Spin Nematic Order Research Articles

Fluctuation-induced spin nematic order in magnetic charge ice

A pyrochlore charge ice consisting of two magnetic cation types gives rise to a Heisenberg magnet with an interesting type of long-range correlated bond disorder. If the antiferromagnetic exchange among cations of the same type dominates over the interspecies exchange, closed antiferromagnetic chains following the fully packed loops of like cations emerge as nonlocal effective low-temperature degrees of freedom. Monte Carlo simulations reveal a fluctuation driven first-order phase transition from an algebraically correlated spin liquid to a long-range ordered spin nematic, which mean-field theory captures with very high accuracy. The global O(3) Heisenberg symmetry is suddenly reduced to a Z2 Ising symmetry as the Néel vectors of the strongly antiferromagnetically correlated loops align with each other to optimize the entropy of their thermal fluctuations. Interestingly, the nematic transition is found to be sensitive to the size statistics of the cation loops and thus provides a direct thermodynamic probe of otherwise elusive structural properties. In turn this sensitivity offers a structural route to engineering the nematic phase stability. Published by the American Physical Society 2024

Dynamics of quantum spin-nematics: Comparisons with canted antiferromagnets

The identification of spin-nematic states is challenging due to the absence of Bragg peaks. However, the study of dynamical physical quantities provides a promising avenue for characterizing these states. In this study, we investigate the dynamical properties of spin-nematic states in three-dimensional quantum spin systems in a magnetic field, using a two-component boson theory that incorporates magnons and bimagnons. Our particular focus lies on the dynamical spin structure factor at zero temperature and the nuclear magnetic resonance (NMR) relaxation rate at finite temperatures. Our findings reveal that the dynamical structure factor does not exhibit any diverging singularity across momentum and frequency while providing valuable information about the form factor of bimagnon states and the underlying structure of spin-nematic order. Furthermore, we find a temperature dependence in the NMR relaxation rate proportional to T3 at low temperatures, similar to canted antiferromagnets. A clear distinction arises as there is no critical divergence of the NMR relaxation rate at the spin-nematic transition temperature. Our theoretical framework provides a comprehensive understanding of the excitation spectrum and the dynamical properties of spin-nematic states, covering arbitrary spin values S and encompassing site and bond nematic orders. Additionally, we apply the same methodology to analyze these dynamical quantities in a canted antiferromagnetic state and compare the results with those in the spin-nematic states. Published by the American Physical Society 2024

Spontaneous dimerization, spin-nematic order, and deconfined quantum critical point in a spin-1 Kitaev chain with tunable single-ion anisotropy

Kitaev-type spin chains have been demonstrated to be fertile playgrounds in which exotic phases and unconventional phase transitions are ready to appear. In this work, we use the density-matrix renormalization-group method to study the quantum phase diagram of a spin-1 Kitaev chain with a tunable negative single-ion anisotropy (SIA). When the strength of the SIA is small, the ground state is revealed to be a spin-nematic phase, which escapes conventional magnetic order but is characterized by a finite spin-nematic correlation because of the breaking spin-rotational symmetry. As the SIA increases, the spin-nematic phase is taken over by either a dimerized phase or an antiferromagnetic phase through an Ising-type phase transition, depending on the direction of the easy axis. For large enough SIA, the dimerized phase and the antiferromagnetic phase undergo a ``Landau-forbidden'' continuous phase transition, suggesting new platform of deconfined quantum critical point in spin-1 Kitaev chain.

