Frustrated Spin Systems Research Articles

Noncollinear phase of the antiferromagnetic sawtooth chain

The antiferromagnetic sawtooth chain is a prototypical example of a frustrated spin system with vertex-sharing triangles, giving rise to complex quantum states. Depending on the interaction parameters, this system has three phases, of which the gapless noncollinear phase (for strongly coupled basal spins and loosely attached apical spins) has received little theoretical attention so far. In this work, we comprehensively investigate the properties of the noncollinear phase using large-scale tensor network computations which exploit the full SU(2) symmetry of the underlying Heisenberg model. We study the ground state both for finite systems using the density-matrix renormalization group (DMRG) as well as for infinite chains via the variational uniform matrix-product state (VUMPS) formalism. Finite temperatures and correlation functions are tackled via imaginary or real time evolutions, which we implement using the time-dependent variational principle (TDVP). We find that the noncollinear phase is characterized by a low-momentum peak and a diffuse tail for the apex-apex correlations. Deep into the phase, the pattern sharpens into a peak indicating a 90∘ spiral. The apical spins are soft and highly susceptible to external perturbations; they give rise to a large number of gapless magnetic states that are polarized by weak fields and cause a long low-temperature tail in the specific heat. The dynamic spin-structure factor exhibits additive contributions from a two-spinon continuum (excitations of the basal chain) and a gapless peak at k=π/2 (excitations of the apical spins). Small temperatures excite the gapless states and smear the spectral weight of the k=π/2 peak out into a homogeneous flat-band structure. Our results are relevant, e.g., for the material atacamite Cu2Cl(OH)3 in high magnetic fields. Published by the American Physical Society 2025

Auxiliary-field Monte Carlo method for frustrated spin systems

Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.

Strong Antiferromagnetic Interactions in the Binuclear Cobalt(II) Complex with a Bridged Nitroxide Diradical

A binuclear cobalt–radical complex formed by the reaction of Co(hfac)2·2H2O (hfac = hexafluoroacetylacetonate) with the 2,2-bis(1-oxyl-3-oxide-4,4,5,5-tetramethylimidazolinyl) biradical (BR) has been synthesized. The complex {(hfac)CoII(BN)CoII(hfac)} crystallizes in the triclinic space group P1¯ : C34H28Co2F24N4O12, a = 11.1513(5) Å, b = 12.8362(7) Å, c = 18.2903(8) Å, α = 103.061(1)°, β = 100.898(2)°, γ = 102.250(1)°, Z = 2. The compound consists of two non-equivalent pseudo-octahedral CoII ions, each bearing two hfac ancillary ligands bridged by the tetradentate bis-nitroxide (BN). The temperature dependence of the magnetic susceptibility indicates a strong antiferromagnetic exchange between each of the Co2+ ions and the nitroxyl biradical, as well as between the spins within the bridging ligand, forming a spin-frustrated system. Micro-squid investigations, performed on a single crystal of {(hfac)CoII(BN)CoII(hfac)}, reveal a peculiarity of the M(H) graph at temperatures below 0.4 K displaying a step that is a result of ground and first excited levels mixing by the applied magnetic field due to a small energy gap between them, as inferred from ab initio calculation. The latter was also carried out for two models of mononuclear Co2+ complexes in order to obtain a set of initial parameters for fitting the experimental magnetic curves using the Phi program. Moreover, direct CAS(12,10)/def2-TZVP calculations of the magnetic dependences χT(T) and M(H) were performed, which satisfactorily reproduced the experimental ones.

