We have investigated the field-angle behaviors of magnetic excitations under an in-plane magnetic field for proximate Kitaev systems. By employing the exact diagonalization method in conjunction with the linear spin wave theory, we have demonstrated that the magnetic excitation gap in the polarized phase is determined by the magnon excitation at $M$ points and has a strong anisotropy with respect to the field direction in the vicinity of the critical field limit. The specific heat from this magnon excitation bears qualitatively the same anisotropic behaviors as expected one for the non-Abelian spin liquid phase in the Kitaev model and experimentally observed one of the intermediate phases in $α$-RuCl$_3$.
We study the stability of Kitaev quasiparticles in the presence of a perturbing Heisenberg interaction as a Fock space localization phenomenon. We identify parameter regimes where Kitaev states are localized, fractal or delocalized in the Fock space of exact eigenstates, with the first two implying quasiparticle stability. Finite temperature calculations show that a vison gap, and a nonzero plaquette Wilson loop at low temperatures, both characteristic of the deconfined Kitaev spin liquid phase, persist far into the neighboring phase that has a concomitant stripy spin-density wave (SDW) order.…
In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also possess Heisenberg ($J$) and off-diagonal symmetric ($Γ$) couplings. We investigate the quantum ($S = 1/2$) $K$-$J$-$Γ$ model on the honeycomb lattice by a variational Monte Carlo (VMC) method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic f…
Motivated by the magnetic phase transition of a proximate Kitaev system $α$-RuCl$_3$ in the presence of a magnetic field, we study the simplest but essential quantum spin model with the ferromagnetic nearest neighboring (NN) Kitaev interaction and additional antiferromagnetic third NN Heisenberg interaction. Employing both exact diagonalization and density matrix renormalization group methods, we demonstrate that the model shows the magnetic phase transition from the zigzag order phase to the spin polarized phase through an intermediate phase in both cases when an in-plane magnetic field is ap…
Magnetic moments arranged at the corners of a honeycomb lattice are predicted to form a novel state of matter, Kitaev quantum spin liquid, under the influence of frustration effects between bond-dependent Ising interactions. Some layered honeycomb iridates and related materials, such as Na$_{2}$IrO$_{3}$ and $α$-RuCl$_{3}$, are proximate to Kitaev quantum spin liquid, but bosonic spin-wave excitations associated with undesirable antiferromagnetic long-range order mask the inherent properties of Kitaev Hamiltonian. Here, we use $^{63}$Cu nuclear quadrupole resonance to uncover the low energy sp…
$α$-RuCl$_{3}$ is a major candidate for the realization of the Kitaev quantum spin liquid, but its zigzag antiferromagnetic order at low temperatures indicates deviations from the Kitaev model. We have quantified the spin Hamiltonian of $α$-RuCl$_{3}$ by a resonant inelastic x-ray scattering study at the Ru $L_{3}$ absorption edge. In the paramagnetic state, the quasi-elastic intensity of magnetic excitations has a broad maximum around the zone center without any local maxima at the zigzag magnetic Bragg wavevectors. This finding implies that the zigzag order is fragile and readily destabilize…
The Kitaev quantum spin liquid epitomizes an entangled topological state, for which two flavors of fractionalized low-energy excitations are predicted: the itinerant Majorana fermion and the Z2 gauge flux. Detection of these excitations remains challenging, because of their fractional quantum numbers and non-locality. It was proposed recently that fingerprints of fractional excitations are encoded in the phonon spectra of Kitaev quantum spin liquids through a novel fractional-excitation-phonon coupling. Here, we uncover this effect in $α$-RuCl3 using inelastic X-ray scattering with meV resolut…
Proximate quantum spin liquids and Majorana continua in magnetically ordered Kitaev magnets
Technical University of Munich（慕尼黑工业大学）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）