Interplay of the Staggered and Three‐Body Interaction Potentials on the Quantum Phases of a Spin‐1 Ultracold Atom in an Optical Lattice

AbstractThe interplay of the staggered and the three‐body interaction potentials on the quantum phases of a spin‐1 Bose Hubbard model using a mean field approximation (MFA) is studied. In the antiferromagnetic (AF) case, a smaller value of the staggered potential (SP) results in the charge and the spin density wave ordering along with the Mott insulator (MI) and the staggered superfluid (SSF) phases. While the competition between two types of the potential leads to the stabilization of the higher order MI and charge density wave (CDW) phases with increasing three‐body interaction strength. Further, the spin eigenvalue and nematic order parameters are calculated to scrutinize the spin singlet‐nematic formation in the MI and the CDW phases and spin population fractions to analyze the nature of the SSF phase. A signature of the spin density wave (SDW) pattern is also observed in the gapped phase lobes. In case of a purely three‐body interaction, the third and higher order insulating lobes become dominant with increasing staggered potential strength. Subsequently, all MFA phase diagrams are then nicely corroborated with the analytical results obtained using a perturbative expansion corresponding to the AF and ferromagnetic cases.

Spin nematic ordering in the spin-1 chain system

We predict the instability of the quantum spin-1 chain material with respect to the spin nematic ordering. For the spin subsystem the spin nematic order parameter is related to the onset of single-ion spin anisotropy. The ordering is caused by the coupling to the elastic subsystem of the crystal or, for spin-1 ultracold bosons in a one-dimensional optical lattice, by the interaction with a Bose-Einstein condensate. Our conclusions are based on the exact integrability of the spin model and on numerical simulations for the nonintegrable cases. Our exact results also describe the behavior of interacting gluons within the lattice toy Yang-Mills-like model, in which spontaneous and field-induced symmetry breaking can be realized.

Order, disorder, and monopole confinement in the spin- 12 XXZ model on a pyrochlore tube

We study the ground state and thermodynamic properties of the spin-half XXZ model, with an Ising interaction $J_z$ and a transverse exchange interaction $J_{x}$, on a pyrochlore tube obtained by joining together elementary cubes in a one-dimensional array. Periodic boundary conditions in the transverse directions ensure that the bulk of the system consists of corner-sharing tetrahedra, with the same local geometry as the pyrochlore lattice. We use exact diagonalization, the density matrix renormalization group (DMRG), and minimally entangled typical thermal states (METTS) methods to study the system. When $J_z$ is antiferromagnetic ($J_{z}>0$) and $J_x$ is ferromagnetic ($J_{x}<0$), we find a transition from a spin liquid to an XY ferromagnet, which has power-law correlations at $T=0$. For $J_{z}<0$ and $J_{x}>0$, spin-two excitations are found to have lower energy than spin-one at the transition away from the fully polarized state, showing evidence for incipient spin-nematic order. When both interactions are antiferromagnetic, we find a non-degenerate ground state with no broken symmetries and a robust energy gap. The low energy spectra evolve smoothly from predominantly Ising to predominantly XY interactions. In the spin-liquid regime of small $|J_{x}|$, we study the confinement of monopole-anti-monopole pairs and find that the confinement length scale is larger for $J_x<0$ than for $J_{x}>0$, although both length scales are very short. These results are consistent with a local spin-liquid phase for the Heisenberg antiferromagnet with no broken symmetries.

Semi-classical simulation of spin-1 magnets

Theoretical studies of magnets have traditionally concentrated on either classical spins, or the extreme quantum limit of spin-1/2. However, magnets built of spin-1 moments are also intrinsically interesting, not least because they can support quadrupole, as well as dipole moments, on a single site. For this reason, spin-1 models have been extensively studied as prototypes for quadrupolar (spin-nematic) order in magnetic insulators, and Fe-based superconductors. At the same time, because of the presence of quadrupoles, the classical limit of a spin-1 moment is not an $O(3)$ vector, a fact which must be taken into account in describing their properties. In this Article we develop a method to simulate spin-1 magnets based on a $u(3)$ algebra which treats both dipole and quadrupole moments on equal footing. This approach is amenable to both classical and quantum calculations, and we develop the techniques needed to calculate thermodynamic properties through Monte Carlo simulations and classical low-temperature expansion, and dynamical properties, through "molecular dynamics" simulations and a multiple-boson expansion. As a case study, we present detailed analytic and numerical results for the thermodynamic properties of ferroquadrupolar order on the triangular lattice, and its associated dynamics. At low temperatures, we show that it is possible to "correct" for the effects of classical statistics in simulations, and extrapolate to the zero-temperature quantum results found in flavour-wave theory.