Density-matrix mean-field theory

Mean-field theories have proven to be efficient tools for exploring diverse phases of matter, complementing alternative methods that are more precise but also more computationally demanding. Conventional mean-field theories often fall short in capturing quantum fluctuations, which restricts their applicability to systems with significant quantum effects. In this article, we propose an improved mean-field theory, density-matrix mean-field theory (DMMFT). DMMFT constructs effective Hamiltonians, incorporating quantum environments shaped by entanglements, quantified by the reduced density matrices. Therefore, it offers a systematic and unbiased approach to account for the effects of fluctuations and entanglements in quantum ordered phases. As demonstrative examples, we show that DMMFT can not only quantitatively evaluate the renormalization of order parameters induced by quantum fluctuations, but can also detect the topological quantum phases. Additionally, we discuss the extensions of DMMFT for systems at finite temperatures and those with disorders. Our work provides an efficient approach to explore phases exhibiting unconventional quantum orders, which can be particularly beneficial for investigating frustrated spin systems in high spatial dimensions.

Investigation of possible spin frustration in isolated isosceles MnIIINiII2 Triangle: Experimental and theoretical studies

Spin frustration is a unique phenomenon in magnetic complexes, where if a spin fails to lower the total energy of the system, by orienting itself in relative to its adjacent spins, it is termed as frustrated. In this work, we discuss about the magnetic properties, and the possibility of spin frustration in our isolated [NiII2MnIII(hep)4(PhCO2)(N3)2] (Hhep = 2-(2-hydroxyethyl) pyridine) having an isosceles NiII2MnIII triangle. The triangular MnIIINiII2 core was held together by two μ3-alkoxide bridges. The magnetic properties of this complex were studied in detail. Variable temperature magnetic susceptibility data of the complex in the range of 1.8 to 300 K, indicates the presence of stronger intramolecular antiferromagnetic coupling between two NiII centres than between NiII and MnIII ions. Experimentally, the ground state of the complex was found to be a mixture of E(2,0) and E(1,1) states, suggesting our complex to be close to a spin frustrated system which was supported by ab initio calculations.

Quantum spin liquid signatures in monolayer 1T-NbSe2

Quantum spin liquids (QSLs) are in a quantum disordered state that is highly entangled and has fractional excitations. As a highly sought-after state of matter, QSLs were predicted to host spinon excitations and to arise in frustrated spin systems with large quantum fluctuations. Here we report on the experimental observation and theoretical modeling of QSL signatures in monolayer 1T-NbSe2, which is a newly emerging two-dimensional material that exhibits both charge-density-wave (CDW) and correlated insulating behaviors. By using scanning tunneling microscopy and spectroscopy (STM/STS), we confirm the presence of spin fluctuations in monolayer 1T-NbSe2 by observing the Kondo resonance as monolayer 1T-NbSe2 interacts with metallic monolayer 1H-NbSe2. Subsequent STM/STS imaging of monolayer 1T-NbSe2 at the Hubbard band energy further reveals a long-wavelength charge modulation, in agreement with the spinon modulation expected for QSLs. By depositing manganese-phthalocyanine (MnPc) molecules with spin S = 3/2 onto monolayer 1T-NbSe2, new STS resonance peaks emerge at the Hubbard band edges of monolayer 1T-NbSe2. This observation is consistent with the spinon Kondo effect induced by a S = 3/2 magnetic impurity embedded in a QSL. Taken together, these experimental observations indicate that monolayer 1T-NbSe2 is a new promising QSL material.

Prethermal Time-Crystalline Spin Ice and Monopole Confinement in a Driven Magnet.

Studies of systems far from equilibrium open up new avenues for investigating exotic phases of matter. A driven-dissipative frustrated spin system is examined in this study, and we suggest an out-of-equilibrium nonmagnetic phase where the spins do not order but adhere to the ice rule in space and establish a long-range crystalline order in time. In contrast to the conventional spin ice, the dynamics of monopoles is confined due to the nonequilibrium feature of our model. Possible experimental realizations of our model are discussed.

