Controllable Floquet topological phases in the magnetic ladder system

Utilizing both the electric and magnetic fields to manipulate electron dynamics enables the external control of topological states. This study investigates the topological characteristics of a quasi-one-dimensional ladder lattice subjected to a time-periodic electric field and a constant magnetic field. The Floquet topological phases are determined in the high-frequency approximation. In the absence of a magnetic field (φ = 0), the energy band diagram is modulated by the electric field parameter α/ℏω , leading to a topological phase transition when α/ℏω crosses the value of 1. When a magnetic field is present ( ϕ=π ), the topological phase transitions in the ladder model are influenced by both the electric field parameter α/ℏω and the perpendicular hopping t 0, resulting in a diverse range of adjustable topological states. These discoveries offer promising prospects for the utilization of ladder lattice systems with externally modifiable topological properties.


Introduction
The exploration of the quantum Hall effect marked the inception of the era of topological insulators (TIs) [1][2][3], following the groundbreaking research conducted by Thouless et al [4].In situations where the system possesses finite dimensions in the transverse direction, electrons are endowed with topological protection, allowing them to move unidirectionally at each boundary.This system provides a platform for investigating the impact of external fields on internal electronic behavior.When we further reduce the transverse dimension while maintaining a perpendicular magnetic field, intriguing phenomena emerge in the behavior of electrons.A ladder lattice represents a quasi-one-dimensional (1D) system with the narrowest possible transverse width that can still accommodate a perpendicular magnetic field.In this scenario, only one robust quantum number aligns with the electric field, and the resulting topological properties differ from those of the quantum Hall effect described by a Chern number.Nonetheless, nontrivial topological features have been predicted in the ladder system and are manipulable through the magnetic field.Ladder lattices have been successfully realized in various physical systems, including superconducting wire networks [5] and diamond chain lattices [6][7][8].Among these ladder models, the Creutz ladder [9,10] stands out as one of the most well-known examples, capable of capturing the magnetic field's effects within its straightforward structure.The Creutz ladder exhibits robust zero-energy modes owing to Aharonov-Bohm caging [11] and sustains topologically protected edge modes [12].
Prior investigations have primarily focused on the behavior of the ladder lattice in the presence or absence of a magnetic field.In recent years, the theory of topological phase transitions induced by open quantum systems and external fields has garnered widespread attention [13,14], but the system driven by a time-periodic electric field has received comparatively limited attention.However, it is worth noting that the utilization of an alternating current (AC) field offers increased reliability and controllability, thereby enabling the exploration of new physics and facilitating the simulation of electronic dynamics in various domains, including optical systems [15][16][17], acoustic systems [18,19].The system under consideration bears similarities to the AC quantum Hall effect in terms of its configuration, as it has been observed to exhibit robustness against the breaking of time-reversal symmetry and the presence of disorder [20,21].
The treatment of time-periodic systems is typically based on the Floquet theorem [22], and therefore, these types of topological states are referred to as Floquet topological states (FTSs) [23][24][25][26][27]. FTSs offer high controllability due to the ability to easily modify the frequency and amplitude of the electric field.In 1D systems, Floquet TIs have exhibited rich topological phases that may differ significantly from their undriven counterparts.For instance, the 1D extended Su-Schrieffer-Heeger (SSH) model has been shown to generate higher Chern numbers [28,29], and singular phenomena, such as anomalous edge states [30,31] between energy bands with zero Chern number, challenging our conventional understanding of TIs.Given that the electric field is typically introduced in a single direction, Floquet dynamics in 1D TIs have been extensively explored, including systems such as the SSH chain, the Kitaev chain with p-wave superconductivity [32,33], and the 1D Creutz ladder model [34,35].
In this paper, our investigation centers on the examination of the topological characteristics of a ladder lattice, with a specific focus on the hourglass ladder [36], which is subjected to a time-periodic electric field directed along the ladder's orientation.The hourglass ladder is composed of two interconnected diatomic chains and represents a very simple structure capable of accommodating the penetration of a magnetic flux.This lattice encompasses well-established systems such as the SSH chain [37] and the Creutz ladder [9] under certain parameter regimes.Our study delves into the changes in the quasi-energy band and the associated topological invariants as we vary the frequency, electric field strength, and magnetic flux.Particularly, we direct our attention toward the potential topological phase transitions induced by the time-periodic driving under high-frequency conditions.Our findings illuminate the existence of diverse topological phases, which are jointly controlled by the external electric and magnetic fields.These results underscore the presence of entirely distinct topological properties within the Floquet ladder system when compared to static systems, offering intriguing possibilities for the application of quasi-1D ladder systems endowed with controllable topological properties.
The paper is organized as follows.In section 2, we present a comprehensive description of the quasi-1D ladder-type Floquet system.We explore the band structure of the system under various parameters and calculate the corresponding topological invariants.This section is divided into two subsections, where we analyze both the nonmagnetic and magnetic cases separately.Section 3 includes our analysis of the edge states in the structure.Section 4 summarizes the entire paper.Some detailed derivations can be found in the appendix.In appendix A, we present the derivation of the effective Floquet Hamiltonian for the time-dependent system in the high-frequency approximation.In appendix B, we discuss the Fourier transformation from the real-space Hamiltonian to the momentum space.Appendix C provides an analytical analysis for a specific case where two quasi-energy bands touch each other.Appendix D includes a discussion about the symmetry of the system, especially the time-reversal symmetry.