Weakly first-order quantum phase transition between spin-nematic and valence-bond crystal order in a square-lattice SU(4) fermionic model

We consider a model Hamiltonian with two $\mathrm{SU}(4)$ fermions per site on a square lattice, showing a competition between bilinear and biquadratic interactions. This model has generated interest due to possible realizations in ultra-cold-atom experiments and existence of spin-liquid ground states. Using a basis transformation, we show that part of the phase diagram is amenable to quantum Monte Carlo simulations without a sign problem. We find evidence for spin nematic and valence-bond crystalline phases, which are separated by a weak first-order phase transition. A $\mathrm{U}(1$) symmetry is found to emerge in the valence-bond crystal histograms, suggesting proximity to a deconfined quantum critical point. Our results are obtained with the help of a loop algorithm which allows large-scale simulations of bilinear-biquadratic $\mathrm{SO}(N$) models on arbitrary lattices in a certain parameter regime.

Classical spin order near the antiferromagnetic Kitaev point in the spin- 12 Kitaev-Gamma chain

A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent SU(2)$_1$ phase and a rank-1 spin ordered phase with $O_h\rightarrow D_4$ symmetry breaking were identified using non-Abelian bosonization and numerical techniques. However, puzzles near the antiferromagnetic Kitaev region with finite Gamma interaction remained unresolved. Here we focus on this parameter region and find that there are two new phases, namely, a rank-1 ordered phase with an $O_h\rightarrow D_3$ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the $O_h\rightarrow D_3$ symmetry breaking corresponds to the classical magnetic order, but appears in a region very close to the antiferromagnetic Kitaev point where the quantum fluctuations are presumably very strong. In addition, a two-step symmetry breaking $O_h\rightarrow D_{3d}\rightarrow D_3$ is numerically observed as the length scale is increased: At short and intermediate length scales, the system behaves as having a rank-2 spin nematic order with $O_h\rightarrow D_{3d}$ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the $O_h\rightarrow D_3$ symmetry breaking. Finally, there is no numerical signature of spin orderings nor Luttinger liquid behaviors in the Kitaev phase whose nature is worth further studies.

Impact of the Lattice on Magnetic Properties and Possible Spin Nematicity in the S=1 Triangular Antiferromagnet NiGa_{2}S_{4}.

NiGa_{2}S_{4} is a triangular lattice S=1 system with strong two dimensionality of the lattice, actively discussed as a candidate to host spin-nematic order brought about by strong quadrupole coupling. Using Raman scattering spectroscopy we identify a phonon of E_{g} symmetry which can modulate magnetic exchange J_{1} and produce quadrupole coupling. Additionally, our Raman scattering results demonstrate a loss of local inversion symmetry on cooling, which we associate with sulfur vacancies. This will lead to disordered Dzyaloshinskii-Moriya interactions, which can prevent long-range magnetic order. Using magnetic Raman scattering response we identify 160K as a temperature of an upturn of magnetic correlations. The temperature range below 160K, but above 50K where antiferromagnetic correlations start to increase, is a candidate for spin-nematic regime.

Two-photon driven magnon-pair resonance as a signature of spin-nematic order

We theoretically study the nonlinear magnetic resonance driven by intense laser or electromagnetic wave in a fully polarized frustrated magnet near a less-visible spin-nematic ordered phase. In general, both magnons and magnon pairs (two-magnon bound state) appear as the low-energy excitation in the saturated state of spin-nematic magnets. Their excitation energies are usually in terahertz (THz) or gigahertz range. Magnon pairs with angular momentum 2$\hbar$ can be excited by the simultaneous absorption of two photons, and such multi-photon processes occur if the applied THz laser is strong enough. We compute laser-driven magnetic dynamics of a frustrated four-spin system with both magnon ($\hbar$) and magnon-pair (2$\hbar$) like excitations which is analogous to a macroscopic frustrated magnet with a spin nematic phase. We estimate the required strength of magnetic field of laser for the realization of two photon absorption, taking into account dissipation effects with the Lindblad equation. We show that intense THz laser with ac magnetic field of 0.1-1.0 Tesla is enough to observe magnon-pair resonance.

