# Colloquium

## Higher-order topological insulators and superconductors

15 May in 11:30

Piet Brouwer (Free University Berlin, Germany)

Topological insulators combine an insulating bulk with gapless states at their surfaces. This talk introduces "higher-order topological insulators", which are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at corners or at edges of the crystal. I'll show that such "higher-order boundary states" are a generic boundary manifestation of the nontrivial bulk topology, if the bulk topology relies on a crystalline symmetry for its protection.

## Towards an analytic description of periodic anomalous waves in nature via the focusing NLS model (joint work with P. G. Grinevich)

7 May in 11:30

Paolo Maria Santini (Dept. of Physics, University of Roma "La Sapienza")

The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. We first show how the finite gap method adapts to the NLS Cauchy problem for a generic periodic initial perturbation of the unstable background solution, in the case of a finite number N of unstable modes, allowing one to construct the solution, to leading order, in terms of elementary functions of the initial data. In particular, if N=1, one obtains the analytic quantitative description of a Fermi-Pasta-Ulam recurrence of AWs already observed in real (water wave and nonlinear optics) and numerical experiments. Then we present the analytic description of the O(1) effect of a small loss or gain on the dynamics of periodic AWs, in full agreements with recent water wave and numerical experiments. If time remains, we shall discuss analogies and differences with AWs in other contexts: on the Ablowitz-Ladik lattice and in the massive Thirring model relativistic field theory (with F. Coppini).

## Charge and entropy transport in strontium titanate

17 April in 11:30

Kamran Behnia (CNRS/ESPCI, PSL Research University, Paris, France)

The ferroelectric instability in pristine strontium titanate is aborted by quantum fluctuations. Therefore, the static electric permittivity saturates to an extremely large value at low temperature and the effective Bohr radius approaches a micron. In this context, removing a tiny fraction of oxygen atoms turns the system to a dilute metal with a sharp Fermi surface and a superconducting instability. The focus of this talk will be charge and entropy transport by electrons in this dilute metal and their general implications.

The temperature dependence of the resistivity of this dilute metal at low-temperature implies that electron-electron scattering can generate a T-square resistivity even in absence of Umklapp scattering with a single Fermi pocket. The magnitude of resistivity at a low temperature would imply a mean-free-path too short to be physically plausible. Combined with the temperature dependence of the Seebeck coefficient, this implies that non-degenerate electrons become heavier with warming. We will see that these observations cannot be explained by available polaronic theories. There is a whole family of dilute metals emerging from doped quantum paraelectrics with similar properties.

The temperature dependence of the resistivity of this dilute metal at low-temperature implies that electron-electron scattering can generate a T-square resistivity even in absence of Umklapp scattering with a single Fermi pocket. The magnitude of resistivity at a low temperature would imply a mean-free-path too short to be physically plausible. Combined with the temperature dependence of the Seebeck coefficient, this implies that non-degenerate electrons become heavier with warming. We will see that these observations cannot be explained by available polaronic theories. There is a whole family of dilute metals emerging from doped quantum paraelectrics with similar properties.

## Effective function theory on Riemann surfaces and applications.

21 February in 11:30

Bogatyrev A.B. (INM RAS)

Many model physical and engineering problems admit closed form solutions in terms of function-theoretic objects on Riemann surfaces or spaces of their moduli. We consider issues of effective and robust calculation of such objects (Abelian integrals,
their periods, linear and quadratic differentials, meromorphic functions ...) for surfaces of a high genus (greater than one).
Examples of solving several problems will be given.

## Electric dipole moment searches using storage rings

24 January in 11:30

Frank Rathmann (Institut f. Kernphysik, Forschungszentrum Juelich, Germany)

