The suppression of finite size effect within a few lattice sites

Boundary modes localized on the boundaries of a finite-size lattice experience a finite size effect (FSE) that could result in unwanted couplings, crosstalks and formation of gaps even in topological boundary modes. It is commonly believed that the FSE decays exponentially with the size of the system and thus requires many lattice sites before eventually becoming negligibly small. Here we consider a two-dimensional strip geometry that is periodic along one direction and truncated along the other direction, in which we identify a special type of FSE of some boundary modes that apparently vanishes at some particular wave vectors along the periodic direction. Meanwhile, the number of wave vectors where the FSE vanishes equals the number of lattice sites across the strip. We analytically prove this type of FSE in a simple model and prove this peculiar feature. We also provide a physical system consisting of a plasmonic sphere array where this FSE is present. Our work points to the possibility of almost arbitrarily tunning of the FSE, which facilitates unprecedented manipulation of the coupling strength between modes or channels such as the integration of multiple waveguides and photonic non-abelian braiding.


I. INTRODUCTION
A bound state can be trapped by a barrier.When the width and height of this barrier are not infinitely large, there is some probability that the state can tunnel through this barrier.These size-dependent phenomena are commonly called the finite size effect (FSE).FSE is ubiquitous for both quantum and classical waves.The barrier here can be a potential barrier such as a quantum dot [1,2] or originates from a band gap material [3,4].Besides the intrinsic absorption loss, the probability of tunneling determines the lifetime of these trapped states, which is crucial for quantum information processing [5][6][7][8].Control over the tunneling probability also enables manipulating interaction between different trapped states to generate various entangled quantum states [9][10][11].Meanwhile, fine-tuning of the coupling coefficient is a key requirement in programmable photonic simulators [12], non-abelian braiding of photons [13] and quantum computers [14], especially when nanostructure are considered.
Similar to trapped states, propagating states such as the waveguide modes also exhibit FSE [15].The tails of waveguide modes extend outside the waveguide with a length scale characterized by the penetration depth.If two waveguides are placed within the penetration depth of each other, there will be unavoidable intercoupling.The crosstalk between waveguides limits the integration of multiple waveguide channels into a compact device [16].Boundary (hinge) modes localized on the boundaries of a strip (hinge of a bar) geometry of a periodic lattice can also be regarded as waveguide modes.
With the recent explosive growth of research in topological physics [17][18][19][20], topological boundary modes and hinge modes associated with nontrivial bulk topologies have attracted a lot of attention due to their robustness against disorder and fabrication imperfections [21][22][23][24][25].However, these boundary and hinge modes also suffer from the FSE when the width of the system is not large enough [26][27][28][29].
So, even topological edge modes can be gapped if the width is not big enough to stop the coupling of modes localized on the opposite edges of the sample.On the other hand, controlling the FSE of topological boundary modes or hinge modes can achieve new versatile controllability, such as spin flipping [30], electrical switching [31], etc.
In this paper, we present a novel type of FSE of boundary modes.Without loss of generality, we consider the boundary modes on a two-dimensional (2D) strip geometry.The boundary modes localized on opposite sides (if they exist) interact with each other, giving rise to an FSE-induced gap.
Different from the prevailing understanding that the FSE vanishes only when the width of the strip is large enough, we discovered that the FSE could vanish at specific wave vectors (nodes) in a narrow finite-width strip.Interestingly, the number of nodes equals the number of lattices perpendicular to the boundary, i.e., the width of the strip.Using a model Hamiltonian, we analytically solved this system and proved the existence of this novel feature, and then demonstrated a filtering effect by utilizing this unique feature.Our system consists of only in-plane dipole orbitals and hence should represent general physics.Moreover, we provide a physical system composed of a plasmonic sphere array where this peculiar FSE can be observed.
The remainder of this paper is organized as follows.In Sec.II, we compare the difference between the typical FSE and this special type of FSE.We provide a tight-binding model Hamiltonian based on coupled in-plane dipoles x P and y P , and further show the feature of the number of nodes equals the number of lattices along the y-direction ( y N ).In Sec.III, we present the filtering effect by using this unique feature to filter out the component we need.In Sec.IV, we show that this peculiar type of FSE also exists in a plasmonic sphere array where all the coupling terms are taken into consideration.Hence our model is not limited to the tight-binding Hamiltonian.Finally, we summarize in Sec.V.

