In this article, I analyze the symmetries and degeneracies of electron eigenstates in a commensurate collinear antiferromagnet. In a magnetic field transverse to the staggered magnetization, a hidden anti-unitary symmetry protects double degeneracy of the Bloch eigenstates at a special set of momenta. In addition to this `Kramers degeneracy' subset, the manifold of momenta, labeling the doubly degenerate Bloch states in the Brillouin zone, may also contain an `accidental degeneracy' subset, that is not protected by symmetry and that may change its shape under perturbation. These degeneracies give rise to a substantial momentum dependence of the transverse g-factor in the Zeeman coupling, turning the latter into a spin-orbit interaction.
I discuss a number of materials, where Zeeman spin-orbit coupling is likely to be present, and outline the simplest properties and experimental consequences of this interaction, that may be relevant to systems from chromium to borocarbides, cuprates, hexaborides, iron pnictides, as well as organic and heavy fermion conductors.
In this article, I study magnetic response of electron wavefunctions in a commensurate collinear antiferromagnet. I show that, at a special set of momenta, hidden anti-unitary symmetry protects Kramers degeneracy of Bloch eigenstates against a magnetic field, pointing transversely to staggered magnetization. Hence a substantial momentum dependence of the transverse g-factor in the Zeeman term, turning the latter into a spin-orbit coupling, that may be present in materials from chromium to borocarbides, cuprates, pnictides, as well as organic and heavy fermion conductors.
Zeeman split Kramers doublets in spin-supersolid candidate Na$_{2}$BaCo(PO$_{4}$)$_{2}$
School of Physics and Astronomy, University of Edinburgh（爱丁堡大学物理与天文学院）；ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory（STFC卢瑟福阿普尔顿实验室ISIS脉冲中子与缪子源）；NIST Center for Neutron Research（美国国家标准与技术研究院中子研究中心）；Department of Chemistry, Princeton University（普林斯顿大学化学系）
Field-Induced Up-Up-Down State and Frustrated Magnetism in a Non-Kramers Triangular Antiferromagnet
Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学量子科学和工程研究院）；International Quantum Academy（国际量子学院）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；School of Physics and Electronics, Henan University（河南大学物理与电子学院）
Inelastic neutron scattering measurement is performed on a breathing pyrochlore antiferromagnet Ba3Yb2Zn5O11. The observed dispersionless excitations are explained by a crystalline electric field (CEF) Hamiltonian of Kramers ion Yb3+ of which the local symmetry exhibits C3v point group symmetry. The magnetic susceptibility previously reported is consistently reproduced by the energy scheme of the CEF excitations. The obtained wave functions of the ground state Kramers doublet exhibit the planer-type anisotropy. The result demonstrates that Ba3Yb2Zn5O11 is an experimental realization of breathi…
Local multiferroic ordering at room temperature in collinear magnetoelectric antiferromagnets induced by flexo-Zeeman coupling
Helmholtz-Zentrum Dresden-Rossendorf e.V.（德累斯顿-罗森多夫亥姆霍兹中心）；Faculty of Physics of Dresden University of Technology（德累斯顿工业大学物理学院）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料与能源中心）；Department of Applied Physics, KTH Royal Institute of Technology（瑞典皇家理工学院应用物理系）
This article is a brief review of Zeeman spin-orbit coupling, arising in a low-carrier commensurate Néel antiferromagnet subject to magnetic field. The field tends to lift the degeneracy of the electron spectrum. However, a hidden symmetry protects double degeneracy of Bloch eigenstates at special momenta in the Brillouin zone. The effective transverse $g$-factor vanishes at such points, thus acquiring a substantial momentum dependence, which turns a textbook Zeeman term into a spin-orbit coupling. After describing the symmetry underpinnings of the Zeeman spin-orbit coupling, I compare it with…
We have performed electron spin resonance measurements over a wide frequency and magnetic field range on a single crystal of the S=1 quasi-one-dimensional Heisenberg antiferromagnet $\rm Ni(C_5H_{14}N_2)_2N_3(PF_6)$. We observed gapped excitation branches above the critical field ($H_{\rm c}$) where the Haldane gap closes. These branches are analyzed by a phenomenological field-theory using complex-field $ϕ^{4}$ model. A satisfactory agreement between experiment and theory is obtained.
