Nearest-neighbor Exchange Coupling Research Articles

Programmable order by disorder effect and underlying phases through dipolar quantum simulators

In this work, we study two different quantum simulators composed of molecules with dipole-dipole interaction through various theoretical and numerical tools. Our first result provides knowledge of the quantum order by disorder effect of the S=1/2 system, which is programmable in a quantum simulator composed of circular Rydberg atoms in the triangular optical lattice with a controllable diagonal anisotropy. When the numbers of up spins and down spins are equal, a set of subextensive degenerate ground states is present in the classical limit, composed of continuous strings whose configuration enjoys a large degree of freedom. Among all possible configurations, we focus on the stripe (up and down spins aligning straightly) and kinked (up and down spins forming zigzag spin chains) patterns. Adopting the the real space perturbation theory, we estimate the leading order energy correction when the nearest-neighbor spin exchange coupling, J, is considered, and the overall model becomes an effective XXZ model with a spatial anisotropy. Our calculation demonstrates a lifting of the degeneracy, favoring the stripe configuration. When J becomes larger, we adopt the infinite projected entangled-pair state (iPEPS) and numerically check the effect of degeneracy lifting. The iPEPS results show that even when the spin exchange coupling is strong the stripe pattern is still favored. Next, we study the dipolar bosonic model with tilted polar angle which can be realized through a quantum simulator composed of cold atomic gas with dipole-dipole interaction in an optical lattice. By placing the atoms in a triangular lattice and tilting the polar angle, the diagonal anisotropy can also be realized in the bosonic system. With our cluster mean-field theory calculation, we provide various phase diagrams with different tilted angles, showing the abundant underlying phases including the supersolid. Our proposal indicates realizable scenarios through quantum simulators in studying the quantum effect as well as extraordinary phases. We believe that our results indicated here can also become a good benchmark for two-dimensional quantum simulators. Published by the American Physical Society 2024

Critical characteristics of spin − 1/2 four-leg nanotube: Monte Carlo Simulation

In this study, we aim to investigate theoretically the critical characteristics of spin −1/2 four-leg nanotube for the nanotechnology applications. The Ising model simulated by Monte Carlo simulation is adopted with the first exchange coupling J , the second exchange coupling between the second nearest neighbor atomic spins J ′ and the transverse exchange coupling J z , and the external magnetic field h as parameters. Herein, we examine the local properties of spin −1/2 four-leg nanotube by analyzing the magnetization, susceptibility and hysteresis loop behavior as function of J ′ , J Z and h . The obtained results show that the isotropic ‐ 1 / 2 four-leg nanotube Ising model presents critical behavior at ferromagnetic to paramagnetic phase transition where the Curie temperature T C increases according to vertical and second nearest neighbor exchange coupling. Furthermore, under external magnetic field, the magnetization keeps the second order transition and the magnetic susceptibility declines with the increase of applied external magnetic field value. In addition to that, we have found single ferromagnetic hysteresis loops, for which the shape increases by increasing the second nearest neighbor exchange coupling when T = J . Finally, the critical temperature shows perfect mirror symmetry relative to the vertical and increases with the second nearest neighbor exchange coupling.

Two-dimensional Heisenberg model with material-dependent superexchange interactions

The two-dimensional (2D) van der Waals ferromagnetic semiconductors, such as CrI$_3$ and Cr$_2$Ge$_2$Te$_6$, and the 2D ferromagnetic metals, such as Fe$_3$GeTe$_2$ and MnSe$_2$, have been obtained in recent experiments and attracted a lot of attentions. The superexchange interaction has been suggested to dominate the magnetic interactions in these 2D magnetic systems. In the usual theoretical studies, the expression of the 2D Heisenberg models were fixed by hand due to experiences. Here, we propose a method to determine the expression of the 2D Heisenberg models by counting the possible superexchange paths with the density functional theory (DFT) and Wannier function calculations. With this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange coupling constants for the 2D ferromagnetic metal Cr$_3$Te$_6$, which is very different for the crystal structure of Cr atoms in Cr$_3$Te$_6$. The calculated Curie temperature Tc = 328 K is close to the Tc = 344 K of 2D Cr$_3$Te$_6$ reported in recent experiment. In addition, we predict two stable 2D ferromagnetic semiconductors Cr$_3$O$_6$ and Mn$_3$O$_6$ sharing the same crystal structure of Cr$_3$Te$_6$. The similar Heisenberg models are obtained for 2D Cr$_3$O$_6$ and Mn$_3$O$_6$, where the calculated Tc is 218 K and 208 K, respectively. Our method offers a general approach to determine the expression of Heisenberg models for these 2D magnetic semiconductors and metals, and builds up a solid basis for further studies.

