# Topological D-Class Physics with Passive Acoustic Elements

Wenting Cheng  Department of Physics, New Jersey Institute of Technology,
Newark, NJ, USA  Emil Prodan<EMAIL_ADDRESS>Department of Physics, Yeshiva
University, New York, NY, USA  Camelia Prodan<EMAIL_ADDRESS>Department of
Physics, New Jersey Institute of Technology, Newark, NJ, USA

###### Abstract

In this work, we supply an engineering solution that paves the way to the
implementation of the full classification table of topological insulators with
passive acoustic crystals. As an example, we design an acoustic crystal
displaying the full range of characteristics of a topological insulator from
class D, such as non-trivial Chern numbers and an exact particle-hole symmetry
for both bulk and edge dynamics. The key is a particular geometry of the
acoustic resonators that facilitates a large number of independent and equal-
strength couplings, eight to be precise, such that the mode-coupling theory
remains accurate. The latter enables us to implement with high fidelity the
de-complexified tight-binding model generating the topological phases from
class D in 2-dimensions, previously announced in [1]. The successful
realization of a topological acoustic crystal from class D paves the way to
the observation of the Majorana-like modes stabilized by $\pi$-fluxes.

The discovery of topological insulators [2, 3, 4, 5, 6, 7, 8, 9] has
revolutionized the design principles of solid state materials and devices. One
of the most important achievements in the field was the completion of the
classification table that enumerates the strong topological insulators [10,
11, 12, 13]. Their different phases are separated by true phase boundaries
where a localization-delocalization phase transition occurs and they also
display bulk-boundary correspondences that are immune to strong disorder [14].
The entries from the table sample the ten fundamental Altland-Zirnbauer
universality classes [15] of mesoscopic systems, hence the topological phase
transitions have unique signatures in each class. Furthermore, each of the ten
classes supports unique physical phenomena and, for these reasons, it is
important to have reliable materials which sample the entire classification
table. In particular, topological insulators from class D are interesting
because they support majorana excitations, which can be stabilized and braided
by $\pi$-flux Dirac solenoids [16].

Around the time when the classification table was perfected, similar
topological phases were predicted for classical degrees of freedom, both
electromagnetic [17] and mechanical [18]. The second work observed that
quadratic terms like $q\cdot p$ in a Lagrangean break the time-reversal
symmetry of a system and can drive it into a topological Chern insulating
phase. Such terms require active elements and, indeed, the first laboratory
realization of a mechanical Chern insulator was based on active components
that break the time-reversal symmetry via such mechanism [19]. An interesting
alternative for producing mechanical topological phases came soon after in
[20], where just passive but well thought couplings together with doubling the
degrees of freedom creates the topological effect. A design based exclusively
on real couplings has the tremendous advantage that no fine-tuning and
enforcement of any phase delay is required. In acoustics, such designs will
require just in- and out-of phase couplings of the resonators, which can be
achieve with short acoustic bridges. This will free the designs from the use
of chiral tubes [21], which deliver the needed functionality only at one
specific frequency and, as such, make the implementation of the particle-hole
symmetry impossible because the latter manifests over an entire range of
frequencies. Therefore, a challenge was put forward to implement the whole
classification table of topological insulators exclusively with passive
elements and real couplings. In two and three dimensions, to authors’
knowledge, none of the topological phases from the table have been implemented
with classical degrees of freedom, except for those from A and AII classes.

The challenge was met in [1] with a simple theoretical solution. The model
Hamiltonians from the classification table [13] contain complex coefficients,
hence they need to be de-complexified without altering the algebraic
operations. The latter is need in order to preserve both the bulk and boundary
spectra. This was achieved in [1] by replacing $\sqrt{-1}$ by the real matrix
$\big{(}\,^{0}_{1}\ {}^{1}_{0}\big{)}$ and this substitution maps any complex
$H=\sum\hat{w}_{\bm{x},\bm{x}^{\prime}}\otimes|\bm{x}\rangle\langle\bm{x}^{\prime}|$
defined over a lattice ${\mathcal{L}}$ to a real Hamiltonian

$\rho(H)=\sum_{\bm{x},\bm{x}^{\prime}\in{\mathcal{L}}}\begin{pmatrix}{\rm
Re}[\hat{w}_{\bm{x},\bm{x}^{\prime}}]&{\rm
Im}[\hat{w}_{\bm{x},\bm{x}^{\prime}}]\\\ -{\rm
Im}[\hat{w}_{\bm{x},\bm{x}^{\prime}}]&{\rm
Re}[\hat{w}_{\bm{x},\bm{x}^{\prime}}]\end{pmatrix}\otimes|\bm{x}\rangle\langle\bm{x}^{\prime}|,$
(1)

at the expense of doubling the degrees of freedom.