II. MODELS OF THE PECULIAR TYPE OF FSE
Our system is shown in Fig. 1(a) which is periodic along the x-direction and finite along the y-direction.
Each unit cell contains one atom or meta-atom which can support multiple modes such as S , x P , y P , z P , etc.These modes interact and evolve into bands in momentum space and the physics can be captured succinctly using the usual tight-binding description.Inside a band gap, the system may exhibit boundary modes if the parameters are appropriately chosen, as shown schematically in Fig. 1(a) with the red and blue shaded regions denoting the mode profiles.These boundary modes can either be Shockley states [32], Tamm states [33] or originate from topological reasons [17,18,20,21].In Fig. 1(b), we sketch the typical consequences of the FSE for a strip geometry of the system.Here the shaded areas represent the projection of bulk bands, and the blue and red lines represent the dispersion of the boundary modes.If there is no symmetry that forbids the boundary modes to couple, they will couple when they cross each other, always forming mini-gaps.However, if the system possesses mirror symmetry ( y m ) as in our case, the boundary modes on opposite sides of the strip should exhibit the same dispersion when the strip is wide enough.Thus, for a finite size, there will be an interaction between boundary modes and once again form a mini-gap.In other words, the presence of a min-gap as shown in Fig. 1(b) seems unavoidable.
In contrast to Fig. 1(b), we present here a special type of FSE as pictorially shown in Fig. 1(c).Due to the mirror symmetry, the boundary modes can couple to form even and odd modes, as shown respectively by the red and blue lines.They twist with each other and form several nodal points.As will be shown later, the FSE of the boundary modes vanishes at these nodal points (no interaction between boundary modes).Intriguingly, we find that the number of nodes equals the number of lattices along the y-direction ( y N ).Anomalous FSEs has been noted for helical boundary modes in topological insulator where the strength of the FSE decrease non-monotonically with the size [28,29,34], therein the oscillation length is hundreds or thousands of lattice constant.Therefore, the anomalous FSE is significantly different from our case as the FSE here oscillates at the lattice scale.Meanwhile, the two boundary modes in our case are related by mirror symmetry and thus exhibit the same dispersion in the absence of FSE.In contrast, the helical boundary modes in Refs.[26,28,29] exhibit opposite group velocities.
First, we provide a tight-binding model Hamiltonian that exhibits the salient features outlined in Fig. 1(c).We derive this Hamiltonian with the coupled dipole equation [35].We assume each site supports two dipoles x P and y P .For simplicity, we assume that other excitations are either far away in energy or are orthogonal to x P and y P (e.g., the z P dipole).The hopping from a dipole j P at lattice site j R to the dipole i P at i R is given by [36]     2 5 3 G r P r r P with , and here t is introduced only to ensure that the unit of hopping is energy.The " " sign is intentionally introduced here to match the latter simulation of a real system.For convenience, we set 1 t  and r is in a unit of the lattice constant a.To obtain a simple tight-binding model, we truncate the hopping to the next nearest neighbor which are the minimal interactions needed to explain the physics presented in Fig. 1(c).The momentum space Hamiltonian for a periodic system exhibiting 4v Due to the 4v C symmetry, bands are degenerate at  and M .The bands are nondegenerate except for those two high symmetry points (See Appendix A).We note that though we derive the Hamiltonian with the coupled dipole equation, a similar tight-binding Hamiltonian can also describe electronic systems of the same symmetry if p-orbitals ( x P and y P ) are dominant in the energy range of interest.
As we are interested in a strip geometry which is periodic along the x-direction and finite along the ydirection, we provide the projected band structure of the periodic system along the x k direction as shown in Figs.2(a-d) with the light gray background (the projected bulk band continuum) as a reference.Here we can focus on the 0 is quantized as  , and hence a semi-infinite system possesses a topological boundary mode which is located inside the band gap with dispersion connecting the two band edges [37].(See Appendix A) The FSE introduces coupling between two such boundary modes located at the y  and y  boundaries if the system is finite.
Then we proceed to investigate the FSE for a finite number of lattice sites ( y N ) along the y-direction.
First, we start with an extreme case where 1 y N  .This case corresponds to an infinite chain of dipoles.
In As shown in Fig. 2(b), there are now four bands in total and two of them appear inside the bulk gap region, with one even state (red) and one odd state (blue) for the y P component.Interestingly, now the red and blue bands cross with each other twice which is also the number of the coupled chains.As we further increase y N as shown in Figs.2(c) and 2(d), the gray bands start filling up the projection of the bulk periodic bands, and the red and blue bands approach the dispersion of the boundary modes for a semi-infinite system.Still, the number of nodes is always equal to y N .
To prove that the number of nodes between the even and odd modes is equal to y N , we analytically solve the system.(Proof in Appendix B).We first obtain the eigen energy and eigenstates of the boundary mode for a semi-infinite system as and respectively.In Eq. ( 4), the subscript labels the number of dipole chain, and the first (second) element inside the parenthesis represent the x P ( y P ) component.

