The ground state, localization, and chiral edge dynamics of a three-legged bosonic magnetic ladder

Three-legged bosonic/fermionic magnetic ladder, reproducing the main features of magnetic lattice systems, is an ideal model to study the edge-bulk coupling and chiral edge dynamics, which are a hallmark of quantum Hall physics. Here, the ground state transition, localization, and chiral edge (bulk) states of an interacting three-legged bosonic magnetic ladder are studied analytically and numerically. When the system is in a quasi-steady state, using variational analysis, the threshold for the transition from zero momentum state to plane wave state is obtained. The energy spectrum, the ground state diagram, the chiral current of the system are presented, and the chiral current reversal at the state transition point is observed. Furthermore, the localization and its stability in the system are discussed, and rich localized phenomena (diffusion, breather, soliton, and self-trapping) are predicted. Stable soliton/breather prefers to form an edge state, while the self-trapping is favorite to form a bulk state, i.e. localized edge and bulk states are obtained. Particularly, for the unstable soliton in the ground state or metastable state, different kinds of chiral edge/bulk state and edge-bulk coupling are observed. The stability of the localized states, the edge-bulk coupling characteristics, and the chirality of the system depend on the energy band structure of the system. Additionally, a controllable transition between localized edge state and bulk state is realized by quenching the soliton state. We proposed a theoretical evidence to design and manipulate edge-bulk coupling and different kinds of localized/chiral edge (bulk) states in three-legged bosonic magnetic ladder.

Recently, as a quasi-one dimensional optical lattice, n-leg ladder system under artificial magnetic field has attracted much attention both experimentally and theoretically.The magnetic ladder can be realized with many ways in the cold atom system in the experiment.Among them, superlattice and laser-induced tunneling techniques have been used to realize the bosonic ladder system under an artificial magnetic field [28,29].The two-legged magnetic ladder has become an ideal physical model for studying orbital magnetic field effects in low-dimension lattice system, showing novel quantum ground state phases, including superfluids, Mott insulators, vortex lattices, charge-density waves, vortex density waves, and biased ladder phases, etc [30][31][32][33][34][35].Based on the dynamic research of the magnetic ladder system, it is shown that there are also abundant nonlinear dynamic characteristics in this system, e.g.chiral Bloch-Zener dynamics, tight-binding approximation [51], the Hamiltonian for a three-legged bosonic magnetic ladder with leg length l x is ( Here â † l,σ and âl,σ represent the creation and annihilation operators in the lth site respectively, nl,σ = â † l,σ âl,σ is the particle number operator.J and K represent the tunneling strengths along the legs and rungs of the ladder, respectively.ϕ is the magnetic flux of each unit cell, and H.c. stands for the conjugate term.g is the strength of the interatomic interaction, here we study the case with repulsive interatomic interaction (i.e.g > 0).We label these three legs with σ = {−1, 0, 1} and consider each leg has l x sites.Experimentally, the tunneling strength between the rungs K can be tuned by changing the intensity of the laser that generates the optical lattice.The tunneling strengths along the legs J and magnetic flux ϕ can be tuned by adjusting the amplitude and angle of the Raman laser, respectively.Feshbach resonance [52] technique can be used to adjust the strength of the interatomic interaction intensity g.Considering the three-leg degrees of freedom of the ladder as pseudospins, this system is equivalent to a pseudospin-coupled system with spin-1.
In the mean field approximation, a l,σ = ⟨â l,σ ⟩, the Hamiltonian (1) is reduced to According to the Heisenberg equation of motion, i.e. ihda l,σ /dt = ∂H/∂a * l,σ , the discrete nonlinear Schrödinger equation of the system can be obtained as: where K = K/ J and g = g/ J.Note that the time here is redefined as t → [h/ J]t.The interatomic interaction g, the rung-to-leg coupling ratio K, and the magnetic field ϕ all affect the ground state and dynamics of the system.Variational method is well suited for studying the ground state and nonlinear dynamics of the system.We use the following Gaussian trial wave function: Here, s z (t) and s(t) denote spin tensor and average spin polarization , respectively, which can both be observed in experiments [53].α(t) and β(t) are variational parameters of the related phases of the upper and lower pseudospin components (i.e.σ = ±1).R σ (t) and p σ (t) represent the width and momentum of the wave packet, respectively.δ σ (t) and ξ σ (t) represent variety rates of the width and the center-of-mass position of the wave packet, respectively.
The Lagrangian density of the system is defined as Here, R σ , s z and s determine the atomic density of the wave packets on three legs.Even though we assume R 1 = R 0 = R −1 , the particle number on three legs still can be manipulate by parameters s z and s.In addition, we focused on the localized states such as breather, soliton, and self-trapping.In this case, the localization of the three legs is synchronized, i.e. ξ 1 = ξ 0 = ξ −1 , p 1 = p 0 = p −1 , and δ 1 = δ 0 = δ −1 also hold.Therefore, our assumptions are reasonable.Particularly, the key purpose of our variational method with Gaussian trial wave function (4) under this simplification is to obtain the critical conditions for ground-state transition and localization transition.The actual dynamics of the system is obtained by direct numerical simulation of equation ( 3).As will shown below, the theoretical predictions are well confirmed by numerical simulations, which further confirms the rationality and the generality of the results.
Based on this assumption, equation ( 4) is substituted into the Lagrangian density equation, and we have where s z ] − s sin ϕ sin p, and the effective Hamiltonian is Note that, to obtain equations ( 5) and ( 6), we replace the sums over l with integrals.This is reasonable when R > 1.Through the analysis of Euler-Lagrangian equation d(∂L/∂ qi )/dt = ∂L/∂q i , the differential equations of motion related to the variational parameters q i = {R, ξ, p, δ, s, s z , α, β} are calculated: Equation ( 7) characterizes the evolution of the momentum of the atoms, clearly showing that the momentum keeps the initial momentum constant during the evolution (i.e.p = p 0 ).Equations ( 8)- (10) represent the evolution of the width and center-of-mass position of the wave packet.Equations ( 11)- (14) gives the spin dynamics of the system.As will be discussed in the following sections, equations ( 7)-( 14) well describe the ground state and nonlinear dynamics of the system.

