Triangular-lattice Heisenberg Antiferromagnet Research Articles

Gapped quantum spin liquid in a triangular-lattice Ising-type antiferromagnet PrMgAl11O19

In the search of quantum spin liquid (QSL), spin-1/2 triangular-lattice Heisenberg antiferromagnets have always been viewed as fertile soils. Despite the true magnetically ordered ground state, anisotropy has been considered to play a significant role in stabilizing a QSL state. However, the nature and ground state of the most anisotropic case, the triangular-lattice Ising antiferromagnet (TLIAF), remain elusive and controversial. Here, we report specific-heat and thermal-conductivity measurements on a newly discovered Ising-type QSL candidate PrMgAl11O19. At zero field, the magnetic specific heat shows a quadratic temperature dependence. On the contrary, no direct positive magnetic contribution to thermal conductivity is detected, ruling out the presence of mobile gapless fermionic excitations. Further analysis of phonon thermal conductivity reveals that the phonons are strongly scattered by thermally activated magnetic excitations out of a gap, which exhibits a linear dependence with magnetic field. These results demonstrate that the spin-1/2 TLIAF PrMgAl11O19 has a gapped Z2 QSL ground state. Published by the American Physical Society 2024

Complete field-induced spectral response of the spin-1/2 triangular-lattice antiferromagnet CsYbSe2

Fifty years after Anderson’s resonating valence-bond proposal, the spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLHAF) remains the ultimate platform to explore highly entangled quantum spin states in proximity to magnetic order. Yb-based delafossites are ideal candidate TLHAF materials, which allow experimental access to the full range of applied in-plane magnetic fields. We perform a systematic neutron scattering study of CsYbSe2, first proving the Heisenberg character of the interactions and quantifying the second-neighbor coupling. We then measure the complex evolution of the excitation spectrum, finding extensive continuum features near the 120°-ordered state, throughout the 1/3-magnetization plateau and beyond this up to saturation. We perform cylinder matrix-product-state (MPS) calculations to obtain an unbiased numerical benchmark for the TLHAF and spectacular agreement with the experimental spectra. The measured and calculated longitudinal spectral functions reflect the role of multi-magnon bound and scattering states. These results provide valuable insight into unconventional field-induced spin excitations in frustrated quantum materials.

Field-orientation dependence of quantum phase transitions in the S=12 triangular-lattice antiferromagnet Ba3CoSb2O9

Ba$_3$CoSb$_2$O$_9$ approximates the two-dimensional spin-1/2 triangular-lattice Heisenberg antiferromagnet. This compound displays magnetic-field-induced quantum phase transitions, including the 1/3-magnetization-plateau, but its magnetization processes for the magnetic field $H$ parallel and perpendicular to the $c$ axis are different due to the weak easy-plane anisotropy and the weak interlayer antiferromagnetic exchange interaction. To elucidate how the quantum phase transitions change between these two field directions, we measured the field-angle dependence of the magnetization process in Ba$_3$CoSb$_2$O$_9$ using pulsed high magnetic fields. We compared obtained magnetic field-field angle phase diagram with those obtained by the large-size cluster mean-field method combined with a scaling scheme and the semiclassical theory. We also found a narrow 1/3-magnetization plateau and a high-field transition with a small magnetization jump for $H\,{\parallel}\,c$, not observed in the previous studies.

Ground state of the S=12 triangular lattice Heisenberg-like antiferromagnet Ba3CoSb2O9 in an out-of-plane magnetic field

Spin-$1/2$ triangular lattice Heisenberg antiferromagnet has been accepted as an ideal system for quantum magnetism studies and quantum simulations. This system, for which the classical ground state degeneracy is lifted by quantum fluctuations, exhibits a series of novel spin structures for a field applied in-plane and out-of-plane. It has been found that both anisotropy and interlayer interaction play an important role in the stabilization of the spin configurations in a magnetic field. Conversely, the phase transitions and spin-state evolution in a field along various orientations can provide a deep insight into physics of the triangular lattice Heisenberg antiferromagnet system. While the quantum magnetization process in an in-plane field has been studied extensively, the ground state evolution in the field along the $c$ axis requires further investigation. Here we performed high field magnetization and neutron scattering investigations on a model system of spin-$1/2$ triangular lattice Heisenberg antiferromagnet ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$ with field along $c$ axis and with a small offset angle. For $H\ensuremath{\parallel}c$, the magnetization reveals a narrow plateau prompting a UUD-like phase, which could be suppressed by tilting the field away from the $c$ axis. From the neutron data, a phase transition ${\ensuremath{\mu}}_{0}{H}_{c1}\ensuremath{\sim}12$ T is detected and interpreted as a transition from an umbrella to a coplanar phase. Around about 22.5 T (${\ensuremath{\mu}}_{0}{H}_{c2}$) for $H\ensuremath{\parallel}c$, another transition is observed which might be attributed to a transition between the coplanar $V$ and ${V}^{\ensuremath{'}}$ phases based on a comparison with the calculations and previous results. Theoretical calculations using the large-size cluster mean-field plus scaling method predicts a similar phase evolution as the previous semiclassical analysis, and agree with experiment well. The discrepancies between theory and experiment are also discussed, suggesting the physics of a triangular lattice Heisenberg antiferromagnet in a field along $c$ axis has not been fully unraveled.