We report magnetic properties of a 4f-honeycomb iodide SmI3 made up of edge-shared network of SmI6 octahedra. High temperature magnetic susceptibility indicates Γ7 Kramers doublet ground state of Sm3+ (4f5) ions stabilized by the spin-orbit coupling and octahedral crystal electric field, which interact with Sm-I-Sm bond angle nearly 90 degree. Magnetization measurements down to 0.1 K detected antiferromagnetic correlations and an anomaly in the magnetization curve before saturation without a sign of long-range order. Relevance between SmI3 and the antiferromagnetic Kitaev material proposed in …
Most of solid-state spin physics arising from spin-orbit coupling, from fundamental phenomena to industrial applications, relies on symmetry-protected degeneracies. So does the Zeeman spin-orbit coupling, expected to manifest itself in a wide range of antiferromagnetic conductors. Yet, experimental proof of this phenomenon has been lacking. Here, we demonstrate that the Néel state of the layered organic superconductor $κ$-(BETS)$_2$FeBr$_4$ shows no spin modulation of the Shubnikov-de Haas oscillations, contrary to its paramagnetic state. This is unambiguous evidence for the spin degeneracy of…
We explore the spin dynamics emerging from the Néel phase of the chain compound antiferromagnet BaCo2V2O8. Our inelastic neutron scattering study reveals unconventional discrete spin excitations, so called Zeeman ladders, understood in terms of spinon confinement, due to the interchain attractive linear potential. These excitations consist in two interlaced series of modes, respectively with transverse and longitudinal polarization. The latter have no classical counterpart and are related to the zero-point fluctuations that weaken the ordered moment in weakly coupled quantum chains. Our analys…
Using the determinant quantum Monte Carlo method, we investigate the antiferromagnetic phase transition that is induced by the Zeeman field in a disordered interacting two-dimensional Dirac fermion system. At a fixed interaction strength $U$, the antiferromagnetic correlation is enhanced as the magnetic filed increases, and when the magnetic field is larger than a $B_{c}(U)$, the antiferromagnetic correlation shall be suppressed by the increased magnetic field. The impact of Zeeman field $B$, Coulomb repulsion $U$ and disorder $Δ$ is not isolated. The intensity of magnetic field effect on the …
Frustrated quasidoublets without time-reversal symmetry can host highly unconventional magnetic structures with continuously distributed order parameters even in a single-phase crystal. Here, we report the comprehensive thermodynamic and neutron diffraction investigation on the single crystal of TmMgGaO$_4$, which entails non-Kramers Tm$^{3+}$ ions arranged on a geometrically perfect triangular lattice. The crystal electric field (CEF) randomness caused by the site-mixing disorder of the nonmagnetic Mg$^{2+}$ and Ga$^{3+}$ ions, merges two lowest-lying CEF singlets of Tm$^{3+}$ into a ground-s…
Frustration, spin correlations and interplay between competing degrees of freedom are some of the key ingredients that underlie exotic states with fractional excitations in quantum materials. Rare-earth based two dimensional magnetic lattice wherein crystal electric field, spin-orbit coupling, anisotropy and electron correlation between rare-earth moments offer a new paradigm in this context. Herein, we present crystal structure, magnetic susceptibility and specific heat accompanied by crystal electric field calculations on the polycrystalline sample of Ba3RB9O18 (R = Yb, Er) in which R3+ ions…
Anomalous magnetocaloric effects in the quasi-one-dimensional antiferromagnet BaCo$_2$V$_2$O$_8$
Peking University（北京大学）；Huazhong University of Science and Technology（华中科技大学）；Southern University of Science and Technology（南方科技大学）；Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences（中国科学院福建物质结构研究所）
The magnetic ground state and the crystalline electric field level scheme of the triangular lattice antiferromagnet KCeO$_2$ are investigated. Below $T_N =$ 300 mK, KCeO$_2$ develops signatures of magnetic order in specific heat measurements and low energy inelastic neutron scattering data. Trivalent Ce$^{3+}$ ions in the $D_{3d}$ local environment of this compound exhibit large splittings among the lowest three $4f^1$ Kramers doublets defining for the free ion the $J = 5/2$ sextet and a ground state doublet with dipole character, consistent with recent theoretical predictions in M. S. Eldeeb …