Magnetic phase diagram of the linear quantum ferro-antiferromagnet Cs2Cu2Mo3O12

A single-crystal sample of the frustrated quasi one-dimensional quantum magnet Cs$_{2}$Cu$_{2}$Mo$_{3}$O$_{12}$ is investigated by magnetic and thermodynamic measurements.A combination of specific heat and magnetic torque measurements maps out the entire $H$-$T$ phase diagram for three orientations.Remarkably, a new phase emerges below the saturation field, irrespective of the crystal orientation. It is suggested that the presaturation phase represents spin-nematic order or other multi-magnon condensate. The phase diagrams within the long-range ordered dome are qualitatively different for each geometry. In particular, multiple transitions are identified in the field along the chain direction.

Spin-nematic order induced superconductivity

We explore a spin-fermion model with fermion-spin-quadrupolar interaction. In a nematic phase, this interaction reduces to a four-fermion interaction that is the basis of superconductivity. When the coupling constant is positive the superconductivity is a p-wave with spin-parallel–paired fermions. When it is negative the superconductivity is a p-wave and fermions are spin-antiparallel paired. For a system with zero chemical potential, even a very small coupling can bind fermions into a bound state that leads to superconductivity. When the chemical potential is non-zero the system possesses quantum critical transition from the normal spin-nematic phase to the phase where superconductivity coexists with spin-nematicity. The value of the quantum critical fermion-spin-nematicity coupling constant depends on the chemical potential.

Strain-Induced Spin-Nematic State and Nematic Susceptibility Arising from 2×2 Fe Clusters in KFe_{0.8}Ag_{1.2}Te_{2}.

Spin nematics break spin-rotational symmetry while maintaining time-reversal symmetry, analogous to liquid crystal nematics that break spatial rotational symmetry while maintaining translational symmetry. Although several candidate spin nematics have been proposed, the identification and characterization of such a state remain challenging because the spin-nematic order parameter does not couple directly to experimental probes. KFe_{0.8}Ag_{1.2}Te_{2} (K_{5}Fe_{4}Ag_{6}Te_{10}, KFAT) is a local-moment magnet consisting of well-separated 2×2 Fe clusters, and in its ground state the clusters order magnetically, breaking both spin-rotational and time-reversal symmetries. Using uniform magnetic susceptibility and neutron scattering measurements, we find a small strain induces sizable spin anisotropy in the paramagnetic state of KFAT, manifestly breaking spin-rotational symmetry while retaining time-reversal symmetry, resulting in a strain-induced spin-nematic state in which the 2×2 clusters act as the spin analog of molecules in a liquid crystal nematic. The strain-induced spin anisotropy in KFAT allows us to probe its nematic susceptibility, revealing a divergentlike increase upon cooling, indicating the ordered ground state is driven by a spin-orbital entangled nematic order parameter.

Hidden SU(2) symmetries, the symmetry hierarchy, and the emergent eightfold way in spin-1 quantum magnets

The largest allowed symmetry in a spin-1 quantum system is an $SU(3)$ symmetry rather than the $SO(3)$ spin rotation. In this work, we reveal some $SU(2)$ symmetries as subgroups of $SU(3)$ that, to the best of our knowledge, have not previously been recognized. Then, we construct $SU(2)$ symmetric Hamiltonians and explore the ground-state phase diagram in accordance with the $SU(3)\supset SU(2)\times U(1)$ symmetry hierarchy. It is natural to treat the eight generators of the $SU(3)$ symmetry on an equal footing; this approach is called the eight-fold way. We find that the spin spectral functions and spin quadrupole spectral functions share the same structure, provided that the elementary excitations are flavor waves at low energies, which serves as a clue to the eight-fold way. An emergent $S=1/2$ quantum spin liquid is proposed to coexist with gapful spin nematic order in one of the ground states. In analogy to quantum chromodynamics, we find the gap relation for hydrodynamic modes in quantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okubo formula.

