United States

Excitation by light pulses enables the manipulation of phases of quantum condensed matter. Here, we photoexcite high-energy holon-doublon pairs as a way to alter the magnetic free energy landscape of the Kitaev-Heisenberg magnet α-RuCl3,[1,2] with the aim to dynamically stabilize a proximate spin liquid phase. The holon-doublon pair recombination through multimagnon emission [3] is tracked through the time-evolution of the magnetic linear dichroism originating from the competing zigzag spin ordered ground state. A small holon-doublon density suffices to reach a spin disordered state. The phase transition is described within a dynamic Ginzburg-Landau framework, corroborating the quasistationary nature of the transient spin disordered phase. Our work provides insight into the coupling between the electronic and magnetic degrees of freedom in α-RuCl3 and suggests a new route to reach a proximate spin liquid phase in Kitaev-Heisenberg magnets.[4]References: [1] A. Banerjee et al., Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet, Nat. Mater. 15, 733-740 (2016). [2] H. Takagi et al.,Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1, 264-280 (2019). [3] Z. Lenarcic et al., Ultrafast charge recombination in a photoexcited Mott-Hubbard insulator, Phys. Rev. Lett. 111, 016401 (2013). [4] R.B. Versteeg et al. Nonequilibrium quasistationary spin disordered state in the Kitaev-Heisenberg magnet α-RuCl3, arXiv:2005.14189 [cond-mat.str-el] (2020) 


MMM 2020

Nonequilibrium quasistationary spin disordered state in the Kitaev-Heisenberg magnet α-RuCl3

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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technical paper

Cylindrical magnetic nanowires (CNWs) are promising candidates for the building blocks of 3D information technologies such as shift registers, magnetic recording, spintronics and logic gates1-4. Spin-polarized-current is an energy efficient way to excite magnetization dynamics in planar nanostructures. However, in CNWs this research is doing its first steps5. Magnetization dynamics in a CNW of Permalloy, 100 nm diameter and 1 µm length, is investigated under simultaneous application of electric current and magnetic field by micromagnetic simulations. The magnetization reversal process initiates with the creation of open vortex structures with different rotation senses at the nanowire ends. We conclude that the electric current by itself enlarges or reduces the length of these vortex structures according to the rotational sense of the associated Oersted field. Large enough current densities produce a vortex structure which covers the whole nanowire surface. At the same time, the magnetization in the very core of the nanowire remains the same, i.e. no complete magnetization reversal is possible in the absence of external magnetic field. The simultaneous action of the applied electric current and magnetic field allows the complete control of the vortex structures in terms of setting the polarity and vorticity. The resulting diagram of magnetic states obtained after the application of field and electric current showing the values required for the vorticity and axial magnetization switching is presented in Figure 1. This control is essential for future information technologies based on 3D vertical nanostructures. The presented state diagram will become useful for future experiments on current-induced domain wall dynamics in CNWs.References: 1. S. Parkin and S.-H. Yang, Nature Nanotechnology., Vol. 10, pp. 195 (2015). 2. F. Nasirpouri, et al., Scientific Reports., Vol. 9, pp. 9010 (2019). 3. S. Ruiz-Gómez et al., Scientific Reports., Vol. 8, pp. 16695 (2018). 4. J. Garcia et al., Journal of Magnetism and Magnetic Materials., Vol. 383, pp. 88 (2015). 5. M. Schöbitz et al., Physical Review Letters Vol. 123, pp. 217201 (2019). 

Control of Vortex Structures in Individual Cylindrical Magnetic Nanowires by Simultaneous Action of Applied Magnetic Field and an Electric Current

