Phonon modes in a Möbius band

It is well known that phonon modes become sensitive to the geometry of an object when the phonon wavelengths are comparable to the objects physical length scale. In contrast, the sensitivity of phonon modes toward topology is much less explored and understood. In this paper we discuss the effects of topology on phonon modes using a finite thickness Möbius band of centerline radius a as the model system. The phonon modes are derived using the xyz algorithm based on Riemannian geometry. From the boundary conditions and parity we identify two sets of modes with wave numbers q = n / 2 a described by odd and even integers n. Modes characterized by odd integers have flexural vibrations whereas those characterized by even integers exhibit dilatational and shear/torsional motion. While the phonon dispersion at large wave numbers agrees with that of structures having simple topology (rings and wires), at low frequencies and wave numbers the Möbius topology introduces significant differences. Uniquely, we find three of the four phonon branches do not go to zero frequency with decreasing wave number, but converge on a finite frequency. We identify a new form of vibrational pattern resembling incomplete breathing modes and discuss the ramifications of the modified spectrum, including a local increase in the density of states and the existence of a phonon band gap.


Introduction
Möbius strip topology has fascinated artists and scientists since the middle of the nineteenth century when the structure was first identified by Listing and Möbius [1][2][3][4]. The mechanical properties, equilibrium shape and surface geometry of this unique one-sided, single-boundary structure have been studied extensively [5,6]. In addition, a rich variety of research associated with the topology has been reported, including the stability of soap films [7], novel light polarization schemes [8], numerical calculations of Laplace-Beltrami eigenfunctions [9], and birdcage resonators that show half-integer harmonic behavior when configured as a Möbius strip [10]. Techniques have been developed to fabricate three-dimensional nanostructures with nontrivial topology [11][12][13][14] and the effects of topology on electron transport behavior [15][16][17][18] discussed. More generally, the consequence of topology, and the concept of topological order, has been the subject of extensive research on correlated electron systems, such as the fractional quantum Hall effect [19] and topological insulators [20,21].
At the nanoscale, geometric constraints give rise to phenomena related to phonon confinement, such as specific heat anomalies [22][23][24], the quantization of thermal conductance [25,26], and modifications to the Raman spectra of nano-crystalline materials [27]. The importance of low-frequency flexural modes on the thermal and mechanical properties of graphene sheets has been comprehensively studied [28], and tuning the flexural mode by modifying the geometry of graphene nano-ribbons has been considered as a means to control transport properties [29]. Also, the thermal conductivity has been shown to depend on topology, with Möbius strips displaying a lower conductivity than rings or ribbons; a result attributed to increased phonon-phonon scattering and localization [30].
In this paper we address the effect of topology on phonon modes in the elastic continuum regime. We derive the vibrational spectra of a finite thickness Möbius strip modeled as a rectangular cross-section bar that is twisted axially by π radians and is formed into a continuous structure by linking the ends. The centerline of the structure is assumed to be a perfect circle of radius a. For clarity, we term this model structure a Möbius band for the remainder of the paper. We do not consider chirality because it does not play a role in the phonon properties of individual, isolated Möbius bands. We first set up the model for the band and derive related structural parameters, such as the metric tensors and Christoffel symbols based on Riemannian geometry. We consider the parity and boundary conditions for phonon modes specific to a Möbius band, and find an unusual coexistence of two classes of phonon mode with allowed wavelengths of 2πa divided by either integers or half integers. Following previous work [31], the phonon modes are derived using the xyz-algorithm [32] reformulated in terms of Riemann geometry. We introduce a set of basis functions satisfying the boundary conditions in order to express the phonon mode displacements. A critical consideration is the need for displacement vector matching imposed by the π radian twist of a Möbius band. Comparing the calculated modes to those of a wire and a ring, we find the lowest frequency dispersion branches close to the zone center for a Möbius band have strikingly different characteristics. In the case of a wire (ring), four (two) branches converge to a frequency ω=0 as the wave number approaches zero. We show that for a Möbius band, one branch has ω=0 at the zone center while three branches converge around a finite frequency ω g at zero wave vector. The density of states is enhanced around ω g below which the spectral density becomes very low, giving rise to a phonon band gap not found in other closed structures, for example rings. Details of the phonon spectra close to the zone center are shown to be sensitive to elasticity, but insensitive to geometry. We discuss these sensitivities and conclude that topology rather than geometry is the dominant factor determining the low-frequency phonon spectrum of a Möbius band.