Topological spin liquids in two spatial dimensions are stable phases in the presence of a small magnetic field, but may give way to field-induced phenomena at intermediate field strengths. Sandwiched between the low-field spin liquid physics and the high-field spin-polarized phase, the exploration of magnetic phenomena in this intermediate regime however often remains elusive to controlled analytical approaches. Here we numerically study such intermediate-field magnetic phenomena for two representative Kitaev models (on the square-octagon and decorated honeycomb lattice) that exhibit either Ab…
The leading order nonlinear (NL) susceptibility, $χ_3$, in a paramagnet is negative and diverges as $T \rightarrow 0$. This divergence is destroyed when spins correlate and the NL response provides unique insights into magnetic order. Dimensionality, exchange interaction, and preponderance of quantum effects all imprint their signatures in the NL magnetic response. Here, we study the NL susceptibilities in the proximate Kitaev magnet $α$-RuCl$_3$ which differs from the expected antiferromagnetic behavior. For $T< T_c$ = 7.5 K and field $B$ in the ab-plane, we obtain contrasting NL responses in…
We introduce a matrix-product state based method to efficiently obtain dynamical response functions for two-dimensional microscopic Hamiltonians, which we apply to different phases of the Kitaev-Heisenberg model. We find significant broad high energy features beyond spin-wave theory even in the ordered phases proximate to spin liquids. This includes the phase with zig-zag order of the type observed in $α$-RuCl$_3$, where we find high energy features like those seen in inelastic neutron scattering experiments. Our results provide an example of a natural path for proximate spin liquid features t…
Topological states of matter such as quantum spin liquids (QSLs) are of great interest because of their remarkable predicted properties including protection of quantum information and the emergence of Majorana fermions. Such QSLs, however, have proven difficult to identify experimentally. The most promising approach is to study their exotic nature via the wave-vector and intensity dependence of their dynamical response in neutron scattering. A major search has centered on iridate materials which are proposed to realize the celebrated Kitaev model on a honeycomb lattice - a prototypical topolog…
The quasi two-dimensional Mott insulator $α$-RuCl$_3$ is proximate to the sought-after Kitaev quantum spin liquid (QSL). In a layer of $α$-RuCl$_3$ on graphene the dominant Kitaev exchange is further enhanced by strain. Recently, quantum oscillation (QO) measurements of such $α$-RuCl$_3$ / graphene heterostructures showed an anomalous temperature dependence beyond the standard Lifshitz-Kosevich (LK) description. Here, we develop a theory of anomalous QO in an effective Kitaev-Kondo lattice model in which the itinerant electrons of the graphene layer interact with the correlated magnetic layer …
Excitation by light pulses enables the manipulation of phases of quantum condensed matter. Here, we photoexcite high-energy holon-doublon pairs as a way to alter the magnetic free energy landscape of the Kitaev-Heisenberg magnet $α$-RuCl$_3$, with the aim to dynamically stabilize a proximate spin liquid phase. The holon-doublon pair recombination through multimagnon emission is tracked through the time-evolution of the magnetic linear dichroism originating from the competing zigzag spin ordered ground state. A small holon-doublon density suffices to reach a spin disordered state. The phase tra…
In the quest for realizations of quantum spin liquids, the exploration of Kitaev materials - spin-orbit entangled Mott insulators with strong bond-directional exchanges - has taken center stage. However, in these materials the local spin-orbital j=1/2 moments typically show long-range magnetic order at low temperature, thus defying the formation of a spin-liquid ground state. Using resonant inelastic x-ray scattering (RIXS), we here report on a proximate spin liquid regime with clear fingerprints of Kitaev physics in the magnetic excitations of the honeycomb iridates alpha-Li2IrO3 and Na2IrO3.…
We study the exchange interactions and resulting magnetic phases in the honeycomb cobaltates. For a broad range of trigonal crystal fields acting on Co2+ ions, the low-energy pseudospin-1/2 Hamiltonian is dominated by bond-dependent Ising couplings that constitute the Kitaev model. The non-Kitaev terms nearly vanish at small values of trigonal field Δ, resulting in spin liquid ground state. Considering Na3Co2SbO6 as an example, we find that this compound is proximate to a Kitaev spin liquid phase, and can be driven into it by slightly reducing Δby \sim 20 meV, e.g., via strain or pressure cont…
