on moduli spaces.

I review recent advances in the development of generalized hydrodynamics, a flexible approach to the out-of-equilibrium dynamics of integrable quantum systems. I explain how this methodology has allowed exact calculations of transport in $1D$ system. Then, I consider the out-of-equilibrium dynamics of an interacting integrable system in the presence of an external dephasing noise. In the limit of large spatial correlation of the noise, we developed an exact description of the dynamics of the system based on a hydrodynamic formulation. This results in an additional term to the standard generalized hydrodynamics theory describing diffusive dynamics in the momentum space of the quasiparticles of the system, with a time- and momentum-dependent diffusion constant. Our analytical predictions are then benchmarked in the classical limit by comparison with a microscopic simulation of the non-linear Schrodinger equation, showing perfect agreement. In the quantum case, our predictions agree with state-of-the-art numerical simulations of the anisotropic Heisenberg spin in the accessible regime of times and with bosonization predictions in the limit of small dephasing times and temperatures.

Graphene is the prototypical two-dimensional material. One of main features of two-dimensional materials is the ease with which their properties can be externally modified. Application of strain is one possible way. In this seminar we will review the geometrical description of strains in crystalline materials, with a focus on graphene. Using this method, we will study the form of the electron-lattice interaction. We will compare this model with the description of electrons in strained graphene in terms of a Dirac equation in curved space. An overview of anharmonic lattice effects in two-dimensional materials will also be made.

Topological phases in $3+1D$ are less well understood than their lower dimensional counterparts. A useful approach to the study of such phases is to look at toy models that we can solve exactly. In this talk I will present new results for an existing model for certain topological phases in $3+1D$ (the model was first presented in [1]). This model is based on a generalisation of lattice gauge theory known as higher lattice gauge theory, which treats parallel transport of lines as well as points. I will first provide a brief introduction to higher lattice gauge theory and the Hamiltonian model constructed from it. Then we will look at the simple excitations (both point-like and loop-like) that are present in this model and how these excitations can be constructed explicitly using so-called ribbon and membrane operators. Some of the quasi-particles are confined and we discuss how this arises from a condensation-confinement transition. We will then look at the (loop-)braiding relations of the excitations and finish by examining the conserved topological charges realised by the Higher Lattice Gauge Theory Model.

[1] A Bullivant, M. Calcada et al., Topological phases from higher gauge symmetry in 3+1D, Phys. Rev. B 95, 155118 (2017).

Discrete time crystals is the name given to many-body systems displaying long-time dynamics that is sub-harmonic with respect to a driving frequency. While these were first discussed in closed quantum systems a few years ago, recent work (partly motivated by experiments) has focussed on including non-unitary effects such as due to an external environment ("dissipation").

In this talk I will begin by discussing general features of periodically-driven many-body systems, then concentrate on one of the unitary models for discrete time crystals. Time permitting, I will finally discuss a general framework for subharmonic oscillations stabilised by dissipative dynamics.

A current challenge in condensed matter physics is the realization of strongly correlated, viscous electron fluids. These fluids are not amenable to the perturbative methods of Fermi liquid theory, but can be described by holography, that is, by mapping them onto a weakly curved gravitational theory via gauge/gravity duality. The canonical system considered for realizations has been graphene, which possesses Dirac dispersions at low energies as well as significant Coulomb interactions between the electrons. In this work, we show that Kagome systems with electron fillings adjusted to the Dirac nodes of their band structure provide a much more compelling platform for realizations of viscous electron fluids, including non-linear effects such as turbulence. In particular, we find that in stoichiometric Scandium (Sc) Herbertsmithite, the fine-structure constant, which measures the effective Coulomb interaction and hence reflects the strength of the correlations, is enhanced by a factor of about 3.2 as compared to graphene, due to orbital hybridization. We employ holography to estimate the ratio of the shear viscosity over the entropy density in Sc-Herbertsmithite, and find it about three times smaller than in graphene. These findings put, for the first time, the turbulent flow regime described by holography within the reach of experiments.