Möbius band and coordinate system
We consider a Möbius band modeled as a rectangular cross-section bar of single crystal that is twisted axially by π radians and is formed into a continuous structure by linking the ends as shown figure 1. The center line C denoted by the dashed line is assumed to be a perfect circle of radius a.
We first introduce a coordinate system which rotates along C. X denotes a position on C and s is a length between the position and prescribed origin on C. t, n and b are the tangential unit vector along C, the normal vector defined below, and the unit vector perpendicular to t and n. κ is the curvature and τ is the torsion. All of these are defined and related by the following Frenet-Serrete formulae; n indicates the center of circle and the curvature becomes κ=1/a. b indicates the direction normal to the circle, independent of s, so that τ=0. Using n and b, we introduce a frame comprised of e 0 ≡t and two unit vectors e 1 and e 2 which rotate with increasing s along C. Assuming a constant rate of rotation of the frame and putting θ=s/2a, we define the unit vectors rotating with increasing s by e e n b cos sin sin cos . 2 Supposing that the Möbius band is made of material having a cubic crystal structure, for example copper or gold, the unit vectors e 0 , e 1 and e 2 are set to be parallel to each crystal axis.
To describe the structure and equation of motion of elasticity in terms of Riemannian geometry, the position in the Möbius band is given by 2 connected to the rotating frame as shown in figure 1, the variable ranges are For the system to be physically feasible, the half-wide w and half-height h should satisfy the condi- ) . Geometric parameters such as the metric tensors g ij and Christoffel symbols Γ ijk are given by which are summarized in appendix.

XYZ algorithm
The xyz algorithm developed by Visscher [32] is a powerful method to numerically obtain vibrational modes of a free-standing object. That work showed that the equation of motion of vibrations for a free-standing object becomes the same as the wave equation in a bulk material and that the exact solution automatically satisfies the boundary condition for a free surface. In the present work, we apply the method to a Möbius band. Since the method was originally developed in terms of Euclidean geometry, we reformulate the method using Riemannian geometry [33] .
To begin, we introduce contravariant and covariant displacement vectors u i and u i which are related by The strain tensor ε ij is defined by Assuming Hooke's law, the stress tensor σ ij is associated with ε ij via the stiffness tensor C ijkl as Using the strain and stress tensors, the Lagrangian of a Möbius band yields where g represents the Jacobian and g g ij = | |. From the variational principle and boundary condition for a free surface, we have the following equation for a phonon mode with frequency ω u 0. 12 We express the displacement vector, using a series of bases Φ λ ; and λ stands for λ={l, m, q} (l, m0). In order to set up an equation to obtain the coefficients i c l , we first substitute equations (13) into (12) and multiply by the complex conjugate of F l¢ . Integrating over the volume, we have Equation (15) is reformulated to be the following secular equation Numerically solving equation (18), we obtain phonon spectra of Möbius bands and corresponding displacement vectors.

Parity and boundary condition
Each phonon mode component has a certain parity for inversion of coordinates x 1 and x 2 associated with structural symmetries. Because of the π radian twist in a Möbius band, the parity will also depend on matching the displacement vectors around the band. Since bases are closely related to the parity, we examine the boundary condition to choose a suitable set of bases. As x 0 changes from 0 to 4πa along the centerline C, the frame rotates axially around e 0 by 2π radians and returns to the original orientation. Then the displacement vector matches itself at the original position, in other words all the phonon modes have a period of 4πa; In addition, since the frame rotates by π radians when x 0 increases from 0 to 2πa, the displacement vector at x x x 0, , ). The frame rotation also inverts displacement components u 1 and u 2 . Putting these things together, we have the following boundary conditions for displacement components; Supposing an asymptotic case of extremely large radius in comparison with wavelengthλ/a=1, the phonon modes of a Möbius band will closely resemble those of a rectangular wire as understood from equation (18), which in the limit of a  ¥ reduces to the secular equation for phonon modes in a rectangular wire [31]. A rectangular wire supports four kinds of phonon modes; a dilatational mode referred to as mode I, two flexural modes referred to as modes II and III, and shear/torsional modes referred to as mode IV. The shear and torsional modes are clearly separated only for a square wire, but are mixed for a rectangular wire. The displacement components of these modes are given by where l and m are 0 or 1, giving spatial symmetry of each displacement component. The combinations (l, m) for phonon modes are summarized in table 1. Here we apply the boundary conditions on the phonon modes in a rectangular wire. Putting equations (23) into (19), we have a condition for wave number q as where n is an integer. The wave number q is discretized to be an integer multiple of Since u i changes its sign only for an odd n when x 0 changes from 0 to 2πa, we have the parity for change in x 0 as Putting equations (27) into (26), we have the following relationship of the displacement; Comparing equation (28) with equations (20)- (22), it is found that the sum of powers N=l+m+n for each component must satisfy When n is an odd number(n odd ), l+m becomes odd for u 0 and even for u 1 and u 2 , and vice versa. As seen from table 1, modes II and III satisfy (29)-(31) only for n odd , and modes I and IV satisfy the conditions only when n is an even number (n even ) including 0. From the relationship between the wave number and wavelength q 2p l = and equation (25), the wavelength is given by the circumference 2πa divided by n/2 as a n When n=0, the wavelength becomes infinite and the vibrations become uniform motions. Considering both n odd and n even are possible for phonon modes in Möbius bands, modes II and III have a wavelength of the circumference divided by a half-integer while modes I and IV have that divided by an integer. It is unusual that the wavelength of modes II and III becomes a non-integer multiple of length for closed(ring-like) structures and that two different sets of phonon modes alternate as the wave number changes by 1/2a. These unusual properties are specific to phonon modes of Möbius bands. As the ratio λ/a increases, the curvature and torsion of a Möbius band will gradually modify or couple the phonon modes. Modifications of phonon modes from those of wire will be most important in the low frequency region near the zone center when the wavelengths become comparable to, or larger than, the circumference of a Möbius band, as shown below.

