Magnetic-induced chiral dynamics in an extended two-leg bosonic ladder

The realization and detection of chiral physics with ultracold atomic gases provide a unique path for the exploration of topological phases. Here, we show that the interplay of magnetic field and interacting particles in an extended two-leg ladder leads to rich chiral Bloch dynamics. Considering both the on-site contact interaction and nearest-neighbor interactions, the ground state and Bloch dynamics of the system are studied analytically and numerically. When the system is in the ground state, the threshold and phase diagram for the transition between zero-momentum state and plane-wave state are analytically obtained, showing the nearest-neighbor interactions along the legs and rungs have opposite impact on the ground state transition, providing new opportunity to manipulate the ground state transition. When the ladder is perturbated under an external linear force, chiral dephasing of Bloch oscillations (BOs), i.e. symmetry breaking damped BOs (the damping rate of BOs on one leg is larger than the other), are observed. This chirality is absent for vanishing the magnetic field and atomic interaction. Particularly, the chirality of damped BOs is inversed when the magnetic field (chiral current) is inversed. In addition, the damping of BOs induced by different types of atomic interactions is different, and the strength and damping rate of BOs initialized in different ground states are distinct, offering dynamic ways to detect the different ground states. Furthermore, the persistent chiral Bloch oscillations observed in single particle case is predicted analytically, which is a crucial requirement for observation and application of chiral BOs in nonlinear regime. Our results provide an interesting path towards the exploration of topological atomic superfluids.


Introduction
The interplay between the magnetic field and the optical lattices provides an ideal playground to investigate the fundamental many-body quantum physics, quantum Hall effects and topological insulators.Particularly, the realization of artificial gauge field for neutral atomic gases in optical lattices offers a new experimental paradigm for studying these exotic states of matter in clean and well-controlled environments [1][2][3][4][5][6].The revealing and detection of novel physics in magnetic lattices with ultracold atomic gases is at the frontier of current theoretical and experimental research.However, because of the high dimensionality, it is a challenge to study the physics of ultracold atomic gases in magnetic lattices.Recently, by using the superlattice and laser-induced tunneling techniques or the synthetic dimension method, the magnetic bosonic/fermionic ladder system is successfully realized [7][8][9][10][11].The two-leg ladder has become the simplest model to study the coupled effects of orbital magnetic field and optical lattices in a low dimensional system and has proven to be an important model both in condensed matter [12][13][14][15] and high-energy physics [16].It is shown that the two-leg model can reproduce the main features of magnetic lattice systems, such as the Hofstadter butterfly spectrum and the chiral edge states of the associated Chern insulating phase [8].
In recent years, a great attention is paid to study the ground state of the two-leg ladder system and rich quantum phase transitions are observed, such as: Meissner phase, Vortex phase, superfluid-Mott insulator New J. Phys.26 (2024) 053047 L-L Mi et al