The behavior of any physical system is governed by the order parameter, determined by the geometry of the physical space of the object, namely their dimensionality and curvature. Usually, the effects of curvature are identified using local interactions only, e.g. local spin-orbit- or curvature-induced Rashba and Dzyaloshinskii-Moriya interactions in condensed matter [1]. Lack of the framework, involving both, local and non-local interactions impedes the description of the essentially micromagnetic textures like magnetic domains, skyrmion-bubbles and vortices. Here, we present a micromagnetic theory of curvilinear ferromagnetic shells [2]. New chiral effects, originating from the magnetostatic interaction, can appear in such systems. They manifest themselves even in statics and are essentially nonlocal. This is in contrast to conventional Dzyaloshinskii--Moriya interaction (material intrinsic or curvature-induced, stemming from the exchange). The physical origin is in a non-zero mean curvature of a shell and non-equivalence between the top and bottom surfaces of the shell. To describe the new effects, we split a conventional volume magnetostatic charge into two terms: (i) magnetostatic charge, governed by the tangent to the sample's surface, and (ii) geometrical charge, given by the normal component of magnetization and the mean curvature. We classify the interplay between the symmetry of the shell, its local curvature and magnetic textures and apply the proposed formalism to analyze magnetic textures in corrugated shells with perpendicular anisotropy.References: [1] R. Streubel, J. Lee, D. Makarov et al, J. Phys. D, 49, 363001, (2016); [3] O. V. Pylypovskyi, D. D. Sheka, V. P. Kravchuk et al, Sci. Rep. Vol. 6, p. 23316 (2016); O. M. Volkov, D. D. Sheka, Y. Gaididei et al, Sci. Rep. Vol. 8, p. 866 (2018). [2] D. D. Sheka, O. V. Pylypovskyi, P. Landeros et al., Comm. Phys. 3, 128 (2019), DOI:10.1038/s42005-020-0387-2 

Magnetostatics-Induced Symmetry Breaking Effects in Curvilinear Shells

Ferromagnetic nanotubes generate continuous interest due to non-collinear spin structures which are tailored by the curvature of the ferromagnetic film [1]. If arranged vertically on a chip [2], they allow for high-density arrays and 3D device architectures in spintronics and magnonics. For their realization, coating of filamentary nanotemplates would be advantageous. However, conformal deposition of ferromagnetic metals is still a challenge. We report the fabrication of Ni nanotubes by atomic layer deposition (ALD) applied to single-crystalline semiconductor nanowires. We used nickelocene as a precursor, water as the oxidant agent and an in-cycle plasma enhanced reduction step with hydrogen. The optimized ALD pulse sequence, combined with a post-processing annealing treatment, allowed us to prepare 30 nm thick metallic Ni layers with a low resistivity of 8 μΩcm at room temperature and good conformality both on the planar substrates and nanotemplates [3]. We prepared several micrometer-long Ni nanotubes with diameters ranging from 120 to 330 nm. We report on the correlation between ALD growth and functional properties of individual Ni nanotubes characterized in terms of magneto-transport at room temperature and the confinement of spin wave modes explored by micro-focus Brillouin light scattering. For the relative variation of resistance R of a nanotube in small magnetic fields we find a value of up to 1.3 % which we attribute to the anisotropic magnetoresistance (AMR) effect. This value is larger by about 30 % compared to the best low-field AMR effect of Ni nanotubes studied previously [4,5]. In the ALD process applied to semiconductor nanowires in Refs. [4,5] the plasma enhancement of the hydrogen reduction step was not available. Together with the spin-wave resonances observed in the Ni nanotubes our findings promise Ni-based 3D device architectures for spintronic and magnonic devices prepared by ALD and operated in the GHz frequency regime. Funding by the German Science Foundation DFG via GR1640/5-2 in SPP1538 Spin caloric transport and SNF via grants 163016, BSCGI0157705, and NCCR QSIT is gratefully acknowledged.References: [1] R. Streubel, P. Fischer, F. Kronast et al., J. Phys. D: Appl. Phys., Vol. 49, p. 363001 (2016). [2] Y.-P. Wang, Z.-J. Ding, Q.-X. Liu et al, J. Mater. Chem. C, Vol. 4, p. 11059 (2016). [3] M.C. Giordano, K. Baumgaertl, S. Escobar Steinvall et al., ACS Appl. Mater. Interfaces (2020), https://doi.org/10.1021/acsami.0c06879. [4] D. Rüffer, R. Huber, P. Berberich et al., Nanoscale, Vol. 4, p. 4989 (2012). [5] D. Rüffer, M. Slot, R. Huber et al., APL Mater., Vol. 2, p. 076112 (2014). 