The energy spectrum and ground state transition
As discussed in the following section, our system described by equations ( 7)-( 14) have both extended state and localized state, which depends on initial Hamiltonian and the energy spectrum of the system.In order to better understand the dynamics of the system, we first study the energy band structure of the system.We assume that initially the system is in a quasi-steady state, that is, qi = 0.So we set p = p 0 , s = s 0 , s z = s z0 , R = R 0 , and β 0 = 0, α 0 = 0, δ 0 = 0, ξ 0 = 0. From equation (9), i.e. ξ = 0, the momentum of the wave packet maintains the initial value of p 0 = arctan[ −s0 sin ϕ 1+(cos ϕ−1)sz 0 ]. s 0 and s z0 satisfy α = 0 and β = 0, i.e. s 0 and s z0 are determined by 4e −γ sin ϕ sin then, the initional energy of the system is According to equations ( 15)-( 17), the energy band structure of the system can be numerically obtained.As shown in figure 2, given different g and K values, we plot the energy band structure at different ϕ values.We can find that the rung-to-leg coupling ratio K and the magnetic flux ϕ have great influence on the energy band structure.When K < K ⋆ = 5.4, with the increase of ϕ, the energy band structure changes from single-well to triple-well, and then gradually to double-well structure [as shown in figures 2(a), (b) and (d), (e)].When K ⩾ K ⋆ = 5.4, the energy band structure changes from a single-well to a double-well structure as ϕ increases [as shown in figures 2(c) and (f)].For energy bands with an energy minimum at p 0 = 0, the atoms should condense in the zero momentum state.When the energy minimum is at two non-zero p 0 , the energy band is degenerate and symmetric around p 0 = 0, and the condensate is in a plane wave state.With the increase of ϕ, the band structure gradually transitions from zero momentum state to plane wave state.
To further characterize the phase transition properties and dynamics of the system, we establish observable chiral current, which is used to characterize the different ground states of the system [43,44]: here the local currents on the legs are j l,σ = i(e −iσϕ a † l+1,σ a l,σ − H.c.).The critical condition for transition of energy band structure form double well structure (s 0 ̸ = 0, p 0 ̸ = 0) to triple or single well structure (have s 0 = 0 and p 0 = 0 state ) can be obtained as follows.Combining p 0 = arctan[ −s0 sin ϕ 1+(cos ϕ−1)sz 0 ] and equation (16) gives The critical condition of K is determined by the corresponding s 0 = 0 and s 0 ̸ = 0 at the energy minimum.At the energy minimum, s 0 = 0 means p 0 = 0, and the distribution of particles on the two edge legs of the ladder (σ = ±1) is the same, i.e. the system is in a zero-momentum state.Conversely, s 0 ̸ = 0 represents p 0 ̸ = 0, the particle distribution on the two edge legs is uneven, and the system is in a plane wave state.
Setting s 0 = 0, f = 0, we obtain: By inserting equation ( 21) into equation ( 15), we can get: s z0 in equation ( 21) is determined by equation (22).We find that changing the interatomic interaction g, the magnetic flux ϕ, and the rung-to-leg coupling ratio K will change the energy band structure and phase transition.
As shown in figures 3(a) and (b), the energy spectrum and ground state diagrams in (ϕ − K) and (ϕ − g) planes are obtained by equations ( 21) and (22).When ϕ is relatively small, the energy band is a single-well structure, the energy has only one minimum value at p 0 = 0, and the ground state is in the zero momentum state.When ϕ is large enough, the energy band is a double-well structure, the lowest energy occurs in two opposite momentums, and the ground state is the plane wave state.In the intermediate region, the energy band structure is a triple-well structure, there are three local energy minima, and the ground state is determined by the energy minima competition among them.In addition, the three local energy minima are merged together at the tricritical point (K ⋆ , ϕ ⋆ ), where K ⋆ = 5.4 and ϕ ⋆ = 0.81π.In figure 3 Figures 3(c) and (d) show the relationship between the chiral current j c and the magnetic flux ϕ corresponding to the ground state and the metastable state, visually depicting the transition of the ground state.It can be seen from figure 3(c) that for K = 1, in the zero-momentum state, the chiral current j c > 0 and first increases and then decreases with the increase of ϕ.When the system is in the plane wave state (with nonzero momentum) the chiral current j c < 0 and decreases with the increase of ϕ. Figure 3(d) shows that for K = 5.4, in the zero momentum state, the chiral current j c > 0 and the changing law of j c against ϕ is similar to that of K = 1; while in the plane wave state, the chiral current j c < 0 and first increases and then decreases with the increase of ϕ.That is, when the system is in the state of p 0 = 0, the chiral currents are all greater than zero, while in the state of p 0 ̸ = 0, the chiral currents are all smaller than zero.The chiral current j c changes the sign when the system transforms form zero momentum state to plane wave state.Chiral-current reversal occurs at the state transition point.We also find the magnitude of j c decrease with K.

