^{†}Corresponding author. Email: sunqing@cnu.edu.cn
^{‡}Corresponding author. Email: andrewjee@sina.com
^{*}Project supported by the National Natural Science Foundation of China (Grant Nos. 11347197, 11404225, and 11474205).
Spinwave excitation plays important roles in the investigation of the magnetic phases. In this paper, we study the spinwave excitation spectra of twocomponent Bose gases with spin–orbit coupling in a deep square optical lattice using the spinwave theory. We find that, while the excitation spectrum of the vortex crystal phase is gapless with a linear dispersion in the vicinity of the minimum point, the spectra of the commensurate spiral spin phase and the skyrmion crystal phase are gapped. Significantly, the spin fluctuations strongly destabilize the classical ground state of the skyrmion phase with the appearance of an imaginary part in the eigenfrequencies of spin excitations. Such features of the spin excitation spectra provide further insights into the exotic spin phases.
Typical spin– orbit coupling (SOC) of electrons in a solid state system is due to the relativistic effect. For instance, it can be derived from the Dirac equation of fermions by expanding in terms of the small parameter v/ c ≈ 0.01, which is the ratio of electron speed to light speed in solids. Therefore, the spin– orbit interaction is much weaker than the Coulomb interaction and is generally neglected in the study of condensed matter physics. However, it has been recently noticed that the SOC plays a crucial role in leading to the socalled topological insulating states in solids.^{[1, 2]} Consequently, it attracts many physicists’ attention. In particular, with successful realization of the synthetic gauge field in ultracold quantum gases, ^{[3– 7]} it is discovered that the spin– orbit coupling may also be important in determining the properties of these systems.^{[8– 26]} For example, the SOC can produce various new types of ground states, such as the stripes, the halfvortex phases in the Bose– Einstein condensates, and a bounded state, which is called Rashbon in the Fermi gases.^{[27– 29]} This makes research on SOC in ultracold gases vigorous.
In addition, ultracold gases in optical lattices have been used to simulate a wide range of condensedmatter phenomena, ^{[30, 31]} such as the Mott metal– insulator transition and the effects of strong correlation in the Hubbard model.^{[32]} As is known, at half filling, the low energy physics of these systems can indeed be described by an effective antiferromagnetic spin Hamiltonian in the deep Mott regime.^{[33– 36]} Therefore, the ground states are generally Neel ordered and the corresponding lowlying excitations are antiferromagnetic magnons.
Then, a question naturally arises: What effects will be induced if the SOC exists in ultracold gases loaded in an optical lattice? Recently, there has been considerable interest in exploring the intriguing physics in spin– orbit coupled optical lattices.^{[37– 49]} Many researchers have analyzed the phase diagrams of the spin– orbit coupled lattices in the deep Mott regime.^{[43– 47]} It was shown that one can derive an additional Dzyaloshinskii– Moriya (DM) type of superexchange interaction^{[50, 51]} by using the secondorder perturbation theory. Furthermore, by applying the classical Monte– Carlo simulations, several exotic spin textures, whose existences were previously explored in various solid state materials with the DM interaction, ^{[52– 54]} were also found in ultracold gases.
However, these investigations are not complete in theory. Indeed, up to now, the effects of spin fluctuations on the stabilities of the exotic spin textures in these phases have not been discussed yet. As is well known, strong spin fluctuations in lowdimensional systems can qualitatively change their phase diagrams outlined by the classical simulations. Therefore, one has to consider these fluctuations in order to determine the stability of each possible phase. In the following, we shall proceed to calculate the lowlying spectra of spin fluctuations in these phases. As a result, we are able to determine the parameter region in which each corresponding phase is stable against the spin fluctuations.