Nickel nanotubes with low resistivity and coherent magnetization dynamics for 3D spintronics and magnonics prepared by atomic layer deposition

This work describes a technique for 3D printing of magnetic materials at millimeter and sub-millimeter scales for microrobotic applications using stereolithography. Traditional fabrication techniques to engineer magnetic materials become very challenging at smaller scales. The need for rapid prototyping techniques using materials with tunable magnetic properties is becoming more important [1]. Unfortunately, additive manufacturing of magnets is still in its infancy [2-3]. Most reported techniques that are based on thermoplastic bonded magnets [4-5], filament extrusion [6], binder jetting [7], or laser sintering [8], have limitations for scaling down in size. A better path towards miniaturization is offered by stereolithography [9], as presented in this work. The presented technique combines magnetic particles (NdFeB, Fe or Fe3O4) with different mixing ratios to a flexible resin that is made from aliphatic urethane diacrylate, epoxy aliphatic acrylate and a photo-initiator (Irgacure 819 - Irgacure 184) [10]. 3D printed multi-material structures are presented in Fig. 1a. This work is motivated by the need to engineer complex magnetic field patterns for MEMS actuators [11-12] and micro/soft robotics (Fig. 1B) [13]. Magnetic moment was measured to evaluate dependence of saturation magnetization and coercivity with particle material and particle concentration (Fig. 2a). Torques and forces over printed samples were measured in response to external magnetic fields (Fig. 2b), demonstrating an ability to generate highly complex 3D structures for actuation at micro- to meso-scales.References: [1] G. Chatzipirpiridis et al., “3D Printing of Thermoplastic Bonded Soft and Hard Magnetic Composites: Magnetically Tuneable Architectures and Functional Devices,” Adv. Intell. Syst., vol. 1, no. 6, p. 1900069, 2019. [2] L. Li, B. Post, V. Kunc, A. M. Elliott, and M. P. Paranthaman, “Additive manufacturing of near-net-shape bonded magnets: Prospects and challenges,” Scr. Mater., vol. 135, pp. 100–104, Jul. 2017. [3] V. Popov, A. Koptyug, I. Radulov, F. Maccari, and G. Muller, “Prospects of additive manufacturing of rare-earth and non-rare-earth permanent magnets,” Procedia Manuf., vol. 21, no. March, pp. 100–108, 2018. [4] L. Pigliaru et al., “3D printing of high performance polymer-bonded PEEK-NdFeB magnetic composite materials,” Funct. Compos. Mater., vol. 1, no. 1, p. 4, Dec. 2020. [5] P. P. Wendhausen, C. H. Ahrens, A. B. Baldissera, P. D. Pavez, and J. M. Mascheroni, “Additive manufacturing of bonded NdFeB, process parameters evaluation on magnetic properties,” in 2017 IEEE International Magnetics Conference (INTERMAG), 2017, vol. 53, no. 11, pp. 1–1. [6] B. Khatri, K. Lappe, D. Noetzel, K. Pursche, and T. Hanemann, “A 3D-Printable Polymer-Metal Soft-Magnetic Functional Composite—Development and Characterization,” Materials (Basel)., vol. 11, no. 2, p. 189, Jan. 2018. [7] C. L. Cramer et al., “Binder jet additive manufacturing method to fabricate near net shape crack-free highly dense Fe-6.5 wt.% Si soft magnets,” Heliyon, vol. 5, no. 11, p. e02804, Nov. 2019. [8] C. V. Mikler et al., “Laser Additive Manufacturing of Magnetic Materials,” JOM, vol. 69, no. 3, pp. 532–543, Mar. 2017. [9] L. Lu, P. Guo, and Y. Pan, “Magnetic-Field-Assisted Projection Stereolithography for Three-Dimensional Printing of Smart Structures,” J. Manuf. Sci. Eng., vol. 139, no. 7, pp. 1–7, Jul. 2017. [10] D. K. Patel, A. H. Sakhaei, M. Layani, B. Zhang, Q. Ge, and S. Magdassi, “Highly Stretchable and UV Curable Elastomers for Digital Light Processing Based 3D Printing,” Adv. Mater., vol. 29, no. 15, p. 1606000, Apr. 2017. [11] A. P. Taylor, C. Velez Cuervo, D. P. Arnold, and L. F. Velasquez-Garcia, “Fully 3D-Printed, Monolithic, Mini Magnetic Actuators for Low-Cost, Compact Systems,” J. Microelectromechanical Syst., vol. 28, no. 3, pp. 481–493, Jun. 2019. [12] Y. Kim, G. A. Parada, S. Liu, and X. Zhao, “Ferromagnetic soft continuum robots,” Sci. Robot., vol. 4, no. 33, p. eaax7329, Aug. 2019. [13] J. J. Abbott, E. Diller, and A. J. Petruska, “Magnetic Methods in Robotics,” Annu. Rev. Control. Robot. Auton. Syst., vol. 3, no. 1, pp. 57–90, May 2020. 

