Hybridized quadrupolar excitations in the spin-anisotropic frustrated magnet FeI2
Xiaojian Bai, Shang-Shun Zhang, Zhiling Dun, Hao Zhang, Qing Huang, Haidong Zhou, Matthew B. Stone, Alexander I. Kolesnikov, Feng Ye, Cristian D. Batista, Martin Mourigal
备注：
32 pages including methods and supplementary information, 3 main figures, 9 supplementary figures
Magnetic order is usually associated with well-defined magnon excitations. Exotic magnetic fluctuations with fractional, topological or multipolar character, have been proposed for radically different forms of magnetic matter such as spin-liquids. As a result, considerable efforts have searched for, and uncovered, low-spin materials with suppressed dipolar order at low temperatures. Here, we report neutron-scattering experiments and quantitative theoretical modeling of an exceptional spin-1 system -- the uniaxial triangular magnet FeI2 -- where a bright and dispersive band of mixed dipolar-quadrupolar fluctuations emerges just above a dipolar ordered ground-state. This excitation arises from anisotropic exchange interactions that hybridize overlapping modes carrying fundamentally different quantum numbers. Remarkably, a generalization of spin-wave theory to local SU(3) degrees of freedom accounts for all details of the low-energy dynamical response of FeI2 without going beyond quadratic order. Our work highlights that quantum excitations without classical counterparts can be realized even in presence of fully developed magnetic order.
Spin nematic phase is a phase of frustrated quantum magnets with a quadrupolar order of electron spins. Since the spin nematic order is usually masked in experimentally accessible quantities, it is important to develop a methodology for detecting the spin nematic order experimentally. In this paper we propose a convenient method for detecting quasi-long-range spin nematic correlations of a quadrupolar Tomonaga-Luttinger liquid state of $S=1/2$ frustrated ferromagnetic spin chain compounds, using electron spin resonance (ESR). We focus on linewidth of a so-called paramagnetic resonance peak in …
Electric quadrupolar state in a frustrated quantum magnet $\rm Tb_{2+\it δ}Ti_{2-\it δ}O_7$ has been studied by means of ultrasonic and magnetostriction measurements. A single crystal showed elastic anomaly at about 0.4 K, manifesting a long-range quadrupole ordering. By investigating anisotropy of the magnetoelastic responses, we found a crossover temperature for the strongly correlated quadrupole state, below which the experimental data of elastic constant and magnetostriction become qualitatively different from their calculations based on a single-ion model. We suppose that relatively high …
The low-temperature thermodynamic properties of the frustrated pyrochlore Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been studied using the single crystal of $x=0.005$ sitting in a long range ordered phase in the $x$-$T$ phase diagram. We observed that the specific heat exhibits a minimum around 2 K and slightly increases on cooling, similar to a Schottky-like anomaly for canonical spin ices. A clear specific-heat peak observed at $T_{\rm c} = 0.53$ K is ascribable to the phase transition to a quadrupolar state, which contributes to a relatively large change in entropy, $S \simeq 2.7$ J K$^{-1}$mol$^{…
Geometrically Frustrated Quadrupoles on the Pyrochlore Lattice and Generalized Spin Liquids
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-Saclay（里昂·布里渊实验室，法国原子能和替代能源委员会，法国国家科学研究中心，巴黎萨克雷大学）；Université Grenoble Alpes, CNRS, Institut Néel（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，尼尔研究所）
The frustrated one-dimensional (1D) quantum magnet LiCuSbO$_4$ is one rare realization of the $J_1-J_2$ spin chain model with an easily accessible saturation field, formerly estimated to 12~T. Exotic multipolar nematic phases were theoretically predicted in such compounds just below the saturation field, but without unambiguous experimental observation so far. In this paper we present extensive experimental research of the compound in the wide temperature (30mK$-$300K) and field (0$-$13.3T) range by muon spin rotation ($μ$SR), $^7$Li nuclear magnetic resonance (NMR) and magnetic susceptibility…