A Cu12 Metallacycle Assembled from Four C3-Symmetric Spin Frustrated Triangular Units

Assembling metallacycles with interesting topological arrangements is a critical task for chemists. We report here a novel dodecanuclear CuII compound, [{Cu3L(µ-N3)}4(Py)14]·2Py (Cu12) (where Py = pyridine and [H6L]Cl = tris(2-hydroxybenzylidine)triaminoguanidinium chloride, respectively), with the topology of a cycle accomplished by four two-connecting approximately flat C3-symmetric guanidine-based ligands. Each ligand affords three tridentate metal-binding cavities and the four node-to-node connections through single azido bridges are provided by pairs of metal centers. A theoretical investigation using CASSCF in addition to DFT calculations showed strong antiferromagnetic coupling within the Cu3-triangles, resulting in spin-frustrated systems. However, these calculations were not able to properly reproduce the very weak antiferromagnetic couplings between the triangle units, highlighting the challenge of describing the magnetic behavior of this compound.

Variational analysis in one and two dimensions of a frustrated spin system: chirality and magnetic anisotropy transitions

<abstract><p>We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system where the spin takes values on two disjoint circles of the 2-dimensional unit sphere. This analysis will be carried out first on a one-dimensional lattice and then on a two-dimensional lattice. The energy consists of the sum of a term that depends on nearest and next-to-nearest interactions and a penalizing term related to the spins' magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the $ \Gamma $-limit of scalings of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the $ \Gamma $-limit of a scaling of the energy, we study the geometric rigidity of chirality transitions.</p></abstract>

Spin Transport in Magnetically Ordered Systems: Ferromagnets, Antiferromagnets and Frustrated Systems

In this review, we outline the important results on the resistivity encountered by an electron in magnetically ordered materials. The mechanism of the collision between the electron and the lattice spins is shown. Experiments on the spin resistivity in various magnetic materials as well as the theoretical background are recalled. We focus on our works of 15 years of principally using Monte Carlo simulations. In these works, we have studied the spin resistivity in various kinds of magnetic systems ranging from ferromagnets and antiferromagnets to frustrated spin systems. It is found that the spin resistivity shows a broad peak at the transition temperature in systems with a second-order phase transition, while it undergoes a discontinuous jump at the transition temperature of a first-order transition. New results on the hexagonal-close-packed (HCP) antiferromagnet are also shown in extended detail for the Ising case in both the frustrated and non-frustrated parameter regions.

Skyrmions and hopfions in three-dimensional frustrated magnets

A model of an inversion-symmetric frustrated spin system is introduced which hosts three-dimensional extensions of magnetic Skyrmions. In the continuum approximation this model reduces to a non-linear sigma model on a squashed sphere which has a natural interpolating parameter. At one limit of the parameter the model reduces to a frustrated magnetic system earlier considered by Sutcliffe as a host to Hopfions, and in the other limit it becomes very similar to the 3D Skyrme model. To better understand the relation between Hopfions and 3D Skyrmions a model interpolating between the Faddeev-Niemi model and the Skyrme model is reconsidered and it is shown that energies of the solitons obey a linear BPS bound. The 3D Skyrmions in the frustrated magnetic model are found and compared to the rational map ansatz.

Melting of Magnetization Plateaus for Kagomé and Square-Kagomé Lattice Antiferromagnets

Unconventional features of the magnetization curve at zero temperature such as plateaus or jumps are a hallmark of frustrated spin systems. Very little is known about their behavior at non-zero temperatures. Here we investigate the temperature dependence of the magnetization curve of the kagome lattice antiferromagnet in particular at 1/3 of the saturation magnetization for large lattice sizes of up to N=48 spins. We discuss the phenomenon of asymmetric melting and trace it back to a combined effect of unbalanced magnetization steps on either side of the investigated plateau as well as on the behavior of the density of states across the plateau. We compare our findings to the square-kagome lattice that behaves similarly at low temperatures at zero field, but as we will demonstrate differently at 1/3 of the saturation magnetization. Both systems possess a flat one-magnon band and therefore share with the class of flat-band systems the general property that the plateau that precedes the jump to saturation melts asymmetrically but now with a minimal susceptibility that bends towards lower fields with increasing temperature.