Model and variational approach
The ground state and chiral dynamics in an extended two-leg bosonic magnetic ladder subject to an artificial magnetic field α piercing the each unit cell are studied.The model diagram is shown in figure 1.With a weak additional harmonic trap and a weak linear force applied along the legs, the Hamiltonian of the system with both on-site contact and nearest-neighbor interactions can be written as [7,23,33]: Here, the ân,σ (â † n,σ ) (σ = L, R) represents the bosonic annihilation (creation) operator at the site n in the left or right leg of the ladder.The tunneling strengths along the legs and rungs of the ladder are given by J and K, respectively.Ũ represents the on-site contact interaction of atoms.Ṽ1 and Ṽ2 are nearest-neighbor interactions along the legs and rungs.The on-site contact interaction and the nearest-neighbor interactions determine the main physical properties of the system.If the next nearest-neighbor interaction is considered, the qualitative characteristics of the system remain the same [58].So we only consider on-site contact interaction, nearest-neighbor interactions along the legs and rungs of the ladder.ω represents the strength of the external harmonic trap, and F is the external linear force.h.c is the complex-conjugate expression.In order to avoid the significant change of band dispersion, we consider the case of weak linear force ( F ≪ 1).Here, the harmonic trap is used to prepare the ground state of the system in absence of the external linear force.When the ground state is prepared, the harmonic trap can be removed adiabatically and the linear force can be adiabatically turned on.Then the BOs can be excited.This is in accordance with the experimental studying of the ground state and dynamics of the system.All parameters of the system can be adjusted within a certain range.K can be controlled in an experiment by adjusting the intensity of the laser that produces the lattice potential [7].Ũ can be tuned by the Feshbach resonance technique.The Ṽ1,2 can be tuned by changing the polarizing angle of the external fields relative to the plane of the ladder [23,33].Note that, under the strong atomic interaction, the Mott insulating phase will occur.Here, we consider the weak atomic interactions, and the Mott insulating phase is not considered.The effective magnetic field α can be turned by changing the wavelength of the traveling wave beam or the angle between them.According to experiment [7], we set 0 < K/ J < 4 and 0 < α < π.In addition, we regard the degree of freedom on both legs of the ladder as pseudospin, the system has the same properties as the spin-orbit coupled system [38].By mean-field approximation ân,σ ≃ ⟨â n,σ ⟩ = ψ n,σ , here, the ψ n,σ represents the order parameter of the condensate, and the dimensionless Hamiltonian (1) can be written as: where K = K/ J, U = Ũ/ J, V 1,2 = Ṽ1,2 / J, ω = ω/ J and F = F/ J.The time t has been rescaled by h/ J. Using the Heisenberg equation of motion idψ n,σ /dt = ∂H/∂ψ * n,σ , the nonlinear Schrödinger equation representing the dynamics of the system can be obtained: To study the ground state and chiral dynamics of the system by using the variational method, we choose the Gaussian trial wave function as follows where, ξ σ represents the wave packet center-of-mass position in the corresponding legs, R σ and η σ represent the width of the wave packet and the change rate of the width, p σ is the related momentum of wave packets in the corresponding legs, ϕ represents the phase difference between two legs, s is the population difference between the two legs (the degrees of freedom of the left and right legs represent pseudospin).All of these variational parameters are time dependent.'±' corresponding σ = L or R. The Gaussian trial wave function ( 4) is normalized as ´dn ) . The particle number difference between the two legs is expressed as s = ´dn When all particles transport to the left leg, s is close to 1, while all particles transport to the right leg, s is close to −1, so s satisfies −1 < s < 1.We can use the variational method to get the differential equation of motion from the Lagrange density Here the asterisk represents the complex conjugate, and the dot represents the first derivative with respect to time t.To simplify the calculation, we assume where, the effective Hamiltonian is given by To obtain L and H given by equations ( 5) and ( 6), we replace the sum over n by integral.By solving the Euler-Lagrange equation, d dt ∂L ∂ q − ∂L ∂q = 0, where q(t) = {ξ, R, p, η, s, ϕ}, the dynamic equation of the system can be obtained: The equations ( 7) and ( 8) describe the center-of-mass and momentum of the condensate.Equations ( 9) and ( 10) describe the evolution of wave packets in the process of centroid motion.We note that R oscillates around a mean value (approximate the initial wave packet width R 0 ).The equations ( 11) and ( 12) represent the phase difference and particle number difference between the two legs of the ladder in the process of dynamic evolution.This means the spin dynamics of the system is governed by equations ( 11) and (12).The equations ( 7)- (12) indicate that the spin dynamics and BOs are mutually coupled by magnetic field.In the following work, we mainly study the ground state properties of non-tilted systems and the chiral dynamics of tilted systems.