Phonon modes in a Möbius band
phonon dispersion branches of a wire. In contrast, and of primary interest, at low frequency and wave number the acoustic phonon branches disagree markedly with the dispersion relations of a wire. Figure 3 is a magnified region of figure 2 revealing the detailed spectral distribution of the acoustic branches in the low frequency region near the zone center, and introduces four characteristic frequencies ω a , ω b , ω c and ω d . A noticeable feature is that the frequencies of the two lowest branches associated with modes II and III for qa<10 deviate significantly from the parabolic dispersion relations of a wire and have finite magnitudes of the characteristic frequencies ( figure 3) ω b and ω c , which are comparable to v a g t w » at small wave numbers. Here v t is the sound velocity of transverse waves defined by v C t 44 r = . The frequencies ω b and ω c at qa = 0.5 can be  » for copper, ω g and ω b,c coincide accidentally.
The spectra associated with mode IV also deviate from the linear dispersion relation of a wire and the frequencies become larger than those of mode I, where the frequency ω a at q=0 is estimated as C a 2 a 11 2 w r = .
On the other hand, the spectra associated with mode I almost coincide with the linear dispersion relation of a wire and the frequency at qa=1 is estimated as Y a where Y is Young's modulus, which is given by Y C C C C 2 11 12 )for cubic materials. For copper, ω c is the minimum of these frequencies, and as there are no modes with a lower finite frequency a band gap of characteristic frequency ω c occurs.
In order to consider the details of the band gap we note that ω a , ω b and ω c depend on C 11 , but only ω d depends on Young's modulus suggesting that ω d could possibly be lower than ω a , ω b and ω c for materials other than copper. To parametrize the posibility of a band gap, we introduce the ratio of ω d and ω b For materials with f1 a phonon band gap exists with modes II, III, and IV providing a high density of states around ω b and ω c , below which the spectral density is low. For materials with f<1, mode I becomes the lowest non-zero energy state, somewhat analogous to having a state in the gap. Although ω c would be better than ω b for the definition of f, we use ω b instead of ω c since it is too complicated to express ω c in a compact form. Aside from the case that ω d falls between ω b and ω c like copper, f is a useful parameter to judge the occurrence of a phonon frequency gap. With f=1.05 for aluminum and 1.18 for diamond, we have confirmed the frequency gaps numerically, as shown in figure 4(a) for diamond. On the other hand, gold ( f=0.82) has a mode with finite frequency below ω d as shown in figure 4(b).
Here we mention the dependence of these characteristic frequencies on the Möbius band thickness and width. The frequencies ω a , ω b , ω c and ω d are tolerant to changes of the cross sectional dimensions, although at higher wavenumbers the slopes of spectra of modes II and III depend strongly on thickness. It is apparent, therefore, that the fundamental properties of the frequency gap do not change with the geometry, suggesting they are characteristic of the topology.
Furthermore, we note that the frequency gap does not appear for other closed structures such as rings. Phonon modes of rings are derived by using the boundary condition u x x u a x x 0, , 2 , , the high wave number phonon spectra of a ring agree well with the dispersion relations of wires, similar to the case for the Möbius bands, the ring spectra deviate from those of wire in the low frequency region. However, unlike the Möbius band the ring does not have a frequency gap since the spectra of the two lowest branches are almost the same as modes II and III of wires and tend to ω=0 with decreasing wave number, as shown in figure 5 [35,36]. This further supports the conclusion that the frequency gap is a phenomenon characterisitc of the Möbius band topology.