We implement the spinwave method^{[55– 59]} to derive the global magnetic phase diagram of the superexchange Hamiltonian. In particular, we calculate the spinwave excitation spectra for the XYferromagnet phase, the spiral spin phase, and the spin vortex phase, whose existence has been studied by the previous classical Monte Carlo simulations.^{[43, 44]} We also discuss the stability of a novel phase, the socalled skyrmion crystal phase in bosonic ultracold gases. We show that the possible existing regions for the skyrmion crystal phase are greatly shrunk by the spin fluctuations. More precisely, we find that the spin excitation spectrum of the commensurate spiral spin phase has two minima, which are symmetrical with respect to the k_{y} axis in the momentum space. In addition, it possesses a gap. On the other hand, the spin fluctuations strongly destabilize the classical ground state of the skyrmion phase with the appearance of an imaginary part in the eigenfrequencies of spin excitations. Given that the above exotic spin configurations have also been discovered and attracted great interest, ^{[52– 54]} such features of the spin excitation spectra also provide insights into these spin phases.
This paper is organized as follows. In Section 2, we derive the effective superexchange spin model for the systems of bosonic gases on a square optical lattice. In Section 3, we present the general formalism of the spin wave theory. Then, in Section 4, we analyze the phase diagram and the excitation spectrum of each abovementioned phase in detail. Finally, in Section 5, we summarize our main results and discuss some related issues with experiments.
Let us consider a Bose gas loaded in a twodimensional square optical lattice. The atoms are characterized by two nearlydegenerate states, which are produced by a hyperfine interaction. In the following, for convenience, we shall use the language of (pseudo)spin to describe the local occupancy of these states. Furthermore, we take the synthetic spin– orbit interaction into consideration. Then, the Hamiltonian of the bosonic gas can be written as
Here
As is well known, the filling factor of bosons is also an important parameter to determine the properties of ultracold gases in an optical lattice. In the following, we shall only consider the case of one boson per site on average. Now, we can perform the standard Schrieffer– Wolf transformation H_{eff} = e^{iS}H e^{− iS}^{[61]} to the Hamiltonian in the largeU limit U_{σ σ ′ } ≫ t. It produces an effective Hamiltonian
which captures the lowenergy physics of the original system.^{[43, 44, 62]} Here,
We would like to emphasize that these interactions favor qualitatively different types of spin orders. In fact, the Heisenberg Hamiltonian gives either the ferromagnetic or the antiferromagnetic ordering, while the DM interaction makes the spiral spin ordering possible. In other words, there exists a competition between them. Consequently, their interplay may drive the system to various exotic phases. In the following, we shall first determine the possible existence of several phases in the system, and then analyze their stabilities by applying the spinwave theory.
In the previous works, some authors have explored the classical ground states of Hamiltonian (2) by the classical Monte Carlo simulations.^{[43, 44]} They showed that the classical ground states of this system support a variety of phases, such as the XYferromagnetic (XYFM) phase, the longitudinal ferromagnetic (ZFM) and antiferromagnetic (ZAFM) phases, the spiral spin phase, the 3× 3 skyrmion crystal (SkX) phase, and the 2× 2 spin vortex (VX) phase, as the coupling constants of the Hamiltonian are changed. However, they did not consider the stabilities of these phases. In order to address this issue, one has to take the strong spin fluctuations of the system into consideration, as we will do in the following.
To start with, we shall first calculate the ground state energy of each abovementioned phase in some specified regions of parameters by the meanfield theory and see if the previous phase diagram can be reestablished. Then, we compute the secondorder correction to the energies of these phases. This correction is due to the spinwave excitations. In some cases, it can render the phase derived by the classical methods invalid. More precisely, as we shall show in the following, the excitation spectrum of a specified phase may develop an imaginary part and hence, becomes a complex function of momentum in some parameter regions. It implies that the presumed phase is actually unstable against the spin fluctuations in these regions. To illustrate the procedure, let us take the 2× 2 VX phase for example.
