Topological water wave states in a one-dimensional structure - arXiv

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of water wave systems, and paves the way to potential management of water waves. (a) E-mail: ... systems9-11 in the last three decades. However these ...
Topological water wave state in a one-dimensional structure Zhaoju Yang1, Fei Gao1 and Baile Zhang1,2,* 1

Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore. 2

Centre for Disruptive Photonic Technologies,

Nanyang Technological University, Singapore 637371, Singapore. *To whom correspondence should be addressed: E-mail: [email protected].

Abstract Topological phenomena are new topics in electronic, photonic and phononic systems, but not found in water-wave system up to now. To characterize the topological properties of Bloch wave-functions, one can use topological invariants - Zak phase for one dimensional periodic lattice. Here we demonstrate the topological transition and Zak phases in a periodic shallow water wave system. The band diagram of periodic water tanks has a nontrivial transition point as the variation of tank parameters. The periodic water tanks at two sides of the transition point have topologically nonequivalent band gaps. As a result, interface state can exist at the boundary separating the two distinct kinds of water tanks and the vertical displacement can be enhanced. Our work may pave the way to explore topological properties further in higherdimensional water wave systems.

Introduction Recently, the field of the water waves propagating through periodic structures developed rapidly, such as band gaps of periodic lattice1-3, super-lensing effect4, refraction5 and metamaterials6-8, which are inspired by rapid progress in photonic systems9-11 in the last three decades. However these concepts are still within the traditional methods of manipulating water waves, and isolated with recent topological concepts12-15.

The topological nontrivial band structures of periodic systems have many intriguing physical phenomena, like quantum Hall effect12 and topological insulators14,15. Their special topological band structures give rise to the one-way propagation of quantum electronic waves along boundaries of the samples. Most importantly, the edge states are robust against back-scattering from disorders because of the lack of the backscatter channels. As further developed in photonics16-25 and phononics26-28, the topological states in classical physics have greatly shown the promising potential of manipulating many forms of waves in a nontrivial way. The above topological properties of two- or three- dimensional Bloch bands can be defined by topological invariants, which are calculated in terms of the well-known Berry phase29 during the adiabatic motion of a particle across the Brillouin zone. For one dimensional case, the topological invariant of Bloch bands is the so-called Zak phase13, which was proposed theoretically by J. Zak in 1989. And this topological property could be found in condensed matters provided by Su-Schrieffer-Heeger (SSH) model30 of polyacetylene and linearly conjugated diatomic polymers31. But until 2013, Zak phase has been experimentally measured in condensed matter physics of ultra-cold atoms32 and proposed in optical waveguide system33. Recently, it has been shown that Zak phase was measured in one dimensional periodic acoustic systems34. Interface states existed at the boundary between two different kinds of phononic crystals and acoustic field could be enhanced. The purpose of this letter is to bring the concept of topological states into one dimensional water wave systems. We theoretically demonstrate the existence of topological transition point as varying unit structure parameters, giving rise to two topologically distinct one-dimensional water tanks. And we numerically calculate the Zak phase and confirm the results by analyzing the symmetry properties of band states. Also we perform the simulation of the water wave system and demonstrate the interface state, which indicates the localization of the vertical displacement of water surface.

Results The unit cell of one dimensional periodic water tanks is shown in Fig. 1a. The lattice constant

is a . The relations of the length and lattice constant is a  LA  2LB . Each unit cell has two

tanks with larger cross section with width WA  0.2a and one small tank with width

WB  0.5a . The tanks filled with injected water with small depth h constitute the one dimensional periodic water wave system. Hereafter we only consider irrotational, inviscid, linear and harmonic water waves in the above system, and the governing shallow water wave equation35 is

 h (r )  t2 (r ) g  0 ,

(1)

which is valid on the condition of �ℎ ≪ 1. The  (r ) is the vertical displacement of water surface and g  9.8 m / s 2 is the gravitational acceleration. The Eqn. 1 indicates the linear

dispersion of  / k  gh . The boundary condition of no flow through the water tanks’ walls is n   0 . We can apply Bloch wave theorem and perform the calculations based on finite

element method to obtain the band diagrams. We should note here that since we investigate one dimensional case, in the following we neglect the flat bands because of their non-uniform profile of vertical displacements alone lateral direction similar as in Ref. 34. And in the below results, they are all the band states with uniform field profile along the direction perpendicular to periodic direction.

First, we set lattice constant a =10 cm and uniform water depth h  0.2a . We sweep the parameter LA when momentum k  0 and the result is shown in Fig. 1b. The second and

third band states cross each other with a degeneracy at LA  5.7 cm, which indicates that the gap closing and reopening between second and third band could be similar to the electronic system14,15 and the topological transition point is at LA  5.7 cm for the water wave system.

To investigate the topological property of the transition point, we can extend the concept of Zak phase13, which was first developed from electronic system, to our one dimensional water wave tanks. Zak phase characterizes the topological property of the Bloch bands and can be acquired by integrating Berry phase29 across the Brillouin zone. Considering a given normalized Bloch wave function  n,k (r ) , we can express the Zak phase as

Zak  i 

 /a

 / a

un,k |  k | un,k  dk ,

(2)

where n is band index, k is momentum, a is lattice constant and the periodic Bloch

function un,k (r )  eikr n,k (r ) .