The Standard Model (SM) of Particle Physics is not capable to account for the apparent matter-antimatter asymmetry of our Universe. Physics beyond the SM is required and is either probed by employing highest energies (e.g., at LHC), or by striving for ultimate precision and sensitivity (e.g., in the search for electric dipole moments). Permanent electric dipole moments (EDMs) of particles violate both time reversal (T) and parity (P) invariance, and are via the CPT-theorem also CP-violating. Finding an EDM would be a strong indication for physics beyond the SM, and pushing upper limits further provides crucial tests for any corresponding theoretical model, e.g., SUSY.
Up to now, EDM searches focused on neutral systems (neutrons, atoms, and molecules). Storage rings, however, offer the possibility to measure EDMs of charged particles by observing the influence of the EDM on the spin motion in the ring. Direct searches of proton and deuteron EDMs, however, bear the potential to reach sensitivities beyond 10−29 e⋅cm. Since the Cooler Synchrotron COSY at the Forschungszentrum Jülich provides polarized protons and deuterons up to momenta of 3.7 GeV/c, it constitutes an ideal testing ground and starting point for such an experimental program. Besides the discussion of the achievements of the JEDI collaboration, and the description of an effort to perform a first direct deuteron EDM measurement at COSY, the talk will highlight in addition future technical developments that will pave the way toward EDM searches in dedicated rings. A recent advancement that grew out of the successful work performed by JEDI is the formation of the CPEDM Collaboration, which aims at the design of an EDM prototype ring that could be hosted either at CERN or at COSY, will be discussed as well.

## Non-ergodic delocalized states for efficient population transfer within a narrow band of the energy landscape

20 December 2019 in 15:00

Vadim Smelyanskiy (Google, Los Angeles)

We will review the advances and challenges in the field of quantum combinatorial optimization and closely related problem of low-energy eigenstates and coherent dynamics in transverse field quantum spin glass models. We will discuss the role of collective spin tunneling that gives rise to bands of delocalized non-ergodic quantum states providing the coherent pathway for the population transfer (PT) algorithm: the quantum evolution under a constant transverse field that starts at a low-energy spin configuration and ends up in a superposition of spin configurations inside a narrow energy window. We study the transverse field induced quantum dynamics of the following spin model: zero energy of all spin configurations except for a small fraction of spin configuration that form a narrow band at large negative energy. We use the cavity method for heavy-tailed random matrices to obtain the statistical properties of the low-energy eigenstates in an explicit analytical form. In a broad interval of transverse fields, they are non-ergodic, albeit extended giving rise to a qualitatively new type of quantum dynamics. For large transverse fields »1 the typical runtime of PT algorithm $\sim \sqrt{2^n / \Omega e^r}$ scales with n and Ω as that of the Grover’s quantum search, except for the small correction to the exponent θ ≈ 1/(2). The model we consider is non-integrable. As a result, our PT protocol does not require any fine-tuning of and may be initialized in a computational basis state. We argue that our approach can be applied to study PT protocol in other optimization problems with the potential quantum advantage over classical algorithms.

## Ordered fluctuations: about vestigial order in quantum materials

20 December 2019 in 11:30

Joerg Schmalian (Karlsruhe Institute of Technology)

A hallmark of the phase diagrams of quantum materials is the existence of multiple electronic ordered states. In many cases those are not independent, competing phases, but instead display a complex intertwinement. In this talk, we focus on a realization of intertwined orders with a fluctuation-driven vestigial phase characterized by a composite order parameter. In other words, we are investigating the condensation of fluctuations.

We demonstrate that this concept naturally explains the nematic state in iron-based superconductors and nematic superconductivity in doped topological insulators. In addition we propose a natural mechanism for charge 4e superconductivity with half flux quanta. We present a formalism that provides a framework to understand the complexity of quantum materials based on symmetry, largely without resorting to microscopic models.

We demonstrate that this concept naturally explains the nematic state in iron-based superconductors and nematic superconductivity in doped topological insulators. In addition we propose a natural mechanism for charge 4e superconductivity with half flux quanta. We present a formalism that provides a framework to understand the complexity of quantum materials based on symmetry, largely without resorting to microscopic models.

## Physics at the Edge of a QHE setting

25 October 2019 in 11:30

Yuval Gefen (Weizmann Institute of Science, Israel)

The structure of the edge of a QHE edge is constrained, but not dictated, by the topology of the bulk. Particularly interesting is the class of hole-conjugate fractional QH states. For such states the edge hosts counter-propagating modes that are responsible for quantized electrical and thermal conductance. In the coherent quantum limit renormalized edge modes emerge, which may involve neutral modes with non-trivial exchange statistics. I will discuss the behavior of topology-influenced transport coefficient both in the coherent and the fully equilibrated regimes, topological classification of non-equilibrium shot noise, and the relation to recent experiments.