Ground state transition
Firstly, we study the ground state properties of the non-tilted magnetic ladder system under a weak harmonic trap ω = 0.0025.In this case, F = 0, the ground state of ( 5)-( 12) is q 0 .We assume the center-of-mass position of the condensate is at the origin of the coordinate (ξ 0 = 0).In addition, using quasi-stationary solutions q0 = 0, we obtain ϕ 0 = 0, η 0 = 0, p 0 = arctan(s 0 tan α) and 2e Here, equation ( 14) can be satisfied with s 0 = 0 or f = 0. s 0 = 0 leads to p 0 = 0 , that is, the distribution of particles on two legs of the ladder is the same, and the system is in a zero-momentum state.s 0 ̸ = 0 leads to p 0 ̸ = 0, that is, the distribution of particles on the two legs of the ladder is different, and the system is in the plane-wave state.So the ground state transition of the system is determined by equation ( 14).The ground state wave packet radius R 0 and spin-polarization s 0 can be determined by equations ( 13) and ( 14).Then the critical condition of ground state transition between the zero-momentum state and plane-wave state can be obtained by s 0 = 0 and f = 0, which leads to the threshold When K > K c , the system is in zero-momentum state, while for K < K c , the system will be in plane-wave state.Equation (15) shows that the competition among magnetic field strength α, atomic interactions (on-site contact interaction U and nearest-neighbor interactions V 1,2 ) and rung-to-leg coupling ratio K induces the ground state transition.That is, the ground states of the system can be controlled by changing the above parameters.Particularly, it can be seen from equation ( 15) that the nearest-neighbor interactions along the legs V 1 and rungs V 2 have the opposite effect on the ground state transition.For fixed α, a repulsive coupling between the legs V 2 prevents the transition from zero-momentum state to plane-wave state, while repulsive on-site contact interaction U and nearest-neighbor interactions V 1 promote the transition from zero-momentum state to plane-wave state.
In addition, the transition of the ground state of the system can also be reflected by the change of the energy band structure.The ground state energy of the system can be expressed as: where the initial spin polarization of the particle satisfies the relation s 0 = tan p 0 / tan α.Then equation (16) shows the dispersion relation of the system.The dispersion relation depends on the rung-to-leg coupling ratio, the magnetic flux, and the atomic interactions strength.The phase transition properties and chiral characteristics of the system can also be characterized by chiral currents.We define the chiral current as [35,59] where the local current on the leg is expressed as j n,σ = i(e ∓iα ψ † n+1,σ ψn,σ − h.c).The energy band structure and chiral current are shown in Fig 2 .Given U, K and α, we plot the energy band structure at different V 1 and V 2 in figures 2(a) and (b).It can be find that the V 1 and V 2 have different influence on the energy band structure.With the decreases (increases) of V 1 (V 2 ), the energy band gradually evolves from one minimum to two minimums (see figures 2(a) and (b)).For a fixed K, U and V 1,2 , when α increases, the energy band evolves from one minimum to two minimums (see figure 2(c)).A parabolic structure with only one  minimum in the energy spectrum represents the zero-momentum state (s 0 = 0, p 0 = 0) and the double well structure with two minimums represents the plane-wave state (s 0 ̸ = 0, p 0 ̸ = 0).These two minimums are symmetric about p 0 = 0. Figure 2(d) shows the relationship between the chiral current and the magnetic field under different ground states, which clearly reflects the ground state transition.The chiral current has a inflection point at the phase transition point α = α c , i.e. the chiral current |j c | increases (decreases) with α in the zero-momentum state (plane-wave state).In the zero-momentum state, the chiral current is not affected by K, but in the plane-wave state, the chiral current increases with K. Interestingly, the chiral current reverses at α = 0.When the magnetic field is positive, the chiral current j c > 0, when the magnetic field is negative, the chiral current j c < 0. This indicates that the chiral dynamics of the system can be controlled by the magnetic field.
The ground state diagrams in K − V 1,2 and α − V 1,2 planes given by equation ( 15) are plotted in figure 3. It can be clearly seen that the ground state of the system can be changed by changing the rung-to-leg coupling ration, the strength of the magnetic field, and the nearest-neighbor interactions.The transition from zero-momentum state to plane-wave state occurs with decreasing V 1 (K) or increasing V 2 (α).That means the nearest-neighbor interaction along the legs V 1 promotes the system to be in zero-momentum state, while the nearest-neighbor interaction along the rungs V 2 promotes the system to be in plane-wave state.V 1 and V 2 have opposite effects on the ground state transition.The nearest-neighbor atomic interactions provide more opportunities to manipulate the ground state phase.