Multimaterial and flexible 3D printed magnets with sub-millimeter resolution

The extension of nanomagnetism to three dimensions is proposed as an exciting route to study new physical effects, such as the emergence of new spin textures and magneto-chiral phenomena, with the prospective of being exploited in future technologies [1,2]. Whereas a significant amount of work has been so far devoted to simulating 3D nanostructures, its fabrication and characterization has proven to be very challenging, and very few of these systems have been experimentally investigated. Our group has been recently developing new fabrication and characterization methods specifically tailored for the study of 3D nanostructures, which made it possible to control the advanced motion of domain walls in 3D straight nanowires [3,4]. Here, we will present recent advances in this realm, which now enable us to create and probe nanomagnets with more complex 3D geometries. First, we will present recent work dedicated to the 3D printing of nanomagnets using Focused Electron Beam Induced Deposition (FEBID). Based on the effective FEBID continuum model [5] and proximity effect corrections, we have developed an algorithm with the capability of modelling the deposition and defining beam scanning patterns for arbitrary 3D geometries [6]. The algorithm performance has been demonstrated in different electron microscopes and precursor gases, allowing us to create 3D nanomagnets with a wide range of geometries directly from CAD files (see Fig. 1). Complementing nanofabrication, we will present a new Kerr magneto-optical setup with the ability to probe 3D magnetic nanostructures, by exploiting the dark-field MOKE (dfMOKE), a technique that we have recently developed [3]. The unprecedented flexibility of the system to characterize these newly accessible 3D geometries will be demonstrated, thanks to the combination of mobile optics and full-3D vector magnetic fields (see Fig. 2).References: [1] Fernández-Pacheco, A. et al. Nat. Commun. 8, 15756 (2017). [2] Streubel, R. et al. J. Phys. D. Appl. Phys. 49, 363001 (2016). [3] Sanz-Hernández, D. et al. ACS Nano 11, 11066–11073 (2017). [4] Sanz-Hernández, D. et al. Nanomaterials 8, 483 (2018). [5] Toth, M., Lobo, C., Friedli, V., Szkudlarek, A. & Utke, I. Beilstein J. Nanotechnol. 6, 1518–1540 (2015). [6] Skoric, L. et al. Nano Lett. 20, 184–191 (2020). 

Advanced fabrication and magnetometry of nanostructures with complex 3D geometries

A magnetic elastomer is a type of magneto-active smart material where a magnetic field induces a mechanical response. Several methods have been used to optimize performance by controlling the microstructure, such as magnetic annealing. Optimization can also be achieved by introducing anisotropy with Fused Deposition Modeling (FDM), a 3D-printing process that extrudes viscous material in 1D lines to create 2D layers which build to a 3D part. Indeed, FDM features have been shown to manipulate the magnetic anisotropy of rigid printed parts.1 However, printing flexible composite materials presents additional challenges with FDM. Here, FDM structures were printed with thermoplastic polyurethane (TPU) polymer and either 20wt% iron (Fe), 40wt% Fe, 40wt% carbonyl iron (CI), or 40wt% magnetite (Fe3O4/Mag). In order to determine the relative effect of different parameters on the magnetic properties, a series of structures were printed combining each material type (Fe-TPU, CI-TPU, Mag-TPU) with different aspect ratios (1:1 / 2:1), infill percentages (60% / 80% / 100%), and infill orientations (parallel/perpendicular to aspect ratio). A Vibrating Sample Magnetometer was used to obtain magnetic hysteresis loops for comparing magnetic susceptibility between samples. Results demonstrated that infill orientation had the largest effect on susceptibility, aspect ratio had the next greatest effect, and infill percentage had a more nuanced effect (Fig.1). This shows that FDM allows the magnetic anisotropy to manifest through mesoscale anisotropy of a part’s internal structure. These trends were found to form a continuum of tunable magnetic responses (Fig.2). Additionally, the chosen particulate transfers its unique magnetic signature to the composite material. Overall, the highly customizable and nuanced characteristics of 3D-printed magnetic elastomer structures will allow for its use in emerging magneto-mechanical applications such as magnetic actuation and soft robotics.References: 1. M. Patton, P. Ryan, T. Calascione, N. Fischer, A. Morgenstern, N. Stenger, and B. Nelson-Cheeseman, Additive Manufacturing, Vol. 27, 482-488 (2019) 


