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Square root topological lattice models such as the non-Hermitian 1D Su-Schrieffer-Heeger (SSH) model [1] and the 2D honeycomb lattice [2, 3] have recently been introduced and realised in several platforms [4, 6]. The recipe for the square root construction has paved the way for &lt;i&gt;2&lt;sup&gt;n&lt;/sup&gt;&lt;/i&gt; root models [7], in which the transition from higher to lower roots can be readily made due to the arborescent structure of the resultant lattice by taking the &lt;i&gt;n&lt;/i&gt;-th power of the corresponding Hamiltonian ((&lt;i&gt;H&lt;/i&gt;&lt;sub&gt;&lt;i&gt;roo&lt;/i&gt;t&lt;/sub&gt;)&lt;i&gt;&lt;sup&gt;n&lt;/sup&gt;&lt;/i&gt;) and subtracting a constant energy. The present method is only applicable for the square root operation. Here, we introduce a new procedure which enables us to realise any arbitrary &lt;i&gt;n&lt;/i&gt;-th root of the lattice, including prime number roots, by simply adding &lt;i&gt;n-1&lt;/i&gt; new sites between two parental sites such that the new directional couplings create a closed loop in either the clockwise or anti-clockwise directions (Fig. 1a). While the Hermiticity is preserved in square root operations, the non-reciprocal couplings in the &lt;i&gt;n&lt;/i&gt;-th root method require breaking the Hermiticity so that the arborescent construction can be followed to form (&lt;i&gt;H&lt;sub&gt;root&lt;/sub&gt;&lt;/i&gt;)&lt;i&gt;&lt;sup&gt;n&lt;/sup&gt;&lt;/i&gt; as the direct sum of &lt;i&gt;H&lt;sub&gt;parental&lt;/sub&gt;&lt;/i&gt; and &lt;i&gt;H&lt;sub&gt;residual&lt;/sub&gt;&lt;/i&gt; where &lt;i&gt;H&lt;sub&gt;parental&lt;/sub&gt;&lt;/i&gt; is the original lattice Hamiltonian, which is topologically non-trivial [8], and &lt;i&gt;H&lt;sub&gt;residual&lt;/sub&gt; &lt;/i&gt;represents the direct sum of all the residual parts. This form allows us to transition between roots (Fig. 1b) without the need for a constant energy shift in every operation (Fig. 2). Every root operation results in a multiplication of the energy band structure, which depends on the degree of the root. This multiplication in energy also results in the multiplication of the topological states. In contrast to conventional TIs, the &lt;i&gt;n&lt;/i&gt;-th root version of the topological lattices possesses multiple topological mid-gap states with finite energy as well as non-Hermitian phenomenon such as the non-Hermitian skin effect, which may be useful for various potential applications such as magnetic sensing due to the extensive sensitivity and robustness of the topological states.

![](https://assets.underline.io/uploads/markdown_image/1/image/7985dfc21e709bda9aa9472f8edf38c1.jpg)

Illustration of systematic shifting between (SSH)1/n lattices.

![](https://assets.underline.io/uploads/markdown_image/1/image/c228455f335d23670adca71013893745.jpg)

Real energy spectrum as a function of site index of cubic root (3n) SSH model with 1 unit cell, the transition from n=3 to n=0.