Model and methods
The ladder lattice is a quasi-1D model with the primary cell comprising four lattice sites labeled as a, b, c, d, as illustrated in figures 1(a) and (b).In this model, both intracell and intercell hoppings are taken into account and denoted by t i (i = 0, 1, 2, 3, 4).The ladder lattice can also be viewed as two diatomic chains connected through interchain hoppings.By introducing both a magnetic field perpendicular to the lattice plane and an electric field along the chain direction, the physical processes within the system can be manipulated effectively through external fields.The Hamiltonian governing the system is described as follows: where J ij represents the hopping between site i and j.The inclusion of long-distance hoppings beyond the nearest-neighbor assumption is possible.The presence of a perpendicular-to-plane magnetic field results in the generation of a magnetic flux hϕ through each square.In this scenario, a specific gauge is chosen to modify only the intrachain neighboring hoppings J j,j+1 , where j denotes the horizontal position.For the lower chain, the modification is denoted as J + j,j+1 (or J j,j+1 e iϕ/2 ), while for the upper chain, it is denoted as J − j,j+1 (or J j,j+1 e −iϕ/2 ), respectively.The terms A i f(t) represent the on-site potential induced by the electric field f (t) along the horizontal direction.In this context, we consider the utilization of a time-periodic square pulse [24]:  where ω = 2π/T is the frequency of the time-periodic driving function.As a result, the potential A i can be expressed as where subscript represents the lattice positions in the horizontal direction, and α and β denote the strengths of the electric field on different sites, respectively.The potentials A i are depicted in figure 1(c).
H lattice represents a general quasi-1D model that can be simplified into two separate SSH chains [37] when the interchain hoppings (between the upper and lower chains) vanish, as shown in figure 2(a).Furthermore, the model becomes a Creutz ladder [9] structure when ϕ = π and as depicted in figure 2(b).Moreover, the condition of u 1 = t 1 = t 2 and v 1 = t 3 = t 4 would result in the hourglass ladder [36], as illustrated in figure 2(c).
Considering the influence of a time-periodic electric field, we aim to determine the quasienergy of the system.By applying Floquet theory, we can derive a time-independent effective Hamiltonian under the assumption of high-frequency approximation.This approximation is necessary to prevent interference between different quasienergy bands.However, the external field can bring perturbations between them and thus produce new topological phase transitions as we will show.The impact of the electric field can be observed through the modified intrachain hoppings, denoted by u n and v n , as given in equation (A8).When the interchain hoppings are neglected, the ladder system can be decoupled into two separate SSH chains.In this case, the competition between the forward intracell hopping u n and the forward intercell hopping v n determines the topological state.For a detailed derivation of the Floquet Hamiltonian in the high-frequency approximation, please refer to appendix A.
The effective Hamiltonian can be rewritten in momentum space as , where and In the above, the variable n represents the number of primary cells along the horizontal direction, where n = |n − n 0 | indicates the hopping range.The symbol˜denotes complex conjugation.A detailed description of the Fourier transformation can be found in appendix B. It should be noted that the hoppings between identical sites of different cells can be suppressed by selecting m(α + β) = 2hωq (A6).In this scenario, the diagonal terms in the Hamiltonian become zero, facilitating further analytical solutions.This choice can guarantee the chiral symmetry in SSH model.The energy band structure, which is intricately linked to the Hall conductivity and the underlying topological properties, can be derived from the Hamiltonian above.In the case of 1D, the Zak phase [32,38] of a given energy band can serve as the topological invariant: where |u j k ⟩ is the jth eigenstate of the system's Hamiltonian.The Zak phase is found to be connected to the winding number W by the relationship W ≡ Z/π mod 2 [27].When t i = 0 (i = 0, 1, 2, 3, 4), the model duplicates the SSH chain.Under these conditions, the Hamiltonian can be written as H(∥) = H SSH ⊕ H SSH , where and It can be readily confirmed that it belongs to the AIII topological classification and possesses chiral symmetry S = σ z .The topological phase transition occurs when the condition u n = −ṽ n is satisfied.The topological invariant takes on the values 0 or 1 when only nearest-neighbor hoppings are considered, but when long-range hoppings are considered, it can contain other integer values when the decay length λ is large enough.Additionally, we investigate the case of the Creutz ladder model by setting the corresponding parameters.The model exhibits a topologically nontrivial phase when t 0 < 2u 1 .