Spin Dynamics Simulation of the Z2-vortex Fluctuations

Motivated by the recent quasi-elastic neutron scattering experiment, we extend the spin-dynamics simulation on the triangular-lattice Heisenberg antiferromagnet, to observe a sharp central peak of its energy width $\sim 0.001J$ ($J$ the exchange coupling) of the $Z_2$-vortex origin, consistently with the experiment.

Structures of magnetic excitations in the spin- 12 kagome-lattice antiferromagnets Cs2Cu3SnF12 and Rb2Cu3SnF12

We show the structures of magnetic excitations in spin-1/2 kagome-lattice antiferromagnets Cs$_2$Cu$_3$SnF$_{12}$ and Rb$_2$Cu$_3$SnF$_{12}$ investigated by inelastic neutron scattering in wide energy and momentum ranges. For Cs$_2$Cu$_3$SnF$_{12}$, four single-magnon excitation modes were observed. Low-energy three modes are assigned to be transverse modes and the high-energy fourth mode is suggested to be an amplitude mode. It was found that the broad excitation continuum without a marked structure spreads in a wide energy range from $0.15J$ to approximately $2.5J$ in contrast to the clearly structured excitation continuum observed in the spin-1/2 triangular-lattice Heisenberg antiferromagnet. These findings strongly suggest spinon excitations as elementary excitations in Cs$_2$Cu$_3$SnF$_{12}$. In Rb$_2$Cu$_3$SnF$_{12}$, singlet-triplet excitations from the pinwheel VBS state and their ghost modes caused by the enlargement of the chemical unit cell were clearly confirmed. It was found that the excitation continuum is structured in the low-energy region approximately below $J_{\mathrm{avg}}$ and the almost structureless high-energy excitation continuum extends to approximately $2.6J_{\mathrm{avg}}$. The characteristics of the high-energy excitation continuum are common to both Cs$_2$Cu$_3$SnF$_{12}$ and Rb$_2$Cu$_3$SnF$_{12}$, irrespective of their ground states. The experimental results strongly suggest that the spin liquid component remains in the ground state as quantum fluctuations in Cs$_2$Cu$_3$SnF$_{12}$ and Rb$_2$Cu$_3$SnF$_{12}$.

Universal fluctuating regime in triangular chromate antiferromagnets

We report x-ray diffraction, magnetic susceptibility, heat capacity, $^{1}\mathrm{H}$ nuclear magnetic resonance (NMR), and muon spin relaxation $(\ensuremath{\mu}\mathrm{SR})$ measurements, as well as density-functional band-structure calculations for the frustrated $S=3/2$ triangular lattice Heisenberg antiferromagnet (TLHAF) $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{HCrO}}_{2}$ (trigonal, space group: $R\overline{3}m$). This compound undergoes a clear magnetic transition at ${T}_{\mathrm{N}}\ensuremath{\simeq}22.5$ K, as seen from the drop in the muon paramagnetic fraction and concurrent anomalies in the magnetic susceptibility and specific heat capacity. Local probes (NMR and $\ensuremath{\mu}\mathrm{SR}$) reveal a broad regime with slow fluctuations down to $0.7\phantom{\rule{0.16em}{0ex}}{T}_{\mathrm{N}}$, this temperature corresponding to the maximum in the $\ensuremath{\mu}\mathrm{SR}$ relaxation rate and in the NMR wipe-out. From the comparison with ${\mathrm{NaCrO}}_{2}$ and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{KCrO}}_{2}$, the fluctuating regime and slow dynamics below ${T}_{\mathrm{N}}$ appear to be hallmarks of the TLHAF with $ABC$ stacking. We discuss the role of interlayer frustration, which may have impacted recent spin-liquid candidates with triangular geometry.