2022 Joint MMM-INTERMAG

N-th Root Topological Lattices

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Square root topological lattice models such as the non-Hermitian 1D Su-Schrieffer-Heeger (SSH) model [1] and the 2D honeycomb lattice [2, 3] have recently been introduced and realised in several platforms [4, 6]. The recipe for the square root construction has paved the way for 2n root models [7], in which the transition from higher to lower roots can be readily made due to the arborescent structure of the resultant lattice by taking the n-th power of the corresponding Hamiltonian ((Hroot)n) and subtracting a constant energy. The present method is only applicable for the square root operation. Here, we introduce a new procedure which enables us to realise any arbitrary n-th root of the lattice, including prime number roots, by simply adding n-1 new sites between two parental sites such that the new directional couplings create a closed loop in either the clockwise or anti-clockwise directions (Fig. 1a). While the Hermiticity is preserved in square root operations, the non-reciprocal couplings in the n-th root method require breaking the Hermiticity so that the arborescent construction can be followed to form (Hroot)n as the direct sum of Hparental and Hresidual where Hparental is the original lattice Hamiltonian, which is topologically non-trivial [8], and Hresidual represents the direct sum of all the residual parts. This form allows us to transition between roots (Fig. 1b) without the need for a constant energy shift in every operation (Fig. 2). Every root operation results in a multiplication of the energy band structure, which depends on the degree of the root. This multiplication in energy also results in the multiplication of the topological states. In contrast to conventional TIs, the n-th root version of the topological lattices possesses multiple topological mid-gap states with finite energy as well as non-Hermitian phenomenon such as the non-Hermitian skin effect, which may be useful for various potential applications such as magnetic sensing due to the extensive sensitivity and robustness of the topological states.

![](https://assets.underline.io/uploads/markdown_image/1/image/7985dfc21e709bda9aa9472f8edf38c1.jpg)

Illustration of systematic shifting between (SSH)1/n lattices.

![](https://assets.underline.io/uploads/markdown_image/1/image/c228455f335d23670adca71013893745.jpg)

Real energy spectrum as a function of site index of cubic root (3n) SSH model with 1 unit cell, the transition from n=3 to n=0.

technical paper

The effect of boundaries and how these can be used to influence the bulk behavior in geometrically frustrated systems are both long-standing puzzles, often relegated to a secondary role. In this talk, I will investigate boundary effects in a geometrically frustrated system, namely an artificial colloidal ice, a microscale soft matter analog of a frustrated nanoscale spin ice system. The artificial colloidal ice is realized by confining interacting paramagnetic colloids to a lattice of gravitational double wells [2]. With this system, both via numerical simulations and “proof of concept” experiments, I will demonstrate that boundaries can be engineered to control the bulk behavior in a colloidal artificial ice [3]. I will also show that an antiferromagnetic frontier forces the system to rapidly reach the ground state (GS), as opposed to the commonly implemented open or periodic boundary conditions. Further, strategically placing defects at the corners may be used to generate novel bistable states, or topological strings, which result from competing GS regions in the bulk. The presented results could be generalized to other frustrated micro- and nanostructures where boundary conditions may be engineered with lithographic techniques.

Topological Boundary Constraints in Artificial Colloidal Ice

In recent years we have been focused on the understanding of magnetism in ensembles of 4f metallic nanoparticles (NPs) (R = Tb,Nd, Gd-M = Cu, Al, Y, La). As a result, we have reported that the existence of an ordered magnetic core and a disordered magnetic shell is regularly found in those ensembles. Depending on the stoichiometry (in our case, RX2) it is possible to easily tune the RKKY interactions to provoke the appearance of ferromagnetic or antiferromagnetic (AF) order within the NPs. Moreover, the fact that we are dealing only with thousand of moments per particle leads to use the super- prefix to define correctly the magnetic behaviour [1]. Besides, it is an upsetting fact that both the collective moment dynamics and the role of the crystalline electric field (CEF) in ensembles of NPs have barely been unearthed. To fill in this gap, we are reporting here a complete neutron-based study (Neutron Diffraction, ND, Small-Angle Neutron Scattering, SANS, and INS) on NdCu2 NPs (&lt;D&gt;≈13nm) to depict the skyline of order and disorder at the nanoscale. ND measurements (LLB) point to that the bulk commensurate AF structure is retained within the core. SANS confirms the survival of the AF within the nanoscale, together with a strong emergence of disorder, in the form of a rise in the low-Q region (Q&lt;0.665 Å-1). Magnetic measurements indicate the interactions among the surface moments strong enough to establish a Spin Glass behaviour, leading therefore to a Superantiferromagnetic state. Regarding the CEF panorama, macroscopic specific heat analyses anticipate a broad Schottky contribution, with a maximum located at T∼25 K. Then, microscopic INS measurements (ILL) performed at T=10K (paramagnetic) and T=1.5K (magnetic state) reveal that the splitting of the CEF levels, as well as the spin–wave excitations that emerged below the Néel transition (TN ≈ 6 K) in polycrystalline NdCu2 [2] are maintained in the NPs. Furthermore, we have been able to isolate the INS from the surface, which has allowed us to unravel the role of finite–size effects and microstrain. These two effects lead to a partial inhibition of the transitions from the ground state to the first excited level, as well as a stiffening of the magnon modes [3].