Terahertz Magneto-optical investigation of quadrupolar spin-lattice effects in magnetically frustrated Tb2Ti2O7
National Institute of Chemical Physics and Biophysics（爱沙尼亚化学物理与生物物理研究所）；Université Grenoble Alpes, CNRS, Institut Néel（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，内耳研究所）；ICMMO, Université Paris-Saclay, CNRS（巴黎萨克雷大学，法国国家科学研究中心，分子与材料化学研究所）；Institute of Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心，辐射物理研究所）；Institute of Physics II, University of Cologne（科隆大学，第二物理研究所）；Institute of Solid State Physics, Vienna University of Technology（维也纳工业大学，固体物理研究所）
本研究通过太赫兹光谱和量子计算揭示了磁性 frustrated Tb2Ti2O7 中四极自旋-晶格效应，发现其在 3 K 时存在振动耦合，表明该材料具有磁性双折射特性。
It is generally difficult to experimentally distinguish magnetic multipolar orders in spin systems. Recently, it was proposed that the temperature dependence of the nuclear magnetic resonance relaxation rate 1/T_1 can involve an indirect, but clear signature of the field-induced spin nematic or multipolar Tomonaga-Luttinger (TL) liquid phase [Phys. Rev. B79, 060406(R) (2009)]. In this paper, we evaluate accurately the field and temperature dependence of 1/T_1 in spin-1/2 frustrated J1-J2 chains combining field-theoretical techniques with numerical data. Our results demonstrate that isotherms o…
We have observed geometrical frustration of quadrupolar and magnetic moments in dysprosium tetraboride, DyB4, where the rare-earth sites form a Shastry-Sutherland lattice. Resonant x-ray scattering at the LIII absorption edge of Dy was utilized. Analysis of the energy, polarization, temperature, and azimuthal-angle dependences of the E1 resonance of the (1 0 0) forbidden reflection show that the magnetic and quadrupolar components within the frustrated c plane have a short-range correlation, suggesting that the moments are fluctuating. By contrast, the basic antiferromagnetic component along t…
The $J_1-J_2$ spin chain model with nearest neighbor $J_1$ and next nearest neighbor anti-ferromagnetic $J_2$ interaction is one of the most popular frustrated magnetic models. This model system has been extensively studied theoretically and applied to explain the magnetic properties of the real low-dimensional materials. However, existence of different phases for the $J_1-J_2$ model in an axial magnetic field $h$ is either not understood or has been controversial. In this paper we show the existence of higher order $p>4$ multipolar phase near the critical point $(J_2/J_1)_c=-0.25$. The criter…
In the spin-1/2 frustrated chain with nearest-neighbor ferromagnetic exchange J1 and next-nearest-neighbor antiferromagnetic exchange J2 under magnetic field, magnetic multipolar-liquid (quadrupolar, octupolar, and hexadecapolar) phases are widely expanded from the saturation down to a low-field regime. Recently, we have clarified characteristic temperature and field dependence of the NMR relaxation rate 1/T_1 in the quadrupolar phase. In this paper, we examine those of 1/T_1 in the octupolar phase combining field theoretical method with numerical data. The relevance of the results to quasi on…
We study frustrated spin-1/2 dimer systems in two dimensions with a bilayer structure, where spins are ferromagnetically coupled in dimers. Our model includes frustrated two-spin exchange interactions as well as four-spin interaction. We pay particular attention to the spin nematic phase, which does not exhibit any magnetic (spin-dipole) order but has a spin-quadrupolar long-range order. Employing a perturbation calculation, a mean-field approximation, and a numerical many-variable variational Monte Carlo method, we determine ground-state phase diagrams on various two-dimensional lattices. It …
Vibronic collapse of ordered quadrupolar ice in the pyrochore magnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$
University of Geneva（日内瓦大学）；Université Grenoble Alpes, CNRS, Institut Néel（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，内耳研究所）；Synchrotron SOLEIL（索雷尔同步辐射中心）；ICMMO, Université Paris-Saclay, CNRS（有机与无机分子和材料化学研究所，巴黎萨克雷大学，法国国家科学研究中心）；Institute of Solid State Physics, TU Wien（维也纳工业大学固体物理研究所）