## Dynamical Glass - en route from KAM and FPUT to MBL

6 September 2019 in 11:30

Sergej Flach (Institute for Basic Science, Republic of Korea)

Classical many body interacting systems are typically chaotic (nonzero Lyapunov exponents) and their microcanonical dynamics ensures that time averages and phase space averages are identical (ergodic hypothesis). In proximity to an integrable limit the long- or short-range properties of the network of nonintegrable action space perturbations define the finite time relaxation properties of the system towards Gibbs equilibrium. I will touch upon few analytical results including the KAM theorem, and review a number of computational studies which originate from the pioneering work of Enrico Fermi, John Pasta, Stanislaw Ulam and Mary Tsingou. I will then focus on short range networks which lead to a dynamical glass (DG), using a classical Josephson junction chain in the limit of large energy densities or small Josephson energies. Close to these limits the Josephson coupling between the superconducting grains induces a short-range nonintegrable network in the corresponding action space. I will introduce a set of quantitative measures which lead to the Lyapunov time TΛ, the ergodization time TE, and to a diffusion constant D. In the DG the system fragments into large patches of nonresonant ’integrable’ grains of size l separated by triplets of resonant chaotic patches, all surviving over large times. TE sets the time scale for chaotic dynamics in the triplets. Contrary, TE ≈ l2/D is the much larger time scale of slow diffusion of chaotic triplets. The DG is a generic feature of weakly non-integrable systems with a short range coupling network in action space, and expected to be related to nonergodic quantum metallic states of quantum many-body systems in proximity to a many-body localization phase.

## Two-fluid phenomena in one-dimensional quantum liquids

21 June 2019 in 11:30

Anton Andreev

One-dimensional quantum liquids are commonly treated using the Luttinger liquid theory, which neglects elementary excitations with high energies. In this approximation, in addition to the number of particles, energy, and momentum the liquid possesses another conserved quantity - J, which may be interpreted as the difference between the numbers of right- and left- moving particles. Beyond the Luttinger liquid approximation J is no longer conserved, but its relaxation time is exponentially long at low temperatures.
I will show that as a result, one-dimensional quantum liquids exhibit two-fluid behavior. In particular in a wide frequency interval they support two sound modes that are similar to the first and second sound in superfluid He.

## Minimal excitations states: From time resolved single particle fermionic states for Electron Quantum Optics to Digital communication and music.

24 May 2019 in 11:30

D. Christian Glattli (Nanoelectronics Group, Service de Physique de l’Etat Condensé, CEA Saclay, France)

In the 90’s, an impressive series of works by theoreticians from the Landau Institute on electrons shot noise in quantum conductors [1] and on the statistics of transfer of electrons [2] has leaded to the emergence of the beautiful concept of minimal excitation states [3-5]. These minimal excitation states can be generated by applying voltage pulses on the contact of a conductor to inject short single electron pulses. These states show minimal noise and provide a convenient and clean single electron source for electron optics whose aim is to perform quantum optics tasks with electrons instead of photons. The minimal excitations states, now called levitons, have been produced in recent experiments [6] and have triggered a large number of theoretical works. They have enabled Hong Ou Mandel like experiments [6] with electrons and single electron quantum Tomography [7]. Extension to fractionally charged anyons is possible.

At the root of the minimal excitation property is a specific single side band modulation of the electron wave by the Lorentzian voltage pulse. This property can be applied to classical electromagnetic or acoustic waves for applications in digital communication [8] or in music sound synthesis.

[1] G. B. Lesovik, JETP Letters, 49 (9), 592-594 (1989).

[2] L.S. Levitov, G.B. Lesovik, Charge-transport statistics in quantum conductors, JETP Lett., 55 (9), 555-559 (1992).

[3] A. Ivanov, H.W. Lee, L.S. Levitov, Coherent states of alternating current, Phys. Rev. B 56(11), 6839-6850 (1997); cond-mat/9501040

[4] L.S. Levitov, H. Lee, G.B. Lesovik, Electron Counting Statistics and Coherent States of Electric Current, J. Math. Phys., 37(10), 4845-4866 (1996); cond-mat/9607137.

[5] J. Keeling, I. Klich, and L. S. Levitov, Minimal Excitation States of Electrons in One-Dimensional Wires, Phys. Rev. Lett. 97, 116403 (2006).