Recent experiments on large chains of Rydberg atoms [1] have demonstrated the possibility of realising one-dimensional, kinetically constrained quantum systems. It was found that such systems exhibit surprising signatures of non-ergodic dynamics, such as robust periodic revivals in global quenches from certain initial states. This weak form of ergodicity breaking has been interpreted as a manifestation of "quantum many-body scars" [2], i.e., the many-body analogue of unstable classical periodic orbits of a single particle in a chaotic stadium billiard. Scarred many-body eigenstates have been shown to exhibit a range of unusual properties which violate the Eigenstate Thermalisation Hypothesis, such as equidistant energy separation, anomalous expectation values of local observables and subthermal entanglement entropy. I will demonstrate that these properties can be understood using a tractable model based on a single particle hopping on the Hilbert space graph, which formally captures the idea that scarred eigenstates form a representation of a large $\operatorname{SU}(2)$ spin that is embedded in a thermalising many-body system. I will show that this picture allows to construct a more general family of scarred models where the fundamental degree of freedom is a quantum clock [3]. These results suggest that scarred many-body bands give rise to a new universality class of constrained quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

Describing complex interacting quantum systems is a daunting task. One very fruitful approach to this problem, developed for unitary dynamics, is to represent the Hamiltonian of a system by a large random matrix. This eventually led to the development of the field of quantum chaos. Arguably, one of its most spectacular achievements was the identification of universal signatures of chaos in quantum systems, characterizing the correlations of their energy levels. In this talk, we will focus on the recent application of (non-Hermitian) random matrix theory to open quantum systems, where dissipation and decoherence coexist with unitary dynamics. First, we will discuss a class of stochastic Lindbladians with random Hamiltonian and independent random dissipation channels (jump operators), as a model for the generator of complicated nonunitary dynamics. We will then explain what difficulties arise when combining dissipation with quantum chaos, and how to overcome them. In particular, we discuss a new non-Hermitian random matrix ensemble with eigenvalues on the torus and how it connects to our recent proposal of using complex spacing ratios as a signature of dissipative quantum chaos.

Recently, topological materials and topological effects have elicited a great interest in the photonics community [1]. While condensed-matter phenomena are traditionally described by Hermitian operators, the same is not true in the context of macroscopic electrodynamics where a dissipative response is the rule, not the exception. In this talk, I will discuss how to determine the topological phases of dissipative (non-Hermitian) photonic structures from first principles using a gauge-independent Green function [2, 3]. It is shown that analogous to the Hermitian case, the Chern number can be expressed as an integral of the system Green function over a line parallel to the imaginary-frequency axis. The approach introduces in a natural way the "band-gaps" of non-Hermitian systems as the strips of the complex-frequency plane wherein the system Green function is analytical. I apply the developed theory to nonreciprocal electromagnetic continua and photonic crystals, with lossy and or gainy elements. Furthermore, I discuss the validity of the bulk-edge correspondence in the non-Hermitian case.

L. Lu, J. D. Joannopoulos, M. Soljačić, Topological photonics, Nat. Photonics, 8, 821, (2014).

M. G. Silveirinha, Topological theory of non-Hermitian photonic systems, Phys. Rev. B, 99, 125155, 2019.

The insulating versus conducting behavior of condensed matter is commonly addressed in terms of electronic excitations and/or conductivity. At variance with such wisdom, W. Kohn hinted in 1964 that the insulating state of matter reflects a peculiar organization of the electrons in their ground state, and does not require an energy gap.

Kohn’s theory of the insulating state got a fresh restart in 1999; at the root of these developments is the modern theory of polarization, developed in the early 1990s, and based on a geometrical concept (Berry phase). Since insulators and metals polarize in a qualitatively different way, quantum geometry also discriminates insulators from conductors. A common geometrical “marker”, based on the quantum metric, caracterizes all insulators (band insulators, Anderson insulators, Mott insulators, quantum Hall insulators...); such marker diverges in conductors.

Novel bulk-edge dualities have recently emerged in topological materials from the observation of some phenomenological correspondences. The similarity of these dualities with string theory dualities is very appealing and has boosted a quite significant number of cross field studies.

We analyze the bulk-edge dualities in the integer quantum Hall effect, where due to the simpler nature of planar systems the duality can be analyzed by powerful analytic techniques. The results show that the correspondence is less robust than expected. In particular, it is highly dependent of the type of boundary conditions of the topological material. We introduce a formal proof of the equivalence of bulk and edge approaches to the quantization of Hall conductivity for metallic plates with local boundary conditions. However, the proof does not work for non-local boundary conditions, like the Atiyah-Patodi-Singer boundary conditions, due to the appearance of gaps between the bulk and edge states.

Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. A prominent example is provided by the chiral magnetic effect in three-dimensional Weyl semimetals, which stems from the chiral anomaly. Here, we reveal a distinct quantum effect, coined parity magnetic effect, which is induced by the parity anomaly in a four-dimensional topological semimetal. Upon preserving time-reversal symmetry, the spectrum of our model is doubly degenerate and the nodal (Dirac) points behave like $\mathbb{Z}_2$ monopoles. When time-reversal symmetry is broken, while preserving the sublattice (chiral) symmetry, our system supports spin-3/2 quasiparticles and the corresponding Dirac-like cones host tensor monopoles characterized by a $\mathbb{Z}$ number, the Dixmier-Douady invariant. In both cases, the semimetal exhibits topologically protected Fermi arcs on its boundary. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.

Topological phase transitions were discovered by Berezinskii-Kosterlitz-Thouless in the 70's. They describe intriguing phase transitions for classical spins systems such as the plane rotator model (or $XY$ model). I will start by reviewing how this phase transition arises in cases such as:

the $XY$ model (spins on $\mathbb{Z}^2$ with values in the unit circle)

the integer-valued Gaussian Free Field (or $\mathbb{Z}$-ferromagnet)

Abelian Yang-Mills on $\mathbb{Z}^4$

I will then connect topological phase transitions to a statistical reconstruction problem concerning the Gaussian Free Field and will show that the feasibility of the reconstruction undergoes a KT transition.

This is a joint work with Avelio Sepúlveda (Lyon) and the talk will be based mostly on the preprint: https://arxiv.org/abs/2002.12284

Symmetry protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap. Here, we introduce a new notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions. More specifically, given a symmetric boundary termination we say two bulk Hamiltonians belong to distinct boundary obstructed topological phases (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but cannot be deformed to each other in the open system without closing the gap at at least one high symmetry surface. BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus but, similar to SPTs, they are associated with boundary signatures in open systems such as surface states or fractional corner charges. In contrast to SPTs, these boundary signatures are not anomalous and can be removed by symmetrically adding lower dimensional SPTs on the boundary, but they are stable as long as the spectral gap at high-symmetry edges/surfaces remains open. We show that the double-mirror quadrupole model of [Science, 357(6346), 2018] is a prototypical example of such phases, and present a detailed analysis of several aspects of boundary obstructions in this model. In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges. We also provide a general framework to study boundary obstructions in free-fermion systems in terms of Wannier band representations (WBR), an extension of the recently-developed band representation formalism to Wannier bands. WBRs capture the notion of topological obstructions in the Wannier bands which can then be used to study topological obstructions in the boundary spectrum by means of the correspondence between the Wannier and boundary spectra. This establishes a form of bulk-boundary correspondence for BOTPs by relating the bulk band representation to the boundary topology.

Poisson local statistics of eigenvalues is widely accepted as a necessary signature of Anderson localization, but so far has been rigorously established only for random systems. We will argue that this paradigm is wrong, and the reality is a lot more complex and interesting, by presenting both rigorous results for the Harper and Maryland models and numerics for other quasiperiodic and similar models with localization. We will also discuss a conjecture on what the distribution is in the general ergodic situation.

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. For integrable models, novel hydrodynamic approaches based on a quasiparticle picture emerged as a new platform allowing for a quantitative understanding of quantum information dynamics in quantum many-body systems. Remarkably, this gives fresh insights on how thermodynamics emerges in isolated out-of-equilibrium quantum systems.

I will start by reviewing this new unifying framework. I will then discuss several applications to entanglement-related quantities, such as entanglement entropies, mutual information, logarithmic negativity. I will also show how the framework allows to study the interplay between quantum information dynamics and transport of local conserved quantities. Finally, I will derive some simple bounds on the quantum information scrambling in out-of-equilibrium systems.

