Photo-induced Entanglement in a Magnonic Floquet Topological Insulator

Photo-induced Entanglement in a Magnonic Floquet Topological Insulator

Satyaki Kar Institute of Physics, Bhubaneswar-751005, India satyaki.phys@gmail.com    Banasri Basu Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata - 700108, India. sribbasu@gmail.com
Abstract

When irradiated via high frequency circularly polarized light, the stroboscopic dynamics in a Heisenberg spin system on a honeycomb lattice develops a next nearest neighbor (NNN) Dzyaloshinskii-Moriya (DM) type termowerre (), making it a magnonic Floquet topological insulator. We investigate the entanglement generation and its evolution on such systems - particularly an irradiated ferromagnetic XXZ spin- model in a honeycomb lattice as the system parameters are optically tuned. In the high frequency limit, we compute the lowest quasi-energy state entanglement in terms of the concurrence between nearest neighbor (NN) and NNN pair of spins and witness the entanglement transitions occurring there. For the easy axis scenario, the unirradiated system forms a product state but entanglement grows between the NNN spin pairs beyond some cut-off DM strength. Contrarily in easy planar case, NN and NNN spins remain already entangled in the unirradiated limit. It then goes through an entanglement transition which causes decrease (increase) of the NN (NNN) concurrences down to zero (up to some higher value) at some critical finite DM interaction strength. For a high frequency of irradiation and a suitably chosen anisotropy parameter, we can vary the field strength to witness sudden death and revival of entanglement in the Floquet system. Both exact diagonalization and modified Lanczos techniques are used to obtain the results upto 24 site lattice. We also calculate the thermal entanglement and obtain estimates for the threshold temperatures below which non-zero concurrence can be expected in the system.

I Introduction

Recently there is an upsurge of interest in realizing and utilizing quantum information aspects of various quantum many body systems (QMBS). Built at the interface of quantum information science, condensed matter theory, statistical physics, quantum field theory, the study of many-body entangled states rapidly has become a very active topic of research. In this respect, quantum entanglement plays a crucial role in the highly efficient quantum computation and quantum information processingvidal (); horodechi (). With the rapid development of the experimental process on quantum control, there is a rapidly growing interest in entanglement generation. Thus the quantification of entanglement has found a key place in quantum information process applications.

On the other hand, recent study of Dirac and topological magnons in solid-state magnetic systemsonose (); ren (); chisnell (); owerre0 (); fransson () is expected to open a challenging avenue towards magnon spintronics and magnon thermal devices. Since the magnons are charge-neutral quasiparticles, it is believed that the magnon quantum computing will offer a favorable pathway for eliminating the difficulties posed by charged electronsadrianov (); khitun () and as such the magnonic devices would be more efficient in quantum memory and information storagesimon (); tanji (); specht (); wang ().

In this context, our focus is to study the entanglement in a magnonic system that is irradiated via a strong periodic circularly polarised light. There are many measures of entanglement, such as entanglement entropy, entanglement of formation, purity or negativity capturing the quantum correlation within an interacting system by different means. Von Neumann entropy gives a standard measure for entanglement of pure states. But for a generic mixed state, an entanglement entropy can give non-zero values for each of the subsystems even if there is no entanglement. So in those cases it is the entanglement of formation that gives a true measure of entanglement.wooters () For a given purity or mixedness, with different possible combination of pure states that the state can collapse into, it is the entanglement of formation that gives the minimum number of singlets required to create the mixed density of states. This is a monotonically increasing function of an argument called concurrence C, with , which by itself can also be regarded as a measure for entanglement in a mixed state, for example, between a pair of qubits within a multi-qubit large system wooters (). In fact this is an entanglement monotone which is zero for separable states and unity for Bell states (four maximally entangled 2-qubit states). For a pair of qubits, concurrences are well defined as will be described later. Our analysis shows their tunability in terms of the frequency of irradiation and provides significant control over the quantum information processing.