Resonating valence bond theory of anomalous spin dynamics of spin- 12 triangular lattice Heisenberg antiferromagnet and its application to Ba3CoSb2O9

Although it is well accepted that the spin-$\frac{1}{2}$ triangular lattice Heisenberg antiferromagnet(TLHAF) has a long range ordered ground state, a thorough understanding of its spin dynamics is still missing. While the linear spin wave theory(LSWT) predicts three branches of magnon mode in the magnetic Brillouin zone(MBZ), the 1/S expansion at the next order is found to break down in a large portion of the MBZ centered around the M point, leaving the fate of the magnon modes there undecided. Recent neutron scattering measurement on Ba$_3$CoSb$_2$O$_9$, an ideal realization of the spin-$\frac{1}{2}$ TLHAF, provides a surprising answer to this issue. Two, rather than three branches of magnon mode are observed around the M point, whose dispersion are strongly renormalized with respect to the LSWT prediction and exhibit pronounced roton-like minimum at the M point. This is accompanied by a strong spin fluctuation continuum at higher energy, inside which two strong and broad spectral peaks of unknown origin are observed. In this work, we propose a simple picture for these spectral anomalies by invoking the resonating valence bond(RVB) physics in the description of the ground state of the system. We find that the roton-like minimum in the magnon dispersion can be explained by the coupling between the collective spin fluctuation and the continuum of Dirac spinon excitation moving in a $\pi$-flux background. We also propose that the two broad peaks in the continuum can be understood respectively as the Landau damped third magnon mode and the Landau damped longitudinal mode. Such a picture can be verified by studying the polarization character of the various spectral features.

Spin-wave study of entanglement and Rényi entropy for coplanar and collinear magnetic orders in two-dimensional quantum Heisenberg antiferromagnets

We use modified linear spin-wave theory (MLSWT) to study ground-state entanglement for a length-$L$ line subsystem in $L\times L$ square- and triangular-lattice quantum Heisenberg antiferromagnets with coplanar spiral magnetic order with ordering vector $\mathbf{Q}=(q,q)$ and $N_G=3$ Goldstone modes, except if $q=\pi$ (collinear order, $N_G=2$). Generalizing earlier MLSWT results for $q=\pi$ to commensurate spiral order with $s\geq 3$ sublattices ($q=2\pi r/s$ with $r$ and $s$ coprime), we find analytically for large $L$ a universal and $n$-independent subleading term $(N_G/2)\ln L$ in the R\'{e}nyi entropy $S_n$, associated with $L^{1/2}$ scaling of $\lambda_0$ and $\lambda_{\pm q}$, with $\lambda_0\neq \lambda_{\pm q}$ for spiral order; here $\{\lambda_{k_y}\}$ are the $L$ mode occupation numbers of the entanglement Hamiltonian. The term $(3/2)\ln L$ in $S_n$ agrees with a nonlinear sigma model (NLSM) study of $s=3$ spiral order ($q=2\pi/3$). These and other properties of $S_n$ and $\lambda_{k_y}$ are explored numerically for an anisotropic nearest-neighbor triangular-lattice model for which $q$ varies in the spiral phase.

Large- S limit of the large- N theory for the triangular antiferromagnet

Large-$S$ and large-$N$ theories (spin value $S$ and spinor component number $N$) are complementary, and sometimes conflicting, approaches to quantum magnetism. While large-$S$ spin-wave theory captures the correct semiclassical behavior, large-$N$ theories, on the other hand, emphasize the quantumness of spin fluctuations. In order to evaluate the possibility of the non-trivial recovery of the semiclassical magnetic excitations within a large-$N$ approach, we compute the large-$S$ limit of the dynamic spin structure of the triangular lattice Heisenberg antiferromagnet within a Schwinger boson spin representation. We demonstrate that, only after the incorporation of Gaussian ($1/N$) corrections to the saddle-point ($N=\infty$) approximation, we are able to exactly reproduce the linear spin wave theory results in the large-$S$ limit. The key observation is that the effect of $1/N$ corrections is to cancel out exactly the main contribution of the saddle-point solution; while the collective modes (magnons) consist of two spinon bound states arising from the poles of the RPA propagator. This result implies that it is essential to consider the interaction of the spinons with the emergent gauge fields and that the magnon dispersion relation should not be identified with that of the saddle-point spinons.