The Kondo lattice model, augmented by a Zeeman term, serves as a useful model of a Kondo insulator in an applied magnetic field. A variational mean field analysis of this system on a square lattice, backed up by quantum Monte Carlo calculations, reveals an interesting separation of magnetic field scales. For Zeeman energy comparable to the Kondo energy, the spin gap closes and the system develops transverse staggered magnetic order. The charge gap, however, survives up to a higher hybridization energy scale, at which point the canted antiferromagnetism is suppressed and the system becomes meta…
The low-temperature spin dynamics of the orthorhombic TbFeO$_3$ perovskite has been studied. It has been found that the inelastic neutron scattering (INS) spectrum contains two modes corresponding to different sublattices in the compound. The iron subsystem orders antiferromagnetically at T$_N$ = 632 K and exhibits the high-energy magnon dispersion. Magnetic dynamics of this subsystem has been described using the linear spin wave theory and our solution yields sizable anisotropy between in-plane and out-of-plane exchange interactions. This approach was previously used to describe the magnon di…
For quasi-one dimensional quantum spin systems theory predicts the occurrence of a confinement of spinon excitation due to interchain couplings. Here we investigate the system SrCo2V2O8, a realization of the weakly-coupled Ising-like XXZ antiferromagnetic chains, by terahertz spectroscopy with and without applied magnetic field. At low temperatures a series of excitations is observed, which split in a Zeeman-like fashion in an applied magnetic field. These magnetic excitations are identified as the theoretically predicted spinon-pair excitations. Using a one dimensional Schrödinger equation wi…
We use strong coupling expansions to calculate the entropy function $S(T,h)$, the magnetization $M(T,h)$, and the double occupancy factor $D(T,h)$ for the half-filled triangular lattice Hubbard model as a function of temperature $T$ and Zeeman field $h$, for various values of the Hubbard parameter ratio $U/t$. These calculations converge well for temperatures larger than the exchange parameter $J=\frac{4t^2}{U}$ for moderate to large $U/t$ values. Setting $μ=U/2$ suffices to obtain the density of half filling within a fraction of one percent at all temperatures studied for $U/t \geq 8$. We dis…
Magnetic ground state of a Jeff = 1/2 based frustrated triangular lattice antiferromagnet
Indian Institute of Technology Madras（马德拉斯理工学院）；Sungkyunkwan University（成均馆大学）；Simon Fraser University（西蒙菲莎大学）；Institute for Basic Science（基础科学研究院）；TRIUMF（TRIUMF实验室）
研究了 Ba4YbReWO12 中 Jeff = 1/2 状态的磁基态，通过穆松放松和比热实验发现其具有动态无序基态，无长程磁序或自旋冻结现象。
We report spin-split Landau levels of quasi-two-dimensional Dirac fermions in a layered antiferromagnet EuMnBi$_2$, as revealed by interlayer resistivity measurements in a tilted magnetic field up to $\sim$35 T. The amplitude of Shubnikov-de Haas (SdH) oscillation in interlayer resistivity is strongly modulated by changing the tilt angle of the field, i.e., the Zeeman-to-cyclotron energy ratio. The effective $g$ factor estimated from the tilt angle, where the SdH oscillation exhibits a phase inversion, differs by approximately 50% between two antiferromagnetic phases. This observation signifie…
Motivated by recent experimental processes, we systemically investigate strongly correlated spin-1 ultracold bosons trapped in a three-dimensional optical lattice in the presence of an external magnetic field. Based on a recently developed bosonic dynamical mean-field theory (BDMFT), we map out complete phase diagrams of the system for both antiferromagnetic and ferromagnetic interactions, where various phases are found as a result of the interplay of spin-dependent interaction and quadratic Zeeman energy. For antiferromagnetic interactions, the system demonstrates competing magnetic orders, i…