[6] Minimal-excitation states for electron quantum optics using levitons, J. Dubois, T. Jullien, F. Portier, P. Roche, A. Cavanna, Y. Jin, W. Wegscheider, P. Roulleau & D. C. Glattli, Nature, 502, 659–663 (2013).

[7] Quantum tomography of an electron, T. Jullien, P. Roulleau, B. Roche, A. Cavanna, Y. Jin & D. C. Glattli, Nature, 514, 603–607 (2014).

[8] Power Spectrum Density of Single Side Band CPM Using Lorenztian Frequency Pulses, Haïfa Farès, D. Christian Glattli, Yves Louet, Jacques Palicot, Preden Roulleau, and Christophe Moy, IEEE Wireless Communications Letters, 6 (6), 786-789, (2017).

At the root of the minimal excitation property is a specific single side band modulation of the electron wave by the Lorentzian voltage pulse. This property can be applied to classical electromagnetic or acoustic waves for applications in digital communication [8] or in music sound synthesis.

[1] G. B. Lesovik, JETP Letters, 49 (9), 592-594 (1989).

[2] L.S. Levitov, G.B. Lesovik, Charge-transport statistics in quantum conductors, JETP Lett., 55 (9), 555-559 (1992).

[3] A. Ivanov, H.W. Lee, L.S. Levitov, Coherent states of alternating current, Phys. Rev. B 56(11), 6839-6850 (1997); cond-mat/9501040

[4] L.S. Levitov, H. Lee, G.B. Lesovik, Electron Counting Statistics and Coherent States of Electric Current, J. Math. Phys., 37(10), 4845-4866 (1996); cond-mat/9607137.

[5] J. Keeling, I. Klich, and L. S. Levitov, Minimal Excitation States of Electrons in One-Dimensional Wires, Phys. Rev. Lett. 97, 116403 (2006).

[6] Minimal-excitation states for electron quantum optics using levitons, J. Dubois, T. Jullien, F. Portier, P. Roche, A. Cavanna, Y. Jin, W. Wegscheider, P. Roulleau & D. C. Glattli, Nature, 502, 659–663 (2013).

[7] Quantum tomography of an electron, T. Jullien, P. Roulleau, B. Roche, A. Cavanna, Y. Jin & D. C. Glattli, Nature, 514, 603–607 (2014).

[8] Power Spectrum Density of Single Side Band CPM Using Lorenztian Frequency Pulses, Haïfa Farès, D. Christian Glattli, Yves Louet, Jacques Palicot, Preden Roulleau, and Christophe Moy, IEEE Wireless Communications Letters, 6 (6), 786-789, (2017).

## Hydrodynamic approach to electronic transport

26 April 2019 in 11:30

Boris Narozhny (KIT)

The last few years have seen an explosion of interest in hydrodynamic effects in interacting electron systems in ultra-pure materials. One such material, graphene, is not only an excellent platform for the experimental realization of the hydrodynamic flow of electrons, but also allows for a controlled derivation of the hydrodynamic equations on the basis of kinetic theory. The resulting hydrodynamic theory of electronic transport in graphene yields quantitative predictions for experimentally relevant quantities, e.g. viscosity, electrical conductivity, etc. In this talk I will review recent theoretical advances in the field, compare the hydrodynamic theory of charge carriers in graphene with relativistic hydrodynamics and recent experiments, and discuss applications of hydrodynamic approach to novel materials beyond graphene.

Видео (в связи с ошибкой видео записано не до конца)

## On reconstructing nonlinearly encrypted signals corrupted by noise.

12 April 2019 in 11:30

Yan V. Fyodorov (King's College London)

I consider the problem of reconstructing a source vector from its encrypted image corrupted by an additive Gaussian noise.
Assuming encryption to be given by a random Gaussian mapping, the reconstruction problem in the framework of the Least Square Scheme turns out to be equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model.
As a measure of the quality of the signal reconstruction one can use the mean overlap between the original signal and its recovered image.
Thi overlap is analysed in the framework of Parisi scheme of Replica Symmetry Breaking. If the mapping is quadratic, there exists a threshold in the noise-to-signal ratio beyond which the reconstruction is impossible. The behaviour close to the threshold is controlled by the replica symmetry breaking mechanism and is characterized by a nontrivial exponent 3/4.

