The Schwinger boson theory provides a natural path for treating quantum spin systems with large quantum fluctuations. In contrast to semiclassical treatments, this theory allows us to describe a continuous transition between magnetically ordered and spin liquid states, as well as the continuous evolution of the corresponding excitation spectrum. The square lattice Heisenberg antiferromagnet is one of the first models that was approached with the Schwinger boson theory. Here we revisit this problem to reveal several subtle points that were omitted in previous treatments and that are crucial to further develop this formalism. These points include the freedom for the choice of the saddle point (Hubbard-Stratonovich decoupling and choice of the condensate) and the $1/N$ expansion in the presence of a condensate. A key observation is that the spinon condensate leads to Feynman diagrams that include contributions of different order in $1/N$, which must be accounted for to get a qualitatively correct excitation spectrum. We demonstrate that a proper treatment of these contributions leads to an exact cancellation of the single-spinon poles of the dynamical spin structure factor, as expected for a magnetically ordered state. The only surviving poles are the ones arising from the magnons (two-spinon bound states), which are the true collective modes of an ordered magnet.
The Schwinger boson theory provides a natural path for treating quantum spin systems with large quantum fluctuations. In contrast to semiclassical treatments, this theory allows us to describe a continuous transition between magnetically ordered and spin liquid states, as well as the continuous evolution of the corresponding excitation spectrum. The square lattice Heisenberg antiferromagnet is one of the first models that was approached with the Schwinger boson theory. Here we revisit this problem to reveal several subtle points that were omitted in previous treatments and that are crucial to further develop this formalism. These points include the freedom for the choice of the saddle point (Hubbard-Stratonovich decoupling and choice of the condensate) and the 1/N expansion in the presence of a condensate. A key observation is that the spinon condensate leads to Feynman diagrams that include contributions of different order in 1/N, which must be accounted for to get a qualitatively correct excitation spectrum. We demonstrate that a proper treatment of these contributions leads to an exact cancellation of the single-spinon poles of the dynamical spin structure factor, as expected for a magnetically ordered state. The only surviving poles are the ones arising from the magnons (two-spinon bound states), which are the true collective modes of an ordered magnet.
Large-S and large-N theories (spin value S and spinor component number N) are complementary, and sometimes conflicting, approaches to quantum magnetism. While large-S spin-wave theory captures the correct semiclassical behavior, large-N theories, on the other hand, emphasize the quantumness of spin fluctuations. In order to evaluate the possibility of the nontrivial recovery of the semiclassical magnetic excitations within a large-N approach, we compute the large-S limit of the dynamical spin structure factor of the triangular lattice Heisenberg antiferromagnet within a Schwinger boson spin representation. We demonstrate that, only after the incorporation of Gaussian (1/N) corrections to the saddle-point ($N=\ensuremath{\infty}$) approximation, we are able to exactly reproduce the linear spin-wave theory results in the large-S limit. The key observation is that the effect of 1/N corrections is to cancel out exactly the main contribution of the saddle-point solution, while the collective modes (magnons) consist of two spinon bound states arising from the poles of the random phase approximation propagator. This result implies that it is essential to consider the interaction of the spinons with the emergent gauge fields and that the magnon dispersion relation should not be identified with that of the saddle-point spinons.
We compute the zero temperature dynamical structure factor S(q,ω) of the triangular lattice Heisenberg model using a Schwinger Boson approach that includes the Gaussian fluctuations (1/N correction) of the saddle point solution. While the ground state of this model exhibits a well-known 120∘ magnetic ordering, the excitation spectrum revealed by S(q,ω) has a strong quantum character that is not captured by low-order 1/S expansions. The low-energy magnons consist of two spinon bound states confined by gauge fluctuations of the auxiliary fields, which can produce internal structure of the magnon peaks. In addition, the high-energy continuum of spinon modes extends up to about three times the single-magnon bandwidth.