Localized states both in σ and l directions
Next, based on variational equations, we explore localization phenomena such as diffusion, breather, soliton, and self-trapping in the system.When the system parameters meet certain conditions, the phenomenon of localization both in σ and l directions may occur.According to equation ( 6), the group velocity v g of the wave packet can be obtained by taking the first derivative of the Hamiltonian H with respect to p: The reciprocal of the effective mass m * is obtained from the second-order derivative of the Hamiltonian H to p: Then equation ( 23) can be simplified as: When m * > 0, i.e. {cos p[1 + (cos ϕ − 1)s z ] − s sin ϕ sin p} > 0, if m * → ∞, the wave packet group velocity satisfies v g → 0, and the phenomenon of self-trapping will occur.But no soliton solution exists when the effective mass is positive.When m * < 0, i.e. {cos p[1 + (cos ϕ − 1)s z ] − s sin ϕ sin p} < 0, there may be soliton solution.With this condition, breathing state can be obtained near the soliton solution.Under different physical parameters, the three-legged bosonic magnetic ladder system may have abundant localization phenomena.Now we discuss in detail the localized states under different conditions.

Localized states transition
Based on energy conservation, the Hamiltonian satisfies H = H 0 in the dynamic process.The characteristic of the self-trapping phenomenon is that the wave packet is confined near the initial position, and the wave packet width eventually tends to a certain value.In this case, as time evolves, the group velocity of the wave packet becomes zero, and the wave packet width remains constant after a short period of change.This can be obtained by using equation (6).Then by H = H 0 , the maximum width of the wave packet is We consider the situation of ṡ = 0, ṡz = 0, α = 0 and β = 0, i.e. there is no runge current.By using equation ( 16) (i.e.β = 0) and equation ( 26) we obtain: In addition, using equation ( 15) (i.e.α = 0), equations ( 26) and ( 27) we have: where, s z0 in equation ( 27) is determined by equation (28).When diffusion occurs, the wave packet is dispersed in the lattice.For m * > 0, that is, {cos p[1 + (cos ϕ − 1)s z ] − s sin ϕ sin p} > 0, if the width of the wave packet tends to infinity, diffusion will occur.In this case, as t → ∞, we have R → ∞, δ → 0, so ξ = {2s 0 cos p 0 sin ϕ + 2[1 + (cos ϕ − 1)s z0 sin p 0 ]} ̸ = 0.In addition, the center-of-mass position and the wave packet width will change with time.According to equation ( 6), when the system has diffusion phenomenon, we can get For the convenience of discussion, we set Therefore, the critical condition of transitional between the self-trapping phenomenon and diffusion phenomenon can be obtained by H 0 = 0. Combining with equations ( 6), ( 26) and ( 27), the critical condition can be easily obtained, and it can be simplified as: If f 1 > 0, the self-trapping occurs, and the wave packet stops expanding near the initial position.On the contrary, if f 1 < 0, the phenomenon of diffusion occurs, and the wave packet will spread out over time.