We identify fingerprints of a proximate quantum spin-liquid (QSL), observable by finite-temperature dynamical thermal transport within a minimal version of the idealized Kitaev model on a two-leg ladder, if subjected to inevitably present Heisenberg couplings. Using exact diagonalization and quantum typicality, we uncover (i) an insulator-conductor crossover induced by fracton recombination at infinitesimal Heisenberg coupling, (ii) low- and high-energy signatures of fractons, which survive far off the pure QSL point, and (iii) a non-monotonous current life-time versus Heisenberg couplings. Gu…
Quantum spin liquids are long-range entangled states of matter with emergent gauge fields and fractionalized excitations. While candidate materials, such as the Kitaev honeycomb ruthenate $α$-RuCl$_3$, show magnetic order at low temperatures $T$, here we demonstrate numerically a dynamical crossover from magnon-like behavior at low $T$ and frequencies $ω$ to long-lived fractionalized fermionic quasiparticles at higher $T$ and $ω$. This crossover is akin to the presence of spinon continua in quasi-1D spin chains. It is further shown to go hand in hand with persistent typicality down to very low…
We report two surprising results regarding the nature of the spatial broken symmetries in the two-dimensional (2D), quarter-filled band with strong electron-electron interactions. First, in direct contradiction to the predictions of one-electron theory, we find a coexisting ``bond-order and charge density wave'' (BCDW) insulating ground state in the 2D rectangular lattice for all anisotropies, including the isotropic limit. Second, we find that the BCDW further coexists with a spin-density wave (SDW) in the range of large anisotropy. Further, in contrast to the interacting half-filled band, in…
Frustrated magnets with strong spin-orbit coupling promise to host topological states of matter, with fractionalized excitations and emergent gauge fields. Kitaev's proposal for a honeycomb-lattice Majorana spin liquid has triggered an intense search for experimental realizations, with bond-dependent Ising interaction being the essential building block. A prime candidate is $α$-RuCl$_3$ whose phase diagram in a magnetic field is, however, not understood to date. Here we present conclusive experimental evidence for a novel field-induced ordered phase in $α$-RuCl$_3$, sandwiched between the zigz…
Quantum matter provides an effective vacuum out of which arise emergent particles not corresponding to any experimentally detected elementary particle. Topological quantum materials in particular have become a focus of intense research in part because of the remarkable possibility to realize Majorana fermions, with their potential for new, decoherence-free quantum computing architectures. In this paper we undertake a study on high-quality single crystal of $α-RuCl_3$ which has been identified as a material realizing a proximate Kitaev state, a topological quantum state with magnetic Majorana f…
$α$-RuCl$_3$ is drawing much attention as a promising candidate Kitaev quantum spin liquid. However, despite intensive research efforts, controversy remains about the form of the basic interactions governing the physics of this material. Even the sign of the Kitaev interaction (the bond-dependent anisotropic interaction responsible for Kitaev physics) is still under debate, with conflicting results from theoretical and experimental studies. The significance of the symmetric off-diagonal exchange interaction (referred to as the $Γ$ term) is another contentious question. Here, we present resonan…
We study the ordered moment direction in the extended Kitaev-Heisenberg model relevant to honeycomb lattice magnets with strong spin-orbit coupling. We utilize numerical diagonalization and analyze the exact cluster groundstates using a particular set of spin coherent states, obtaining thereby quantum corrections to the magnetic anisotropy beyond conventional perturbative methods. It is found that the quantum fluctuations strongly modify the moment direction obtained at a classical level, and are thus crucial for a precise quantification of the interactions. The results show that the moment di…