At the very outset, let us reiterate here that recent trend shows plenty of work on optical lattices involving Dirac plasmonsweick (), Dirac magnonsbalatsky () or photonic topological insulatorskhanikaev () that have Dirac like bosonic spectrum. When dynamics is studied in such systems in presence of time-varying fields, plethora of exotic phenomena such as defect productions, dynamical freezing, dynamical phase transition or entanglement generationamit (); ksrmp (); kar1 (); kar2 () can be expected. Particularly for a periodic quench, one can use the Floquet theoryeckart () for stroboscopic evolutionnori () of the system, which results in an effective static Hamiltonian out of the originally dynamical system.

A Dirac system shows interesting dynamical features upon light irradiationlindner (); lindner2 (); moessner (); light (); light0 (); light1 (); light2 (); light3 (); light4 (); debu (); debu2 (); aniruddha (). An irradiated field can lead to nontrivial Floquet systems like Floquet topological insulators (FTI)moessner (), as can be seen, for example in an irradiated semiconductor quantum welllindner () or a 3D topological insulatorlindner2 (). We know that a ferromagnetic Heisenberg spin- (FMHS) model with next nearest neighbor (NNN) Dzyaloshinskii -Moriya interaction (DMI) in a honeycomb lattice, under a linear spin wave approximation (LSWA), turns out to be a magnonic equivalent of the Haldane model - the famous primitive toy model to show topological transitionspantaleon (). Interestingly, this can also be achieved via irradiation with high frequency circularly polarized lightowerre (). The resulting Floquet Hamiltonian develops easily tunable synthetic laser-induced NNN DMI in addition to a FMHS with modified anisotropy. Within LSWA, the model behaves like a bosonic Haldane model enabling the system to emerge as topologically nontrivial at intermediate frequencies of the irradiation.

In this paper, we probe the entanglement characteristics of such Floquet model, born out of irradiating the spin system - both in their topologically trivial and non-trivial limits. In the infinite frequency limit (which is equivalent to zero DMI), the resulting lowest quasi-energy state is a ferromagnetic product state and hence unentangled, unless the anisotropy is of easy planar type. But with moderately high frequencies (but not small ones, as discussed in Appendix A for which other higher order terms from the high frequency expansion of the Floquet Hamiltonian also become relevant), the system can become entangled due to generation of the DMI term, as an antisymmetric DM exchange interaction can excite entanglement and teleportation fidelityzhang () in the system. This is a short-range interacting system and thus the entanglement transitions does not coincide with the topological transition that occurs as soon as DMI term is brought in. However, deep within the topological phase, system shows finite entanglement, irrespective of the value of anisotropy.

While dealing with low temperature entanglement of these systems, we not only need information of the lowest energy state but that of low energy excitations as well. Following that, a measurement on thermal entanglement is very effective in this context. We compute thermal concurrences in our Floquet model and notify its behavior at the various low temperatures.

For numerical computation, we use diagonalization methods like exact diagonalization (for small lattices with L=6 and 12) and a modified Lanczos techniquedag () (for L=18 and 24) and obtain the concurrences there from.

The article is organized as follows. In section II, we start with the Hamiltonian formulation of the problem. In section III, we introduce concurrences in the Floquet model and discuss briefly how to compute that. Section IV details our results and the corresponding discussion and finally in section V, we conclude our work.

Ii Hamiltonian Formulation

A ferromagnetic spin-1/2 XXZ model is given as

(1)

When such system is irradiated with light, the electric field () of the light interacts with the spin moments () yielding time periodic Aharonov-Casher phasescasher () between sites and in the lattice. This paves way for a Floquet analysis resulting in an effective static Hamiltonian for the dynamic system. Particularly for high frequency circularly polarized irradiation with , a high frequency expansion can lead us to a Floquet Hamiltonian given as,

(2)

The details of the calculation can be found in the Appendix A. Notice that the spin anisotropy gets altered from to thereby changing the spin anisotropy parameter in the Floquet model. Here is -th order Bessel’s function of first kind with (see Appendix-A for definition of the parameters). Furthermore, an additional NNN DMI term sets in having amplitude where . So for very large , this is essentially zero and can only become significant otherwise. This DMI term acts as a complex NNN hopping term, like in a spinless Haldane model and is the reason behind its topological nontriviality. Here is a prefactor for hopping between sites and and for sublattice of the system.

