Schwinger-boson Mean-field Theory Research Articles

Schwinger-boson mean-field study of the spin- 1/2 J1−J2−Jχ model in the honeycomb lattice: Thermal Hall signature

We theoretically investigate, within Schwinger-boson mean-field theory, the transition from a gapped ${\mathbb{Z}}_{2}$ quantum spin liquid, in a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg spin-$\frac{1}{2}$ system in a honeycomb lattice, to a chiral ${\mathbb{Z}}_{2}$ spin-liquid phase under the presence of time-reversal symmetry-breaking scalar chiral interaction (with amplitude ${J}_{\ensuremath{\chi}}$). We numerically obtain a phase diagram of this ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{\ensuremath{\chi}}$ system, where different ground states are distinguished based on the gap and the nature of excitation spectrum, topological invariant of the excitations, the nature of spin-spin correlation, and the symmetries of the mean-field parameters. The chiral ${\mathbb{Z}}_{2}$ state is characterized by the nontrivial Chern number of the excitation bands and lack of long-range magnetic order, which leads to a large thermal Hall coefficient.

Schwinger boson theory of the J1, J2=J3 kagome antiferromagnet

We study the kagome antiferromagnet for quantum spin-1/2 with first J1, second J2 and third J3 neighbour exchanges, along the J2 = J3 = J line. We use Schwinger-boson mean-field theory for the precise determination of the phase diagram, and two different rewritings of the Hamiltonian to build an intuition about the origin of the transitions. The spin liquid obtained at J = 0 remains essentially stable over a large window, up to J = 1/3, because it is only weakly frustrated by the J term. Then at J = 1/2, the intermediate Z2 spin liquid condenses into a long-range chiral order because of the change of nature of local magnetic fluctuations. As a side benefit, our Hamiltonian rewriting offers an exact solution for the ground state of our model on a Husimi cactus.

Why the Schwinger boson mean field theory fails to describe the spin dynamics of the triangular lattice antiferromagnetic Heisenberg model?

We find that the Schwinger boson mean field theory (SBMFT) supplemented with Gutzwiller projection provides an exceedingly accurate description for the ground state of the spin- triangular lattice antiferromagnetic Heisenberg model (spin- TLHAF). However, we find the SBMFT fails even qualitatively in the description of the dynamical behavior of the system. In particular, the SBMFT fails to predict the Goldstone mode in the magnetic ordered phase. We show that the coherent peak in the two-spinon continuum in the presence of spinon condensate should not be interpreted as a magnon mode. The SBMFT also predicts incorrectly a gapless longitudinal spin fluctuation mode in the magnetic ordered phase. We show that these failures are related to the following facts: (1) spinon condensation fails to provide a consistent description of the order parameter manifold of the 120 degree ordered phase. (2) There lacks in the SBMFT the coupling between the uncondensed spinon and the spinon condensate, which breaks both the spin rotational and the translational symmetry. (3) There lacks in the SBMFT the rigidity that is related to the no double occupancy constraint on the spinon system. We show that such failures of the SBMFT is neither restricted to the spin- TLHAF nor to the magnetic ordered phase. We proposed a generalized SBMFT to resolve the first two issues and a new formalism to address the third issue.

Q = 0 order in quantum kagome Heisenberg antiferromagnet

We have studied the nearest neighbor Heisenberg model with added Dzyaloshinskii–Moriya interaction using Schwinger boson mean-field theory considering the in-plane component as well as out-of-plane component. Motivated by the experimental result of vesignieite that the ground state is in a Q = 0 long-range order state, we first looked at the classical ground state of the model and considered the mean-field ansatz which mimics the classical ground state in the large S limit. We have obtained the ground-state phase diagram of this model and calculated properties of different phases. We have also studied the above model numerically using exact diagonalization up to a system size N = 30. We have compared the obtained results from these two approaches. Our results are in agreement with the experimental result of the vesignieite.

Topological nematic spin liquid on the square kagome lattice

The ground state of the spin$-1/2$ kagome antiferromagnet remains uncertain despite decades of active research. Here we step aside from this debated question to address the ground-state nature of a related, and potentially just as rich, system made of corner-sharing triangles: the square-kagome lattice (SKL). Our work is motivated by the recent synthesis of a distorted SKL compound mentioned in [Morita & Tohyama, J. Phys. Soc. Japan 87, 043704 (2018)]. We have studied its spin$-1/2$ $J_{1}$-$J_{2}$ phase diagram with an unrestricted Schwinger boson mean-field theory (SBMFT). We show that, in addition of agreeing with previous studies, three original phases appear: two incommensurate orders and a topological quantum spin liquid with weak nematicity. The topological order is characterized by fluxes on specific gauge-invariant quantities and the phase is stable under anisotropic perturbations relevant for experiments. Finally, we provide dynamical structure factors of the reported phases that could be observed in inelastic neutron scattering.