In addition, soliton solutions may exist when m * < 0. When the soliton appears, the wave packet width remains constant with time, and the wave packet group velocity keeps the initial value and does not change with time, i.e.Ṙ = 0, δ = 0, ξ = const for t → ∞.Combining with equation ( 8) and Ṙ = 0 one gets δ = 0. Considering β = 0, i.e. equation ( 16), we can derive that K is determind by: Beyond that, based on equation (15) (i.e.α = 0) and equation (31), one can obtain where, s z0 in equation ( 31) is determined by equation (32).According to Ṙ = 0, δ = 0 and equation ( 6), we derive the critical condition for appearance of the soliton state: When condition equation ( 33) is satisfied, the width of the wave packet does not change with time, and the center-of-mass position of the wave packet moves at a constant group velocity.In regions between f 2 > 0 and f 3 < 0 or the region with f 3 > 0 breather phenomenon occurs, where the width of the wave packet oscillates with time, but the group velocity does not change.Equation (31) guarantees that the spin dynamics of the system is inhibited, and both the spin tensor and spin polarization maintain their initial values s z = s z0 and s = s 0 .
We have analytically analyzed the localization phenomenon in three-legged magnetic ladder.These results show that the magnetic filed, the interatomic interaction, the spin tensor, and the average spin polarization all have strong influence on the dynamics of the system.Equations ( 26)- (33) predict the critical conditions for the occurrence of diffusion, self-trapping, soliton, and breather, which give the dynamic phase diagram of the system.Since the phase diagram is richer when m * < 0, we mainly focus on the phase diagram for m * < 0. Figure 4 shows the phase diagram in the (g − ϕ) plane for various s z0 .The phase diagram is divided into four regions.The solid line represents the soliton solution, and the two sides near the soliton solution represent the region where breather exists.Below the dash-dotted line indicates the region where diffusion occurs.It is noted that the dashed line in the phase diagram is given by the numerical simulation, which distinguishes the region of the self-trapping and the region of the breathing.With the increase of s z0 , the diffusion area first decreases and then increases, while the self-trapping region extends to a small g region, which means the appearance of self-trapping requires smaller interatomic interaction intensity as s z0 increases.
In order to verify the phase diagram given in figure 4, we select four points A, B, C, and D in figure 4(b), and the wave packet dynamics given by the numerical simulation of equation ( 3) using the fourth-order Runge-Kutta method with the parameters marked by four points are shown in figure 5.The black, red, and green lines represent the dynamic evolution of the wave packets on the three legs of the ladder {1, 0, −1}.As the interatomic interaction increases, diffusion, breather, soliton, and self-trapping occur successively.This result is consistent with the theoretical analysis.Interestingly, the soliton or breather are always localized at the system's edges 2 ), i.e. localized edge states observed, which are completely absent when ϕ = 0 and are the analog to the current-carrying edge states in the integer quantum Hall effect [54].However, when the self-trapping phenomenon occurs in the three-legged bosonic magnetic ladder system, most of the atoms will quickly concentrate on a narrowed bulk state in three-legged ladder (i.e.|a l,1 . That is, soliton or breather is favorite to form an edge state, while the self-trapping is favorite to form a bulk state.In other words, according to our theory, we can manipulate localized edge and bulk states in controllable way.In addition, through numerical simulations, we find that the solitons in three-legged ladder suffers from the instability, so it is of great significance to discuss the stability of solitons.Next, we discuss the stability of solitons.