The so-called V15 molecule is formed of 15 spins 1/2 antiferromagnetically coupled. The resultant spin is equal to 1/2. Contrary to what is expected at first sight, this half-integer spin is gapped. We show that this is a consequence of the multi-spin character of the molecule. Time reversal symmetry (Kramers theorem) is preserved due to the existence of a groundstate degeneracy larger than 2. Magnetization experiments have been performed at different temperatures, sweeping-field rates and couplings with the cryostat. Their interpretation in terms of the Landau-Zener-Stuckelberg model gives ev…
Nd2Hf2O7, belonging to the family of geometrically frustrated cubic rare earth pyrochlore oxides, was recently identified to order antiferromagnetically below T_N = 0.55 K with an all-in/all-out arrangement of Nd3+ moments, however with a much reduced ordered state moment. Herein we investigate the spin dynamics and crystal field states of Nd2Hf2O7 using muon spin relaxation (muSR) and inelastic neutron scattering (INS) measurements. Our muSR study confirms the long range magnetic ordering and shows evidence for coexisting persistent dynamic spin fluctuations deep inside the ordered state down…
We present synchrotron x-ray diffraction, neutron powder diffraction and time-of-flight inelastic neutron scattering measurements on the rare earth pyrochlore oxide Nd2Zr2O7 to study the ordered state magnetic structure and cystal field states. The structural characterization by high-resolution synchrotron x-ray diffraction confirms that the pyrochlore structure has no detectable O vacancies or Nd/Zr site mixing. The neutron diffraction reveals long range all-in/all-out antiferromagnetic order below T_N ~ 0.4 K with propagation vector k = (0 0 0) and an ordered moment of 1.26(2) μ_B/Nd at 0.1 …
We report on magneto-optical studies of the quasi-two-dimensional van der Waals antiferromagnet FePS$_3$. Our measurements reveal an excitation that closely resembles the antiferromagnetic resonance mode typical of easy-axis antiferromagnets, nevertheless, it displays an unusual, four-times larger Zeeman splitting in an applied magnetic field. We identify this excitation with an $|S_z|=4$ multipolar magnon -- a single-ion 4-magnon bound state -- that corresponds to a full reversal of a single magnetic moment of the Fe$^{2+}$ ion. We argue that condensation of multipolar magnons in large-spin m…
We suggest that electron spin resonance (ESR) experiment can be used as a probe of spinon excitations of hypothetical spin-liquid state of frustrated antiferromagnet in the presence of asymmetric Dzyaloshinskii-Moriya (DM) interaction. We describe assumptions under which the ESR response is reduced to the response of 2D electron gas with Rashba spin-orbit coupling. Unlike previous treatments, the spin-orbit coupling, Δ_{SO}, is not assumed small compared to the Zeeman splitting, Δ_Z. We demonstrate that ESR response diverges at the edges of the absorption spectrum for ac magnetic field perpend…
Simple mean-field approach for frustrated antiferromagnets on hexagonal lattices, aimed to describe the high-temperature part of the temperature-magnetic field phase diagram, is proposed. It is shown, that an interplay between modulation vector symmetry, Zeeman energy and magneto-dipolar interaction leads to stabilization of the triple-$Q$ skyrmion lattice in a certain region of the phase diagram. Corresponding analytical expressions for phase boundaries are derived. It is argued that the developed theory can be applied for the description of the high-temperature part of the phase diagram obse…
THz magnetization dynamics is excited in ferrimagnetic thulium iron garnet with a picosecond, single-cycle magnetic field pulse and seen as a high-frequency modulation of the magneto-optical Faraday effect. Data analysis combined with numerical modelling and evaluation of the effective Lagrangian allow us to conclude that the dynamics corresponds to the exchange mode excited by Zeeman interaction of the THz field with the antiferromagnetically coupled spins. We argue that THz-pump IR-probe experiments on ferrimagnets offer a unique tool for quantitative studies of dynamics and mechanisms to co…