Adiabatic demagnetization cooling well below the magnetic ordering temperature in the triangular antiferromagnet KBaGd(BO3)2

Crystal structure, thermodynamic properties, and adiabatic demagnetization refrigeration (ADR) effect in the spin-$\frac{7}{2}$ triangular antiferromagnet $\mathrm{KBaGd}{({\mathrm{BO}}_{3})}_{2}$ are reported. With the average nearest-neighbor exchange coupling of 44 mK, this compound shows magnetic order below ${T}_{N}=263$ mK in zero field. The ADR tests reach the temperature of ${T}_{min}=122\phantom{\rule{0.16em}{0ex}}\mathrm{mK}$, more than twice lower than ${T}_{N}$, along with the entropy storage capacity of 192 mJ ${\mathrm{K}}^{\ensuremath{-}1}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ and the hold time of more than 8 h in the PPMS setup, both significantly improved compared to the spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{\mathrm{Yb}}^{3+}$ analog. We argue that $\mathrm{KBaGd}{({\mathrm{BO}}_{3})}_{2}$ shows a balanced interplay of exchange and dipolar couplings that together with structural randomness and geometrical frustration shift ${T}_{min}$ to well below the ordering temperature ${T}_{N}$, therefore facilitating the cooling.

Spin direction tunable topological transition in two-dimensional frustrate antiferromagnetic triangular lattice T-FeO2 monolayer

Recently, numerous two-dimensional (2D) magnetic materials are predicted with various promising properties, whereas noncollinear antiferromagnetic 2D materials are rarely reported. In this paper, we predicted a stable 2D noncollinear antiferromagnetic triangular lattice T-FeO2. The ground state is 120° antiferromagnetic with stronger next nearest neighbor exchange coupling than that of nearest neighbor exchange coupling because of the RKKY interaction. Our Monte Carlo simulations reveal that the magnetic phase transition is from a 120° antiferromagnetic state to a stripy state and then to a paramagnetic state with increasing temperature. In addition, by tuning the spin direction from an in plane antiferromagnetic state to a canted weak ferromagnetic state, a nontrivial topological phase transition could be induced. Our investigation about magnetic property and nontrivial topological phase transition is very promising for antiferromagnetic spintronics.

Structural, electronic, and magnetic properties of CrTe2

Two-dimensional chromium ditelluride (${\mathrm{CrTe}}_{2}$) is a promising ferromagnetic layered material that exhibits long-range ferromagnetic ordering in the monolayer limit. The formation energies of the different possible structural phases (1T, 1H, 2H) calculated from density functional theory (DFT) show that the 1T phase is the ground state, and the energetic transition barriers between the phases, calculated by the nudged elastic band method, are large, on the order of 0.5 eV. The self-consistent Hubbard $U$ correction parameters are calculated for all the phases of ${\mathrm{CrTe}}_{2}$. The calculated magnetic moment of 1T-${\mathrm{CrTe}}_{2}$ with $\ensuremath{\ge}2$ layers lies in the plane, whereas the magnetic moment of a monolayer is out-of-plane. Band filling and tensile biaxial strain cause the magnetic moment of a monolayer to switch from out-of-plane to in-plane, and compressive biaxial strain in a bilayer causes the magnetic moment to switch from in-plane to out-of-plane. The magnetic anisotropy is shown to originate from the large spin-orbit coupling (SOC) of the Te atoms and the anisotropy of the exchange coupling constants ${J}_{xy}$ and ${J}_{z}$ in an XXZ type Hamiltonian. Renormalized spin wave theory using experimental values for the magnetic anisotropy energy and Curie temperatures provides a range of values for the nearest neighbor exchange coupling.