Stability of soliton and chiral extended state
Now we use the method of linear stability analysis to study the stability of soliton under small perturbation.We consider small perturbation as: Among them, µ indicates the chemical potential, the complex perturbation amplitudes u l,σ and ν l,σ are much smaller than the soliton state a s l,σ , and λ measures the growth rate of the perturbation.Substituting the above disturbance state into equation ( 3), keeping the linear terms u l,σ and ν l,σ , the linearized equations around the soliton state can be obtained: According to the linear stability criterion, a soliton state is stable when all eigenvalues {λ} are pure imaginary numbers and unstable if one or more eigenvalues have non-zero real part.The eigenvalue problem equation ( 35) is solved using the Fourier collocation method [55].We find that the linear stability analysis is consistent with the results of the dynamic evolution of solitons over time.
In figures 6-8, we present the relationship between the maximum growth rate of the perturbation and the chemical potential of the soliton state.At a certain µ, when MaxRe(λ) = 0, the soliton is stable, and when MaxRe(λ) > 0, the soliton is unstable.In figures 6-8, the inset on the left side of each figure is the energy band structure corresponding to the soliton state, and the position of the soliton on the energy band and the corresponding chemical potential are marked by magenta dots.The right inset of each figure shows the dynamic evolution of the wave packet of the solitons with time.Through numerical simulations, we find that the stability of solitons depends on the energy band structure.The solitons initialized in an energy band with only one energy minimum deviated from the ground state are stable when s z > 0.5 (i.e. the solitons characterized with an edge state, .The stable solitons always behave with unsymmetry localized edge states (i.e.symmetry broken between the two edges, see figure 6), while the unstable solitons diffuse as they propagating unidirectionally (see figure 7).In particular, as expected, solitons initialized in the ground state or metastable state (i.e.soliton located at local energy minima) are all unstable, and extended states occur (see figure 8).Interestingly, both chiral and trivial extended states are observed.When the ground state solitons are initialized in a zero-momentum state p 0 = 0 (i.e.s = 0), equation ( 4) indicates that a symmetrical edge state   That is, chiral edge states occur and the bulk is decoupled from the edges.This behavior is the analog to a Harper-Hofstadter Hamiltonian [3,4,42,56].However, when the solitons are located at local energy minimum with p 0 ̸ = 0 (i.e.s ̸ = 0), equation (4) shown that an unsymmetrical edge state with In this case, edge-bulk coupling and chiral current in both edge and bulk are observed.But different from the case of p 0 = 0 (see figures 8(a)-(c)), the soliton in σ = 1 leg expands in leftwards, while the soliton in σ = 0 leg expands in rightwards.That is, chiral-current reversal along edge is observed (j c < 0).This is in good agreement with the predictions shown in figures 3(c) and (d): when the system is in the zero-momentum state, the chiral current j c > 0, when the system is in the plane wave state (with nonzero momentum) the chiral current j c < 0. Particularly, if the energy band has two minimums (see figure 8(f)), the bulk is coupled with each edge of two edges and chiral currents in three legs persist.The chirality of unstable solitons shown in figure 8 further confirms the ground state properties of the system.Interestingly, the localization, edge-bulk coupling and chirality of the system strongly depends on the energy spectrum of the system, which can be used to design and manipulate different kinds of edge-bulk coupling, localized and chiral edge (bulk) states in three-legged magnetic ladder.