The chiral Bloch oscillations and spin dynamics in tilted magnetic ladder
When the external harmonic trap is removed (ω = 0) and the magnetic ladder system is tilted (F ̸ = 0), the presence of nonzero F breaks the discrete translation symmetry and leads to a variation of the central quasi-momentum p.Since the group velocity changes sign when the quasi-momentum exceeds the Brillouin zone, the wave packet will show an oscillatory behavior in real space, BOs occur.The BOs are intraband dynamics with weak F, and it ensures that the particles do BOs at the lowest energy band.According to equation ( 8), the momentum of the particle satisfies the classical motion equation Because the spin and momentum of the particles are locked by the magnetic field, the change of momentum will also cause the change of spin dynamics.These dynamic characteristics can directly distinguish different states of the system.At this time, the external linear force plays the role of restoring force.Equation (18) shows that spin-momentum locking persists during the BOs.The ideal BOs can be obtained in absence of interatomic interaction.The atomic interaction can lead to dephasing and broadening of the condensate in momentum space, which causes dephasing and damping of BOs in coordinate space.The oscillation of center-of-mass of the condensate will be damped and the damping rate is proportional to the strength of atomic interaction [50,52].Therefore, reducing the interaction strength can slow down the decoherence of the condensate, and the damping of BOs will also be weakened.Here, we consider the weakly repulsive interatomic interactions to more obviously observe the BOs (0 < U < 1 and 0 < V 1,2 < 1).In order to study the BOs started at different ground states, the fourth-order Runge-Kutta method is used to direct numerical simulation of equation ( 3).Initially, the ground state wave packet ( 4) is initialized at n = 0 lattice site and the harmonic trap is adiabatically removed (linearly removed), while F is adiabatically turned on.The oscillatory mode of the Gaussian wave packet started at zero-momentum and plane-wave states are given by figures 4 and 5.In the two-leg ladder system, the degrees of freedom of the left and right legs are equivalent to pseudo-spin.The BOs are accompanied by spin dynamics.The spin and momentum of the particles are strongly coupled through the magnetic field (see equation (18)).This efficient spin-orbit interaction locks spin and momentum and leads to obvious chiral characteristics.Interestingly, the interplay of magnetic field and atomic interactions results in some new chiralities: (1) The chirality induced by spin-momentum locking observed in single particle case still exist, i.e. when the wave packet moves away from (goes back to) the initial position, most of atoms mainly populate the right (left) leg.(2) Due to the influence of the atomic interactions, the dephasing and destruction of BOs take place when t > t c .Particularly, the destruction of the BOs on two legs is asymmetric.When t > t c , on the right leg, the wave packet moves away from the initial position to the maximum displacement with strong dephasing, while the wave packet on the left leg goes back to the initial position with weak dephasing.Chiral destruction of BOs takes place.We will shown in figure 6 that this chirality will reverse as the magnetic field reverse and disappear in the absence of the magnetic field and atomic interactions.(3) The destruction of BOs by U, V 1 and V 2 is different.The destruction caused by the nearest-neighbor interaction V 1 along the legs is the strongest, while the destruction caused by the nearest-neighbor interaction V 2 along the rungs is the weakest (see figures 4(a2), (b2), (c2) and 5(a2), (b2), (c2)).( 4) Comparing the figures 4 and 5, we find that the BOs started at two different ground states have distinct characters.When the atoms move back to the initial position, the BOs started in the plane-wave state stays long time at the initial position.Particularly, the amplitude of BOs in plane-wave state is smaller than that in the zero-momentum state, and the destruction of BOs in plane-wave state is more early and significant.
Those chiral features provide dynamical ways to distinguish the different ground states of the system.
To have a deep insight into the physical mechanism of the chiral BOs observed in figures 4 and 5, we study the BOs of the condensates expressed by the dynamics of center-of-mass of wave packets.The the system will also reverse with α, i.e. when the magnetic field α is negative, the chiral destruction of BOs will be reversed, the destruction of BOs on the left leg is stronger than that on the right leg.To verify this, we perform numerical simulation of equation ( 3) with α = 0 and α = −0.3π.The results are shown in figure 6.As α = 0, the particles are uniformly distributed on two legs throughout, and the destruction and damping of BOs on two legs are exactly the same (see figures 6(b1)-(b3)).No chirality observed.In particular, when the sign of the magnetic field is opposite, the destruction rate of BOs on two legs is reversed (see figures 6(a1)-( a2) and (c1)-( c2)).This further illustrates that the magnetic field is the key factor leading to the chiral reversal of the system.The above numerical results are completely consistent with our theoretical analysis.We conclude that the magnetic field results in the chiral destruction of the BOs.
The difference of the amplitudes and damping rate of BOs started in two ground states can be understood as follows.As predicted by equations ( 19) and ( 20), we expect that the maximum spin polarization s in two ground states should be different, resulting in the distinct properties of BOs.Figures 4-6 indicate that the damping rate of BOs inversely to the BOs, which depends on α.In order to obtain the maximum of BOs and spin polarization, we assume R = R 0 .Then, from equation (7) and equations ( 11) and ( 12), we have where p 0 = arctan(s 0 tan α).Equation (21) shows that the amplitude of BOs depends on the external linear force F, magnetic field strength α, and initial polarization s 0 .This further shows that the amplitude of BOs started at the zero-momentum (p 0 = 0) and plane-wave (p 0 ̸ = 0) states are different.Particularly, in the plane-wave state, the amplitude of BOs on the two legs is different.The analytical solution of the spin polarization can be obtained by linearizing equations ( 11) and (12).Spin dynamics is the superposition of driving oscillation induced by the linear force F and inherent oscillation induced by the coupling along the rungs of the ladder K.The driving oscillation plays a dominant role, so we ignore the inherent oscillation.The maximum amplitude of spin polarization can be obtained as equation (22).Equation (22) shows that the magnetic field α is the key factor to excite spin dynamics and the amplitude of spin polarization is related to the rung-to-leg coupling ration K, the external linear force F and initial polarization s 0 .That means the amplitude of spin polarization started at the zero-momentum (s 0 = 0) and plane-wave (s 0 ̸ = 0) states are also different.This is clearly shown in figure 7. ξ max decreases with α and F. s max increases with α and it tends to saturation in the plane-wave state.Larger spin polarization in plane-wave state results in weak BOs with strong damping rate (see equation (20)).s max is not affected by F. In the zero-momentum state, ξ max is not affected by K, and in the plane-wave state, K has a weak effect on ξ max .s max decreases with K. Particularly, as α increases, ξ max and s max have a inflection point at the phase transition point, which indicates that ξ max and s max can be used as additional parameters to identify the phase transition.
In order to further verify our theoretical predictions, we provide numerical simulation of the dynamics of the center-of-mass of the wave packet as ξ σ (t) = ´dn(ψ * n,σ nψ n,σ )/ ´dn(ψ * n,σ ψ n,σ ).The spin dynamics of the system is expressed as s(t) = N L − N R .The effects of different parameters on the Bloch dynamics and spin dynamics of the system are discussed in detail.The results are shown in figures 8 and 9.The BOs are accompanied by spin dynamics.In presence of atomic interaction, the BOs are damped oscillations.The damping caused by the nearest-neighbor interaction along the legs is the largest, while the nearest-neighbor interaction along the rungs contributes the least to the damping (see figures 8(b1)-(b3) and 9(b1)-( b3)).The damping rate of BOs on two legs is different, i.e., the damping rate on right leg is stronger than that on left leg.This phenomenon reverses with the magnetic field (see the dotted lines shown in figures 8(d3), (e3) and 9(d3), (e3)), where, the damping rate of BOs on left leg is stronger than that on right leg when α < 0. Accordingly, antiphase oscillation of spin polarization s takes place.F and α have strong effect on the amplitude of BOs, which are decrease with F and α, while K has very weak effect.Comparing figures 8 and 9 one finds that the amplitude (damping rate) of BOs in plane-wave state is smaller (larger) than that in zero-momentum state.Particularly, the amplitude of BOs on two legs is different in plane-wave state, while it is the same in zero-momentum state.Interestingly, the spin dynamics in plane-wave state is stronger than that in zero-momentum state.The spin dynamics is enhanced with the increase of α, i.e., s max increases with α, which is more obvious in the zero-momentum state (see figure 8(f3)).s max reaches its maximum and is saturated in plane-wave state, where the spin mixing dynamics is the strongest (see figures 9(f1)-(f3)).The strong spin mixing dynamics is accompanied with strong damping of BOs.Those results are well predicted by equation (20), where the damping rate depends on s.The analytical results and the numerical results are in good agreement.