Deformed spin- 12 square lattice in antiferromagnetic NaZnVOPO4(HPO4)

We report the structural and magnetic properties of a new spin-$\frac12$ antiferromagnet NaZnVOPO$_4$(HPO$_4$) studied via x-ray diffraction, magnetic susceptibility, high-field magnetization, specific heat, and $^{31}$P nuclear magnetic resonance (NMR) measurements, as well as density-functional band-structure calculations. While thermodynamic properties of this compound are well described by the $J_1-J_2$ square-lattice model, \textit{ab initio} calculations suggest a significant deformation of the spin lattice. From fits to the magnetic susceptibility we determine the averaged nearest-neighbor and second-neighbor exchange couplings of $\bar J_1\simeq -1.3$ K and $\bar J_2\simeq 5.6$ K, respectively. Experimental saturation field of 15.3\,T is consistent with these estimates if 20\% spatial anisotropy in $J_1$ is taken into account. Specific heat data signal the onset of a magnetic long-range order at $T_{\rm N} \simeq 2.1$ K, which is further supported by a sharp peak in the NMR spin-lattice relaxation rate. The NMR spectra mark the superposition of two P lines due to two noneqivalent P sites where the broad line with the strong hyperfine coupling and short $T_1$ is identified as the P(1) site located within the magnetic planes, while the narrow line with the weak hyperfine coupling and long $T_1$ is designated as the P(2) site located between the planes.

Robust quantum entanglement and teleportation in the tetrapartite spin-1/2 square clusters: Theoretical study on the effect of a cyclic four-spin exchange

The whole entanglement measure so-called geometric Π4 average of tangles and bipartite entanglement of the antiferromagnetic spin-1/2 XXX Heisenberg model on a tetranuclear square cluster with cyclic four-spin interaction are rigorously examined by the help of thermal negativities. The model comprises two nearest-neighbor exchange couplings J1 and J2 such that J1≫J2. When the cyclic four-spin exchange is zero, the maximum value of whole entanglement Π4 is achieved at low enough temperatures and relatively high magnetic fields (B≈J1). Also, maximum bipartite entanglement between pair spins with exchange coupling J1 is achievable at high temperature and high magnetic field. This quantity remains alive for sufficiently high temperature and high magnetic field values comparable with the relevant exchange coupling J1. A nonzero value of the cyclic four-spin exchange notably enhances the degree of the whole entanglement, while it weakens the bipartite entanglement degree. We demonstrate that the whole entanglement reaches an unconventional minimum at a special parameter region of cyclic four-spin exchange almost ten order of magnitude smaller than J1, where the system is in a quantum antiferromagnetic state. The real complex [Cu4L4(H2O)4](ClO4)4 as a strong antiferromagnetic tetranuclear square compound provides us an experimental representative to estimate the strength of the whole and bipartite entanglements at high enough temperature. It is demonstrated that the entanglement negativities of this complex are yet depend on the considered cyclic four-spin interaction even though its magnitude is significantly smaller than J1. We realize that the possibility of quantum teleportation through a couple of discrete Cu4II complexes would solely happen for the magnetic fields lower than a critical value. It is also demonstrated that enhancing the cyclic four-spin exchange results in diminishing the average fidelity. Moreover, we conclude that the quantum Fisher information investigated for the pair of spins with exchange interaction J1 reaches its minimum close to the critical magnetic field at which a ground-state phase transition occurs from highly entangled state to separable one.