Iii Concurrence in Floquet model

In order to compute the concurrence of the ground state as well as low energy excitations of a system, we need the full energy spectrum of the problem and we use numerical diagonalization of the Hamiltonian matrix to serve that purpose. As we deal with a Floquet model here, we look out for the Floquet quasi-energy spectrum and particularly, the lowest quasi-energy state and concurrences there in.

Figure 1: (a) Honeycomb lattice containing sublattices A and B. Unit vectors and (see Appendix A) are shown as well. Site numbering, implementing (b) PBC and (c) OBC, in a brickwall lattice (which is topologically equivalent to a honeycomb lattice) of size L = 18. The odd (even) numbered sites fall within the sublattice A (B).

We first briefly describe the lattice - its site numbering and its bond connections, that are necessary to identify different interaction pairs. A honeycomb lattice (see cartoon in Fig. 1(a)) can more conveniently be described using a brick-wall lattice. First we use periodic boundary condition (PBC) and the site numbering are given accordingly. The example for site lattice can be seen in Fig.1(b). For computing concurrence between NN and NNN pairs, we considered the numbered pairs (3,4) and (3,11) respectively. Please note here that this numbering is not unique and we only need to ensure that the numbering and boundary conditions do not break the symmetry of the lattice and treat each of the hexagonal plaquettes equally. In order to see the effect on entanglement at the edges, later we also consider finite systems using open boundary conditions (OBC) along directions. As can be seen from Fig.1(c), this amounts to pair of (zigzag) edges parallel to -direction while the system effectively extends to infinity along following the usual PBC (like in a nanoribbon).

Our paper deals with systems of lattice size and 24 respectively. Due to numerical constraints, we use exact diagonalization only for smaller 6 and 12 site lattices while for and , we use a modified Lanczos techniquedag (). This latter method search for the ground state (a lowest quasi-energy state, in this case) starting from a random state , with nonzero overlap with the ground state. This is then acted upon by the Floquet Hamiltonian to obtain the state , which is orthogonal to . The Hamiltonian, in its representation spanned by the basis states and , is then diagonalized. The lowest eigenstate is renamed as and iterations are continued until the true minimum energy state is obtained. See Ref.dag, for details.

Given the state, we can now compute the concurrence between NN or NNN spins. Let’s call the spin- basis vectors as ’s, in terms of which we can write the eigenstates of our Hamiltonian as and let denote the -th eigenvalue. The ground state density matrix will then be given by . We can also compute the thermal density matrix which, in the canonical ensemble, is given by , with .

As the system is bipartitioned into subsystems and , we can write and the reduced density matrix in the subsystem will be given as

(3)

Our subsystem consists of a pair of spins, especially NN or NNN pairs, that we consider here. We should mention here that our work involves bi-partite entanglement alone and does not deal with entangled states (like a GHZ state or a W state) corresponding to further partitioning of systems.

Now let us look at the definition of quantum concurrence. For a 2-qubit system, the pure state contains a measure of concurrence , where is the time reversed state of . For a spin system, a time reversed state is the spin-flipped state and for a spin-1/2 (2-qubit) system it is given by . When we have a general mixed state, full information of the wave function is not available and the time-reversed density matrix is obtained instead, as , to compute the concurrence. Here denotes the reduced density matrix at the reduced 2-qubit level. Concurrence becomes a function of and can be shownwooters () to be given by

(4)

where ’s denote the square root of eigenvalues of in descending order.