Quenching of localized state and transition between localized edge state and qusai-bulk state
We have considered the localized states both in l and σ directions, i.e. ṡ = 0 and ṡz = 0.When equations (27) and (31) are satisfied, the critical conditions for the transition of localized states are obtained, and the evolution of these localized states with time is obtained through numerical simulation.Now we consider the case where there is a spin exchange, i.e. ṡ ̸ = 0 and ṡz ̸ = 0.In this case, we perform a direct numerical simulation of equation ( 3) by quenching K from the soliton state K = K 2 to a different K = K f .Since K f ̸ = K 2 , then ṡ ̸ = 0 and ṡz ̸ = 0, the spin dynamics of the system is excited.Then the polarization of the system and the dynamics of the soliton will be modified.We discuss the spin dynamics and wave packet dynamics through the numerical simulation of equation ( 3) using both sudden quenching and linear quenching of K.In particular, we find that the linear quenching of K can induce a controllable transition between a localized edge state ( s z → 1) and a stable bulk state ( s z → 1 2 )., there is no rung currents and spin dynamics is not excited (i.e.s z = s z0 and s = s 0 ).If K ̸ = K 2 , rung currents occur (i.e.s z ̸ = s z0 and s ̸ = s 0 ), the spin dynamics of the system is excited.It can be seen from figures 9(a) and (d) that s z and s evolve with time in the form of damped oscillations, and eventually the system will reach new polarization states of s z = s z ̸ = s z0 and s = s ̸ = s 0 .When K f > K 2 , s z and s oscillate with larger amplitudes and longer durations, and s z < s z0 , s < s 0 , the system is transformed from an initial edge state to a quasi-bulk state.Conversely, when K f < K 2 , the s z and s oscillate with smaller amplitudes and shorter durations, and s z > s z0 , s > s 0 , the system is transformed from an initial edge state to a narrowed edge state.The quenched wave packet dynamics under different K f is shown in figure 10.Numerical simulation results show that although K f ̸ = K 2 , the quasi-soliton or breather state can still be maintained.In the case of K f < K 2 , as K f decreases, the final wave packet amplitude of σ = 1 leg increases, and the wave packet amplitude of σ = 0 leg decreases.This indicates that more atoms gather in the edge of the system when K f is smaller.In the case of K f > K 2 , the amplitude of the final wave packet of σ = 1(σ = 0) leg decreases (increases) with the increases of K f , that is, a quasi-bulk state occurs.Note that, for sudden quenching, the new polarized states characterized by s z and s are not controllable.That is, sudden quenching cannot achieve a controllable transition to the expected polarized states.