Coexistence of antiferromagnetism and superconductivity in Mn/Nb(110)

We report on the structural and magnetic properties of single and double atomic layers of Mn on a clean and unreconstructed Nb(110) substrate. Low-temperature scanning tunneling spectroscopy measurements reveal a proximity-induced superconducting state in the Mn thin films, which are found to grow pseudomorphically on the Nb surface. Spin-polarized scanning tunneling microscopy measurements reveal a c(2x2) antiferromagnetic order in the Mn layers, with an out-of-plane spin-orientation. First-principles density functional theory calculations confirm the experimentally observed magnetic state, which is understood as the consequence of a strong intra- and inter-layer nearest-neighbor antiferromagnetic exchange coupling. These results are expected to be of importance for the design and the investigation of novel superconducting antiferromagnetic spintronic systems.

Thermal transport properties of Heisenberg antiferromagnet on honeycomb lattice: The effects of anisotropy

We study the thermal conductivity of a spin-1/2 two-dimensional anisotropic antiferromagnet on honeycomb lattice in the presence of antiferromagnetic long range ordering. The conductivity has been studied along zigzag direction. Next nearest neighbor exchange coupling has been added to the model Hamiltonian. The possible effects of spin–orbit coupling are investigated by adding Dzyaloshinskii–Moriya interaction to the model Hamiltonian. Such interaction creates anisotropic effects in the model Hamiltonian. The original antiferromagnetic model hamiltonian is mapped to a bosonic model via linear spin wave theory in the context of Holstein Primakoff transformations. The Green’s function approach is applied to obtain the energy spectrum of quasi-particle excitations responsible for thermal transport. We have found the temperature dependence of static thermal conductivity in the magnetic long range ordering phase for various next nearest neighbor coupling constant and anisotropy parameter. Furthermore we have studied the temperature dependence of the static thermal conductivity for various Dzyaloshinskii-Moriya interaction strength and next nearest neighbor coupling constants. Our results show that each curve for temperature dependence of thermal conductivity includes a peak so that the height of peak reduces with magnetic field however the temperature positions of peaks is independent of magnetic field value. Moreover the increase of anisotropic parameter leads to enhance thermal conductivity values. Also the next nearest neighbor coupling constant improves thermal conductivity of localized electrons on honeycomb structure . The exponential behavior for temperature dependence of thermal conductivity at low temperatures is the novel feature in thermal conductivity results. • Study of thermal conductivity of honeycomb lattics. • Study of anisotropic parameter on conductivity. • Study of magnetic field effects on conductivity.

MoP3SiO11 : A 4d3 honeycomb antiferromagnet with disconnected octahedra

We report the crystal structure and magnetic behavior of the $4d^3$ spin-$\frac32$ silicophosphate MoP$_3$SiO$_{11}$ studied by high-resolution synchrotron x-ray diffraction, neutron diffraction, thermodynamic measurements, and ab initio band-structure calculations. Our data revise the crystallographic symmetry of this compound and establish its rhombohedral space group ($R\bar 3c$) along with the geometrically perfect honeycomb lattice of the Mo$^{3+}$ ions residing in disconnected MoO$_6$ octahedra. Long-range antiferromagnetic order with the propagation vector $\mathbf k=0$ observed below $T_N=6.8$ K is a combined effect of the nearest-neighbor in-plane exchange coupling $J\simeq 2.6$ K, easy-plane single-ion anisotropy $D\simeq 2.2 $ K, and a weak interlayer coupling $J_c\simeq 0.8$ K. The 12% reduction in the ordered magnetic moment of the Mo$^{3+}$ ions and the magnon gap of $\Delta\simeq 7$ K induced by the single-ion anisotropy further illustrate the impact of spin-orbit coupling on the magnetism. Our analysis puts forward single-ion anisotropy as an important ingredient of $4d^3$ honeycomb antiferromagnets despite their nominally quenched orbital moment.