The co-existence of spatial and non-spatial symmetries together with appropriate commutation/anticommutation relations between them can give rise to static higher-order topological phases, which host gapless boundary modes of co-dimension higher than one. Alternatively, space-time symmetries in a Floquet system can also lead to anomalous Floquet boundary modes of higher co-dimensions, with different commutation/anticommutation relations with respect to non-spatial symmetries. In my talk I will review how these dynamical analogs of the static HOTI's emerge, and also show how a coherently excited phonon mode can be used to support non-trivial Floquet higher-order topological phases. If time allows, I will also review recent work on Floquet engineering and band flattening of twisted-bilayer graphene.

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. I will explain how logarithmic derivatives of the Fredholm determinants can be expressed in terms of a $2\times 2$ Riemann-Hilbert problem.

This Riemann-Hilbert representation can be used to derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal, and it reveals a remarkable connection with a family of unbounded solutions to the Korteweg-de Vries (KdV) equation and with an integro-differential version of the Painlevé II equation.

The eigenstate thermalization hypothesis (ETH) is a cornerstone in our understanding of quantum statistical mechanics. The extent to which ETH holds for nonlocal operators (observables) is an open question. I will address this question using an analogy with random matrix theory. The starting point will be the construction of extremely non-local operators, which we call Behemoth operators. The Behemoths turn out to be building blocks for all physical operators. This construction allow us to derive scalings for both local operators and different kinds of nonlocal operators.

I will give an overview of my work on topological methods in condensed matter physics almost 40 years ago. Include will be Homotopy and $\operatorname{TKN}^2$ integers, holonomy and Berry's phase and quarternions and Berry's phase for Fermions. If time allows, I'll discuss supersymmetry and pairs of projections.

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

Emergence of anomalous transport laws in deterministic interacting many-body systems has become a subject of intense study in the past few years. One of the most prominent examples is the unexpected discovery of superdiffusive spin dynamics in the isotropic Heisenberg quantum spin chain with at half filling, which falls into the universality class of the celebrated Kardar-Parisi-Zhang equation. In this talk, we will theoretically justify why the observed superdiffusion of the Noether charges with anomalous dynamical exponent $z=3/2$ is indeed superuniversal, namely it is a feature of all integrable interacting lattice models or quantum field theories which exhibit globally symmetry of simple Lie group $G$, in thermal ensembles that do not break $G$-invariance. The phenomenon can be attributed to thermally dressed giant quasiparticles, whose properties can be traced back to fusion relations amongst characters of quantum groups called Yangians. Giant quasiparticles can be identified with classical solitons, i.e. stable nonlinear solutions to certain integrable PDE representing classical ferromagnet field theories on certain types of coset manifolds. We shall explain why these inherently semi-classical objects are in one-to-one correspondence with the spectrum of Goldstone modes. If time permits, we shall introduce another type of anomalous transport law called undular diffusion that generally occurs amongst the symmetry-broken Noether fields in $G$-invariant dynamical systems at finite charge densities.

In the presence of a strong magnetic field, and for an integer filling of the Landau levels, Coulomb interactions favor a ferromagnetic ground-state. It has been shown already twenty years ago, both theoretically and experimentally, that when extra charges are added or removed to such systems, the ferromagnetic state becomes unstable and is replaced by spin textures called Skyrmions. We have generalized this notion to an arbitrary number $N$ of internal states for the electrons, which may correspond to the combination of spin, valley, or layer indices. The first step is to associate a many electron wave-function, projected on the lowest Landau level, to any classical spin texture described by a smooth map from the plane to the projective space $\mathbb{CP}^{N-1}$. In the large magnetic field limit, we assume that the spin texture is slowly varying on the scale of the magnetic length, which allows us to evaluate the expectation value of the interaction Hamiltonian on these many electron quantum states. The first non trivial term in this semi-classical expansion is the usual $\mathbb{CP}^{N-1}$ non-linear sigma model, which is known to exhibit a remarkable degeneracy of the many electron states obtained from holomorphic textures. Surprisingly, this degeneracy is not lifted by reintroducing quantum fluctuations. It is eventually lifted by the sub-leading term in the effective Hamiltonian, which selects a hexagonal Skyrmion lattice and therefore breaks both translational and internal $SU(N)$ symmetries. I will show that these optimal classical textures can be interpreted in an appealing way using geometric quantization.