Motivated by the complex behavior of the frustrated magnet $β$-TeVO$_4$, we study an anisotropic Heisenberg model for coupled spin-$1/2$ zigzag chains. Using cluster mean field approach to capture quantum correlations we find, upon reducing temperature in the absence of applied field, (i) a partially ordered state, (ii) a collinear antiferromagnetic phase, and (iii) an elliptical spiral state characterized by finite vector chirality. For finite fields, we find metamagnetic response close to saturation magnetization. We show via explicit calculations that the quadrupolar order parameter is fini…
The ground states of the frustrated pyrochlore oxide Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$, sensitively depending on the small off-stoichiometry parameter $x$, have been studied by specific heat measurements using well characterized samples. Single crystal Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ boules grown by the standard floating zone technique are shown to exhibit concentration ($x$) gradient. This off-stoichiometry parameter is determined by precisely measuring the lattice constant of small samples cut from a crystal boule. Specific heat shows that the phase boundary of the electric quadrupolar state has a…
We have studied the hidden long-range order (LRO) of the frustrated pyrochlore magnet Tb$_2$Ti$_2$O$_7$ by means of specific-heat experiments and Monte-Carlo (MC) simulations, which has been discussed as the LRO of quadrupole moments inherent to the non-Kramers ion of Tb$^{3+}$. We have found that the sharp specific-heat peak is collapsed into a broad hump by magnetic fields above 0.3 T for $H//[001]$. This result, qualitatively reproduced by MC simulations, suggests that a field-induced magnetic state overcomes the quadrupolar LRO state, as a similar case of a classical spin ice. The present …
Elucidating the nature of spin excitations is important to understanding the mechanism of superconductivity in the iron pnictides. Motivated by recent inelastic neutron scattering measurements in the nearly 100% detwinned BaFe$_{2}$As$_{2}$, we study the spin dynamics of an $S=1$ frustrated bilinear-biquadratic Heisenberg model in the antiferromagnetic phase with wavevector $(π,0)$. The biquadratic interactions are treated in a dynamical way using a flavor-wave theory in an $SU(3)$ representation. Besides the magnon (dipolar) excitations, the biquadratic interactions give rise to quadrupolar e…
Frustrated rare-earth pyrochlore titanates, Yb$_2$Ti$_2$O$_7$, and Tb$_2$Ti$_2$O$_7$ have been proposed as promising candidates to realize quantum spin ice (QSI). Multiple exotic quantum phases, including Coulombic ferromagnet, quantum valence-bond solid, and quadrupolar ordering, have been predicted to emerge in the QSI state upon application of a (111)-oriented external magnetic field. Here, we report on the primal successful layer-by-layer growth of ultra-thin films of frustrated quantum pyrochlores, R$_2$Ti$_2$O$_7$ (R = Er, Yb, and Tb), along the (111) direction. We confirm their high cry…
Frustrated spin systems can show phases with spontaneous breaking of spin-rotational symmetry without the formation of local magnetic order. We study the dynamic response of the spin-nematic phase of one-dimensional spin-1/2 systems, characterized by slow large-distance decay of quadrupolar correlations, by numerically computing one-spin and two-spin dynamical structure factors at zero temperature using time-dependent density matrix renormalization group methods. We interpret the results in terms of an effective theory of gapped magnon excitations interacting with a quasi-condensate of bound m…
A mixed Heisenberg spin chain with frustrated side chains is investigated by numerical and perturbational calculations. A frustration-induced quantum partially polarized ferrimagnetic phase and a nonmagnetic spin quadrupolar phase are found adjacent to the conventional Lieb-Mattis type ferrimagnetic phase or the nonmagnetic singlet cluster solid phases. The partially polarized ferrimagnetic phase has an incommensurate spin structure. Similar structures are commonly found in other frustration-induced partially polarized ferrimagnetic phases. Numerical results also suggest a series of almost cri…