Order and disorder : The magnetic and crystalline electric field skyline of NdCu2 nanoparticles

The extent of magnetic ordering in highly-frustrated thermally active artificial kagome spin ice is limited by the lowest achievable blocking temperature, as the moments freeze before ordering is achieved [1]. A more substantial degree of magnetic ordering can be achieved by either lowering the blocking temperature of the individual nanomagnets or by increasing the critical transition temperature of the system. First, we pursue an approach of introducing interfacial Dzyaloshinskii-Moriya interactions (DMI), which lowers the blocking temperature by a factor of five, while keeping the transition temperature unchanged [2]. We demonstrate that a seven-ring kagome structure consisting of 30 nanomagnets with DMI can be thermally annealed into its ground state. Furthermore, the spin-ice correlations extracted from extended kagome lattices are found to exhibit the quantitative signatures of long-range charge-order, thereby giving experimental evidence for the theoretically predicted continuous transition to a charge-ordered state. Second, we find that slight modifications of the nanomagnets at the vertices in the artificial kagome ice allow us to control the energy landscape which leads to an increase of the critical temperatures [3]. These results provide paths for the tuning of magnetic ordering in artificial spin systems.

Long-range Magnetic Ordering in Artificial Kagome Spin Ice

The frustrated quantum spin systems exhibit various exotic magnetic phenomena. One of interesting phenomena is the magnetization plateau, which corresponds to the quantization of the magnetization. The S=1/2 distorted diamond spin chain is a typical one-dimensional frustrated systems. The previous theoretical study, using the numerical diagonalization of finite-size culsters and the level spectroscopy analysis[1], indicated the S=1/2 distorted diamond spin chain can exhibit the 1/3 magnetization plateau due to two different mechanisms, depending on the ratio of the exchange interactions. Azurite Cu3(OH)2(CO3)2 is a good candidate material of the distorted diamond spin chain. The high-field magnetization measurement successfully revealed that azurite exhibits the 1/3 magnetization plateau[2]. Recently as another good candidate material alumoklyuchevskite K3Cu3AlO2(SO4)4 was discovered[3]. In this compound ferromagnetic exchange interactions exist as well as antiferromagnetic ones, while azurite has only antiferromagnetic ones. On the other hand, if the anisotropic ferromagnetic interaction is introduced to the distorted diamond spin chain, another mechanism of the 1/3 magnetization plateau would be expected, based on the previous theoretical study on the mixed spin chain[4]. Thus it would be interesting to study the model of the distorted diamond spin chain with the anisotropic ferromagnetic interaction, to consider the magnetization plateau. In the present work, we investigated this model using the numerical diagonalization and the level spectroscopy analysis and obtained the phase diagram at 1/3 of the saturation magnetization, which includes the plateau phase due to a new mechanism for the distorted diamond spin chain.