We study the inelastic light scattering response in two- (2D) and three-dimensional (3D) Kitaev spin-liquid models with \ms band structures in the symmetry classes BDI and D leading to protected gapless surface modes. We present a detailed calculation of the resonant Raman/Brillouin scattering vertex relevant to iridate and ruthenate compounds whose low-energy physics is believed to be proximate to these spin-liquid phases. In the symmetry class BDI, we find that while the resonant scattering on thin films can detect the gapless boundary modes of spin liquids, the non-resonant processes do not…
The magnetic field response of the Mott-insulating honeycomb iridate Na$_{2}$IrO$_{3}$ is investigated using torque magnetometry measurements in magnetic fields up to 60 tesla. A peak-dip structure is observed in the torque response at magnetic fields corresponding to an energy scale close to the zigzag ordering ($\approx 15~K$) temperature. Using exact diagonalization calculations, we show that such a distinctive signature in the torque response constrains the effective spin models for these classes of Kitaev materials to ones with dominant ferromagnetic Kitaev interactions, while alternative…
Magnetic properties of the substitution series Ru$_{1-x}$Cr$_x$Cl$_3$ were investigated to determine the evolution from the anisotropic Kitaev magnet $α$-RuCl$_3$ with $J_{\rm eff} = 1/2$ magnetic Ru$^{3+}$ ions to the isotropic Heisenberg magnet CrCl$_3$ with $S = 3/2$ magnetic Cr$^{3+}$ ions. Magnetization measurements on single crystals revealed a reversal of the magnetic anisotropy under doping, which we argue to arise from the competition between anisotropic Kitaev and off-diagonal interactions on the Ru-Ru links and approximately isotropic Cr-Ru and isotropic Cr-Cr interactions. In addit…
We study a quantum spin Kitaev model with zigzag edges to clarify the effects of anisotropy in the exchange couplings on the spin propagation. We simulate the spin and Majorana dynamics triggered by a magnetic pulse, using the real-space time-dependent Majorana mean-field theory. When the anisotropy is small, the dispersion of the itinerant Majorana fermions remains gapless, where the velocity of the spin propagation matches the group velocity of the itinerant Majorana fermions at the nodal points. On the other hand, in the gapped system with a large anisotropy, the spin propagation is strongl…
The Kitaev material $α$-RuCl$_3$ is among the most prominent candidates to host a quantum spin-liquid state endowed with fractionalized excitations. Recent experimental and theoretical investigations have separately revealed the importance of both the magnetoelastic coupling and the magnetic anisotropy, in dependence of the applied magnetic field direction. In this combined theoretical and experimental research, we investigate the anisotropic magnetic and magnetoelastic properties for magnetic fields applied along the main crystallographic axes as well as for fields canted out of the honeycomb…
We have investigated the longitudinal thermal conductivity of $α$-RuCl$_{3}$, the magnetic state of which is considered to be proximate to a Kitaev honeycomb model, along with the spin susceptibility and magnetic specific heat. We found that the temperature dependence of the thermal conductivity exhibits an additional peak around 100 K, which is well above the phonon peak temperature ($\sim$ 50 K). The higher-temperature peak position is comparable to the temperature scale of the Kitaev couplings rather than the Néel temperatures below 15 K. The additional heat conduction was observed for all …
Crossover and Changeover in Spin-1 Kitaev-$Γ$ Chain with Uniaxial Single-ion Anisotropy
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Nankai University（南开大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；Lanzhou University（兰州大学）；Zhejiang University（浙江大学）
We present an augmented parton mean-field theory which (i) reproduces the $exact$ ground state, spectrum, and dynamics of the quantum spin liquid phase of Kitaev's honeycomb model; and (ii) is amenable to the inclusion of integrability breaking terms, allowing a perturbation theory from a controlled starting point. Thus, we exemplarily study dynamical spin correlations of the honeycomb Kitaev quantum spin liquid within the $K-J-Γ$ model which includes Heisenberg and symmetric-anisotropic (pseudo-dipolar) interactions. This allows us to trace changes of the correlations in the regime of slowly …