We present a comprehensive experimental and theoretical study of the pyrochlore Tb$_2$Ge$_2$O$_7$, an exemplary realization of a material whose properties are dominated by competition between magnetic dipolar and electric quadrupolar correlations. The dipolar and quadrupolar correlations evolve over three distinct regimes that we characterize via heat capacity, elastic and inelastic neutron scattering. In the first regime, above $T^*=1.1$ K, significant quadrupolar correlations lead to an intense inelastic mode that cannot be accounted for within a scenario with solely magnetic dipole-dipole c…
Investigation of materials that exhibit quantum phase transition provides valuable insights into fundamental problems in physics. We present neutron scattering under pressure in a triangular-lattice antiferromagnet which has a quantum disorder in the low-pressure phase and a noncollinear structure in the high-pressure phase. The neutron spectrum continuously evolves through the critical pressure; a single mode in the disordered state becomes soft with the pressure, and it splits into gapless and gapped modes in the ordered phase. Extended spin-wave theory reveals that the longitudinal and tran…
Frustration in quantum spin systems promote a variety of novel quantum phases. An important example is the frustrated spin-$1$ model on the square lattice with the nearest-neighbor bilinear ($J_1$) and biquadratic ($K_1$) interactions. We provide strong evidence for a nematic spin liquid phase in a range of $K_1/J_1$ near the SU(3)-symmetric point ($J_1 = K_1$), based on the linear flavor-wave theory and extensive density matrix renormalization group calculation. This phase displays no spin dipolar or quadrupolar order, preserves translational symmetry but spontaneously breaks $C_4$ lattice ro…
We present the phase diagram of the frustrated ferromagnetic S=1/2 Heisenberg J_1-J_2 chain in a magnetic field, obtained by large scale exact diagonalizations and density matrix renormalization group simulations. A vector chirally ordered state, metamagnetic behavior and a sequence of spin-multipolar Luttinger liquid phases up to hexadecupolar kind are found. We provide numerical evidence for a locking mechanism, which can drive spiral states towards spin-multipolar phases, such as quadrupolar or octupolar phases. Our results also shed light on previously discovered spin-multipolar phases in …
An SU(2) slave boson formulation of bond-type spin nematic orders is developed in frustrated ferromagnets, where the spin nematic states are described as the resonating spin-triplet valence bond (RVB) states. The d-vectors of spin-triplet pairing ansatzes play the role of the directors in the bond-type spin quadrupolar states. The low-energy excitations around such spin-triplet RVB ansatzes generally comprise the (potentially massless) gauge bosons, massless Goldstone bosons, and spinon individual excitations. Extending the projective symmetry group argument to the spin-triplet ansatzes, we sh…
Recently, it has been shown that spin nematic (quadrupolar) or higher multipolar correlation functions exhibit a quasi long-range order in the wide region of the field-induced Tomonaga-Luttinger-liquid (TLL) phase in spin-1/2 zigzag chains. In this paper, we point out that the temperature dependence of the NMR relaxation rate 1/T_1 in these multipolar TLLs is qualitatively different from that in more conventional TLLs of one-dimensional quantum magnets (e.g., the spin-1/2 Heisenberg chain); 1/T_1 decreases with lowering temperature in multipolar TLL. We also discuss low-energy features in spin…
We investigate the collective behavior of orbital angular momentum in the spin ferromagnetic state of a Mott insulator with $t_{2g}$ orbital degeneracy. The frustrated nature of the interactions leads to an infinite degeneracy of classical states. Quantum effects select four distinct orbital orderings. Two of them have a quadrupolar order, while the other states show in addition weak orbital magnetism. Specific predictions are made for neutron scattering experiments which might help to identify the orbital order in YTiO$_3$ and to detect the elementary orbital excitations.