In the first step, we perform the following transformation:
where i denotes the spin primitive cell of the specified phase and μ is the sublattice index, as shown in Fig. 1(a). The matrix Â _{μ } reads
It represents the rotation of each spin in the Hamiltonian from the local z axis to a new direction (θ , ϕ ), as shown in Fig. 1(b).
Next, we apply the Holstein– Primakoff transformation^{[63]} by substituting the spin operators
Here, the spin angles θ _{μ } and ϕ _{μ } are determined by the spin configurations of the specified meanfield ground state. For example, in the 2× 2 VX phase, we have θ _{1} = θ _{2} = θ _{3} = θ _{4} = π / 2, ϕ _{1} = 3π / 4, ϕ _{2} = 7π / 4, ϕ _{3} = π / 4, and ϕ _{4} = 5π / 4. In addition, (μ , ν ) and (α , β ) denote two pairs of nearestneighbor sites in the primitive cell along the
by keeping only the terms of twoboson operators in the expansion of the transformed effective Hamiltonian. Here, W_{μ }, M_{μ , μ ′ }, and N_{μ , μ ′ } are the coefficient matrices depending on the meanfield spin configuration. The explicit expressions of the matrices are given in Appendix A. Then, by using the Fourier transformation, we are able to rewrite H_{2} as
where
Furthermore, after substituting the Bogoliubov transformation into Eq. (6), we determine the transformation matrix
Consequently, by diagonalizing H_{MN}Λ with an invertible matrix
In the following section, we will study the possible existence and stability of the XYFM phase, ZFM phase, ZAFM phase, spiral spin phase, 3× 3 SkX phase, and 2× 2 VX phase in ultracold gases on a square optical lattice by the aboveoutlined procedure. As is shown below, although this procedure can be easily applied to most of these phases, it is rather involved for the 3× 3 skyrmion crystal phase. The main difficulty is due to the complicated spin configuration of this phase. It contains nine sublattices. In other words, according to the local spin orientations in the 3× 3 skyrmion phase, the whole lattice can be divided into nine separate sublattices. In each of them, all the localized spins point in a specified direction. A primitive spin cell of this phase is shown in Fig. 2(b).
First, we compute the groundstate energy H_{0} for each possible spin configuration listed in the last section and determine the phase diagram by choosing the lowest energy configurations in different parameter regions. Then, we compare our results with the previous ones derived by the classical Monte Carlo calculations. Our phase diagram is shown by the solid lines in Fig. 2(a). It is in good agreement with the Monte– Carlo simulations given in Ref. [43].
Next, we calculate the spinwave excitation spectrum of each phase and see if it is stable against the spin fluctuations. In fact, we find that the ZAFM and 2× 2 VX phases are stable. However, the 3× 3 SkX phase is pronouncedly affected by the spin fluctuations. Actually, in two parameter regions, i.e., the white areas in Fig. 2(a), the spinwave excitation spectral functions have imaginary parts and hence, render the corresponding phases unstable. On the left side of the commensurate 4× 1 spiral phase, the unstable parameter region implies the emergence of other commensurate spiral orders or incommensurate spiral phases. While the right unstable parameter region may support new states, which need to be further investigated in future. Now, let us analyze the spin excitation spectrum of each phase in a more detailed manner.
We first consider the weak spin– orbit coupling regime in which the parameter α in Table 1 is small. In this case, the Heisenberg interaction in Hamiltonian (2) is dominant and the DM interaction can be treated as a perturbation. In particular, if we set α = 0, the effective Hamiltonian becomes the standard ferromagnetic XXZ model on a twodimensional square lattice.^{[64]} The phase diagram of this model is for λ < 1, and all the spins are aligned in the XY plane. More precisely, the system is in the XYFM phase. In this phase, the Holstein– Primakoff transformation can be easily applied. A direct calculation yields
Next, we take the effect of the DM interaction into consideration. We find that the excitation spectrum remains gapless and linear near ∣ k∣ ∼ 0 for the XYFM phase. However, it becomes asymmetric with respect to k_{x} = 0, as shown by the dashed line of λ = 0.8 and α = 0.1π in Fig. 3. A similar dispersion occurs along the k_{y} axis. In the meantime, the gap of the ZFM phase is decreased upon the increased strength of the SOC. However, both the spin excitation spectra of the XYFM and ZFM phases will become imaginary as the SOC strength α is further increased. It indicates that these phases are destabilized. This region of parameters is characterized by the white area on the left side of the spiral phase in Fig. 2(a).