Microscopic evidence of a quantum magnetization process in the S=12 triangular-lattice Heisenberg-like antiferromagnet Ba3CoSb2O9

We report results of the neutron scattering investigations of frustrated quantum magnet ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$ in magnetic fields up to 25.9 T. Contrary to other materials, ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$ exhibits properties typical of an ideal $S=\frac{1}{2}$ triangular lattice Heisenberg antiferromagnet, making it a perfect model system for testing theoretical predictions. In this work, we looked into the magnetization process in ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$ on a microscopic scale with a magnetic field applied in plane. A sequence of magnetic phase transitions, including the new high-field phase at 22.5 T reported recently has been followed at low temperatures as a function of field and modeled using the large-size cluster mean-field plus scaling method. Showing good agreement with the model, our results bridge the theory and the experiment providing microscopic information about the high-field spin ordering in $S=\frac{1}{2}$ triangular lattice Heisenberg-like antiferromagnet ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$.

Avoided quasiparticle decay from strong quantum interactions

Quantum states of matter---such as solids, magnets and topological phases---typically exhibit collective excitations---phonons, magnons, anyons. These involve the motion of many particles in the system, yet, remarkably, act like a single emergent entity---a quasiparticle. Known to be long-lived at the lowest energies, common wisdom says that quasiparticles become unstable when they encounter the inevitable continuum of many-particle excited states at high energies. Whilst correct for weak interactions, we show that this is far from the whole story: strong interactions generically stabilise quasiparticles by pushing them out of the continuum. This general mechanism is straightforwardly illustrated in an exactly solvable model. Using state-of-the-art numerics, we find it at work also in the spin-$\frac{1}{2}$ triangular lattice Heisenberg antiferromagnet (TLHAF) near the isotropic point---this is surprising given the common expectation of magnon decay in this paradigmatic frustrated magnet. Turning to existing experimental data, we identify the detailed phenomenology of avoided decay in the TLHAF material Ba$_3$CoSb$_2$O$_9$, and even in liquid helium---one of the earliest instances of quasiparticle decay. Our work unifies various phenomena above the universal low-energy regime in a comprehensive description. This broadens our window of understanding of many-body excitations, and provides a new perspective for controlling and stabilising quantum matter in the strongly-interacting regime.

Non-coplanar Model States in Quantum Magnetism Applications of the High-Order Coupled Cluster Method

Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases frac{1}{2} le S le 5 . For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Delta ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S le 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.

Two-temperature scales in the triangular-lattice Heisenberg antiferromagnet

The anomalous thermodynamic properties of the paradigmatic frustrated spin-1/2 triangular lattice Heisenberg antiferromagnet (TLH) has remained an open topic of research over decades, both experimentally and theoretically. Here we further the theoretical understanding based on the recently developed, powerful exponential tensor renormalization group (XTRG) method on cylinders and stripes in a quasi one-dimensional (1D) setup, as well as a tensor product operator approach directly in 2D. The observed thermal properties of the TLH are in excellent agreement with two recent experimental measurements on the virtually ideal TLH material Ba$_8$CoNb$_6$O$_{24}$. Remarkably, our numerical simulations reveal two crossover temperature scales, at $T_l/J \sim 0.20$ and $T_h/J\sim 0.55$, with $J$ the Heisenberg exchange coupling, which are also confirmed by a more careful inspection of the experimental data. We propose that in the intermediate regime between the low-temperature scale $T_l$ and the higher one $T_h$, the gapped "roton-like" excitations are activated with a strong chiral component and a large contribution to thermal entropies, which suppress the incipient 120$^\circ$ order that emerges for temperatures below $T_l$.

Two-step melting of three-sublattice order in S = 1 easy-axis triangular lattice antiferromagnets

We consider $S=1$ triangular lattice Heisenberg antiferromagnets with a strong single-ion anisotropy $D$ that dominates over the nearest-neighbour antiferromagnetic exchange $J$. In this limit of small $J/D$, we study low temperature ($T \sim J \ll D$) properties of such magnets by employing a low-energy description in terms of hard-core bosons with nearest neighbour repulsion $V \approx 4J + J^2/D$ and nearest neighbour unfrustrated hopping $t \approx J^2/2D$. Using a cluster Stochastic Series Expansion (SSE) algorithm to perform sign-problem-free quantum Monte Carlo (QMC) simulations of this effective model, we establish that the ground-state three-sublattice order of the easy-axis spin-density $S^z(\vec{r})$ melts in zero field ($B=0$) in a {\em two-step} manner via an intermediate temperature phase characterized by power-law three-sublattice order with a temperature dependent exponent $\eta(T) \in [\frac{1}{9}, \frac{1}{4}]$. For $\eta(T) < \frac{2}{9}$ in this phase, we find that the uniform easy-axis susceptibility of an $L \times L$ sample diverges as $\chi_L \sim L^{2-9 \eta}$ at $B=0$, consistent with a recent prediction that the thermodynamic susceptibility to a uniform field $B$ along the easy axis diverges at small $B$ as $\chi_{\rm easy-axis}(B) \sim B^{-\frac{4-18\eta}{4-9\eta}}$ in this regime.