The non-magnetic ladder
Given that the hourglass ladder encompasses the SSH chain and Creutz ladder, our current focus revolves around analyzing the topological properties induced by a time-periodic electric field in this model.Our investigation begins with the hourglass ladder at ϕ = 0. Prior studies have examined similar models without the presence of an electric field [39][40][41].However, our interest lies in examining the effects of the electric field, specifically exploring the Floquet topological properties of the hourglass ladder.
When the interchain hopping parameter t 0 is set to zero, the Hamiltonian exhibits chiral symmetry S = σ z ⊗ σ 0 (D2).It can be readily verified that SH(k)S = −H(k), which is a fundamental requirement for the system to possess chiral symmetry.As the value of t 0 increases, the energy bands ν 1 and ν 2 are elevated, while bands ν 3 and ν 4 are lowered.The winding number W can be calculated for each band.It is observed that the topological numbers of bands ν 1 and ν 2 are the same, as are the indices of bands ν 3 and ν 4 .When a high-frequency time-periodic field is introduced, the quasi-energy band structure inherits this characteristic.
In figure 3, the quasi-energy band structure is plotted as a function of the increasing electric field strength α/hω, and the corresponding topological invariant for different bands is provided.Regardless of the electric field strength, bands ν 1 and ν 2 have the same value of W, as do bands ν 3 and ν 4 .However, as the electric field strength changes, a topological phase transition occurs in bands ν 1 and ν 2 .Although obtaining an analytical solution to the eigenvalue problem of the Hamiltonian is challenging in general cases, it can be proven that  bands ν 1 and ν 2 touch at a single point when α/hω = 1 and k = 0 (see appendix C).This topological phase transition is entirely attributed to the external field.Interestingly, this point represents a Dirac-like point in the extended parameter space spanned by k and α/hω.In principle, synthetic dimensions can be employed to define and explore topological properties beyond the current lattice dimension.Examples include the 4D quantum Hall effect [42], synthetic gauge fields [43], and synthetic Weyl points [44].

The magnetic ladder
When a magnetic field (ϕ = π) is introduced to the hourglass ladder system, quantum interference effects become significant even in the absence of an electric field.Previous research has identified a topological phase transition in the Creutz ladder as the value of t 0 is varied [9].The Hamiltonian in this system exhibits chiral symmetry S = σ 0 ⊗ iσ y (D3), which is reflected in the symmetric forms of the energy bands, as depicted in figures 4 and 5.With the addition of a time-periodic electric field, the quasi-energy bands are influenced by both the electric field and the magnetic field through the complex potentials.
Due to the high frequency of the time-periodic field, there is no overlap between different quasi-energy bands.As a result, we anticipate the presence of a topological phase transition by manipulating the value of t 0 while keeping the electric field parameter fixed.Figure 4 illustrates the quasi-energy band structure of the magnetic hourglass ladder under an electric field with a strength of α = 2.5hω.It is observed that there are four separated bands, among which the lowest and highest bands exhibit a consistent topological number as t 0 increases.However, the middle two bands would touch at a point when t 0 = 0.55 and k = 0, where a topological phase transition occurs for them.This feature resembles that of the Creutz ladder.
If we fix the value of t 0 and change the strength of the external electric field, we can clearly see the topological phase transition induced by the electric field.In figure 5, we give two special cases: (a) t 0 = 0.25 and (b) t 0 = 0.65.The different band is plotted in different colors and the vertical width represents the band width.It is seen that there is no band overlapping except for some touching points, and each touching point is associated with a topological phase transition for two corresponding touching bands.The topological invariant for bands ν 1 and ν 4 are same, so are ν 2 and ν 3 .The detailed values are listed below in the corresponding area.From these two cases, we can observe the significant influence of the electric field, which strongly modifies the topological properties of the 1D ladder.