## Correlation-induced localization

5 April 2019 in 11:30

Vladimir Kravtsov

Conventional Anderson localization is due to destructive interference of
matter waves described by local random Hamiltonians. Correlations in
random diagonal elements of such a Hamiltonian are known to favor
delocalization. Recently systems with non-local Hamiltonians become
experimentally accessible. We consider two families of such random
matrix Hamiltonians with correlations in the long-range hopping terms
and demonstrate that localization is enhanced and the wave function
ergodicity is progressively degrading as the correlations become stronger.
We review the localization/delocalization criteria of Mott and Anderson
and show that the former is the sufficient criterion of weak ergodicity
and the latter is the sufficient criterion of localization. The fact
that these two criteria are not complimentary is the reason why the
non-ergodic extended (multifractal) states may exist when neither the
Mott, nor the Anderson criterion is fulfilled.

We suggest a new class of random matrix models (Toeplitz RMT) with translation-invariant hopping integrals and identify the character of eigenfunction and eigenvalue statistics in them. We formulate the principles of level statistics if the type of eigenfunction statistics is known both in the coordinate and in the momentum basis and demonstrate that for the Toeplitz RMT the ergodic delocalization in the coordinate space may coexist with the Poisson level statistics.

Finally, we suggest a matrix-inversion trick that allows to identify uniquely the type of eigenfunction statistics and prove the absence of delocalized states in the bulk of spectrum of long-range Hamiltonians with deterministic (fully correlated) hopping.

We suggest a new class of random matrix models (Toeplitz RMT) with translation-invariant hopping integrals and identify the character of eigenfunction and eigenvalue statistics in them. We formulate the principles of level statistics if the type of eigenfunction statistics is known both in the coordinate and in the momentum basis and demonstrate that for the Toeplitz RMT the ergodic delocalization in the coordinate space may coexist with the Poisson level statistics.

Finally, we suggest a matrix-inversion trick that allows to identify uniquely the type of eigenfunction statistics and prove the absence of delocalized states in the bulk of spectrum of long-range Hamiltonians with deterministic (fully correlated) hopping.

## Recent theoretical developments in the integer quantum Hall effects

22 February 2019 in 11:30

Ferdinand Evers (Regensburg University)

The quantum Hall effects belong to the most striking phenomena in condensed matter physics. Despite of intensive theoretical efforts over the last three decades, important aspects of the quantum Hall transitions are still not fully understood. In particular, there is still no consensus concerning the critical field theory and the corresponding scaling properties of the observables near and at the plateau transition.

Notwithstanding this status, the last ten years have seen considerable progress in understanding basic properties of scaling near localization-delocalization transitions that have implications also for the quantum Hall transition. These concern, e.g., the important topic of corrections to scaling, the wavefunction statistics and higher-order multifractality. A brief review of these developments will be offered in the first part of the talk.

The second part of the talk will be devoted to the spin quantum Hall effect, which is a variant of the integer quantum Hall effect (class A) taking place in the Altland-Zirnbauer class C. For this transition several critical exponents are known analytically and can serve as reference points. Therefore, this transition provides an ideal testbed for the qualitative predictions made by analytical theories. Analytical predictions will be confronted with most recent numerical results.

Notwithstanding this status, the last ten years have seen considerable progress in understanding basic properties of scaling near localization-delocalization transitions that have implications also for the quantum Hall transition. These concern, e.g., the important topic of corrections to scaling, the wavefunction statistics and higher-order multifractality. A brief review of these developments will be offered in the first part of the talk.

The second part of the talk will be devoted to the spin quantum Hall effect, which is a variant of the integer quantum Hall effect (class A) taking place in the Altland-Zirnbauer class C. For this transition several critical exponents are known analytically and can serve as reference points. Therefore, this transition provides an ideal testbed for the qualitative predictions made by analytical theories. Analytical predictions will be confronted with most recent numerical results.

## Noisy quantum measurements: just a nuisance or fundamental physics?

7 December 2018 in 11:30

Wolfgang Belzig

Weak, almost non-invasive quantum measurements differ from the standard text book example of strongly invasive, projective measurements, since they leave the measured system basically unchanged. This opens the path to measure e.g. non-commuting observables and at the same time poses several open questions: Which order of operators is measured? Can quantum tests like Bell or Leggett-Garg be reformulated? What time scales are involved in the measurement process? We will address some basic properties of weak measurements leading to surprises like apparent spontaneous time-reversal symmetry breaking or the possibility of engineered detectors to tailor the measured quantum correlations.