Spin-induced strongly correlated magnetodielectricity, magnetostriction effect, and spin-phonon coupling in helical magnet Fe3(PO4)O3

Helical magnets are extremely promising, as they have fascinating magnetic, electric, and phononic properties. Here, we report a spectrum of simultaneously occurring and highly entangled intriguing phenomena induced by helical spin ordering in a noncentrosymmetric and spin-frustrated system ${\mathrm{Fe}}_{3}(\mathrm{P}{\mathrm{O}}_{4}){\mathrm{O}}_{3}$. These phenomena include magnetodielectric effect in the form of a frequency-independent pronounced dielectric peak, clear magnetostriction effect manifested as a dramatic downturn in the thermal variation of lattice parameters, and strong spin-phonon coupling (which displays a unique anomalous hardening and softening of various phonon modes) at temperatures as high as ${T}_{N}=163\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The observed dielectric peak is seemingly associated with a structural distortion via the strong magnetostriction effect.

Possible coexistence of short-range resonating valence bond and long-range stripe correlations in the spatially anisotropic triangular-lattice quantum magnet Cu2 (OH) 3NO3

We show that short-range resonating valence bond correlations and long-range order can coexist in the ground state (GS) of a frustrated spin system. Our study comprises a comprehensive investigation of the quantum magnetism on the structurally disorder-free single crystal of ${\mathrm{Cu}}_{2}{(\mathrm{OH})}_{3}{\mathrm{NO}}_{3}$, which realizes the $s$ = 1/2 Heisenberg model on a spatially anisotropic triangular lattice. Competing exchange interactions determined by fitting the magnetization measured up to 55 T give rise to an exotic GS wave function with the coexistence of the dominant short-range resonating valence bond correlations and weak long-range stripe order (ordered moment ${M}_{0}=|\ensuremath{\langle}{s}_{i}^{z}\ensuremath{\rangle}|\ensuremath{\sim}0.02$). At low temperatures, a first-order spin-flop transition is visible at $\ensuremath{\sim}$1--3 T. As the applied field further increases, another two magnetic field induced quantum phase transitions are observed at $\ensuremath{\sim}$14--19 and $\ensuremath{\sim}$46--52 T, respectively. Simulations of the Heisenberg exchange model show semiquantitative agreement with the magnetic-field modulation of these unconventional phases, as well as the absence of visible magnetic reflections in neutron diffraction, thus supporting the GS of the spin system of ${\mathrm{Cu}}_{2}{(\mathrm{OH})}_{3}{\mathrm{NO}}_{3}$ may be approximate to a quantum spin liquid. Our study establishes structurally disorder-free magnetic materials with spatially anisotropic exchange interactions as a possible arena for spin liquids.

Dzyaloshinskii-Moriya interaction induced magnetoelectric coupling in a tetrahedral molecular spin-frustrated system

We have investigated magnetoelectric coupling in the single-molecule magnet $\mathrm{Mn}_{4}\mathrm{Te}_{4}(\mathrm{P}\mathrm{Et}_{3})_{4}$ with tetrahedral spin frustration. Our density functional studies found that an electric dipole moment can emerge with various non-collinear spin orderings. The forms of spin-dependent dipole are determined and consistent with that in non-centrosymmetric magnets driven by the Dzyaloshinskii-Moriya interaction. Writing a parameterized spin Hamiltonian, after solving for eigenvalues and eigenstates we quantified the magnetoelectric coupling by calculating the thermal average of the electric and magnetic susceptibilities, which can be influenced by external magnetic and electric fields, respectively. The quadratic relations are expected to be observable in experiments.

Bridging cumulative and noncumulative geometric frustration response via a frustrated N -state spin system

The resolution of geometric frustration in systems with continuous degrees of freedom often involves a cooperative inhomogeneous response and superextensive energy scaling. In contrast, the frustration in frustrated Ising-like spin systems is resolved uniformly. In this work we bridge between these two extremes by studying a frustrated model composed of $N$-state spins, and varying $N$. The expected cooperative response, observed for large $N$, is strongly attenuated as $N$ is reduced, in a nontrivial way. Moderate $N$ values show unique topological-like phases not observed before in frustrated models.