Linear quenching of K
If the rung-to-leg coupling ratio K is sufficiently slowly quenched from K = K 2 to K f , we expect that controllable transition between different polarized states can be realized.When K is slowly quenched from K 2 to K f , equation (31) indicates that a new polarized state with s = s f and s z = s zf can be obtained.This suggests that a controlled transition from an initial polarized state to the expected polarized state can be realized by slowly quenching K.For given parameters R 0 , g and ϕ, equations ( 31)-(33) determine the relationship between s f and s zf with K f .The results presented in figures 9(c) and (f) show that both s zf and s f decrease with the increase of K f .That is, a smaller K f can realize a higher polarized state, and for a sufficiently small K f , a fully polarized state with s f → 1 and s zf → 1 can be realized, i.e. a pure edge state can be realized.However, when K f increases, s f → 0 and s zf → 1 2 , an expected quasi-bulk state can be observed.Here, we use a linear quenching of K formulated as: where K 2 is given by equation ( 31), K f is the final rung-to-leg coupling ratio value after quenching, and τ q is the quenching time.Figures 9(b) and (e) show the time variation of s z and s via linear quenching of K with τ q = 10, respectively, which correspond to the cases with sudden quenching of K shown in figures 9(a) and (d).We find that, for a give K f , the spin tensor s z is manipulated from s z0 = 0.75 to a final s z = s zf and spin polarization s is manipulated from s 0 = 0.71 to a final s = s f given by equation (31).That is, linear quenching achieves a controllable polarization transition.Figures 9(b) and (e) illustrate that, unlike the sudden quenching of K, the linear quenching of K results in a smoother transition of spin tensor and spin polarization.Figure 11 presents the time evolution of the wave packets for the linear quenching with different K f .The results show that stable soliton or breather can be maintained for all K f .A pure edge state for K f = 0.25 and a bulk state with K f = 5 are observed.
The realization of a higher polarized state by linear quenching with small K f can be understood as follows.As shown in figure 9, the strength of spin dynamics (i.e. the oscillation amplitudes of s z and s ) depends on the rung-to-leg coupling ratio K f and quenching method.Small K f means weak spin dynamics and higher polarization [as predicted in figures 9(c) and (f)].For fixed K f , sudden quenching excites strong spin dynamics, while the linear quenching as an adiabatic slow quenching method excites very weak spin dynamics, especially for small K f .Strong spin dynamics results in spin mixing and favors the uniform distribution of particle numbers on the three legs (i.e.bulk state).Therefore, a higher polarized state is realized by linear quenching with small K f .
The results of sudden quenching and linear quenching of K show that both quenching methods can maintain soliton and breather dynamics.Linear quenching of K realizes a controllable transition between the localized edge state and localized bulk state.

Conclusion
In conclusion, we have investigated the ground state and chiral nonlinear dynamics of an interacting three-legged bosonic ladder with magnetic flux using variational analysis and numerical simulation.Considering the combined effect of interaction and synthetic gauge field, the ground state diagram, the chiral current, the localization, and chiral edge (bulk) states of the system are presented.The dependence of the stability of the localized states, the edge-bulk coupling characteristics, and the chirality of the system on the energy band structure of the system is revealed explicitly.The physical mechanism that causes the ground state transition, localization and chiral edge (bulk) state in the system is discussed in detail.A controllable transition from the localized edge state to the localized bulk state is realized.These results provide a theoretical evidence for the experimental observation and manipulation of different kinds of edge-bulk coupling and localized/chiral edge (bulk) states in three-legged bosonic magnetic ladder, paving the possible way for the investigation of different forms of edge and bulk 2D topological matter in bosonic atomic systems.

Figure 1 .
Figure 1.Schematic diagram of the three-legged magnetic ladder.The blue balls stand for the atoms on the chain.Here J and K represent the tunneling strength along the legs and rungs of the ladder, respectively.The magnetic flux of each cell is ϕ, and the interatomic interaction intensity is represented by g.

Figure 2 .
Figure 2. Energy band structures for different magnetic fluxes.The three columns correspond to K = 0.5, K = 1, and K = 5.4, respectively.The two rows correspond to g = 0.5 and g = 1, respectively.(a)-(f) The red lines indicate the band structure at the transition point from a single minimum to three local energy minimas.The dashed lines indicate the band structure at the transition point from the zero momentum state to the plane wave state.The cyan lines indicate the band structure at the transition point from three local energy minima to two local energy minima.
(a), when K < K ⋆ , equation (21) determines the critical transition condition of the band structure from three local energy minima values to two local energy minima values.When K ⩾ K ⋆ , equation (21) determines the critical transition conditions from the zero momentum state to the plane wave state.In figure 3(b), equation (21) only determines the critical transition condition of the energy band structure from three local energy minima values to two local energy minima values (the dotted line in the upper part of the figure).