Thermal Conductivity of Localized Electrons on Magnetic Ordered Monolayer Graphene

We study the thermal conductivity of a spin-1/2 two-dimensional anisotropic antiferromagnet on honeycomb lattice in the presence of antiferromagnetic long range ordering. The conductivity has been studied along zigzag direction. Next nearest neighbor exchange coupling has been added to the model Hamiltonian. The possible effects of spin-orbit coupling are investigated by adding Dzyaloshinskii-Moriya interaction to the model Hamiltonian. Such interaction creates anisotropic effects in the model Hamiltonian. The original antiferromagnetic model hamiltonian is mapped to a bosonic model via linear spin wave theory in the context of Holstein Primakoff transformations. The Green's function approach is applied to obtain the energy spectrum of quasi-particle excitations responsible for thermal transport. We have found the temperature dependence of static thermal conductivity in the magnetic long range ordering phase for various next nearest neighbor coupling constant and anisotropy parameter. Furthermore we have studied the temperature dependence of the static thermal conductivity for various Dzyaloshinskii-Moriya interaction strength and next nearest neighbor coupling constants.

Competing electronic orders on a heavily doped honeycomb lattice with enhanced exchange coupling

Motivated by recent discovery of correlated insulating and superconducting behavior in twisted bilayer graphene, we revisit graphene's honeycomb lattice doped close to the van Hove singularity, using the truncated unity functional renormalization group approach. We consider an extended Hubbard model on the honeycomb lattice including on-site and nearest-neighbor Coulomb repulsions, and nearest-neighbor ferromagnetic exchange and pair hopping interactions. By varying the strength of the nearest-neighbor exchange coupling and Coulomb repulsion as free parameters, we present rich ground-state phase diagrams which contain the spin-triplet $f$-wave and spin-singlet chiral $d$-wave superconducting phases, the commensurate and incommensurate spin- and charge-density-wave phases, and the ferromagnetic phase. In the absence of the exchange coupling and for small value of the nearest-neighbor repulsion, the four-sublattice spin-density-wave phase is generated right around the van Hove filling, while the chiral $d$-wave superconductivity emerges slightly away from it. Surprisingly, the chiral $d$-wave superconductivity is strongly suppressed by weak nearest-neighbor exchange coupling in our calculations. We argue that this suppression might be one of the reasons why the chiral superconductivity proposed for doped graphene has not yet been observed experimentally.

Magnetically induced metal-insulator transition in Pb2CaOsO6

We report on the structural, magnetic, and electronic properties of two new double-perovskites synthesized under high pressure; Pb2CaOsO6 and Pb2ZnOsO6. Upon cooling below 80 K, Pb2CaOsO6 simultaneously undergoes a metal--insulator transition and develops antiferromagnetic order. Pb2ZnOsO6, on the other hand, remains a paramagnetic metal down to 2 K. The key difference between the two compounds lies in their crystal structure. The Os atoms in Pb2ZnOsO6 are arranged on an approximately face-centred cubic lattice with strong antiferromagnetic nearest-neighbor exchange couplings. The geometrical frustration inherent to this lattice prevents magnetic order from forming down to the lowest temperatures. In contrast, the unit cell of Pb2CaOsO6 is heavily distorted up to at least 500 K, including antiferroelectric-like displacements of the Pb and O atoms despite metallic conductivity above 80 K. This distortion relieves the magnetic frustration, facilitating magnetic order which in turn drives the metal--insulator transition. Our results suggest that the phase transition in Pb2CaOsO6 is spin-driven, and could be a rare example of a Slater transition.

Extremely well isolated two-dimensional spin- 12 antiferromagnetic Heisenberg layers with a small exchange coupling in the molecular-based magnet CuPOF