Controlling Magnetic Properties of 3D-Printed Magnetic Elastomer Structures via Fused Deposition Modeling

In geometrically frustrated magnetism, the very nature of the ground state of Tb2Ti2O7, has remained a long standing conundrum. In this pyrochlore material, no conventional spin-ice or long-range magnetic order is stabilized, even at very low temperatures [1]. Quantum fluctuations are suspected of being at the origin of such an exotic quantum phase, yet so far no complete description is available. Using neutron scattering [2] and synchrotron-based terahertz spectroscopy [3], it has been shown that spin/lattice couplings are present in form of hybrid Tb crystal-field / phonon modes that are present across a broad temperature range from 200 K down to 6 K. This so called vibronic process affects the electronic ground state that can no longer be described solely by electronic wave functions and involve quadrupolar degrees of freedom associated with the rare earth magnetic element in its local environnement. We present here new magneto-optical investigation of vibronic spin-lattice coupling effects. We apply a static magnetic field along the cubic <111> direction while probing with linearly polarized THz radiation. Through the Zeeman effect, the magnetic field enhances splitting within the low-energy Tb crystal field transitions revealing new details in our THz spectra. Complementary magneto-optical quantum calculations including quadrupolar terms show that indeed vibronic effects are required to describe our observations at 3 K. A further prediction of our theoretical model is the presence of a novel magneto-optical birefringence as a result of this vibronic process. Together, our results [4] reveal the significance of considering quadrupolar spin-lattice effects when describing the spin-liquid ground state of Tb2Ti2O7. They also highlight the potential for future magneto-optical investigations to probe complex materials where spin /lattice couplings are present and unravel new magneto-optical activity in the THz range.References: [1] J. S. Gardner, B. D. Gaulin, A. J. Berlinsky et al., Phys. Rev. B 64, 224416 (2001) [2] T. Fennell, M. Kenzelmann, B. Roessli et al., Phys. Rev. Lett. 112, 017203 (2014) [3] E. Constable, R. Ballou, J. Robert et al., Phys. Rev. B 95, 020415(R) (2017) [4] K. Amelin, Y. Alexanian, U. Nagel et al, to be published 

THz Magneto-Optical Investigation of Quadrupolar Spin-Lattice Effects in Magnetically Frustrated Tb2Ti2O7

The spin nematic phase, which is a kind of multipole phases, has attracted a lot of interest in the field of the strongly correlated electron systems, as well as the quantum spin liquid phase. Using the numerical exact diagonalization, the density matrix renormalization group (DMRG) calculation, and the finite-size scaling analysis, it is found that some spin nematic and spin liquid phases are induced by external magnetic field in the anisotropic and frustrated quantum spin systems. In our previous work[1], it was found that a field-induced nematic phase appears at some critical field in the anisotropic spin ladder. The nematic phase is characterized by the power-law decay in the correlation function of the second-order spin moment. In addition at some higher critical field a quantum phase transition can occur to the conventional field-induced Tomonaga-Luttinger liquid. Several typical magnetization curves calculated by DMRG are presented. Recently the field-induced nematic phase was observed on the frustrated spin ladder system[2]. So we studied on a frustrated spin ladder system[3], using the numerical diagonalization and DMRG. As a result, it was found that several exotic quantum phases, including the spin-nematic liquid phase. We also reported several interesting phase diagrams of this model and some related systems[4,5]. In the present paper, we investigate the S=1 antiferromagnetic chain with the biqadratic interaction, as another candidate that exhibits the field-induced spin nematic liquid pahse. The numerical diagonalization study of finite-size clusters indicates that the spin nematic liquid pahse appears in this model under sufficiently strong magnetic field. The phase diagram in the plane of the magnetic field and the biquadratic interaction is presented.References: [1] T. Sakai, T. Tonegawa and K. Okamoto, Physica Status Solidi B 247, 583 (2010) . [2] N. Buttgen et al., Phys. Rev B 90, 134401 (2014) . [3] T. Hikihara, T. Tonegawa, K. Okamoto and T. Sakai, J. Phys. Soc. Jpn. 86, 054709 (2017). [4] T. Tonegawa, T. Hikihara, K. Okamoto, S. C. Furuya, and T. Sakai, J. Phys. So. Jpn. 87, 104002 (2018). [5] T. Sakai, K. Okamoto and T. Tonegawa, Phys. Rev. B 100, 054407 (2019) 