Figure 2: and versus at limit for and respectively.
Figure 3: Concurrence versus (in units of ) for(a) and (b) . In (b), results are obtained using modified Lanczos technique whereas in (a) both exact diagonalization (circles) and Lanczos (lines) results are shown.

Iv Results and Discussion

First we study the effect of anisotropy or at limit, the anisotropic Heisenberg ferromagnetic spin system alone. Within the lowest quasi-energy state, we probe the concurrence between nearest neighbor sites () and that between next-nearest neighbor sites () of the Floquet model. See Fig.2 for results of C vs. for lattices with and . Physically, variation of can be achieved by varying , keeping fixed. We see that concurrence becomes nonzero abruptly at signalling an entanglement transition. This feature remains intact in the thermodynamic limit as well, as shown via finite size scaling in Appendix B. However, the jump/discontinuity reduces as lattice size is increased.

Next we consider the situation as the DMI is turned on, with decrease of to finite large values. We see the concurrence to appear and then increase gradually beyond some cut-off values for whereas always remains zero. That cut-off remains the same as long as the Ising anisotropy remains. For , the system is already entangled at . There is also a cut-off strength in this case, beyond which perishes and shoots up and keep on increasing to reach a plateau finally. We should mention here that for a fixed , is inversely proportional to and as such can be replaced with to visualize the frequency dependence of the concurrences more clearly. Accordingly, Fig.3 shows variation of concurrences in terms of in units of the prefactor .

Now let us explain the results of concurrence that we get. At very large frequency, the DMI term becomes negligible and the XXZ model of the Floquet Hamiltonian shows finite entanglement as soon as the anisotropy becomes easy planar. Product state of the easy axis FM turns into a entangled state with moments oriented in the spin- planekarkeola (). Finally for , the ground state still has no product state form as no direction in plane is preferred for the spin-moments. As a result, the entanglement, emerging from , gradually saturates to a finite value for large . With (or, ), there is no direct interaction between NNN spins. Heisenberg point being the critical point, spin correlation is at its peak for closest spins, which then decays as the distance between the spins is increased. Hence both NN and NNN spins are very much correlated as well as entangled at . Moreover, as the entanglement producing spin-fluctuation terms appear only between NN spins, we find whenever they are nonzero. We should emphasize here that as spin exchange between NN pairs entangles them more, we see to increase steadily with beyond the Heisenberg point. This pushes for steady decrease possibly due to spin conservation or the monogamy of entanglementbal (). Let us add here that the bump in observed for at is a finite size effect which gets wiped off significantly in the plot corresponding to and . Notice that it does not appear for as PBC makes this a special case where the NN and 3rd NN sites often become identical.

Now as the DMI term is turned on, due to decrease of the frequency of irradiation from very large values, the system becomes topological. However, it takes some finite values to get the Floquet system with easy axis anisotropy to become entangled for NNN spin pairs. This is because the NNN spin fluctuation terms oppose the FM ordering and it takes a finite threshold to disrupt that ordering and set in entanglement. On the other hand the NN pairs never get entangled by introduction of this complex NNN hopping term. The easy plane ferromagnet, which had finite and , shows decrease and increase in entanglement with for NN and NNN pairs respectively. The DMI term is a precursor of the spin-orbit coupling in the system and it favors spin canting. As this term acts between NNN pairs, a strong indicates a larger correlation among the NNN pairs. But it also competes with NN spin exchange term and let the correlation between the NN pairs perish gradually. Fig.3(a) shows the concurrence results for where both exact diagonalization and Lanczos results are shown which fairly matches for the values of anisotropies considered. For larger site lattice, we use Lanczos method and obtain same qualitative results as shown in Fig.3(b). Notice that for easy planar case, shows two smooth branches connected by a jump/discontinuity in the middle (however, such high frequency branch vanishes for very large , as also can be seen in Fig.4(a) which shows for small ’s). Using finite size scaling analysis, we have seen this to exist even in the thermodynamic limit (see Appendix B). This is a phase transition in which a redressing of the spins develop within the spin- plane. An easy planar ferromagnet already has entangled NN and NNN spin-pairs even without any DMI term. With finite , DMI is brought in which opposes the existing NNN spin ordering (and that enhances accordingly due to monogamy of entanglement) resulting in slight reductions in with . On the other hand, the low or large branch of the plot appears beyond the cut-off strength, like in the easy axis case, and characterizes increase in NNN spin correlation (in its DMI induced new spin ordering) with or .