We report on a comprehensive characterization of the newly synthesized Cu$^{2+}$-based molecular magnet [Cu(pz)$_2$(2-HOpy)$_2$](PF$_6$)$_2$ (CuPOF), where pz = C$_4$H$_4$N$_2$ and 2-HOpy = C$_5$H$_4$NHO. From a comparison of theoretical modeling to results of bulk magnetometry, specific heat, $\mu^+$SR, ESR, and NMR spectroscopy, this material is determined as an excellent realization of the 2D square-lattice $S=1/2$ antiferromagnetic Heisenberg model with a moderate intraplane nearest-neighbor exchange coupling of $J/k_\mathrm{B} = 6.80(5)$ K, and an extremely small interlayer interaction of about 1 mK. At zero field, the bulk magnetometry reveals a temperature-driven crossover of spin correlations from isotropic to $XY$ type, caused by the presence of a weak intrinsic easy-plane anisotropy. A transition to long-range order, driven by the low-temperature $XY$ anisotropy under the influence of the interlayer coupling, occurs at $T_\mathrm{N} = 1.38(2)$ K, as revealed by $\mu^+$SR. In applied magnetic fields, our $^1$H-NMR data reveal a strong increase of the magnetic anisotropy, manifested by a pronounced enhancement of the transition temperature to commensurate long-range order at $T_\mathrm{N} =2.8$ K and 7 T.

Magnetic properties of the one-dimensional S=32 Heisenberg antiferromagnetic spin-chain compound Na2Mn3O7

We have performed magnetic susceptibility $(\ensuremath{\chi})$, heat capacity $({C}_{p})$, and nuclear magnetic resonance (NMR) measurements on the layered compound ${\mathrm{Na}}_{2}{\mathrm{Mn}}_{3}{\mathrm{O}}_{7}$ in which the magnetic ${\mathrm{Mn}}^{4+}$ ions constitute a maple-leaf lattice. At high temperature, $\ensuremath{\chi}$ follows the Curie-Weiss (CW) law with CW temperature as high as 152 K and ${\mathrm{Mn}}^{4+}$ ions are found to be in the high-spin state $(S=\frac{3}{2})$. With decreasing temperature, $\ensuremath{\chi}$ passes through a broad maximum at 115 K and decreases rapidly at low temperature. A similar broad peak appears at around 62 K in the magnetic contribution to heat capacity $({C}_{m})$. The temperature dependence of $\ensuremath{\chi}$ and ${C}_{m}$ are the characteristics of an one-dimensional Heisenberg antiferromagnetic spin chain with a nonmagnetic ground state. The obtained values of spin-gap energy $({\mathrm{\ensuremath{\Delta}}}_{s})$ and nearest-neighbor antiferromagnetic exchange coupling $(J)$ are 28 and 27 K, respectively. There is no evidence of a long-range magnetic ordering down to 2 K in heat capacity data, in agreement with spin-singlet ground state. These observations are further corroborated by $^{23}\mathrm{Na}$ NMR studies through measurements of both the NMR shift and the spin-lattice relaxation rate.

Dynamical and static spin conductivities in the Heisenberg model on honeycomb lattice: The effects of magnetic long range ordering

We have studied both dynamical and static spin conductivities of Heisenberg antiferromagnet on honeycomb lattice in the presence of a magnetic long range ordering. The effects of spatial anisotropy as weak Dzyaloshinskii–Moriya interaction and next nearest neighbor exchange coupling on the behaviors of conductivities are discussed. A sublattice antiferromagnetic long range ordering has been considered for localized electrons on honeycomb lattice structure. Using Holstein–Primakoff bosonic transformations, the behaviors of spin transport properties have been studied by means of excitation spectrum of mapped bosonic gas. We have found the temperature dependence of static spin conductivity in the field induced gapped spin-polarized phase for various Dzyaloshinskii–Moriya interaction strengths. Furthermore we have studied the frequency dependence of dynamical spin conductivity for various Dzyaloshinskii–Moriya interaction strengths and different next nearest neighbor coupling constants. We find that the height of peak in the temperature dependence of static spin conductivity increases upon increasing the anisotropy parameter. The static spin conductivity is found to be monotonically increasing with anisotropy parameter due to increase of the energy gap in the excitation spectrum. Furthermore we have studied the temperature dependence of the spin conductivity for different next nearest neighbor coupling constants.