We study the spin-one Kitaev model on the honeycomb lattice in the presence of single-ion anisotropies. We consider two types of single ion anisotropies: A $D_{111}$ anisotropy which preserves the symmetry between $X$, $Y$, and $Z$ bonds but violates flux conservation and a $D_{100}$ anisotropy that breaks the symmetry between $X$, $Y$, and $Z$ bonds but preserves flux conservation. We use series expansion methods, degenerate perturbation theory, and exact diagonalization to study these systems. Large positive $D_{111}$ anisotropy leads to a simple product ground state with conventional magnon…
Contrasting ground states of quantum magnets with the integer and half-integer spin moments are the manifestation of many-body quantum interference effects. In this work, we investigate the distinct nature of the integer and half-integer spin quantum spin liquids in the framework of the Kitaev's model on the honeycomb lattice. The models with arbitrary spin quantum numbers are not exactly solvable in contrast to the well-known quantum spin liquid solution of the spin-1/2 system. We use the tensor network wavefunctions for the integer and half-integer spin quantum spin liquid states to unveil t…
In recent years, several magnetic Mott insulators with strong spin-orbit couplings were suggested to be proximate to the Kitaev quantum spin liquid (QSL) whose one of the most exciting features is the fractionalization of spin excitations into itinerant Majorana fermions and static $Z_2$ fluxes. Motivated by the emergence of this plethora of 4d and 5d transition metal Kitaev materials and by the fact that some level of disorder is inevitable in real materials, here we study how the Kitaev QSL responds to various forms of disorder, such as vacancies, impurities, and bond randomness. First, we a…
The honeycomb magnet $α-$RuCl$_3$ is a prime candidate material for realizing the Kitaev quantum spin liquid (QSL), but it shows long-range magnetic order at low temperature. Nevertheless, its broad inelastic neutron scattering (INS) response at finite frequency has been interpreted as that of a 'proximate QSL'. A moderate in-plane magnetic field indeed melts the residual zigzag order, giving rise to peculiar intermediate field phases before the high-field polarized state. In INS measurements the low-frequency spin waves disappear, leading to a broad scattering continuum in the field-induced i…
Mott insulators with strong spin-orbit coupling display a strongly anisotropic response to applied magnetic fields. This applies in particular to Kitaev materials, with $α$-RuCl$_3$ and Na$_2$IrO$_3$ representing two important examples. Both show a magnetically ordered zigzag state at low temperatures, and considerable effort has been devoted to properly modeling these systems in order to identify routes towards realizing a quantum spin liquid. Here, we investigate the relevant Heisenberg-Kitaev-$Γ$ model primarily at elevated temperatures, focusing on the characteristic anisotropy between the…
The honeycomb $K-Γ$ model is known to have both deconfined spin liquid and confined phases. We study here a confinement transition between the Kitaev spin liquid and a dimerized phase in the limit of strong bond anisotropy. By partially projecting out Majorana states we are able to map the model onto a model of weakly coupled Ising chains in a transverse field. Within this mapping the ordered Ising phase corresponds to the condensation of $Z_2$ fluxes, or confinement. Our results may improve our understanding of the extensively studied spin liquid candidate material $α$-RuCl$_3$, where $K-Γ$ i…
In the pseudospin-$\frac{1}{2}$ honeycomb Mott insulators with strong spin-orbit coupling, there are two types of bond-dependent exchange interactions, named Kitaev ($K$) and $Γ$, leading to strong frustration. While the ground state of the Kitaev model is a quantum spin liquid with fractionalized excitations, the ground state of the $Γ$ model remains controversial. In particular, the phase diagram of the $KΓ$ model with ferromagnetic $K$ and antiferromagnetic $Γ$ interactions has been intensively studied because of its relevance to candidate materials such as $α$-RuCl$_{3}$. Numerical studies…
Magnetic anisotropy is crucially important for the stabilization of two-dimensional (2D) magnetism, which is rare in nature but highly desirable in spintronics and for advancing fundamental knowledge. Recent works on CrI$_3$ and CrGeTe$_3$ monolayers not only led to observations of the long-time-sought 2D ferromagnetism, but also revealed distinct magnetic anisotropy in the two systems, namely Ising behavior for CrI$_3$ versus Heisenberg behavior for CrGeTe$_3$. Such magnetic difference strongly contrasts with structural and electronic similarities of these two materials, and understanding it …