Figure 4: (Color online) Variation of (black) and (red/grey) as a function of on a size lattice for (a) and (b) . The inset shows the variation of and with .
Figure 5: (top)Thermal Concurrence for at . Threshold temperature for (bottom-left) and (bottom-right) in the plane.

Our results on -vs- plots describe variation of concurrences as the is varied, keeping fixed. But it is also useful to look at the variation of concurrences with (see that is proportional to the electric field amplitude ) for fixed large values. In Fig.4 we can see such variations for two different sets of . In Fig.4(a) and within the range of shown, easy planar anisotropy is experienced by the Floquet system. It shows that a large anisotropy can push to zero even at the unirradiated limit . In Fig.4(b), we consider a comparatively small , yet being large compared to and . It shows transition from easy planar to easy axis anisotropy (see the inset). With increase of , the anisotropy changes from easy planar to easy axis type (beyond ) and that makes to go to zero for all larger values, whereas the behavior of demonstrates aptly the sudden death and revival of entanglementsudden-death () as it remains zero only if the is less than the cut-off value as mentioned in discussion pertaining to Fig.3.

iv.1 Thermal Entanglement

We know that the ground state is realized at zero temperature and in practice, low energy excitations also need to be taken into account to understand the low temperature phenomena in a system. Thus in our case, its wise to take a look at the thermal entanglement that corresponds to entanglement properties at a finite temperature. Fig.5 shows the results of thermal concurrences and in a size system obtained for various values, and for . Our thermal entanglement results show that temperature causes the entanglement measure in the system to wear off and with high temperature, the thermal fluctuation leads the system towards complete unentanglement. As further quantification, we compute the threshold temperature above which there is no concurrence possible in the Floquet states. In bottom panel of Fig.5, we show the variation of for and in a plane. It shows that a large ( much larger than unity) keeps the NN spins entangled upto some appreciably large values, if the irradiation born term is not very strong. On the other hand, a large makes the NNN spins entangled with appreciably large values when easy planar is not very large.

Figure 6: Variation of fidelity of the lowest quasi energy state between the edge-less and edged configurations corresponding to and at . The inset shows the same plot zoomed in around f=1.
Figure 7: ( (black) and (red/gray) versus at the edges (solid lines) and within the bulk (dashed lines) of a size lattice with PBC along and OBC along for and for (a) and (b) respectively.
Figure 8: (top) and (Bottom) versus , for fixed values, within the bulk (black lines) and at the edges (red lines) on a size lattice with PBC along and OBC along .