We present a comprehensive study of a three-orbital lattice model suitable for the layered iridate Sr2IrO4. Our analysis includes various on-site interactions (including Hubbard and Hund's) as well as compressive strain, and a Zeeman magnetic field. We use a self-consistent mean field approach with multiple order parameters to characterize the resulting phases. While in some parameter regimes the compound is well described by an effective J=1/2 model, in other regimes the full multiorbital description is needed. As a function of the compressive strain, we uncover two quantum phase transitions:…
We discuss possible magnetic structures in UPt$_3$ based on our analysis of elastic neutron-scattering experiments in high magnetic fields at temperatures $T<T_N$. The existing experimental data can be explained by a single-{\bf q} antiferromagnetic structure with three independent domains. For modest in-plane spin-orbit interactions, the Zeeman coupling between the antiferromagnetic order parameter and the magnetic field induces a rotation of the magnetic moments, but not an adjustment of the propagation vector of the magnetic order. A triple-{\bf q} magnetic structure is also consistent with…
We study the transverse dynamical susceptibility of an antiferromagnetic spin$-1/2$ chain in presence of a longitudinal Zeeman field. In the low magnetization regime in the gapless phase, we show that the marginally irrelevant backscattering interaction between the spinons creates a non-zero gap between two branches of excitations at small momentum. We further demonstrate how this gap varies upon introducing a second neighbor antiferromagnetic interaction, vanishing in the limit of a non-interacting "spinon gas". In the high magnetization regime, as the Zeeman field approaches the saturation v…
We report a $^{63}$Cu nuclear quadrupole resonance (NQR) study of the vortex state for an aligned polycrystalline sample of a slightly overdoped high-$T_c$ superconductor YBa$_2$Cu$_3$O$_{7-δ}$ ($T_{c}\sim$92 K) at a low magnetic field of 96 mT along the c axis, near a lower critical field $H_{c1}$. We observed the frequency distribution of the nuclear spin-lattice relaxation time $^{63}T_1$ in the Zeeman-perturbed $^{63}$Cu NQR spectrum below $T_c$. The characteristic behavior of 1/$^{63}T_1$, taking the minimum values with respect to temperature and frequency, indicates the significant role …
Magnetic van der Waals (vdW) materials offer a fantastic platform to investigate and exploit rich spin configurations stabilized in reduced dimensions. One tantalizing magnetic order is the interlayer antiferromagnetism in A-type vdW antiferromagnet, which may be effectively modified by the magnetic field, stacking order and thickness scaling. However, atomically revealing the interlayer spin orientation in the vdW antiferromagnet is highly challenging, because most of the material candidates exhibit an insulating ground state or instability in ambient conditions. Here, we report the layer-dep…
The rare-earth triangular-lattice chalcogenide is a great platform for exploring both spin liquids and novel magnetic orders with anisotropic spin interactions and magnetic frustrations. Here, we report the thermodynamic and neutron scattering measurements of rare-earth triangular-lattice chalcogenide KErSe$_{2}$, using single-crystal samples. Our experiments revealed a long-range stripe order below 0.2 K. Although the magnetic order was three-dimensional, magnetic excitations exhibited negligible modulation along the z direction, indicating very weak interlayer coupling. Furthermore, magnetic…
The fraction of the paramagnetic phase surviving at the impurity-induced antiferromagnetic order transition of the doped spin-Peierls magnet Cu(1-x)Mg(x)GeO3 (x < 5%) is found to increase with an external magnetic field. This effect is qualitatively explained by the competition of Zeeman energy and exchange interaction between local antiferromagnetic clusters
We report on THz absorption spectroscopy combined with high magnetic fields of polycrystalline RCrO3 (R= Pr, Sm, Er) aiming understanding spin wave resonances at their low temperature magnetic phases. Our measurements show that the temperature, and the implicit anisotropies at which the Cr3+ spin reorientation at TSR takes place, are determinant on the ferromagnetic-like (FM) and the antiferromagnetic-like (AFM) spin modes being optically active. It is found that they are dependent on Rare Earth 4f moment and ion size. We also studied temperature and field dependence of crystal field levels in…