Spontaneous spin-nematic ordering in a spin-chain system

The spontaneous onset of the spin-nematic order in spin-chain systems is studied. The ordering persists due to distortions of either magnetic ions, or ligands taking part in the superexchange between magnetic ions. The spin-nematic order yields a nonzero biaxial magnetic anisotropy for the spin chain, violating the U(1) symmetry. The influence of the external magnetic field and nonzero temperature is considered. We predict how the spin-nematic ordering can manifest itself in the temperature and magnetic field behavior of magnetoacoustic characteristics.

Effects of an attractive three-body interaction on a spin-1 Bose Hubbard model

We study the effects of an attractive three-body interaction potential on a spin-1 ultracold Bose gas using the mean field approach (MFA). For an antiferromagnetic interaction, the third Mott insulator (MI) lobe is predominantly affected, where it completely engulfs the second and the fourth MI lobes at large values of interaction strength. However, no significant change is observed beyond the fourth MI lobe. The formation of the spin singlet (nematic) MI phase and the different order of phase transitions to the superfluid phase have been carefully scrutinized with the help of spin eigenvalues and the spin nematic order parameter. In the ferromagnetic case, the phase diagram shows similar features to that of a scalar Bose gas. We have compared our results on the MFA phase diagrams for both types of interaction potential via a perturbation expansion in both cases.

The driving mechanism of the d-wave orbital order in the iron-based superconductors

We study the driving mechanism and the form of the orbital order in the electronic nematic phase of the iron-based superconductors (IBSs) within the random phase approximation of a 5-band model. We find the magnetic correlation energy of the system can be significantly improved when an orbital order of the d-wave form is spontaneously generated. On the other hand, the magnetic correlation energy increases as one introduce either an on-site or an extended s-wave orbital order. More specifically, we find that the on-site orbital order is disfavored by the Hund’s rule coupling and the extended s-wave orbital order is disfavored by the stripy magnetic correlation pattern in the IBSs. Our work indicates that the orbital order and the spin nematic order in the IBSs develop in a cooperative fashion in the electronic nematic phase of the IBSs and should be both understood as a component of a composite order.

Intertwined Vestigial Order in Quantum Materials: Nematicity and Beyond

A hallmark of the phase diagrams of quantum materials is the existence of multiple electronic ordered states, which, in many cases, are not independent competing phases, but instead display a complex intertwinement. In this review, we focus on a particular realization of intertwined orders: a primary phase characterized by a multi-component order parameter and a fluctuation-driven vestigial phase characterized by a composite order parameter. This concept has been widely employed to elucidate nematicity in iron-based and cuprate superconductors. Here we present a group-theoretical framework that extends this notion to a variety of phases, providing a classification of vestigial orders of unconventional superconductors and density waves. Electronic states with scalar and vector chiral order, spin-nematic order, Ising-nematic order, time-reversal symmetry-breaking order, and algebraic vestigial order emerge from one underlying principle. The formalism provides a framework to understand the complexity of quantum materials based on symmetry, largely without resorting to microscopic models.

Spin nematics next to spin singlets

We provide a route to generate nematic order in a spin-1/2 system. Unlike the well-known magnon-binding mechanism, our spin nematics requires neigher the frustration effect nor a spin polarization in a high field or in the vicinity of a ferromagnet, but instead appears next to the spin singlet phase. We start from a state consisting of a quantum spin-1/2 singlet dimer placed on each site of a triangular lattice, and show that inter-dimer ring exchange interactions efficiently dope the SU(2) triplets that itinerate and interact, easily driving a stable singlet state to either Bose Einstein condensates or a triplet crystal, some hosting a spin nematic order. A variety of roles the ring exchange serves include the generation of a bilinear-biquadratic interaction between nearby triplets, which is responsible for the emergent nematic order separated from the singlet phase by a first order transition.