Spin-1/2 Triangular-Lattice Heisenberg Antiferromagnet with \(\sqrt{3} \times \sqrt{3} \)-Type Distortion — Behavior around the Boundaries of the Intermediate Phase

The S = 1/2 triangular-lattice Heisenberg antiferromagnet with distortion is investigated by the numerical-diagonalization method. The examined distortion type is \(\sqrt{3} \times \sqrt{3} \). We study the case when the distortion connects the undistorted triangular lattice and the dice lattice. For the intermediate phase reported previously in this system, we obtain results of the boundaries of the intermediate phase for a larger system than those in the previous report and examine the system size dependence of the boundaries in detail. We also report the specific heat of this system, which shows a marked peak structure related to the appearance of the intermediate state.

Spontaneous Magnetization of the Spin-1/2 Heisenberg Antiferromagnet on the Triangular Lattice with a Distortion

The spin-1/2 triangular-lattice Heisenberg antiferromagnet with a -type distortion is studied by the numerical-diagonalization method. We examine this model between the two cases, one is the undistorted triangular lattice and the other is the model on the honeycomb lattice with isolated spins. When the distortion is controlled, we find a nontrivial region where the ground states shows spontaneous magnetization; the magnitude increases gradually as the distortion is larger and is smaller than one third of the saturated magnetization.

Magnetization Process of the Spin-1/2 Triangular-Lattice Heisenberg Antiferromagnet with Next-Nearest-Neighbor Interactions — Plateau or Nonplateau —

An $S=1/2$ triangular-lattice Heisenberg antiferromagnet with next-nearest-neighbor interactions is investigated under a magnetic field by the numerical-diagonalization method. It is known that, in both cases of weak and strong next-nearest-neighbor interactions, this system reveals a magnetization plateau at one-third of the saturated magnetization. We examine the stability of this magnetization plateau when the amplitude of next-nearest-neighbor interactions is varied. We find that a nonplateau region appears between the plateau phases in the cases of weak and strong next-nearest-neighbor interactions.

Gapless spin excitations in the [formula omitted] Kagome- and triangular-lattice Heisenberg antiferromagnets

The S=1/2 kagome- and triangular-lattice Heisenberg antiferromagnets are investigated using the numerical exact diagonalization and the finite-size scaling analysis. The behaviour of the field derivative at zero magnetization is examined for both systems. The present result indicates that the spin excitation is gapless for each system.

Collective and local excitations in Ba2CoTeO6 : A composite system of a spin-1/2 triangular-lattice Heisenberg antiferromagnet and a honeycomb-lattice J1−J2 Ising antiferromagnet

We report the results of multifrequency high-magnetic-field electron-spin resonance (ESR) measurements on the highly frustrated antiferromagnet ${\mathrm{Ba}}_{2}{\mathrm{CoTeO}}_{6}$. This compound is magnetically composed of two subsystems A and B, which are described as a spin-1/2 triangular-lattice Heisenberg antiferromagnet and a honeycomb-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Ising antiferromagnet, respectively. ${\mathrm{Ba}}_{2}{\mathrm{CoTeO}}_{6}$ undergoes successive magnetic phase transitions at ${T}_{\mathrm{N}1}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}12.0$ K and ${T}_{\mathrm{N}2}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}3.0$ K. For a magnetic field $H$ parallel to the $c$ axis, subsystem B exhibits successive metamagnetic transitions with magnetization plateaus at one-third and one-half of the saturation magnetization. Below ${T}_{\mathrm{N}2}$, we observed collective ESR modes for $H\phantom{\rule{0.16em}{0ex}}\ensuremath{\parallel}\phantom{\rule{0.16em}{0ex}}c$, which are characteristic of a triangular-lattice Heisenberg antiferromagnet with weak easy-plane anisotropy. We also observed a local excitation mode, which can be assigned as a single flip of the Ising-like spin of subsystem B. From a detailed analysis of the collective and local ESR modes, combined with the magnetization process, we determined the magnetic parameters of subsystems A and B, and confirmed that the two subsystems are almost decoupled.