Edge state
The nontrivial topological states exhibit edge states in a finite-sized system, which is a valuable property for experimental observations and practical applications.To demonstrate this, we aim to confirm the presence of edge states and, more importantly, observe the distinct edge states associated with different topological bands.In the following analysis, we consider a finite hourglass ladder with a length of L = 15 unit cells.
In figure 6, we present the calculation of edge modes in a non-magnetic finite hourglass ladder for two values of the driven electric field.The value of α/hω is chosen as −2.5 and 2.5, which correspond to the topological invariant of band ν 1 as W = 1.In the energy band spectrum, two distinct edge modes have emerged, one at the high-energy position of +2t 0 and the other at the low-energy position of −2t 0 , resembling that in the trimer chains [45].To account for the long-range interaction, we appropriately set the decay length λ = 2.5 in our calculations.For an infinite lattice, different electric field strengths lead to different topological numbers for the energy bands, resulting in a varying number of edge states in a finite lattice.In panels (a) and (b), there are two degenerate edge modes at different energy positions.The corresponding modes (in the same colors) in real space for these three cases are plotted in figures 6(c) and (d), respectively.The horizontal axis 'Index' means the atom's position ordered from left to right unit cell and in each unit cell from site a to site d (shown in figure 1).
For the magnetic hourglass ladder driven by the electric field, the quantum interference would produce more complex modifications to the topological states as stated before.The finally modified energy band structure determines where the edge states appear.In figure 7, we give two cases, (a) t 0 = 0.1 and (b) t 0 = 1, under the same electric field strength α/hω = 1.2.It is seen that the edge modes appear at the zero-energy position in case (a), whereas edge modes disappear at zero-energy gaps but appear at both high-energy and  low-energy positions.The corresponding edge states are plotted in figures 7(c) and (d) in the same colors and shapes.The horizontal axis is the atom's position in the same order same to that in figure 6.

Conclusion
In this research, we explore the Floquet topological characteristics of a quasi-1D ladder-type lattice system.Our primary focus is on understanding how the quasi-energy band structure is influenced by the electric field in the high-frequency approximation.Notably, in the absence of a magnetic field, a substantial increase in the strength of the time-periodic field alone can induce a noteworthy topological phase transition.This transition results in the emergence of a Dirac-like point within the parameter space defined by the momentum k and the field parameter α/hω.When a magnetic field is introduced, the topological properties become intricate due to the combined quantum interference arising from both the electric and magnetic fields.In such a scenario, topological phase transitions can be triggered either by variations in the structural parameter t 0 or, more significantly for practical applications, by manipulating the external field strength.In conclusion, the ladder-type lattice system showcases a wealth of controllable topological phase transitions that are easily achievable through external field manipulation.This feature makes it a promising prototype for investigating the application of quasi-1D systems with controllable topological properties, particularly for advancements in miniaturization.

Appendix A. The effective Floquet Hamiltonian
To handle this periodicity, we extend the system to the Floquet-Hilbert space: F = H ⊗ L T , where H is the time-independent space and L T is the time-dependent space.In the Floquet-Hilbert space, the corresponding Hamiltonian is denoted as H f , satisfying In the H space, corresponding to the fact that in the rotating frame, we obtain the transformed Floquet Hamiltonian equation by introducing the unitary operator where the choice of the unitary operator is K = e −i/h ´Hdriven (t)dt .The Floquet Hamiltonian is obtained as where ∆A ij = A i − A j and F(t) = ´f(t)dt.We consider the use of time-periodic square wave pulses where T = 2π/ω.The Hamiltonian can be expressed by a power series expansion in 1/ω.In the high-frequency regime(ω ≫ J), Floquet modes exhibit little variation over a short period of time, so the zeroth-order approximation is enough.The zeroth-order approximation is just the zeroth Fourier transform of the Hamiltonian H(t) [47][48][49], i.e. or where the integral Finally, we derive a time-independent effective Hamiltonian with the renormalized hoppings [24]: where λ represents the hopping strength between different lattice sites with the distance of d ij along the chain, and λ is the decay length.To make the diagonal terms in the Hamiltonian zero, we choose the parameter m(α + β) = 2hωq (where m = ±1, 2, 3, • • • ).Here, q = 0, 1, 2, 3, • • • represents the zero points of a function defined at equidistant points in real space, which cancels out all long-range hoppings between the same sublattice.
Set q = J = 1,therefore the renormalized hoppings are and