The long-lived chiral Bloch oscillations
As discussed above, the atomic interaction leads to the damping and destruction of BOs, which limits the observation and application of BOs.When considering the atomic interaction, can long-lived BOs be achieved in the magnetic ladder system?The equation (20) shows that for fixed R, the damping rate is proportional to D and depends on U, V 1,2 and s, where the interaction between atoms plays a dominant role.The damping rate induced by U, V 1 and V 2 is different.In order to completely eliminate the damping of the system, it is necessary to adjust the values of U and V 1,2 so that they can compensate each other, and then a long-lived BOs can be achieved.So, setting D = 0, the condition of long-lived BOs can be obtained Equation ( 23) is the compensation relationship required to achieve long-lived BOs. Figure 10 is the phase diagram in the U − V 2 plane for achieving long-lived BOs.The white area does not satisfy the compensation relationship (23), where the damping of BOs appears.This phenomenon has been discussed in detail in the previous section.Here, we focus on long-lived BOs in shadow region.In the case of zero magnetic field, the spin polarization is zero, and only at the black dotted line a long-lived BOs can be achieved.In order to verify the theoretical analysis, numerical simulation of equation ( 3) is carried out by taking the parameters made in the shadow area of figure 10, and the results are shown in figure 11.When the condition ( 23) is satisfied, the effective atomic interaction is cancelled.The long-lived chiral BOs on both two legs are obtained.In this case, L-L Mi et al    the spin-momentum locking induced chirality of BOs still persist, i.e. when the wave packet moves away from (goes back to) the initial position, most of atoms mainly populate the right (left) leg.However, the asymmetric BOs on two legs observed in figures 4-9 disappears.The amplitudes of BOs on two legs are the same in both zero-momentum and plane-wave states.We conclude that the coupling of atomic interaction and magnetic field results in the asymmetric BOs.However, the BOs started in two states are still different.The BOs amplitude started in the zero-momentum state is larger than that in plane-wave state, while the spin dynamics started in the plane-wave state is stronger than that in the zero-momentum state (see figures 11(a1)-( d1) and (a2)-(d2)).The numerical results obtained from equation ( 3) are in full agreement with the analytical results, which fully proves the accuracy of the compensation condition (23).