Magnetization Plateau of the Distorted Diamond Spin Chain with Anisotropic Ferromagnetic Interactions

Dipolar Spin Ices (DSI) are geometrically frustrated magnetic materials, where the moments are arranged on the sites of a pyrochlore lattice [1]. The ground-state obeys the ice-rule, where two magnetic moments point to the center of the tetrahedral unit cell and two point out. If the rule is broken, i.e. one moment is switched, a net magnetic charge is observed at the center of the tetrahedron, referred to as magnetic monopoles. The trace of the switched magnetic moments is referred to as a Dirac string, which has magnetic monopoles at its ends. However, these materials require cryogenic conditions for the stability of monopoles. To study the magnetic monopoles on more suitable platform, artificial spin ices (ASI) consisting of elongated magnetic islands were considered [2]. Their 2D nature lifts the degeneracy of the ground state, as not all ice-rule obeying vertices have equal energies. Due to this differentiation between energetic states of the vertices, the Dirac strings bind the monopoles and store energy, limiting the mobility within the lattice. Here, we combine the advantages of DSI and 2D ASI, and present a 3D ASI lattice [3], where 3D ellipsoids with nearly perfect ising behavior are arranged on an Ice Ih crystal, where the angle between the ellipsoids is always θ = 109.5°, see Fig. 1. This symmetry recovers the lifted degeneracy of the ground state and enables tension-free Dirac strings. Based on the definitions of unbound magnetic monopoles [4], we show that the mobility threshold for magnetic charges is by 2 eV lower than their unbinding energy, while the propagation barrier is constant. The energy of the system does not increase with the length of the Dirac string, showing its tension-free nature. Furthermore, we apply external global fields and steer one magnetic monopole in the desired direction without creating additional magnetic charges, see Fig. 2. Finite temperature micromagnetic analysis reveals that the magnetic monopoles within the 3D ASI lattice are stable and steerable at room temperature.

![](https://assets.underline.io/uploads/markdown_image/1/image/7329a0465c413b41f6b95412aa66de49.jpg)

Fig.1: 3DASI lattice with 3D and top view.

![](https://assets.underline.io/uploads/markdown_image/1/image/ebc78830ac4545aedb5992bd9d0f1124.jpg)

Fig. 2: Averaged magnetization and applied field with respect to time, and the schematic magnetic configurations before and after propagation of monopole.

Tension-free Dirac Strings and Steered Magnetic Charges in 3D Artificial Spin Ice

Lithographically patterned arrays of magnetostatically interacting nanostructures are now well-established tools for investigating the physics of frustrated magnets [1]. Reaching a low-energy manifold where intriguing phenomena occur is challenging experimentally, and the effective temperature of the system under study must be lowered efficiently. So far, two main approaches were used in practice. Either the artificial systems are rotated in a decaying magnetic field [2], or their magnetization dynamics are activated by a thermal treatment [3]. Even though both protocols were employed successfully in previous works, there is a fundamental difference between them. If one can intuitively understand that thermal fluctuations may be a source of stochasticity, one might wonder whether field demagnetization protocols are not instead a source of determinism, especially when the experiments are conducted well below the blocking or Curie temperature of the nanomagnets. The main purpose of this presentation is to answer the following questions: What happens if the very same athermal lattice is field demagnetized several times? Would we get the same arrested state repeatedly and deterministically, or are the successive arrested configurations statistically decorrelated because stochasticity drives the physics? Here we investigate the response of an athermal, artificial square ice [4] submitted to successive field demagnetization protocols. We show that each resulting magnetic state differs substantially from the previous one, but not entirely. Analyzing these states via spin and vertex density maps, we find that the field demagnetization protocol we apply is essentially a stochastic process, despite quenched disorder, which is unambiguously present in our arrays, acts as a source of determinism. We also provide a comparison with conventional two-dimensional square spin system [2] submitted to the very same field history and show that quenched disorder plays a less important role. Finally, we demonstrate that the spin and vertex density maps are a convenient tool to probe and reveal quenched disorder in an artificial spin system, as they allow to spot the regions that remain unaffected between successive field protocols.

Is a field demagnetization protocol applied to an a priori athermal, artificial square ice magnet, a stochastic process?