Cluster size effects in the magnetic properties of [formula omitted]-[formula omitted] alloys

A spin-1/2 Ising model, defined in the body centered cubic lattice, is used to describe some of the thermodynamic properties of Fep-Alq alloys, with p+q=1. The model assumes, besides the nearest-neighbor exchange coupling, the existence of further next-nearest-neighbor superexchange interactions, where the latter ones depend on the aluminum atoms cluster size. The Ising system so considered is studied by employing Monte Carlo simulations, using a hybrid algorithm consisting of one single-spin Metropolis move together with one single-cluster Wolff algorithm allied, in addition, with single histograms procedures and finite-size scaling techniques. Quite good fits to the experimental results of the ordering critical temperature, as a function of Al concentration in the range 0⩽q<0.7, are obtained and compared to more recent theoretical approaches done on the same alloys.

Spin transport properties of anisotropic Heisenberg antiferromagnet on honeycomb lattice in the presence of magnetic field

We have addressed the spin conductivity of a two-dimensional anisotropic antiferromagnet on honeycomb lattice in the presence of a longitudinal magnetic field. A spatial anisotropy in the form of a weak Dzyaloshinskii-Moriya interaction is considered in the model Hamiltonian. Next nearest neighbor exchange coupling has been added to the model Hamiltonian. We have investigated both dynamical and static spin conductivities in terms of the excitation spectrum by means of a hard core bosonic representation. The effects of next-nearest-neighbor coupling and the Dzyaloshinskii-Moriya interaction on the spin transport properties have also been studied via the bosonic model by a Green’s function approach. We have found the temperature dependence of static spin conductivity in the field induced gapped spin-polarized phase for various magnetic field and anisotropy parameters. Furthermore we have studied the magnetic field dependence of the static spin conductivity for various Dzyaloshinskii-Moriya interaction strength and the frequency dependence of the dynamical spin conductivity for various next nearest neighbor coupling constants. We find that the peak in the static spin conductivity moves to higher temperature upon increasing the magnetic field at fixed anisotropy parameter. The static spin conductivity is found to be monotonically decreasing with magnetic field due to increase of the energy gap in the excitation spectrum. Furthermore we have studied the temperature dependence of the spin conductivity for different magnetic field and various next nearest neighbor coupling constants.

Coexisting spin resonance and long-range magnetic order of Eu in EuRbFe4As4

Magnetic excitations and magnetic structure of EuRbFe$_4$As$_4$ were investigated by inelastic neutron scattering (INS), neutron diffraction, and random phase approximation (RPA) calculations. Below the superconducting transition temperature $T_\text{c}=36.5$~K, the INS spectra exhibit the neutron spin resonances at $Q_\text{res}=1.27(2)$~$\text{\AA}^{-1}$ and $1.79(3)$~$\text{\AA}^{-1}$. They correspond to the $\mathbf{Q}=(0.5,0.5,1)$ and $(0.5,0.5,3)$ nesting wave vectors, showing three dimensional nature of the band structure. The characteristic energy of the neutron spin resonance is $E_\text{res}=17.7(3)$~meV corresponding to $5.7(1)k_\text{B}T_\text{c}$. Observation of the neutron spin resonance mode and our RPA calculations in conjunction with the recent optical conductivity measurements are indicative of the $s_\pm$ superconducting pairing symmetry in EuRbFe$_4$As$_4$. In addition to the neutron spin resonance mode, upon decreasing temperature below the magnetic transition temperature $T_\text{N}=15$~K, the spin wave excitation originating in the long-range magnetic order of the Eu sublattice was observed in the low-energy inelastic channel. Single-crystal neutron diffraction measurements demonstrate that the magnetic propagation vector of the Eu sublattice is $\mathbf{k}=(0, 0, 0.25)$, representing the three-dimensional antiferromagnetic order. Linear spin wave calculations assuming the obtained magnetic structure with the intra- and inter-plane nearest neighbor exchange couplings of $J_1/k_\text{B}=-1.31$~K and $J_c/k_\text{B}=0.08$~K can reproduce quantitatively the observed spin wave excitation. Our results show that superconductivity and long-range magnetic order of Eu coexist in EuRbFe$_4$As$_4$ whereas the coupling between them is rather weak.