We present results on entropy and heat-capacity of the spin-S honeycomb-lattice Kitaev models using high-temperature series expansions and thermal pure quantum (TPQ) state methods. We study models with anisotropic couplings $J_z=1\ge J_x=J_y$ for spin values 1/2, 1, 3/2, and 2. We show that for $S>1/2$, any anisotropy leads to well developed plateaus in the entropy function at an entropy value of $\frac{1}{2}\ln{2}$, independent of $S$. However, in the absence of anisotropy, there is an incipient entropy plateau at $S_{max}/2$, where $S_{max}$ is the infinite temperature entropy of the system.…
We study the Kitaev-Heisenberg-$Γ$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped $Z_2$ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudo-dipolar, $Γ$, limit). We find that the paramagnetic phase obtained in the large $Γ$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by an equal superposition of differently oriented stacked one-dimensional $Z…
Polarization-resolved Raman spectroscopy was performed and analyzed from large, high quality, mono-domain single crystal of α-RuCl3, a proximate Kitaev quantum spin liquid. Spectra were collected with laser polarizations parallel and perpendicular to the honeycomb plane. Pairs of nearly degenerate phonons were discovered and show either a 4-fold or 2-fold polarization angle dependence in their Raman intensity, thereby providing evidence to definitively assign the bulk crystal point group as C2h. The low frequency continuum that is often attributed to scattering from pairs of Majorana fermions …
Emergence of quantum spin liquid and spin-flop phase in Kitaev antiferromagnets in a [111] magnetic field
College of Physics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学物理学院）；Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT（工业和信息化部航空航天信息材料与物理重点实验室（南航））；School of Physics and Institute for Advanced Study in Physics, Zhejiang University（浙江大学物理学院与高等研究院）
We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality CrI$_3$ single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with $K \sim -5.2$ meV, 25 times larger than the Heisenberg exchange $J \sim -0.2$ meV, and responsible for opening the $\sim$5 meV gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy $Γ\sim -67…
The spin-1/2 Heisenberg kagome antiferromagnet is one of the paradigmatic playgrounds for frustrated quantum magnetism, with an extensive number of competing resonating valence bond (RVB) states emerging at low energies, including gapped and gapless spin liquids and valence bond crystals. Here we revisit the crossover from this quantum RVB phase to a semiclassical regime brought about by anisotropic Kitaev interactions, and focus on the precise mechanisms underpinning this crossover. To this end, we introduce a simple parametrization of the classical ground states (GSs) in terms of emergent Is…
Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the $Γ'$ terms. To fully understand the effect of the $Γ'$ interactions, we investigate the quantum $K$-$Γ$-$Γ'$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z$_2$ quantum spin liquids (QSLs) are found at $Γ'>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that…
We study the ground state properties of the Kitaev-Heisenberg model in a magnetic field and explore the evolution of spin correlations in the presence of non-magnetic vacancies. By means of exact diagonalizations, the phase diagram without vacancies is determined as a function of the magnetic field and the ratio between Kitaev and Heisenberg interactions. We show that in the (antiferromagnetic) stripe ordered phase the static susceptibility and its anisotropy can be described by a spin canting mechanism. This accounts as well for the transition to the polarized phase when including quantum flu…
The Kitaev spin liquid, a ground state of the bond-dependent Kitaev model in a honeycomb lattice has been a centre of attraction, since a microscopic theory to realize such an interaction in solid-state materials was discovered. A challenge in real materials though is the presence of the Heisenberg and another bond-dependent Gamma interactions detrimental to the Kitaev spin liquid, and there have been many debates on their relative strengths. Here we offer a strategy to extract the Kitaev interaction out of a full microscopic model by utilizing the symmetries of the Hamiltonian. Two tilted mag…
