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An experimental realization of the Haldane conjecture of a gapped spin liquid is of great interested among condensed matter scientists. The conjecture is implemented in the chains of Heisenberg integer spins bounded by identical intrachain exchange interactions &lt;i&gt;J&lt;/i&gt; and could be realized in uniform quasi-one-dimensional integer spin compounds [1]. The Haldane phase has to compete with magnetic anisotropy and long-range ordered phases, depending on the ratio of interchain &lt;i&gt;J&#39;&lt;/i&gt; and intrachain &lt;i&gt;J&lt;/i&gt; exchange interactions and uniaxial &lt;i&gt;D&lt;/i&gt; and rhombic &lt;i&gt;E&lt;/i&gt; single-ion anisotropy. The critical ratios J&#39;/J and D/J for Haldane phase, easy axis long ordered AFM and easy plane long ordered XY-AFM were first discovered by Sakai and Takahashi [2]. Many compounds, both organic and inorganic, have been studied to verify this idea, some of them have been identified as belonging to the Haldane sector of the Sakai-Takahashi phase diagram.&lt;br/&gt;In this work, the position on Sakai-Takahashi phase diagram was determined for the novel integer spin S = 1 chain compound, strontium nickel selenite chloride Sr&lt;sub&gt;2&lt;/sub&gt;Ni(SeO&lt;sub&gt;3&lt;/sub&gt;)&lt;sub&gt;2&lt;/sub&gt;Cl&lt;sub&gt;2&lt;/sub&gt;. XRD measurements showed that the crystal structure of the compound contains the NiO&lt;sub&gt;4&lt;/sub&gt;Cl&lt;sub&gt;2&lt;/sub&gt;-octahedra bridged by the SeO&lt;sub&gt;3&lt;/sub&gt;-pyramids into uniform spin chains. These chains are well separated by strontium cations and cavities of selenium’s lone pair of electrons that could lead to 1D-dimensionality of magnetic subsystem. In magnetic measurements it was determined that strontium nickel selenite chloride, Sr&lt;sub&gt;2&lt;/sub&gt;Ni(SeO&lt;sub&gt;3&lt;/sub&gt;)&lt;sub&gt;2&lt;/sub&gt;Cl&lt;sub&gt;2&lt;/sub&gt;, is a spin S = 1 chain system which passes through a correlations regime at &lt;i&gt;T&lt;/i&gt;&lt;sub&gt;max&lt;/sub&gt; ~ 12 K to long-range order at &lt;i&gt;T&lt;/i&gt;&lt;sub&gt;N&lt;/sub&gt; = 6 K. The values of &lt;i&gt;J&#39;&lt;/i&gt;/&lt;i&gt;J&lt;/i&gt; = 0.083 and &lt;i&gt;D&lt;/i&gt;/&lt;i&gt;J&lt;/i&gt; = 0.357 place this compound into a hitherto unoccupied XY-AFM sector of the Sakai-Takahashi phase diagram.&lt;br/&gt;The results was recently published in Scientific reports journal [3].&lt;br/&gt;Support by the P220 program of Government of Russia through the project 075-15-2021-604 is acknowledged. We also acknowledge the support of the HLD-HZDR, member of the European Magnetic Field Laboratory (EMFL).

![](https://assets.underline.io/uploads/markdown_image/1/image/ac2674811e3113c6c930d976a3efd7a0.jpg)

Sakai-Takahashi phase diagram for uniform spin-1 chain compounds, novel compound Sr2Ni(SeO3)2Cl2 is also marked.

2022 Joint MMM-INTERMAG

Magnetic properties of S = 1 spin chain in Sr2Ni(SeO3)2Cl2: XY-antiferromagnet at Sakai-Takahashi phase diagram

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.