Then we choose three unit tanks with only different parameter of LA , which is tank A ( LA  3.0 cm), tank B ( LA  5.7 cm) and tank C ( LA  8.0 cm). We numerically calculated the band structures and show the results in Fig. 2a-c, respectively. We can see that in three panels of Fig. 2., each one has a band gap between first and second bands. This band gap does not close during the parameter sweeping, which indicates that the first band gap of the three structures in Fig. 2 are topologically identical. However, Fig. 2a, c shows the existence of second band gap between second and third bands and the Fig. 2b shows the band closing in agreement with the Fig. 1b. After numerically calculating the Zak phase of the second bands by using a discretized form34 of Eqn. 2, we gain 0 and π for periodic tank A and C, as shown in blue numbers in the two panels. As we know, the topological property of the band gap

depends only on the summation of the Zak phase of all the bands below the gap14,15. In view of above results (topologically identical first band gap and different Zak phase of second bands), we know that the second band gaps in Fig. 2a, c are topological non-equivalently. To further confirm the topological difference of the band gaps in Fig. 2a, c, we also analyze the Zak phase from the symmetry properties34,36 of band states, which means that if the states of one band at the center and edge of the Brillouin zone have the same symmetry (even or odd function with respect to center inversion plane of each unit), the Zak phase of this band will be zero. Otherwise the Zak phase of this band is π. The profiles of the vertical displacement of water surface of second bands of water tank A, C are shown in Fig. 3a, b respectively. The red

(blue) lines indicate the eigen-function profile at k  0 ( k   / a ). For periodic tank A, the profiles are both even function with respect to the tank center point, which means the Zak phase of the second band is zero. Whereas for periodic tank C, the profile is even (odd) function at the center (edge) of the Brillouin zone. So the Zak phase acquires π. These results are in agreement with our numerical calculations shown above.

Nontrivial Zak phases underlie the existence of protected interface states. In our precious water wave system, there should be an interface state localized at the boundary between topologically different periodic water tanks. For two semi-infinite periodic water tanks of tank A and tank C separated by an interface, we calculated the band diagram as shown in Fig. 4a. We find that there is one single band state inside the second band gap and by checking the field profile we know this state localized at the interface. By contrast, we choose a different structure, which

also has two water tanks but with LA  3 cm and LA  4 cm. In view of the previous result in Fig. 1b, we know that they are topologically identical to each other. The band diagram in Fig. 4b tells us there is no localized interface states inside the second band gap. To demonstrate the interface states, we put a water wave source with frequency 4.2 Hz at the boundary (X=0) separating the two semi-infinite water tanks (left: tank A, right: tank C) as shown in Fig. 4c. The red-blue pattern shows the positive and negative water surface displacement of the interface state and they decay rapidly into the bulks. In Fig. 4d, we plot the vertical displacement of water surface of 20 unit cells (left: 10 tanks A and right: 10 tanks C) along the periodic direction. The black line shows that the displacement of water surface has strong localization at the interface (X=0). We should note that, if we simulate the periodic water tanks with the parameters as shown in Fig. 4b, field localizations can still be found because of the lack of bulk channels inside the band gap. However, this is not localized interface state. The interface state demonstrated in Fig. 4a,c,d has an enhancement of vertical displacement with the order of magnitude int

2

tri  102 over the trivial case in Fig. 4b. 2

Discussion In summary, we investigated the topological transition point and extended the Zak phase13 into one dimensional periodic water wave systems. We also demonstrated the interface states in one specific water tank configuration. Similar methods could be applied to investigate the topological properties of Bloch bands at higher frequencies. This kind of experiment should be quite easy to perform. For this paper, we use lattice constant 10 cm and we can simply 3D print the water tank units and bond them together by strong glues. The parameters can be scaled up or down if needed. The measurement of Zak phase could use similar reflection-phase and

symmetry method34. The reflection phases and symmetry properties of wave functions can be directly observed by taking photos (high speed camera if needed). Also the interface state field pattern of vertical displacement can be observed directly from photos. The results here bring the concept of topology into water wave systems in one dimension and may pave the way to explore topological states further in higher dimensions.

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Figure 1 | Water tank structure and topological transition. a, the three dimensional view of one unit cell of water tank with lattice constant a. b, topological transition point at LA  5.7 cm.

Parameters: lattice constant a  10 cm, length relation with lattice constant a  LA  2LB , width WA  0.2a and WB  0.5a , water height h  0.2a .

Figure 2 | Band structures of different periodic tanks. Band structures of one-dimensional periodic water tanks with length parameters a, LA  3.0 cm. b, LA  5.7 cm. c, LA  8.0 cm. The first band gaps in three panels are topologically identical. The second band gaps in panel a, c are topologically different. The blue numbers in panel a, c represent the Zak phase. Panel b is the band diagram at transition point. The other parameters are the same as in Figure 1.

Figure 3 | Symmetry of bands at center and edge of Brillouin zone. a, (b,) The eigenfunction profile of Tank A (C) along periodic direction. The red (blue) lines show the profile at the center (edge) of the Brillouin zone. Two lines in panel a have both even symmetry, whereas lines in panel b show different symmetry property, which means that Zak phase is 0 (π) in the left (right) panel. The other parameters are the same as in Figure 1.

Figure 4 | Interface state. The band structure of two kinds of interface with a, semi-infinite periodic tank LA  3.0 cm and LA  8.0 cm. b, LA  3.0 cm and LA  4.0 cm. There is an

interface state localized inside the second band in pane a, whereas no interface state in panel b. c, the structure for demonstrating interface state. Red-blue pattern shows positive and negative water surface displacement distributions. The operating frequency is 4.2 Hz. d, strong localization of vertical displacement. The other parameters are the same as in Figure 1.