Vibrational patterns
From the finding that the phonon spectra of a Möbius band match the dispersion relations of wires at high frequencies or large wave numbers, we expect that the vibrational patterns of Möbius bands will also resemble those of wires. Figure 6 shows vibrational patterns at points A, B, C and D denoted in figure 2. It is apparent that they exhibit dilatational(A), torsional(D) and flexural motions(B and C) corresponding to those in the rectangular wire [31]. Here we pay attention to B and C whose wavelengths are λ=2πa/(19/2). We may say that the twisted structure of Möbius bands absorbs or compensates for the phase difference caused by the half of wavelength, making such a wavelength realizable. Vibrational motions of modes at low frequencies are more specific to the Möbius band. Figure 7 shows vibrational patterns for modes (a) at q=0, (b) and (c) at qa=1/2 and (d) at qa=1 denoted in figures 3 and 4. Because of gentle spatial variations of displacement vectors for small wave numbers, the arrows express the dynamical motions. Mode (a) exhibits uniform axial-torsion along the circumference. Although the motion is common to the torsional mode of wire, the frequency ω a is finite, as is the case for phonon modes in a ring referred to as 'torsional modes' in reference [36] (see figure 8(a)). Mode (d) is longitudinal waves whose wavelength matches the circumference λ=2πa, and leads to a mode with ω=0 at q=0. Thus the motion as well as the spectra coincide with mode I of wires. We note that a ring does not support such a mode leading to ω=0. The vibrational patterns of modes (b) and (c) at qa=1/2 with λ=4πa show new forms of vibration that resemble incomplete breathing motions, which are in contrast to the breathing mode of a ring shown in figure 8(b).

Summary and discussions
The phonon modes in a Möbius band with finite thickness have been derived. The Möbius band has been modeled as a rectangular bar axially twisted by π radians whose ends are perfectly linked. The phonon spectra and corresponding displacement vectors are obtained by means of the xyz-algorithm that we have extended in terms of Riemannian geometry. As a result of mode parity and boundary conditions, we find that a Möbius band supports two distinct sets of allowed phonon wavelengths characterized by odd and even integers. The wavelengths are given by λ=2πa/(n/2), with odd integers corresponding to flexural modes II and III, and even    integers to dilatational and shear/torsional modes I and IV. Importantly, this leads to integer and half-integer wavelength relationships being allowed, with the lowest frequency flexural mode (n = 1) having a wavelength of twice the center line circumference and the lowest finite frequency dilatational mode (n = 2) having a wavelength equal to the circumference. It also follows that the nature of the modes alternate as the wave number changes by 1/2a. A structure that supports modes with integer and half-integer wavelength relationships is unusual and is a direct consequence of the nontrivial topology.
We have also shown that phonon modes with wavelengths much shorter than the center-line circumference coincide with the dilatational, shear/torsional and two flexural modes of a rectangular wire. In this limit the vibrational patterns closely match those of wires, with the twisted structure of a Möbius band accounting for the half of wavelength phase difference. Major differences between phonon spectra of the Möbius band, wire and ring appear in the low frequency region near the zone center. For the Möbius band, three of the four acoustic branches do not go to ω=0 with decreasing wave number; instead they converge toward a finite frequency around which the number of states locally increases giving rise to the possibility of a phonon band gap. We have addressed the robustness if the gap towards geometry and have noted that a gap does not occur in other closed structures, such as rings that have two branches with ω=0 at q=0. We have parameterized details of the gap in terms of characteristic frequencies that depend on the elasticity of the material. Similar to other mechanisms that modify the phonon spectral distribution, we expect that the distorted spectral distribution resulting from the topology will give rise to anomalous thermal properties, which will be discussed elsewhere.
Finally, we emphasize that the topology plays a significant role in the phonon modes. The phonon band gap is one of the significant characteristics caused by the Möbius topology. Finite systems, such as nano particles, also have phonon spectral gaps. In these cases the band gap is caused by phonon confinement, which is a size dependent effect [24]. Very interestingly, a ring does not have a phonon band gap in spite of the finite size since the lowest acoustic branch begins at ω=0 and increases parabolically with respect to wavenumber. The difference in spectra between the nano particles and rings is caused by the difference in topology, i.e. the existence of a hole. Likewise, the difference between the Möbius bands and rings is also due to the difference in topology, i.e. twisted or not-twisted. Thus, topology substantially affects the phonon modes in a finite system and introduces effects that are comparable with, or larger than, those originating from geometry.

Acknowledgments
We would like to express our appreciation to Miles P Blencowe for his helpful comments.