iv.2 Results for finite geometries with edges

As the presence of the DMI term brings in topological nontriviality to the Floquet problem, we need to pay special attention to the edges. Hence we consider finite size clusters with PBC along and OBC along so as to produce nano-ribbon geometries with zig-zag edges at the top and bottom along direction and try to quantify the edge correlations in the system. First we calculate the fidelity ( wave-function overlap) between the lowest quasi-energy states of an edge-less (with PBC along and ) and edged (with PBC along , OBC along ) system (for L=18) as is optically tuned keeping fixed. A few results are shown in Fig.6 for isotropic as well as easy planar configurations. The oscillating behavior appears due to the presence of Bessel’s functions within the Hamiltonian parameters. At Heisenberg point, we witness in the unirradiated limit, implying identical states for the edged and edge-less configurations. Hence, like any property, entanglement measures also do not change by merely bringing in such edges. But larger anisotropies (like , as shown in Fig.6) causes states to differ even in the unirradiated limit and we get different measures for concurrences for edge spin pairs and bulk spin pairs (see Fig.7). We should mention here that this fidelity calculation can only indicate identical or non-identical lowest quasi-energy state entanglement behaviors depending on whether or . For , we obtain in general and different entanglement measures can be expected for the edged configuration. As long as , our results show full fidelity (see Fig.6 and the corresponding entanglement match in Fig.7). We can say that for those cases, edge states are not present in the lowest quasi-energy states of the FTI. Other than those points, we witness both and to reach larger values at edges as compared to that in bulk, in the nanoribbon geometry considered (see Fig.7-8). Here we look for one possible explanation of such behavior. These points corresponds to and thus the wave-functions at the edges are more likely to differ from that from an edge-less configuration. So we can say that the lowest quasi-energy state do contribute to the edge states, which indicates gaplessness of the spectrum. Now we know that the entanglement entropy of a short-ranged gapped system show areal law behavior for the ground state entanglementhastings () whereas for a gapless system a logarithmic correction is added to that with prefactor proportional to the central charge of the corresponding conformal field theory at the critical pointcala (). This makes entanglement at the gapless point to be higher than that of a gapped regime. We find that the entanglement measure of two qubit concurrence, that we calculate here, also demonstrates similar behavior and produces larger concurrences at the edges than within the bulk.

V Conclusion

In this work, we have studied spin-spin entanglement in a Floquet system arising out of a FMHS model in a honeycomb lattice irradiated via circularly polarized light. Though this work can be termed as a simple study of entanglement for a spin-1/2 XXZ model with NNN DMI on a honeycomb lattice, the easy synthetic tunability of the Floquet system makes this work stand out firmly of the rest for we have the freedom to adjust the parameters of the Hamiltonian. We find that just by varying the amplitude and frequency of irradiation, and not directly modifying the anisotropy or DMI strengths as such, can lead to a plethora of interesting findings, in 2-spin ground-state as well as thermal concurrences. Firstly, when is very large (as compared to and ), the DMI contribution is negligible and increasing the field strength reduces the spin anisotropy . For the easy planar scenario, reduction of comes with decrease (increase) of . This occurs as quantify interactions between NN spin-pairs as well as due to the monogamy of entanglement. Across the Heisenberg point corresponding to the Floquet model, a transition develops from entangled to unentangled NN and NNN spin pairs. If the original spin anisotropy is barely above unity, high frequency irradiation can make the system unentangled producing separable product states in the lowest energy eigenfunctions.

Now as the frequency becomes intermediate so as to make appreciable, the system becomes topological. We see no coincidence between topological and entanglement transitions occurring there. In fact, this is not surprising as our working model obtained from the Floquet theory, comprises of short-range interaction/spin fluctuations alone and hence unlike in long-range entangled fractional Hall systemsmaria (), we don’t see any immediate entangling or disentangling as the DMI term is turned on. However, we notice interesting nontrivial entanglement features in presence of the DM term. The easy axis Floquet FMHS system produces non-zero beyond a cut-off value, as the DM term competes with the NN spin flip term of the Hamiltonian. For the easy-planar case, both and are nonzero without a DM term. Here also shoots up to a higher value beyond a cut-off while reduces down to zero. When we vary the field strength (which is proportional to ), we find that we can choose to have convenient parameters so that sudden-death and revival/rebirth of entanglement can be observed. This is due to transition of between easy axis and easy planar type which show different entanglement behaviors. Furthermore, we study thermal concurrence to demonstrate how system entanglement steadily decreases with the temperature.