   
2 2 cos / 4 2 cos and with N being the normalization constant, 1 , and To solve for the FSE for the boundary modes in a finite system, we use the eigenfunction of the semiinfinite system as basis.With perturbation theory to the first order, we obtain that the hopping strength between the two boundary modes localized on opposite boundaries is Thus, to the first-order approximation, the frequencies of the even and odd boundary modes are the first-order approximation we take.With the increase of y N where the prediction of E  works better, red + and blue square approach each other.

III. FILTERING EFFECT
The boundary modes can be utilized as waveguide channels to guide waves.In our system, the coupling strength between two boundary modes on opposite sides vanishes at these nodes and thus each boundary mode can be confined to one side of the waveguide as it propagates.As the coupling between boundary modes localized on two boundaries is a function of x k and y N , we can use this unique feature to filter out the component we need.In Fig. 3, we demonstrate this effect.The system consists of four chains ( 4 y N  ) and the wave function is assumed to be initially only on the 4 th chain as sketched in Fig. 3(a) with the amplitude given by where we assume a Gaussian package with center wave vector, width and center position given by

IV. Plasmonic sphere array with this FSE
Up to this point, we used a tight-binding model with next-nearest neighbor coupling to reveal the physics of such a peculiar FSE.We then demonstrate that such a novel effect exists with full wave simulations when all the coupling terms are considered.As our system only requires the symmetry imposed by the x P and y P modes in a square lattice, the FSE discussed herein should be quite universal.Possible candidates are photonic crystals, phononic crystals, cold atoms, and 2D materials with properly chosen orbitals.Below we show that this peculiar type of FSE also exists in a plasmonic sphere array.We consider an array of plasmonic spheres with lattice constant with 3 3    I being the 3 3  identity matrix and Note here, when the wavelength is much larger than the lattice constant, i.e., 0 1 k r = ,   ik r e  .Thus we get the quasi-static limit [40-42]  in Fig. 4) as the interaction of the plasmonic spheres with the free propagating wave is taken into consideration.Except for that, the dispersion is quite similar to Fig. 2(c).Here we can also find three nodes (marked by the red dots) between the middle two bands.Hence we have numerically demonstrated the fact that the nodes of FSE are not limited to the tight-binding Hamiltonian in Eq. ( 2).
The underlying physics should be universal even in the presence of the light cone when we consider x P and y P orbitals in a square lattice.