Schwinger boson perturbation theory for spin-S Kitaev-Heisenberg magnets: phase diagram and dynamical response near Kitaev spin liquids
We develop a Schwinger boson perturbative framework for the spin-S Kitaev-Heisenberg model. The spin-liquid saddle point of the pure Kitaev model is used as the unperturbed state, and magnetic instabilities and dynamical spin correlations are evaluated around this saddle point. We decompose the Hamiltonian exactly into two parts by exploiting the Klein duality intrinsic to the Kitaev-Heisenberg model. We perform the random-phase approximation by taking the part invariant under the duality transformation as the unperturbed term and treating the remaining term, which changes sign under it, as the perturbation. We determine the phase boundaries separating the quantum spin-liquid regimes from the adjacent magnetically ordered phases for S=1/2, 1, 3/2, and 2. The spin-liquid regions shrink rapidly with increasing S and become very narrow at S=2. We also compute the spin dynamics of the spin-S Kitaev-Heisenberg model, focusing on S=1, and find that the dressed dynamical spin structure factor retains a broad two-spinon continuum at finite energies, while the low-energy spectral weight softens at the ordering wave vectors of the adjacent magnetic phases. Our framework thus enables thermodynamic-limit calculations of spin dynamics near magnetic instabilities.
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-12triangular lattice antiferromagnetic Heisenberg model (spin-12TLHAF). 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-12TLHAF 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.
The Kitaev model is an exactly solvable model that exhibits a quantum spin liquid as its ground state. While this model was originally proposed as an S = 1 / 2 spin model on a honeycomb lattice, extensions to higher-spin systems have recently attracted attention. In contrast to the S = 1 / 2 case, such higher- S models are not exactly solvable and remain poorly understood, particularly for spin excitations at finite temperatures. Here, we focus on the S = 1 Kitaev model, which has been proposed to host bosonic quasiparticles. We investigate this model using Schwinger boson mean-field theory, where bosonic spinons are introduced as fractional quasiparticles by extending bond operators to address anisotropic spin interactions. We determine the mean-field parameters that realize a quantum spin liquid in both ferromagnetic and antiferromagnetic Kitaev models. Based on this mean-field ansatz, we calculate the dynamical and equal-time spin structure factors. We find that, when one uses the conventional decoupling scheme based on Wick decoupling with respect to spinons to calculate spin correlations, the resultant spin structure factors exhibit a momentum dependence that is not consistent with the sign structure expected from the exchange interaction: They possess a strong spectral weight indicating ferromagnetic (antiferromagnetic) correlations in the antiferromagnetic (ferromagnetic) Kitaev model. To resolve this issue, we propose an alternative scheme for evaluating the spin correlations, which is based on decoupling with respect to the bond operators. We demonstrate that, in our scheme, this discrepancy is removed, and the momentum dependence of the spin structure factors is consistent with the sign of the exchange constant. We also calculate the temperature evolution of the dynamical spin structure factor and find that the continuum observed at zero temperature splits into two distinct structures as the temperature increases, which can be understood in terms of the bandwidth narrowing of spinons. Finally, we clarify the origin of why the two distinct decoupling schemes result in different momentum dependences of the spin structure factors and discuss their relationship to results obtained in previous studies.
Large-N SU(4) Schwinger boson theory for coupled-dimer antiferromagnets
We develop a systematic large-N expansion based on the Schwinger boson representation of SU(4) coherent states of dimers for the paradigmatic spin-1/2 bilayer square lattice Heisenberg antiferromagnet. This system exhibits a quantum phase transition between a quantum paramagnetic state and a Néel order state, driven by the coupling constant g=J′/J, which is defined as the ratio between the inter-dimer J′ and intra-dimer J exchange interactions. We demonstrate that this approach accurately describes static and dynamic properties on both sides of the quantum phase transition. The critical coupling constant gc≈0.42 and the dynamic spin structure factor reproduce quantum Monte Carlo results with high precision. Notably, the 1/N corrections reveal the longitudinal mode of the magnetically ordered phase along with the overdamping caused by its decay into the two-magnon continuum. The present large-NSU(N) Schwinger boson theory can be extended to more general cases of quantum paramagnets that undergo a quantum phase transition into magnetically ordered states.