An experimental realization of the Haldane conjecture of a gapped spin liquid is of great interested among condensed matter scientists. The conjecture is implemented in the chains of Heisenberg integer spins bounded by identical intrachain exchange interactions J and could be realized in uniform quasi-one-dimensional integer spin compounds [1]. The Haldane phase has to compete with magnetic anisotropy and long-range ordered phases, depending on the ratio of interchain J' and intrachain J exchange interactions and uniaxial D and rhombic E single-ion anisotropy. The critical ratios J'/J and D/J for Haldane phase, easy axis long ordered AFM and easy plane long ordered XY-AFM were first discovered by Sakai and Takahashi [2]. Many compounds, both organic and inorganic, have been studied to verify this idea, some of them have been identified as belonging to the Haldane sector of the Sakai-Takahashi phase diagram. In this work, the position on Sakai-Takahashi phase diagram was determined for the novel integer spin S = 1 chain compound, strontium nickel selenite chloride Sr2Ni(SeO3)2Cl2. XRD measurements showed that the crystal structure of the compound contains the NiO4Cl2-octahedra bridged by the SeO3-pyramids into uniform spin chains. These chains are well separated by strontium cations and cavities of selenium’s lone pair of electrons that could lead to 1D-dimensionality of magnetic subsystem. In magnetic measurements it was determined that strontium nickel selenite chloride, Sr2Ni(SeO3)2Cl2, is a spin S = 1 chain system which passes through a correlations regime at Tmax ~ 12 K to long-range order at TN = 6 K. The values of J'/J = 0.083 and D/J = 0.357 place this compound into a hitherto unoccupied XY-AFM sector of the Sakai-Takahashi phase diagram. The results was recently published in Scientific reports journal [3]. Support by the P220 program of Government of Russia through the project 075-15-2021-604 is acknowledged. We also acknowledge the support of the HLD-HZDR, member of the European Magnetic Field Laboratory (EMFL).

![](https://assets.underline.io/uploads/markdown_image/1/image/ac2674811e3113c6c930d976a3efd7a0.jpg)

Sakai-Takahashi phase diagram for uniform spin-1 chain compounds, novel compound Sr2Ni(SeO3)2Cl2 is also marked.

technical paper

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

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.

N-th Root Topological Lattices

Topolectrical (TE) circuits have not only bridged the seemingly different disciplines of electrical circuits and topological condensed matter but also allowed the realization of highly tunable lattice systems with desired topological properties based on the parameters of the corresponding TE circuit lattices [1, 2]. The frequency dependence of the electrical inter-node couplings (i.e., the admittance of capacitors and inductors) suggests that the circuits can be switched between the insulating, semimetallic, and Chern insulator phases by varying the frequency of the alternating signal [3]. Here, we propose a TE circuit lattice that exhibits this switching within a single circuit set-up. We consider a 2-dimensional (2D) TE circuit containing two types of nodes A and B (Fig. 1a) with complex nearest-neighbor couplings (combination of capacitive and resistive) realized via negative impedance converters (NICs) (Fig. 1b) and capacitive next-nearest neighbor (NNN) couplings. The critical frequency of the circuit depends only on the grounding inductor and NNN capacitor (Fig. 1c). In the absence of resistive coupling, the TE circuit respects time-reversal symmetry (TRS) and hosts insulating and semimetallic phases when the frequency is less than, and greater or equal to the critical frequency, respectively. Furthermore, the 2D Weyl points in the semimetallic phase are connected by topological edge states in the admittance spectra (Fig. 2b,e). However, a non-zero Rd breaks TRS and a bulk bandgap appears. Chiral edge states emerge under open boundary conditions and the circuit exhibits the quantum anomalous Hall effect at ω &gt; ωc (Fig. 2f). Moreover, the Chern number of the system changes signs when the frequency crosses the critical value.

![](https://assets.underline.io/uploads/markdown_image/1/image/335a98dd245cf287b314d00c26f10710.jpg)

a. Schematic of frequency-dependent Chern circuit. b. NIC and grounding connections. c. System Hamiltonian and critical frequency.

![](https://assets.underline.io/uploads/markdown_image/1/image/c73aa987cfe48dc4e49ba3714769b396.jpg)

a-f. Admittance dispersion of a finite system at different frequencies. The top and bottom rows correspond to zero and non-zero resistive couplings, respectively. Common parameters: C1=1.5 mF, C2=1 mF and Lm=1 nH.