Quantum Spin Nematic Liquid in the S=1 Antiferromagnetic Chain with the Biquadratic Interaction

Spin liquids are disordered states of matter that still fluctuate at low temperature. In many cases, the ground state of spin liquids is expected to be ordered, for example because the dipolar interaction lifts the (quasi)degeneracy between low-energy spin configurations. However, this is not what is observed experimentally, and many systems dynamically freeze in a liquid phase [1] before the ordering takes place. In spin ice compounds or artificial magnets, ground state configurations are then generally out of reach for practical reasons. In this presentation, we will show how dynamical freezing can be bypassed in an artificial kagome ice. We will illustrate the efficiency of our method by demonstrating that the a priori dynamically inaccessible ordered ground state of the dipolar kagome ice is imaged reproducibly in real space, and at room temperature. The central idea of this work is to replace the spin degree of freedom by a micromagnetic knob, which can be adjusted by a proper design of the vertex geometry. Our strategy is to imprint the desired spin configuration, vertex by vertex, in a kagome array made of connected nanomagnets, with a notch at each vertex site. Each notch locally lifts the energy degeneracy between the six states of a given vertex satisfying the ice rule constraint [2], allowing to reach efficiently any desired spin microstate using a field demagnetization protocol. This concept will be demonstrated both numerically and experimentally, and we will show how we can access all predicted phases of the dipolar kagome ice [3, 4, 5].References: [1] I. A. Chioar, B. Canals, D. Lacour, Phys. Rev. B 90, 220407(R) (2014) [2] J. D. Bernal and R. H. Fowler, J. Chem. Phys. 1, 515 (1933) [3] W. R. Branford, S. Ladak, D. E. Read, Science 335, 1597-1600 (2012) [4] B. Canals, I. A. Chioar, V. D. Nguyen, Nature com. 7, 11446 (2016) [5] G.-W. Chern, P. Mellado and O. Tchernyshyov, Phys. Rev. Lett. 106, 207202 (2011). 

Bypassing Dynamical Freezing and Accessing Low-energy Microstates in Artificial Kagome Ice

The nearest neighbour antiferromagnetic Ising model on the kagome lattice is well known for its high frustration and macroscopic ground state degeneracy [1,2]. Recently, new experimental realizations of Ising-like spin systems have been developed - so-called artificial spin systems - based on mesoscopic structures of chirally coupled nanomagnets [3]. On the square lattice, such systems have proven to behave as nearest neighbour Ising antiferromagnets. Because of frustration, realizations of kagome lattice Ising antiferromagnets with these mesoscopic structures (Fig.1) are much more challenging to study. From the theoretical side as well, such models still present some difficulties [4,5]. I will describe how to tackle classical frustrated spin systems using Monte Carlo techniques [5] and tensor network methods [4,6,7] (Fig.2), focusing on the kagome lattice nearest neighbour Ising antiferromagnet in an external magnetic field, and I will compare the experimental and numerical results.References: [1] K. Kanô and S. Naya, Prog. Theor. Phys., Vol. 10, p. 158 (1953) [2] A. Sütö, Z. Phys. B, Vol. 44, p. 121 (1981) [3] Z. Luo et. al., Science, Vol. 363, p. 1435 (2019) [4] B. Vanhecke, J. Colbois, L. Vanderstraeten, et. al., arXiv:2006.14341 (2020) [5] G. Rakala and K. Damle, Phys. Rev. E Vol. 96, 023304 (2017) [6] V. Zauner-Stauber, L. Vanderstraeten, et. al. Phys. Rev. B Vol. 97, 045145 (2018) [7] L. Vanderstraeten, B. Vanhecke, and F. Verstraete, Phys. Rev. E Vol. 98, 042145 (2018) 


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