Cu-based hybrid organic-inorganic perovskites represent an intriguing class of quasi-two-dimensional layered materials. They are well known to behave as nearly ideal two-dimensional (2D) Heisenberg magnets, making them potentially suited for spintronic applications. However, the nature of spin wave propagation in these materials and how it is affected by the presence of various anisotropic exchange contributions warrants further investigation. Here we examine the static and dynamic magnetic properties of (C6H5CH3CH2NH3)2CuCl4, (S-C6H5CH3CH2NH3)2CuCl4, (C8H9NH3)2CuCl4, and (C6H5CH3CH2NH3)2CuBr4 single crystals using SQUID magnetometry and ferromagnetic resonance (FMR) spectroscopy, respectively. Static magnetization and susceptibility measurements reveal the effects of organic cation and halide substitution on critical ordering behavior. Dynamic FMR measurements show spin wave propagation in the paramagnetic region above the critical temperature as a result of strong intralayer exchange coupling.

Static and Dynamic Magnetic Properties of Cu-Based Organic-Inorganic Hybrid Perovskite Single Crystals

Long before the recent fascination with 2D materials, the critical behavior and universality scaling of phase transitions in low-dimensional systems was a topic of great interest. Particularly intriguing is the case of magnetic order in two dimensions, once excluded in systems with continuous symmetry by the Hohenberg-Mermin-Wagner theorem. While an out-of-plane anisotropy has been shown to lift this restriction and stabilize magnetic order in van der Waals monolayers -as demonstrated in CrI3, CrBr3 and Fe3GeTe2- [1], this proof has remained elusive for a 2D magnet with in-plane rotational symmetry. In this presentation, we highlight the discovery of such an atomically-thin easy-plane system, namely a CrCl3 monolayer on Graphene/6H-SiC (0001) grown by molecular-beam epitaxy. Employing in-situ X-ray magnetic circular dichroism, we demonstrate intrinsic ferromagnetic ordering in the monolayer with a critical scaling characteristic of a 2D-XY system [2]. We highlight the role of the in-plane hexagonal crystalline symmetry and negligible substrate interaction to attain an ideal easy-plane (XY) instead of the more ubiquitous easy-axis (Ising) behaviour. These observations point towards the first realization of a finite-size Berezinskii-Kosterlitz-Thouless (BKT) phase transition in a large-area, quasi-freestanding monolayer magnet with a XY universality class; and further constitutes a suitable platform to study exotic phenomena inherent of easy-plane magnets, like superfluid spin transport or in-plane topological spin textures such as merons.

![](https://assets.underline.io/uploads/markdown_image/1/image/4377cea9c0aad6be1933c4cfdb831960.jpg)

Schematic crystal structure of CrCl3/Graphene/6H-SiC layers in top view and cross-section configurations, along with the in-situ RHEED pattern of monolayer CrCl3 grown by on the Graphene substrate, along Γ-M direction. On the right, the atomically resolved STM image of the CrCl3 monolayer results in a moiré pattern from the superposition of hexagonal unit cells of CrCl3 and graphene. The Fourier-transformed image yields a twist angle of 23.8 degree.

![](https://assets.underline.io/uploads/markdown_image/1/image/65ce4e4a29ae04b4d066d19641ce963e.jpg)

XMCD values at zero magnetic field (remanence) as a function of temperature. The critical scaling fit close to the phase transition is shown, determining the exponent β=0.227, consistent with a 2DXY magnetic universality class.