Figure 3 .
Figure 3. (a) and (b) The energy spectrum and ground state diagram in the (ϕ − K) plane and (ϕ − g) plane.(c) and (d) The relationship between chiral current jc and magnetic flux ϕ.The insets in (a) and (b) show the schematic diagrams of the band structures in different regions, where the bands consist of one, two and three minima, respectively.The yellow region represents the zero momentum state, and the red region represents the plane wave state.The tricritical points are indicated by the symbols ⋆.The insets in (c) and (d) show the corresponding energy spectrum with different chiral currents, and the magenta balls indicate the states in which the system occupies.In (c) and (d), the black and red lines represent jc > 0 and jc < 0 respectively.

Figure 4 .
Figure 4.The dynamical phase diagram in the (g − ϕ) plane with different sz.The dashed line shows the boundary between self-trapping and breather.The solid line represents the soliton solution.The dash-dotted line is used to distinguish the self-trapping and diffusion regions.The other parameters are R0 = 5, p0 = 0.5π.

5 .
(a)-(d) The time evolution of the wave packet of points A, B, C, and D marked in figure and the black, red, and green lines illustrate three components σ = {1, 0, −1}, respectively.
(a) and (b)), and unstable when s z < 0.5 [i.e. the solitons with a quasi-bulk state, |a l,0 | 2 > |a l,1 | 2 , |a l,−1 | 2 , see figure 7(a)].For a soliton initialized in an energy band with three local energy minima deviated from the local minima, the stability of the soliton gradually changes from stable (see figures 6(c) and (d)) to unstable (see figures 7(b) and (c)) as the depth of the two degenerate symmetrical wells in the energy band increases.But solitons initialized in an energy band with only two degenerated energy minima are all unstable (see figure 7(d))

Figure 6 .
Figure 6.The relationship between the maximum perturbation growth rate of stable solitons and the chemical potential.The insets on the left are the energy band structure, and the insets on the right are the evolution of the solitons with time.The magenta dots mark the position of the solitons in the energy band structure and the corresponding chemical potential.

Figure 7 .
Figure 7.The relationship between the maximum perturbation growth rate of unstable solitons and the chemical potential.Others are the same as in figure6.

Figure 8 .
Figure 8.The relationship between the maximum perturbation growth rate of ground state and metastable state solitons and the chemical potential.Arrows indicate the direction of the wave packet expansion.Others are the same as in figure 6.

Figure 9 .
Figure 9.Time evolution of the spin tensor sz and spin polarization s for sudden quenching (the first column) and linear quenching (the second column) of K from K = K2 to different K f .The variation of final spin tensor s zf and spin polarization s f against K for different values of ϕ after the linear quenching (the third column).Other parameters: R0 = 5, p0 = 0.5π, ϕ = 0.25π, g = 1.17 in the second and third columns.

5. 1 .
Sudden quenching of KWe consider the soliton marked by point C in figure4(b).When K = K 2 , the soliton state will occur.We suddenly quench the value of K from K = K 2 to different K f values, and the other system parameters are the same as used in figure5(c).Figures9(a) and (d) show the time evolution of the s z and s under different K f , respectively.They depict the excitation of spin dynamics at different K f .If K = K 2 , as shown by the magenta lines in figures 9(a) and (d)

Figure 10 .
Figure 10.The time evolution of the wave packet with sudden quenching of the soliton marked at point C in figure 4(b) from K = K2 to different K f .The black, red, and green lines illustrate the wave packet of different σ = {1, 0, −1}, respectively.

Figure 11 .
Figure 11.Same as in figure 10, but with linear quenching of K.