Thermal-transport studies of kagomé antiferromagnets

Searching for the ground state of a kagomé Heisenberg antiferromagnet (KHA) has been one of the central issues of condensed-matter physics, because the KHA is expected to host spin-liquid phases with exotic elementary excitations. Here, we show our longitudinal () and transverse () thermal conductivities measurements of the two kagomé materials, volborthite and Ca kapellasite. Although magnetic orders appear at temperatures much lower than the antiferromagnetic energy scale in both materials, the nature of spin liquids can be captured above the transition temperatures. The temperature and field dependence of is analyzed by spin and phonon contributions, and large sample variations of the spin contribution are found in volborthite. Clear changes in are observed at the multiple magnetic transitions in volborthite, showing different magnetic thermal conduction in different magnetic structures. These magnetic contributions are not clearly observed in low- crystals of volborthite, and are almost absent in Ca kapellasite, showing the high sensitivity of the magnetic excitation in to the defects in crystals. On the other hand, a clear thermal Hall signal has been observed in the lowest- crystal of volborthite and in Ca kapellasite. Remarkably, both the temperature dependence and the magnitude of of volborthite are found to be very similar to those of Ca kapellasite, despite of about an order of magnitude difference in We find that of both compounds is well reproduced, both qualitatively and quantitatively, by spin excitations described by the Schwinger-boson mean-field theory applied to KHA with the Dzyaloshinskii–Moriya interaction. This excellent agreement demonstrates not only that the thermal Hall effect in these kagomé antiferromagnets is caused by spins in the spin liquid phase, but also that the elementary excitations of this spin liquid phase are well described by the bosonic spin excitations.

Rapid filling of the spin gap with temperature in the Schwinger-boson mean-field theory of the antiferromagnetic Heisenberg kagome model

Using Schwinger-boson mean-field theory, we calculate the dynamic spin structure factor at low temperatures $0<T\ll J$ for the spin-$1/2$ antiferromagnetic Heisenberg kagome model, within the gapped $\mathbb{Z}_2$ spin liquid phase Ansatz. We find that the spectral gap rapidly fills with temperature, with robust low-energy spectral weight developing by a temperature of $\Delta/3$, where the spin gap is $2\Delta$ (i.e., $\Delta$ is the spinon gap), before any appreciable rise in spinon density or change in zero-temperature mean-field parameters. This is due to deconfinement of spinons which leads to terms suppressed only by $\exp(-\Delta/T)$. At still higher temperatures, the spinon density increases rapidly leading to a breakdown of the Schwinger-boson mean-field approach. We suggest that if the impurity-free spectral functions can be obtained through neutron scattering experiments on kagome herbertsmithites, temperature dependence of the subgap weight can provide distinct signatures of a $\mathbb{Z}_2$ quantum spin liquid.

Failure of the Schwinger boson approach in the description of the ground state in the spatially anisotropic Heisenberg model

Density matrix renormalization group (DMRG) and Schwinger boson mean field theory (SBMFT) are used to study the behavior of the ground state energy in the integer spin two-dimensional anisotropic J 1x -J 1y -J 2 Heisenberg model in the square lattice with second neighbors interactions, where the influence of the critical parameters (D c vs. J 2 ) have been verified on continuum spin conductivity (AC conductivity). We get a different behavior for the ground state energy using both methods, principally in the limit where the force of the parameters is larger, where the SBMFT theory is expected not to become accurate.

Spin Thermal Hall Conductivity of a Kagome Antiferromagnet.

A clear thermal Hall signal (κ_{xy}) was observed in the spin-liquid phase of the S=1/2 kagome antiferromagnet Ca kapellasite [CaCu_{3}(OH)_{6}Cl_{2}·0.6H_{2}O]. We found that κ_{xy} is well reproduced, both qualitatively and quantitatively, using the Schwinger-boson mean-field theory with the Dzyaloshinskii-Moriya interaction of D/J∼0.1. In particular, κ_{xy} values of Ca kapellasite and those of another kagome antiferromagnet, volborthite, converge to one single curve in simulations modeled using Schwinger bosons, indicating a common temperature dependence of κ_{xy} for the spins of a kagome antiferromagnet.