## Wonders of viscous electronics

19 October 2018 in 11:30

Gregory Falkovich (Weizmann Institute of Science, Israel)

Quantum-critical strongly correlated systems feature universal collision-dominated collective transport. Viscous electronics is an emerging field dealing with systems in which strongly interacting electrons flow like a fluid. Such flows have some remarkable properties never seen before. I shall describe recent theoretical and experimental works devoted, in particular, to a striking macroscopic DC transport behavior: viscous friction can drive electric current against an applied field, resulting in a negative resistance, recently measured experimentally in graphene. I shall also describe conductance exceeding the fundamental quantum-ballistic limit, field-theoretical anomalies and other wonders of viscous electronics. Strongly interacting electron-hole plasma in high-mobility graphene affords a unique link between quantum-critical electron transport and the wealth of fluid mechanics phenomena.

## Quantum electrodynamics of heavy ions and atoms

5 October 2018 in 11:30

Vladimir Shabaev (St. Petersburg State University)

The present status of the QED theory of heavy ions and atoms is reviewed. The theoretical predictions for the Lamb shifts, the hyperfine splittings, and the bound-electron g factors of highly charged few-electron ions are compared with available experimental data. Special attention is paid to tests of QED at strong-coupling regime and determination of fundamental constants. The current status of studying the parity nonconservation effects with heavy atoms is also reported. Recent results on the charge-transfer and pair-creation probabilities in low-energy heavy-ion collisions are presented. Prospects for tests of QED at supercritical fields are discussed.

## Superconductivity that breaks time-reversal symmetry and its experimental manifestations

28 September 2018 in 11:30

Victor Yakovenko (University of Maryland)

References:

- [1] X. Gong, M. Kargarian, A. Stern, D. Yue, H. Zhou, X. Jin, V. M. Galitski, V. M. Yakovenko, and J. Xia, Science Advances 3, e1602579 (2017), arXiv:1609.08538
- [2] P. M. R. Brydon, D. S. L. Abergel, D. F. Agterberg, and V. M. Yakovenko, arXiv:1802.02280

## Quantum Many-Body Physics of Qubits

22 June 2018 in 11:30

Leonid Glazman (Yale University)

## Eukaryotic cell polarity and protein sorting

27 April 2018 in 11:30

Andrea Gamba, Politecnico di Torino

## Chiral magnetic crystals

23 March 2018 in 11:30

Markus Garst (TU Dresden)

The weak Dzyaloshinskii-Moriya interaction in chiral cubic magnets like MnSi, FeGe or Cu2OSeO3 twists the magnetization on long length scales resulting in spatially periodic magnetic textures — magnetic crystals. There exist especially magnetic crystals with a one- and two-dimensional periodicity corresponding to the magnetic helix and the topologically non-trivial skyrmion lattice, respectively. In this talk, we provide an overview of their properties. In particular, we discuss the crystallization process of these magnetic crystals that is characterized by strongly correlated chiral paramagnons that drive the transition first-order [1,2]. This fluctuation-induced first-order transition is well described by a theory put forward by Brazovskii. We will introduce the magnon band structure and their non-reciprocal properties in the presence of a magnetic field [3,4]. For the skyrmion lattice, this band structure is topological and characterized by finite Chern numbers that can be attributed to the formation of magnon Landau levels due to an emergent orbital magnetic field [5,6,7]. Finally, we will discuss domain walls of helimagnets that share similarities with grain boundaries consisting of disclination and dislocation defects of the helimagnetic order [8].

Video

References:

- [1] M. Janoschek et al. Phys. Rev. B 87, 134407 (2013).
- [2] A. Bauer, M. Garst and C. Pfleiderer, Phys. Rev. Lett. 110, 177207 (2013).
- [3] M. Kugler et al. Phys. Rev. Lett. 115, 097203 (2015)
- [4] T. Weber et al. arXiv:1708.02098
- [5] C. Schütte and M. Garst, Phys. Rev. B 90, 094423 (2014).
- [6] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler, Nat. Mater. 14, 478 (2015).
- [7] M. Garst J. Waizner, and D. Grundler, J. Phys. D: Appl. Phys. 50, 293002 (2017)
- [8] P. Schoenherr et al. Nat. Phys. in press, arXiv:1704.06288

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