In the middle region of Fig. 2(a), the DM interaction is strong, comparing with the Heisenberg interaction. The competition between these interactions leads to a spiral spin phase. To determine the stability of this phase, we consider a commensurate 4× 1 periodic structure with all the spins lying in the z– y plane. Notice that, in this configuration, the angles θ _{i} between the spins and the z axis are specified by θ _{1} = θ _{2}, θ _{3} = θ _{4}, and θ _{1} + θ _{3} = π .
Then, we calculate the excitation spectrum of this phase by the spinwave analysis. Our results are shown in Figs. 4(a) and 4(e). One can easily see that there exists a spin gap in the diagram. In other words, the Goldstone mode is absent in this phase. Moreover, the gap has two symmetric minima at (± k_{0}, 0), as shown in Fig. 4(a). It is due to the fact that the spin configuration of the spiral spin phase is symmetric with respect to the y axis. By varying parameter α , we find that the energy gap gradually closes as the DM interaction increases and the excitation spectrum becomes linear in the vicinity of these minimal points. At the same time, the 4× 1 periodic structure becomes unstable. It suggests the emergence of other commensurate spiral orders or incommensurate spiral phases.
Next, let us consider the large spin– orbit coupling regime around α = π /2, in which the spinconserved hopping of particles is suppressed. The Wilson loop which is characterized by
Finally, we study the 3× 3 skyrmion crystal phase, which is located in the phase diagram between the 4× 1 spiral spin phase and the 2× 2 VX phase. As is mentioned at the end of Section 3, the spin configuration of this phase consists of nine sublattices, which are characterized by an upspin at the center and others lying in the xy plane. By the spinwave theory, we calculate its spin excitation spectrum. Our results are shown in Figs. 4(b) and 4(f). Obviously, the spectrum also has an energy gap. Interestingly, we find that the spin fluctuations dramatically destabilize this classical ground state and make the skyrmion crystal phase unstable in part of the classical phase diagram determined by the Monte Carlo simulations, ^{[43]} where the eigenfrequencies of spin excitations exhibit an imaginary part. It may imply the appearance of a novel state in the concerned region. However, more specific description on the possible new states in this regime is beyond the current spinwave analysis, and one may use more sophisticated unbiased quantum Monte– Carlo simulations.
We study the exotic magnetic phases of twocomponent Bose gases with spin– orbit coupling on a deep square optical lattice by using the spinwave theory. In particular, we systematically investigate the spinwave excitation spectra for the XYferromagnet, spiral spin, spin vortex, and skyrmion crystal phases. We find that while the excitation spectrum of the vortex crystal phase has no gap and is linear in the vicinity of the center of the Brillouin zone, the spectra of the commensurate spiral spin phase and the skyrmion crystal phase possess energy gaps. We also discuss the stability of the novel skyrmion crystal phase and show that the possible existing regions for the skyrmion crystal phase are greatly shrunk by the spinwave excitations. In Ref. [44], several methods for detecting spin textures in the Mott insulator phase have been proposed, such as noise correlations in timeofflight (TOF) images and optical Bragg scattering. Finally, we would like to emphasize that similar spin textures, which are discussed in the present work, can also be induced by the DM type of interactions in some solid state materials. Therefore, we believe that the current investigation should provide further insights for the study of these materials.
We would like to thank Dr. Zhang X F for many helpful discussions.
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