Spin-Nernst effect in the paramagnetic regime of an antiferromagnetic insulator

We theoretically investigate a pure spin Hall current driven by a longitudinal temperature gradient, i.e., the spin Nernst effect (SNE), in a paramagnetic state of a collinear antiferromagnetic insulator with the Dzyaloshinskii-Moriya interaction. The SNE in a magnetic ordered state in such an insulator was proposed by Cheng et al. [R. Cheng, S. Okamoto, and D. Xiao, Phys. Rev. Lett. 117, 217202 (2016)]. Here we show that the Dzyaloshinskii-Moriya interaction can generate a pure spin Hall current even without magnetic ordering. By using a Schwinger boson mean-field theory, we calculate the temperature dependence of SNE in a disordered phase. We also discuss the implication of our results to experimental realizations.

Nematic quantum phases in the bilayer honeycomb antiferromagnet

The spin$-1/2$ Heisenberg antiferromagnet on the honeycomb bilayer lattice is shown to display a rich variety of semiclassical and genuinely quantum phases, controlled by the interplay between intralayer frustration and interlayer exchange. Employing a complementary set of techniques, comprising spin rotationally invariant Schwinger boson mean field theory, bond operators, and series expansions we unveil the quantum phase diagram, analyzing low-energy excitations and order parameters. By virtue of Schwinger bosons we scan the complete range of exchange parameters, covering both long range ordered as well as quantum disordered ground states and reveal the existence of an extended, frustration induced lattice nematic phase in a range of intermediate exchange unexplored so far.

Spontaneous dimerization and moment formation in the Hida model of the spin-1 kagome antiferromagnet

The Hida model, defined on honeycomb lattice, is a spin-1/2 Heisenberg model of aniferromagnetic hexagons (with nearest-neighbor interaction, $J_A>0$) coupled via ferromagnetic bonds (with exchange interaction, $J_F <0$). It applies to the spin-gapped organic materials, m-MPYNN.X (for $\mathrm{X}=$I, BF4, ClO4), and for $|J_F|\gg J_A$, it reduces to the spin-$1$ kagom\'e Heisenberg antiferromagnet (KHA). Motivated by the recent finding of trimerized singlet (TS) ground state for spin-1 KHA, we investigate the evolution of the ground state of Hida model from weak to strong $J_F/J_A$ using mean-field triplon analysis and Schwinger boson mean-field theory. Our triplon analysis of Hida model shows that its uniform hexagonal singlet (HS) ground state (for weak $J_F/J_A$) gives way to the dimerized hexagonal singlet (D-HS) ground state for all $J_F$'s beyond a moderate $J_F/J_A$. From the Schwinger boson calculations, we find that the evolution from the uniform HS phase for spin-1/2 Hida model to the TS phase for spin-1 KHA happens through two quantum phase transitions: 1) the spontaneous dimerization transition at $J_F/J_A \sim -0.28$ from the uniform HS to D-HS phase, and 2) the moment formation transition at $J_F/J_A \sim -1.46$, across which the pair of spin-1/2's on every FM bond begin to express as a bound moment that tends to spin-1 for large negative $J_F$'s. The TS ground state of spin-1 KHA is thus adiabatically connected to the D-HS ground state of the Hida model. Our calculations clearly imply that the \ce{m-MPYNN.X} salts realize the D-HS phase at low temperatures, which can be ascertained through neutron diffraction.

Schwinger-boson mean-field study of the J1−J2 Heisenberg quantum antiferromagnet on the triangular lattice

We use Schwinger boson mean field theory (SBMFT) to study the ground state of the spin-$S$ triangular-lattice Heisenberg model with nearest ($J_1$) and next-nearest ($J_2$) neighbor antiferromagnetic interactions. Previous work on the $S=1/2$ model leads us to consider two spin liquid Ans\"{a}tze, one symmetric and one nematic, which upon spinon condensation give magnetically ordered states with 120$^{\circ}$ order and collinear stripe order, respectively. The SBMFT contains the parameter $\kappa$, the expectation value of the number of bosons per site, which in the exact theory equals $2S$. For $\kappa=1$ there is a direct, first-order transition between the ordered states as $J_2/J_1$ increases. Motivated by arguments that in SBMFT, smaller $\kappa$ may be more appropriate for describing the $S=1/2$ case qualitatively, we find that in a $\kappa$ window around 0.6, a region with the (gapped $Z_2$) symmetric spin liquid opens up between the ordered states. As a consequence, the static structure factor has the same peak locations in the spin liquid as in the 120$^{\circ}$ ordered state, and the phase transitions into the 120$^{\circ}$ and collinear stripe ordered states are continuous and first-order, respectively.