V. SUMMARY
In summary, we analytically solved a next-nearest-neighbor hopping model to investigate the FSE of boundary modes.The boundary modes of a finite-width strip twist around each other, intersecting at nodes and the number of nodes equals the number of lattices across the strip.This FSE leads to a special type of filtering effect where only the components with wave vectors match those of the nodes preserved.Meanwhile, this peculiar FSE is general and also presents in the plasmonic sphere array.
Our model is based on just in-plane dipole orbitals and needs no further assumption, and the FSE discussed here can be found in electronic waves, classical waves and cold atoms.Our work points to the possibility of getting rid of the formation of gaps due to FSE with properly chosen orbitals and lattice, and thus opens a feasible way for integrating multiple waveguides into a compact device.
Meanwhile, the series   where N is the normalization constant, 1 The wave function satisfies where Eq. (B8) is used in the last step.Noting that 1 0 d  [see Fig.

x k  region as the 0 xk
 region is simply related by time- reversal symmetry.Now we have a band gap region between the two bands and the band gap the hopping strength is zero thus the FSE vanishes, and then there is a nodal point on the even and odd boundary bands.It can be proved that E proof is also provided in Appendix B).To check that the zeros of E  indeed predict the number of nodes, we provide the locations of boundary band nodes (red +) and the zeros of E  (blue square) in Fig.2(e).The number of the red + exactly equals the number of the blue square for every y N , and the difference comes from

k
as given by Eq. (B3).All the lattice sites are set as lossless (orange) except the first column which is absorptive with onsite energy 1 0.2i    .We solve the time-dependent Schrodinger equation.For simplicity, we set 1  h in the simulation.With the propagating of the wave package, the unwanted components are coupled to the first column and get absorbed eventually.Figure3(b)gives the normalized x k component as a function of evolution time.At 0 t  , it exhibits a Gaussian shape centered at 0 x k as given by Eq. (9).With the increase of t, the Gaussian shape gradually evolves to two sharp peaks at / 2 x k a   0.22 and 0.28, which are exactly the x k values at which the two boundary modes decoupled.

.
The coupled dipole equation is

1 a
vacuum with c being the speed of light,    is the electric dipolar term of the Mie coefficients[39].W  is the dyadic Green's function, and   0 B k r , and 0 1

44 ]
and W  reduces to the hopping in Eq.(1).The coupled dipole equation can be reformulated as ext In a periodic system, eig  is a function of  and the Bloch wave vector.For a passive system, to the extinction of the driving field, and the width of an extinction peak on  is proportional to the mode quality.[43,44]Hence the summation of all the imaginary parts of the eigen-polarizabilities, i.e., resonance peak shows the dispersion of such a plasmonic sphere array.In addition to the dispersion, this plot automatically shows the presence of light cone (marked by the black dashed line FIG. 1.A peculiar type of FSE.(a) The coupling of two boundary modes (denoted by the red and blue shaded regions) localized on opposite sides of a strip geometry of a square lattice with x P and y P orbitals.The lower right inset shows the reciprocal space.(b-c) Sketches of two different types of FSEs, where the FSE is finite for all x k in (b) and vanishes at several nodes for certain x k s in (c).
of x k and the energy of the electromagnetic waves.Here the black dashed line marks the position of the light cone, and the red dots highlight the positions of nodes on the boundary modes.In this plot, the lattice constant is

.
FIG. 6.(a) Comparing of  obtained from Eq. (B3) for a semi-infinite system and the numerically

.
The first few n p functions are provided in Figs.7(b) and 7(c).We can clearly see the oscillation feature and observe that n p exhibits 1 n zeros.

FFF 1 y  and 1 y
FIG. 8. i F defined in Eq. (B14a) with To obtain more detailed features of n x k , which gives the essential reason for the emergence of nodes.And Eq. (B9) proves that n p possesses 1 n zeros for with E being half of the energy difference between the even and odd modes.E  also equals the hopping strength between the boundary modes localized on different boundaries.If we assume that E  is a small number and can thus treat it as a perturbation.Then to the first order of E  , we have exhibit no zeros, and combined with Eq. (B7), we find that the zeros of E  is the same as the function (the numerator without some trivial constants)