Thermal Hall responses in frustrated honeycomb spin systems

We study geometrical responses of magnons driven by a temperature gradient in frustrated spin systems. While Dzyaloshinskii-Moriya (DM) interactions are usually incorporated to obtain geometrically nontrivial magnon bands, here we investigate thermal Hall responses of magnons that do not rely on the DM interactions. Specifically, we focus on frustrated spin systems with sublattice degrees of freedom and show that a nonzero Berry curvature requires breaking of an effective $PT$ symmetry. According to this symmetry consideration, we study the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{2}^{\ensuremath{'}}$ Heisenberg models on a honeycomb lattice as a representative example and demonstrate that magnons in the spiral phase support the thermal Hall effect once we introduce a magnetic field and asymmetry between the two sublattices. We also show that driving the magnons by a temperature gradient induces spin current generation (i.e., magnon spin Nernst effect) in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{2}^{\ensuremath{'}}$ Heisenberg models.

NMR evidence against a spin-nematic nature of the presaturation phase in the frustrated magnet SrZnVO(PO4)2

Using $^{31}$P nuclear magnetic resonance (NMR) we investigate the recently discovered presaturation phase in the highly frustrated two-dimensional spin system SrZnVO(PO$_4$)$_2$ [F. Landolt et al., Phys. Rev. B 104, 224435 (2021)]. Our data provide two pieces of evidence against the presumed spin-nematic character of this phase: i) NMR spectra reveal that it hosts a dipolar spin order and ii) the 1/$T_1$ relaxation rate data recorded above the saturation field can be fitted by the sum of a single-magnon term, exponential in the gap, and a critical second-order term, exponential in the triple gap, leaving no space for a nematic spin dynamics, characterized by a double-gap exponential. We explain the unexpectedly broad validity of the simple fit and the related critical spin dynamics.

Real-space imaging of phase transitions in bridged artificial kagome spin ice

In frustrated spin systems, magnetic phase transitions underpin the formation of exotic, frustration-driven magnetic phases. Of great importance is the ability to manipulate these transitions to access specific phases, which in turn provides a means to discover and control novel phenomena. Artificial spin systems incorporating lithographically fabricated arrays of dipolar-coupled nanomagnets that enable real-space observation of the magnetic configurations provide such an opportunity. In particular, the kagome spin ice is predicted to exhibit two phase transitions, one of which is to a low-temperature phase whose long-range ground-state order has not been observed experimentally. To achieve this ordered state, we change the global symmetry of the artificial kagome system by selectively tuning the near-field nanomagnet interactions through nanoscale bridges at the lattice vertices. By precisely tuning the interactions, we are able to quantify the influence of frustration on the phase transition, finding that the driving force for spin and charge ordering depends on the degeneracy strength at the vertex.

Gapless quantum spin liquid and global phase diagram of the spin-1/2 [formula omitted] square antiferromagnetic Heisenberg model

The nature of the zero-temperature phase diagram of the spin-1/2J1-J2 Heisenberg model on a square lattice has been debated in the past three decades, and it remains one of the fundamental problems unsettled in the study of quantum many-body theory. By using the state-of-the-art tensor network method, specifically, the finite projected entangled pair state (PEPS) algorithm, to simulate the global phase diagram of the J1-J2 Heisenberg model up to 24×24 sites, we provide very solid evidences to show that the nature of the intermediate nonmagnetic phase is a gapless quantum spin liquid (QSL), whose spin-spin and dimer-dimer correlations both decay with a power law behavior. There also exists a valence-bond solid (VBS) phase in a very narrow region 0.56≲J2/J1≤0.61 before the system enters the well known collinear antiferromagnetic phase. We stress that we make the first detailed comparison between the results of PEPS and the well-established density matrix renormalization group (DMRG) method through one-to-one direct benchmark for small system sizes, and thus give rise to a very solid PEPS calculation beyond DMRG. Our numerical evidences explicitly demonstrate the huge power of PEPS for highly frustrated spin systems. Finally, an effective field theory is also proposed to understand the physical nature of the discovered gapless QSL and its relation to deconfined quantum critical point (DQCP).