Quantum spin liquid and magnetic order in a two-dimensional nonsymmorphic lattice: Considering the distorted kagome lattice of volborthite

The Kagome-lattice-based material, Volborthite, $\mathrm{Cu_3 V_2 O_7 (OH)_2 \cdot 2 H_2 O}$, has been considered as a promising platform for discovery of unusual quantum ground states due to the frustrated nature of spin interactions. Here we explore possible quantum spin liquid and magnetically ordered phases in a two-dimensional non-symmorphic lattice described by $p2gg$ layer space group, which is consistent with the spatial anisotropy of the spin model derived from density functional theory (DFT) for Volborthite. Using the projective symmetry group (PSG) analysis and Schwinger boson mean field theory, we classify possible spin liquid phases with bosonic spinons and investigate magnetically ordered phases connected to such states. It is shown, in general, that only translationally invariant mean field states are allowed in two-dimensional non-symmorphic lattices, which simplifies the classification considerably. The mean field phase diagram of the DFT-derived spin model is studied and it is found that possible quantum spin liquid phases are connected to two types of magnetically ordered phases, a coplanar incommensurate $(q,0)$ spiral order as the ground state and a closely competing coplanar commensurate $(\pi,\pi)$ spin density wave order. In addition, periodicity enhancement of the two-spinon continuum, a signature of symmetry fractionalization, is found in the spin liquid phases connected to the $(\pi,\pi)$ spin density wave order. We discuss relevance of these results to recent and future experiments on Volborthite.

Chiral Spin Liquid on a Kagome Antiferromagnet Induced by the Dzyaloshinskii-Moriya Interaction.

The quantum spin liquid material herbertsmithite is described by an antiferromagnetic Heisenberg model on the kagome lattice with a non-negligible Dzyaloshinskii-Moriya interaction (DMI). A well-established phase transition to the q=0 long-range order occurs in this model when the DMI strength increases, but the precise nature of a small-DMI phase remains controversial. Here, we describe a new phase obtained from Schwinger-boson mean-field theory that is stable at small DMI, and which can explain the dispersionless spectrum seen in the inelastic neutron scattering experiment by Han etal. [Nature (London) 492, 406 (2012)NATUAS0028-083610.1038/nature11659]. It is a time-reversal symmetry breaking Z_{2} spin liquid, with the unique property of a small and constant spin gap in an extended region of the Brillouin zone. The phase diagram as a function of DMI and spin size is given, and dynamical spin structure factors are presented.

Schwinger boson mean-field theory of the kagome Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions

We have studied the effect of the Dzyaloshinskii-Moriya interaction on kagome Heisenberg anti-ferromagnet using Schwinger boson mean field theory (SBMFT). Within SBMFT framework, Messio et al had argued that the ground state of kagome antiferromagnet is possibly a chiral topological spin liquid (Phys. Rev. Lett. 108, 207204 (2012)). Thus, we have computed zero-temperature ground state phase diagram considering the time-reversal breaking states as well as fully symmetric Ans\"atze. We discuss the relevance of these results in experiments and other studies. Finally, we have computed the static and dynamic spin structure factors in relevant phases.

Anomalous spin disordered properties of strongly correlated honeycomb compound In3Cu2VO9

We study the ground-state and finite-temperature magnetic properties of an interlayer frustrated J1 − J2 − Jc Heisenberg model on three-dimensional honeycomb lattice by employing the Schwinger boson mean-field theory, focusing on the low-energy physics in In3Cu2VO9. We find that with the increase of interlayer coupling Jc from 0 to 3.6 meV, the interlayer frustrated system transits from an antiferromagnetic (AFM) phase to a state with intralayer AFM order and interlayer disorder. This spin disordered phase explains not only the intralayer phase transition at TN = 38 K, but also the qualitative behaviors of the intermediate-temperature specific heat and magnetic susceptibility of In3Cu2VO9.

Spin structure factors of chiral quantum spin liquids on the kagome lattice

We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\bm{k} \to -\bm{k}$ symmetry as well as the six-fold rotation symmetry around the $\Gamma$-point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the \emph{cuboc1} state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.

Schwinger boson spin-liquid states on square lattice

We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z_2 gauge field excitations. We classify them by the projective symmetry group method, and find six spin liquid states that are potentially relevant to the J_1-J_2 Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wavevector at Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.

Schwinger boson mean field perspective on emergent spins in diluted Heisenberg antiferromagnets

Using an adaptation of Schwinger boson mean field theory (SBMFT) for nonuniform systems, we study the nature of low-energy spin excitations on the square and Bethe lattice at their percolation threshold. The optimal SBMFT parameters are interpreted as on-site potentials and pairing amplitudes, which enables an explanation of why emergent local moments develop in this system on dilution [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); H.J. Changlani et al., ibid 111, 157201 (2013)] and why the corresponding single particle frequencies are driven to anomalously low values. We discuss how our mean field calculations suggest the strong link between the presence of sublattice imbalance and long range antiferromagnetic order, and why linear spin wave theory is inadequate for capturing this relation. Within the SBMFT framework, we also extract an energy scale for the interaction between emergent moments, which show qualitative agreement with many-body calculations.