Emergent Gauge Flux and Spin Ordering in Magnetized Triangular Spin Liquids: Applications to Hofstadter-Hubbard Model
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料中心）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学北京量子器件重点实验室）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）
Induced quantum magnetism on a triangular lattice of non-Kramers ions in PrMgAl11O19
Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）；Adam Mickiewicz University, Faculty of Physics and Astronomy（亚当·密茨凯维奇大学物理与天文学学院）；Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Adam Mickiewicz University, NanoBioMedical Centre（亚当·密茨凯维奇大学纳米生物医学中心）
We study the magnetic-field effect on a Kondo insulator by exploiting the periodic Anderson model with the Zeeman term. The analysis using dynamical mean field theory combined with quantum Monte Carlo simulations determines the detailed phase diagram at finite temperatures. At low temperatures, the magnetic field drives the Kondo insulator to a transverse antiferromagnetic phase, which further enters a polarized metallic phase at higher fields. The antiferromagnetic transition temperature $T_c$ takes a maximum when the Zeeman energy is nearly equal to the quasi-particle gap. In the paramagneti…
We study a spin-one Heisenberg model with exchange interaction, J, uniaxial single-ion exchange anisotropy, D, and Zeeman coupling to a magnetic field, B, parallel to the symmetry axis. We com- pute the (D/J,B/J) quantum phase diagram for square and simple cubic lattices by combining an- alytical and Quantum Monte Carlo approaches, and find a transition between XY-antiferromagnetic and ferronematic phases that spontaneously break the U(1) symmetry of the model. In the language of bosonic gases, this is a transition between a Bose-Einstein condensate (BEC) of single bosons and a BEC of pairs. O…
In the framework of the weak coupling renormalization group technique we examine the possible instabilities of the extended quasi-one-dimensional electron gas model with both intrachain and interchain electron-electron interactions, including the influence of umklapp scattering and the coupling of spins to a magnetic field. In the limit of purely repulsive intrachain interactions, we confirm the passage from singlet d-wave like superconductivity to an inhomogeneous FFLO state under magnetic field. The passage is accompanied by an anomalous increase of the upper critical field that scales with …
We report the low temperature magnetic properties of Ce$_2$Sn$_2$O$_7$, a rare-earth pyrochlore. Our susceptibility and magnetization measurements show that due to the thermal isolation of a Kramers doublet ground state, Ce$_2$Sn$_2$O$_7$ has Ising-like magnetic moments of $\sim1.18$ $μ_\mathrm{B}$. The magnetic moments are confined to the local trigonal axes, as in a spin ice, but the exchange interactions are antiferromagnetic. Below 1 K the system enters a regime with antiferromagnetic correlations. In contrast to predictions for classical $\langle 111 \rangle$-Ising spins on the pyrochlore…
Motivated by the recent experimental progress on the strong spin-orbit-coupled rare earth triangular antiferromagnet, we analyze the highly anisotropic spin model that describes the interaction between the spin-orbit-entangled Kramers' doublet local moments on the triangular lattice. We apply the Luttinger-Tisza method, the classical Monte Carlo simulation, and the self-consistent spin wave theory to analyze the anisotropic spin Hamiltonian. The classical phase diagram includes the 120-degree state and two distinct stripe ordered phases. The frustration is very strong and significantly suppres…
We study magnetic behaviour of the Yb$^{3+}$ ions on a frustrated pyrochlore lattice in the spinel {\CYS}. The crystal-electric field parameters deduced from high-energy inelastic neutron scattering reveal well-isolated ytterbium ground state doublet with a weakly Ising character. Magnetic order studied by powder neutron diffraction evolves from the $XY$-type antiferromagnetic $Γ_5$ state to a splayed ice-like ferromagnet (both with k=0) in applied magnetic field with $B_c$=3 T. Low-energy inelastic neutron scattering identifies weakly dispersive magnetic bands around 0.72 meV starting at $\mi…