Other than the bulk, we also study a zig-zag edged configuration and probe the effect of high frequency irradiation on that. We find that the lowest quasi-energy state differs due to the development of edges in the easy planar Floquet system and causes the concurrence measures to be higher at the edges as compared to that within the bulk.

Ours is an important piece of work as the concurrence patterns obtained can be useful in extracting quantum information from various QMBS. For example, controlled creation or destruction of entanglement via tuning concurrence of the Floquet states has already been shown for periodically driven coupled flux qubitssanchez (). Structures of entanglement for both surface and bulk states are examined in the topological insulator panigrahi () or the full density matrix of two qubit systems have been measured experimentally and the corresponding concurrence and fidelity computedshulman ().

Our thermal concurrence results also add important insight to the low temperature entanglement behavior in QMBS. Down the line, one can also explore the effect of transverse (normal to easy direction) magnetic field on the spins that sometimes witness enhancement of thermal entanglement with temperaturejafari () (which is not the usual behavior). Besides, it will also be interesting to quantify the quantum coherencechand () or perform the Bell-state measurementbell () on the entangled Floquet states. In short, we believe that the present study may trigger various further analytic as well as experimental researches with possible connection to spintronics and topological computations. We should mention here that experimentally, quantification of concurrence is possible using different protocols for both pure statesentropy (); measureC () and mixed statesentropy () and it will be interesting if that can be pursued for our Floquet system and compared with our numerical results.

Acknowledgements.
SK thanks S. Ghosh, S. Mandal, S. Yarlagadda and K. Sengupta for valuable comments, acknowledges CSIR, India for the financial support and IACS, India for providing computational facilities. Both the authors thank S. Owerre and D. Chowdhury for useful discussion on the work. Besides, the authors also acknowledge the anonymous referees for their comments and suggestions that resulted in considerable improvement of the paper.

Appendix A Floquet Hamiltonian

In a honeycomb lattice, there are two sublattices designated by A and B. Each of those sublattices can be defined using three unit vectors . Let us consider , the length of the NN bonds to be unity. Fig.1a shows a cartoon of the same.

A ferromagnetic XXZ spin- model is given by the Hamiltonian
.

Under a Holstein-Primakoff transformation, this takes the form: with and . Here and . denote the magnon annihilation operators and ’s are Pauli matrices to describe the pseudospins. The energy dispersion becomes that gives degeneracy at the Dirac points . Also notice that the Dirac nodes appear with nonzero energy .

Upon irradiation via circularly polarized light with (with ), an additional phase is added, due to Aharonov-Casher effect, to the amplitude of the spin fluctuation term involving site and :

(5)

where spin moment and are gyromagnetic ratio and Bohr magneton respectively). This brings in the time dependence as

(6)

For studying dynamics using Floquet theory, first the Fourier components of the Hamiltonian are obtained and they are given as

where and denotes the angular orientation of the bond.
For large , we utilize a high frequency expansion which gives an effective stationary Hamiltonian to the problem: .

For the present case, we obtain
and
. This 1st order correction turns out to be . Here for sublattice and . Thus, as long as is not negligible, compared to unity or ( the strength of the two terms of ), we should consider this 1st order correction to the Floquet Hamiltonian. Similar calculations can be seen in Ref. eckart, ; owerre, as well. Lastly, we want to add here that the 2nd order correction is proportional to and if we were to use Eq.2 for , we must be careful not to choose small enough that this term also becomes non-negligible.

Figure 9: Finite size scaling results: Asymptotic values of at limit.

Appendix B Finite size scaling

We did a finite size scaling analysis for concurrences at limit involving and size lattices which shows that the basic feature remains the same other than reducing the absolute values of and to non-zero smaller values (see Fig.9). It is not possible to do such analysis in presence of DMI, as itself shows some size dependence. But we can do finite size scaling analysis for the discontinuous jumps observed in Fig.3 and our calculations show its values to be 0.012 and 0.020 for and respectively. This shows that such jump indeed exist in the asymptotic limit.

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