Conclusions
Two-leg bosonic magnetic ladder, representing the coupled effects of orbital magnetic field and optical lattices, offers an ideal model to study the chiral physics and spin dynamics in ultracold atomic system.However, the previous investigations about the chiral dynamics of magnetic ladder mainly focused on the single particle case.Here, we show that the interplay of magnetic field and interacting particles leads to rich chiral Bloch and spin dynamics.Considering both the on-site contact interaction and the nearest-neighbor interactions, the ground state and Bloch dynamics of the extended interacting two-leg bosonic magnetic ladder are studied analytically and numerically.When the system is in a quasi-steady state, the threshold and phase diagram of ground states is analytically obtained.The physical mechanism for inducing and manipulating the ground state transition is revealed explicitly.Under the perturbation of an external linear force, the interplay of atomic interactions and magnetic field leads to chiral population of two legs, resulting chiral damped Bloch oscillations and spin dynamics.This chirality strongly depends on the ground state and spin dynamics, which can be effectively manipulated by atomic interactions and magnetic field, offering dynamic way to detect the ground state.Furthermore, we show that, when the on-site contact interaction, the nearest-neighbor interactions and spin polarization satisfy a threshold, the persistent chiral BOs take place, which is a crucial requirement for observation and application of BOs.Our results have potential application for the exploration of topological atomic insulating phases and topological atomic superfluids.

Figure 1 .
Figure 1.The sketch of the two-leg ladder system with on-site contact (U) and nearest-neighbor interactions (V1, V2) in the presence of a uniform magnetic flux α.Where L and R represent the left and right legs of the ladder respectively.

Figure 2 .
Figure 2. Energy band structure and chiral current of the ladder system.(a)-(c) with K = 1.5.The other system parameters are U = 0.5 and ω = 0.0025.

Figure 7 .
Figure 7.The variation of the maximum amplitude of BOs and spin polarization with α, F and K. K = 1.5 in the first column, and F = 0.2 in the second column.The other system parameters are U = V1 = V2 = 0.5.

Figure 8 .
Figure 8.The time evolution of wave packet center ξσ and spin polarization s under different parameters started at zero-momentum state.

Figure 9 .
Figure 9. Same as in figure 6 but in the regime of the plane-wave state.

Figure 10 .
Figure 10.The phase diagram for emergence of long-lived BOs in the U − V2 plane.

Figure 11 .
Figure 11.Temporal evolutions of the atomic density for oscillatory mode of a Gaussian wave packet and spin polarization started at the zero-momentum state (The first column, α = 0.2π) and plane-wave state (The second column, α = 0.3π).Here, on-site contact interaction and nearest-neighbor interactions satisfy the compensation relationship(23).V1 = −0.7,V2 = 0.72, U = 0.913 and F = 0.2.The other system parameters are K = 1.5 and F = 0.2.