AC Frequency Modulated Phase Transitions in Topolectrical Circuits

Effect of calcium doping on magnetocaloric properties and critical behaviour of La0.55CaxSr0.45-xMnO3 (x = 0.00, 0.05, 0.1, 0.2 and 0.25) prepared by solid state reaction technique has been reported here. XRD measurements reveal rhombohedral structure for all the samples. Lattice parameters as well as Mn-O-Mn bond angle decreases with increasing Ca content. These structural changes control ferromagnetism in the materials via the double-exchange mechanism [1]. Temperature dependent magnetization measurement reveals a reduction of saturation magnetization and decrease in Curie temperature (TC) from 354 K to 311 K with increase in Ca content. A second order ferromagnetic (FM) to paramagnetic (PM) phase transition of all the samples is confirmed by Arrott plots. The magnetocaloric effect is calculated from the isothermal magnetization measurements using Maxwell’s thermodynamic relations. The maximum entropy change ((-ΔSm)max) and relative cooling power (RCP) are observed to decrease while TC shifted toward the room temperature with increase in Ca content. Therefore the sample La0.55CaxSr0.45-xMnO3 could be used as good solid state refrigerant for magnetic refrigerator near room temperature. Critical behaviour around PM-FM phase transition has been investigated using various methods including modified Arrott plots (MAP) [2], Kouvel–Fisher (KF) [3] method and critical isotherm analysis. The M-H data taken around TC was investigated in the framework of the mean field model, the 3D–Heisenberg model, the 3D–Ising model and tricritical mean field model [2]. The well agreement between MAP and KF methods confirms that the calculated values of the critical exponents are reliable. Calculated critical exponents reveal that the samples with x=0.00, 0.05, 0.1 possess a long-range ferromagnetic interaction belonging to the mean-field model, and x=0.2, 0.25 possess ferromagnetic short-range interactions belonging to the 3D Heisenberg model. Chemical pressure due to substitution of larger Sr ions with smaller Ca ions, cooperative Jahn–Teller distortions, and presence of ferromagnetic clusters all have significant influence on the nature of the FM-PM phase transition in the studied samples.

![](https://assets.underline.io/uploads/markdown_image/1/image/627ea12b43d0694da1615951ce85d79c.jpg)

K-F plot

![](https://assets.underline.io/uploads/markdown_image/1/image/b85b658e9ac6bc92a8595bd77dda4c34.jpg)

Entropy chang vs. T

Effect of Calcium Doping on Magnetocaloric Properties and Critical Behaviour of La0.55CaxSr0.45-xMnO3 around FM-PM Phase Transition

The long-range ordering of two-dimensional magnets with the isotropic short-ranged exchange interaction is destroyed by the fundamentally strong thermal fluctuation of low dimensional systems. We show that the spatial fluctuation, modeled by a random field at the lattice sites, significantly suppresses the long-range ordering as well. By using the random phase approximation along with the quantum statistics of low-energy magnon excitations, we have derived, from the anisotropic Heisenberg model, the magnetization as a function of the temperature in the presence of the random field. We found that the random field effectively decreases the energy gap of the magnon excitations. Since the energy gap is essential for 2D magnetic ordering, the random field would significantly reduce the transition temperature. Another finding is that the phase transition between the ferromagnet and the paramagnet becomes first-order, namely, the magnetization jumps from a finite value in the ferromagnetic phase to nearly zero at the paramagnetic phase. We will present the quantitative relations of above findings with the strength of the random field, and we estimate how these predictions could be experimentally realized. This work is supported by NSF-ECCS-2011331.

Effects of Random Fields on the Phase Transition of a Two-Dimensional Ferromagnet

Strong correlations in metals are known to yield a rich variety of quantum phases [1]. Whether and how they also drive electronic topology is of considerable current interest and, yet, a largely open question. A promising setting arises in heavy fermion systems, which may feature not only strong correlations but also a large spin-orbit coupling. Recently, a Weyl-Kondo semimetal phase has been concurrently discovered in theoretical [2] and experimental [3] studies. The theoretical work was carried out in a Kondo lattice model that is time-reversal invariant but inversion-symmetry breaking. In this talk, I will i) describe the basic characteristics of the Weyl-Kondo semimetal phase, with a particular emphasis on how strong correlations not only produce the Weyl nodes but also pin them near to the Fermi energy, and its experimental signatures, as has been advanced in the material Ce3Bi4Pd3; ii) discuss the extreme topological control of the correlation-driven Weyl nodes [4,5]; and iii) present a design principle – via the cooperation of strong correlations with space-group symmetry--for new Weyl-Kondo semimetal materials [6]. Finally, I will discuss the enrichment of our results for the strong correlation physics, in the form of a global phase diagram, of heavy fermion metals [1]. *This work was done with Sarah Grefe, Hsin-Hua Lai, Lei Chen, Chandan Setty and Haoyu Hu at Rice University and with Silke Paschen, Sami Dzsaber, Andrey Prokofiev, Jennifer Cano and Maia G. Vergniory. It was supported by the Air Force Office of Scientific Research Grant No. FA9550-21-1-0356, the National Science Foundation Grant No. DMR-1920740, and the Robert A. Welch Foundation Grant No. C-1411.

Magnetic properties of S = 1 spin chain in Sr2Ni(SeO3)2Cl2: XY-antiferromagnet at Sakai-Takahashi phase diagram

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Magnetic properties of S = 1 spin chain in Sr2Ni(SeO3)2Cl2: XY-antiferromagnet at Sakai-Takahashi phase diagram

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

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Artificial Spin Ice with Y-shaped Nanomagnets

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