In frustrated magnets when magnetic ordering is suppressed down to low temperature, the formation of a quantum spin liquid becomes a possibility. How such a spin liquid manifests in the presence of conduction electrons is a question with potentially rich physical consequences, particularly when both the localized spins and conduction electrons reside on frustrated lattices. We propose a novel mechanism for symmetry breaking in systems where conduction electrons hybridize with a quantum spin liquid through Kondo couplings. We apply this to the pyrochlore iridate Pr2Ir2O7, which exhibits an anom…
Non-magnetic ground state in A$_2$WCl$_6$ (A = Cs, Rb, K): A face-centered cubic system of spin-orbit-entangled $J$ = 2 states
Max Planck Institute for Solid State Research（马克斯·普朗克固相研究所）；Institute for Functional Matter and Quantum Technologies, University of Stuttgart（斯图加特大学功能物质与量子技术研究所）；National Institute for Materials Science（国立材料研究所）；Synchrotron Radiation Research Center, National Institutes for Quantum Science and Technology（量子科学技术研究机构同步辐射研究中心）；Graduate School of Engineering and Science, Shibaura Institute of Technology（芝浦工业大学理工学研究科）；Nuclear Structure Research Group, RIKEN Nishina Center for Accelerator-Based Science（RIKEN仁科加速器科学中心核结构研究组）；ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory（STFC卢瑟福阿普尔顿实验室ISIS中子和μ子源）；Department of Physics, University of Tokyo（东京大学物理学系）
The hidden ordered state of the frustrated pyrochlore oxide Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ is possibly one of the two electric multipolar, or quadrupolar, states of the effective pseudospin-1/2 Hamiltonian derived from crystal-field ground state doublets of non-Kramers Tb$^{3+}$ ions. These long-range orders are antiparallel or parallel alignments of transverse pseudospin components representing electric quadrupole moments, which cannot be observed as magnetic Bragg reflections by neutron scattering. However pseudospin waves of these states are composite waves of the magnetic-dipole and electri…
We report the properties of a triangular lattice iron-chalcogenide antiferromagnet FeAl$_{2}$Se$_{4}$. The spin susceptibility reveals a significant antiferromagnetic interaction with a Curie-Weiss temperature Θ$_{CW}$ ~ -200K and a spin-2 local moment. Despite a large spin and a large |Θ$_{CW}$|, the low-temperature behaviors are incompatible with conventional classical magnets. No long-range order is detected down to 0.4K. Similar to the well-known spin-1 magnet NiGa$_{2}$S$_{4}$, the specific heat of FeAl$_{2}$Se$_{4}$ exhibits an unusual double-peak structure and a T$^{2}$ power law at low…
Nematic fluctuations occur in a wide range of physical systems from liquid crystals to biological molecules to solids such as exotic magnets, cuprates and iron-based high-$T_c$ superconductors. Nematic fluctuations are thought to be closely linked to the formation of Cooper-pairs in iron-based superconductors. It is unclear whether the anisotropy inherent in this nematicity arises from electronic spin or orbital degrees of freedom. We have studied the iron-based Mott insulators La$_{2}$O$_{2}$Fe$_{2}$O$M$$_{2}$ $M$ = (S, Se) which are structurally similar to the iron pnictide superconductors. …
Non-equilibrium Muon Dynamics Imitating Fractional Spin Excitations in a Quantum Spin Chain
Jožef Stefan Institute（约瑟夫·斯蒂芬研究所）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；ISIS facility, Rutherford Appleton Laboratory（鲁瑟福阿普顿实验室ISIS设施）；Institute of Solid State Physics, Vienna University of Technology（维也纳工业大学固体物理研究所）
Spin correlations of the frustrated pyrochlore oxide Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been investigated by using inelastic neutron scattering on single crystalline samples ($x=-0.007, 0.000,$ and $0.003$), which have the putative quantum-spin-liquid (QSL) or electric-quadrupolar ground states. Spin correlations, which are notably observed in nominally elastic scattering, show short-ranged correlations around $L$ points [$q = (\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$], tiny antiferromagnetic Bragg scattering at $L$ and $Γ$ points, and pinch-point type structures around $Γ$ points. The short-r…
The exotic normal state of iron chalcogenide superconductor FeSe, which exhibits vanishing magnetic order and possesses an electronic nematic order, triggered extensive explorations of its magnetic ground state. To understand its novel properties, we study the ground state of a highly frustrated spin-$1$ system with bilinear-biquadratic interactions using unbiased large-scale density matrix renormalization group. Remarkably, with increasing biquadratic interactions, we find a paramagnetic phase between Néel and stripe magnetic ordered phases. We identify this phase as a candidate of nematic qu…
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…
Spin-1 antiferromagnets are abundant in nature, but few theories or results exist to understand their general properties and behavior, particularly in situations when geometric frustration is present. Here we study the $S=1$ Kagome compound Na$_2$Ti$_3$Cl$_8$ using a combination of Density Functional Theory, Exact Diagonalization, and Density Matrix Renormalization Group methods to achieve a first principles supported explanation of exotic magnetic phases in this compound. We find that the effective magnetic Hamiltonian includes essential non-Heisenberg terms that do not stem from spin-orbit c…