Appendix B. Hamiltonian in momentum space
The real-space effective Hamiltonian equation for the general model is obtained from figure (1) as where J ± n = J n e −n/λ e ±inϕ/2 , u n and v n are given by equation (A8).The symbol˜above means the conjugate operation. Using , the Hamiltonian can be transformed into momentum space Choosing a = 1 2 , the Hamiltonian matrix becomes ) .(B4) However, the situation changes when the system is driven by an AC electric field.For the AC-driven system, there only exists the quasi-energy associated with the operator Ĥ − ih∂ t .So whether the system is time-reversal symmetric or not is determined by the effective Hamiltonian, which is actually a time integral of Ĥ − ih∂ t rather than Ĥ.Let's give a detailed derivation.
According to the Floquet theory, we treat the Hamiltonian by a unitary transformation as The effective Hamiltonian is a time integral of H(t), rather than H(t), over a time period.For the periodic electric field used in our model, the effective Hamiltonian is If we change the periodic electric field by a translation to the effective Hamiltonian becomes where In this case, we have u n (v n ) = u * n (v * n ) and thus the system respects the time-reversal symmetry.It is concluded that the time translation is crucial to the time-reversal symmetry of the effective Hamiltonian for the time-dependent system.However, quasi-energy topological states are the same whichever electric field form is adopted.

Figure 1 .
Figure 1.(a) The schematics of the ladder-type system driven by the time-periodic electric field.(b) The hoppings in a general ladder lattice, which are denoted in the same colors.(c) The onsite potential along the chain direction when the time-periodic square wave is represented by f (t).

Figure 3 .
Figure 3.The band structure of the non-magnetic hourglass ladder model with the increasing electric field strength α/hω.There are four energy bands labeled with ν i (i = 1, 2, 3, 4), two of which (ν1 and ν2) are higher, with the average value of t0, and the other two (ν3 and ν4) are flat bands, degenerate if only the nearest-neighboring hoppings considered, with energy about t0 below zero.The winding number W are the same for bands ν3 and ν4, labeled on their flat bands, so are bands ν1 and ν2, labeled on their projected face at zero.The topological phase transition occurs at α/hω = 1, which is indicated by a vertical section.

Figure 4 .
Figure 4.The quasi energy band structure of the magnetic hourglass ladder model when an electric field of strength α = 2.5hω is added.The four energy bands ν i (i = 1, 2, 3, 4) are drawn in different colors.The topological numbers W are given in the corresponding domains.A section is plotted at t0 = 0.55, where a topological phase transition occurs for bands ν2 and ν3, touching at k = 0.

Figure 5 .
Figure 5.The evolution of quasi band structures with the field strength for the magnetic hourglass ladder when a time-periodic electric field is added, (a) t0 = 0.25 and (b) t0 = 0.7.For a value of α/hω, there are four quasi-energy bands whose energy bandwidths are given in different colors.The touching points accompany with topological phase transitions.The topological numbers are listed below in corresponding areas.

Figure 6 .
Figure 6.Edge mode in the non-magnetic hourglass ladder model with L = 15 unit cells.The energy distribution of the finite lattice is drawn when the driven electric field strength α/hω is set as (a) −2.5, (b) 2.5, respectively.There appear high-energy (+2t0) edge states and relatively low-energy (−2t0) edge states in each case.The corresponding edge states are plotted in (c)-(d), in corresponding colors respectively.The parameter t0 is set as 1.

Figure 7 .
Figure 7. Edge mode in the magnetic hourglass ladder model with L = 15 unit cells.The magnetic parameter is ϕ = π.The driven electric field strength α/hω is set as 1.2.Two cases for (a) t0 = 0.1 and (b) t0 = 1 are considered.The corresponding edge states are plotted in (c) and (d), respectively.
)where J i ,j is given in appendix A. Now the symmetry of the effective Hamiltonian is what we are concerned about.It is obvious thatu n (v n ) ̸ = u * n (v * n ).So we conclude that no time-reversal operator T can be found to fulfill the time-reversal symmetryH (k) = T H * (−k) T −1 and H(k) = −CH * (−k)C −1 .