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents, though we do not know how accurate it is. I will show that the low-energy properties of the Kitaev honeycomb lattice model, such as the shape of Majorana zero modes or the deformations of the correlation length, are faithfully described in terms of Riemann-Cartan geometry. Moreover, I will present how effective axial gauge fields can couple to Majorana fermions, thus giving a unified picture between vortices and lattice dislocations that support Majorana zero modes.

Transport in disordered one-dimensional wires is described by diffusion at short distances/times and by Anderson localization at long distances/times. I will investigate how this picture is altered in a disordered multi-channel wire where some of the channels are topologically protected. This scenario can be realized at the interface between two quantum Hall systems, in a Weyl semimetal in a magnetic field or at the edge of a quantum spin Hall insulator. Technically, the problem is described by a $0+1$-dimensional field theory in the form of a supersymmetric non-linear sigma model with a topological term. I will show how to solve this field theory exactly to obtain DC (static) transport quantities such as DC conductance and shot noise as well as dynamical responses such as diffusion probability of return and correlations of local density of states. I will discuss several surprising findings of this exact solution. First, I find that disorder is much more effective in localizing the diffusive channels in the presence of topologically protected ones. This can be understood in terms of statistical level repulsion by mapping the problem to that of a random matrix ensemble with zero eigenvalues. Second, I find that localization corrections dramatically alter the long time behavior of the return probability in a system described by diffusion+drift equation at the classical level. Finally, I find that in a disordered wire with topologically protected channels, the wave functions display level attraction rather than level repulsion.

The dimer model is a model of uniform perfect matching and is one of the fundamental models of statistical physics. It has many deep and intricate connections with various other models in this field, namely the Ising model and the six-vertex model.

This model has received a lot of attention in the mathematics community in the past two decades. The primary reason behind such popularity is that this model is integrable, in particular, the correlation functions can be represented exactly in a determinental form. This gives rise to a rich interplay between algebra, geometry, probability and theoretical physics.

For graphs with very regular local structures, exact computations of the correlation functions are possible by Kasteleyn theory. R. Kenyon pioneered the development of the subject in this direction by proving that the fluctuations of the height function associated to the dimer model on the square lattice converges to the Gaussian free field (a conformally invariant Gaussian field). However, such computations seem only possible on graphs with special local structures, while the dimer model is supposed to have GFF type fluctuations in a much more general setting.

In this talk, I will give an overview of an ongoing project with N, Berestycki (U. Vienna) and B. Laslier (Paris-Diderot 7) where we establish a form of universality about the GFF fluctuation of the dimer model. Our approach does not use Kasteleyn theory, but uses a mapping known since Temperley-Fisher, which maps the dimer model to uniform spanning trees. Remarkably, as observed by Benjamini, the “winding” of the branches of this spanning tree exactly measures the height function of the dimers. We combine this approach with the developing universal scaling limit results of the uniform spanning trees, revolutionized by Schramm through the discovery of SLE. We show that the continuum “winding” of these continuum limiting spanning trees converge to the GFF and harness from this the universality of the scaling limit. A key input in identifying the limit is the so-called imaginary geometry developed by Miller and Sheffield. In a more recent work, we extend this universality partially to general Riemann surfaces as well.

This talk is based on the following preprints and some works in progress.

The eigenstate entanglement entropy is a powerful tool to distinguish integrable from generic quantum-chaotic Hamiltonians. In integrable models, the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) has a volume-law coefficient that generally depends on the subsystem fraction. In contrast, the volume-law coefficient is maximal (subsystem fraction independent) in quantum-chaotic models. In the seminar I will present an overview of our current understanding of bipartite entanglement entropies in many-body quantum states above the ground states, and contrast analytical predictions with numerical results for eigenstates of physical Hamiltonians.

Periodic driving of quantum systems is attracting interest since we can use it to realize new states of matter with exotic properties. This concept, known as Floquet engineering, has been widely used in cold atoms [1] and recently in solid-state systems [2]. In this talk, I plan to explain Floquet engineering basics using simple examples such a 2D and 3D Dirac electrons in circularly polarized laser fields [2]. Then, to develop a more in-depth and intuitive understanding of Floquet states, I will explain an example of a dynamical Landau quantization realized in oscillating magnetic fields.