Recent developments of magnetism in flatland: Discovery of Intrinsic 2D-XY Ferromagnetism in a van der Waals Monolayer

The quantum dot with electron correlations attached to superconducting leads displays a local quantum phase transition from a spin-singlet (zero phase) to a spin-doublet (pi phase) ground state. We recently showed that one needs to apply an external magnetic field to distinguish the physical properties of the two phases [1]. Most of the properties of the superconducting quantum dot, including the zero-pi transition, can be qualitatively correctly described by a mean-field theory with a two-particle self-consistency [1]. The mean-field description of magnetic fluctuations with static self-energy fails, however, at non-zero temperatures. Linear response theory in the external magnetic field is broken with field-induced poles in the mean-field dynamical transversal susceptibility. Poles in the dynamical susceptibility generate, however, non-analytic singular contributions to the self-energy when going beyond the mean-field approximation. The renormalization of the electron propagators must hence be made dynamical and in the spin-polarized state in order to smooth the singularities in the perturbation expansion and to keep the self-energy and Green functions analytic. The dynamical renormalization due to the electron correlations leads to the broadening of the energies of the in-gap states into bands filling the superconducting gap with increasing temperature even in the limit to zero magnetic field. We demonstrate this behavior on a single impurity Anderson model attached to superconducting leads.

![](https://assets.underline.io/uploads/markdown_image/1/image/61769d37767a655ec56ea1ced4f971aa.jpg)

Thermodynamically induced broadening of the energies of the in-gap states of the superconducting single-impurity Anderson model with the Coulomb repulsion U=Δ, the strength of the hybridization of the dot to the superconducting leads Γ=Δ, and the superconducting gap of the dot (-Δ,Δ) at different temperatures T and zero magnetic field in the energy scale Δ =1.

Failure of the mean-field description of magnetic fluctuations in the superconducting quantum dot

Torque ripple is the main cause of noise and vibration [1]. Mutual electromagnetic torque ripple is produced by a mismatch between excitation current and the shape of back-EMF [2]. So far, the most popular technology to reduce the cogging torque and torque ripple is skewing, including skewing of stator slots or magnet poles [3]. Based on the above analysis, this paper proposes a segment-skew U-shaped permanent magnet motor. The PMSM rotor topology used for vibration and noise research in this paper is shown in Fig. 1. PMSM parameters are shown in Table 1. Table 2 shows the torque ripple of the motor under different number of skew pole segments and angles. The data are the maximum and minimum values of torque ripple at peak current. According to Table 2, we choose a scheme with 5 segments and 5 degrees of skew angle in this paper. The spatial order of electromagnetic force is shown in Table 3. Only the major force wave orders are listed in the Table 3. Where is v the harmonic frequency of armature reaction magnetic field; is harmonic frequency of rotor permanent magnetic field. Fig. 2 compares the torque ripple at the optimum Angle under different segments of the rotor under peak current. Fig. 3 compares the cogging torque of the rotor with and without segmented skew. Fig. 4 compares the line back electromotive force (EMF). Fig. 5 compares the torque ripple of the rotor at rated current and peak current. Fig. 6 compares the torque current curves of the rotor under MTPA and Id=0 control. Fig. 7 compares the external characteristic curves of the motor. Fig. 8 compares the torque ripple under the magnetic weakening characteristics. Fig. 9 compares the radial electromagnetic force harmonics of rotors. Fig. 10 shows the comparison of noise harmonic response curves of the motor at 5000rpm. Fig 11 and Fig 12 show the experimental equipment and results. In this paper, the characteristics of segmented skew and non-segmented skew rotor motors are studied. It is proved that the segmented skew rotor can make the Back-EMF curve more close to the sine wave, suppress the torque ripple, reduce the electromagnetic force generated by the first order tooth harmonics, and reduce the electromagnetic noise.

![](https://assets.underline.io/uploads/markdown_image/1/image/84bc25c69b293c37f062e882d38e8158.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/7b54b4d2e62db22ee1c07b7e5c485139.jpg)

N-th Root Topological Lattices

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Topological Boundary Constraints in Artificial Colloidal Ice

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N-th Root Topological Lattices

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Next from 2022 Joint MMM-INTERMAG

Topological Boundary Constraints in Artificial Colloidal Ice

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Multiferroic Behavior in NH4Al(SO4)2-12H2O: Magnetism and Ferroelectricity Based on Proton Orbital Degeneracies

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