Figure 1: Acoustic crystal implementing the Hamiltonian from Eq. (4) with
passive acoustic elements: (a) The actual domain of the acoustic equation used
in the COMSOL simulation of a system composed of 6 $\times$ 6 unit cells. The
yellow boxes indicate three unit cells $\bm{x}$, $\bm{x}+\bm{a}_{1}$ and
$\bm{x}+\bm{a}_{2}$, to be referred to in panels (c) and (d). (b-d) Schematic
representation of the lattice: (b) Top view of the square lattice with two
observers $O_{1}$ and $O_{2}$ looking along the $\bm{a}_{1}$ and $\bm{a}_{2}$
directions, respectively. Each red dot represents four resonators, which in
this schematic have been stack on top of each other. (c,d) The couplings
implementing $\rho(H)$, as seen by the observers $O_{1}$ and $O_{2}$,
respectively. The blue dashed lines indicate the positive couplings and the
green dashed lines indicate the negative couplings, all of equal strengths.
Each red dot represents one resonator. (e) COMSOL simulation of the resonant
spectrum of the crystal in a 8-cells wide ribbon geometry with periodic (black
markers) and hard (red markers) boundary conditions in the transversal
direction as function of $\Delta H=H_{1}-H_{2}$ defined in Fig. 3 and
representing the mass parameter from Eq. (4). The simulation is in perfect
agreement with the known phase diagram of model (4). (f) Resonant spectrum
corresponding to $\Delta H=0.5$mm, rendered as function of the quasi-momentum
parallel to the ribbon. Bulk spectrum (black dots) and two doubly-degenerate
chiral edge bands (marked in red and blue) are visible, displaying a high
degree of mirror symmetry relative to the mid-gap. (g,h) COMSOL simulations of
the mode profiles corresponding to a frequency inside the bulk gap ($\approx
5040$Hz) and finite geometries with clean edges and with a defect,
respectively. (i) Sound excitation with a source (see arrow) at 5035.1Hz and
with sound-absorbing boundary conditions on the left and lower edges. The
latter filter out one topological channels, leaving only the uni-directional
anti-clockwise sound propagation. (j) Same as (i), only with sound-absorbing
boundary conditions imposed to the right and upper edges, prompting clockwise
unidirectional wave propagation.

Any such Hamiltonian has a built-in $U(1)$-symmetry, since $\rho(H)$
automatically commutes with
$U=\begin{pmatrix}{}^{0}_{-\mathbb{I}}&{}^{\mathbb{I}}_{0}\end{pmatrix}\otimes
I$, and also has a fermionic time reversal symmetry
$\Sigma=U\mathcal{K}=\mathcal{K}U$, $\Sigma^{2}=-1$ ($\mathcal{K}$ = complex
conjugation). The matrix $U$ has spectrum at $\pm 1$ and we let $\Pi_{\pm}$ be
the projectors onto the corresponding spectral subspaces. Now, assume that the
original complex Hamiltonian displays one of the symmetries from the
classification table, $\Theta H\Theta^{-1}=\epsilon H$, $\epsilon=\pm 1$, with
$\Theta$ anti-unitary and $\Theta^{2}=\gamma$, $\gamma=\pm 1$. Such $\Theta$
can always be written as ${\mathcal{K}}W=W{\mathcal{K}}$ with $W$ unitary and,
if $\rho_{\pm}(H)=\Pi_{\pm}\rho(H)\Pi_{\pm}$, the following remarkable
identity holds [1]

$\epsilon\,\rho_{\pm}(H)=\big{(}{\mathcal{K}}\rho(W)J\big{)}\rho_{\pm}(H)\big{(}{\mathcal{K}}\rho(W)J\big{)}^{-1},$
(2)

where
$J=\begin{pmatrix}{}^{0}_{\mathbb{I}}&{}^{\mathbb{I}}_{0}\end{pmatrix}\otimes
I$. In other words, the symmetry projected real Hamiltonian obeys exactly the
same symmetry as the complex operator, which is implemented by

$\tilde{\Theta}={\mathcal{K}}\,\rho(W)J,\quad\tilde{\Theta}^{2}=\gamma.$ (3)

Note that the original symmetries $U$ and $\Sigma$ have little to do with
$\tilde{\Theta}$. Ref. [1] checked the above statement as well as the bulk-
boundary correspondences for all model Hamiltonians from the table up to
dimension three. The dynamics of these Hamiltonians with real coefficients can
be projected on the symmetry sectors $\Pi_{\pm}$ by choosing appropriate
initial conditions or using specialized sources [1]. More practical ways to
project onto the symmetry sectors will be discussed below.

The above principles have been around for several years, but experimental
implementations resisted to show up. Although optimal, the map (1) leads to
relatively complex crystal structures. For orientation, the Kane-Mele model
with Rashba coupling [3, 4], hence the true topological insulator from AII
class, requires eight resonators per repeating cell [1] (please see note [22])
and eight connections per resonator. In this work, we make the first important
step towards the experimental implementation of the full classification table
with passive components, by demonstrating a topological acoustic crystals from
class D (see Fig. 1), built on the principles described above. There are two
main challenges here: 1) opening a topological gap that carries a non-trivial
Chern number and 2) enforcing the particle-hole symmetry associated with the
class D, for both bulk and boundaries. Our work demonstrates that full control
over both aspects can be achieved. Let us stress that the passage from class A
to class D is significant because the particle-hole symmetry needs to be
enforced over a significant range of frequencies. The engineering challenge is
to ensure that mode-coupling theory [23], which we use to connect with the
theoretical tight-binding models, is accurate. This is difficult when large
number of connections are needed.

If $\sigma_{i}$ denote Pauli’s matrices and
$S_{j}|\bm{x}\rangle=|\bm{x}+\bm{a}_{j}\rangle$ represent the shift operators,
with $\bm{a}_{j}$, $j=1,2$, being the generators of the lattice (Fig. 1(b)),
the model generating the class D in 2-dimensions reads [13]

$H=\tfrac{1}{\imath}\sum_{j=1,2}\sigma_{j}\otimes(S_{j}-S_{j}^{\dagger})+\sigma_{3}\otimes\big{[}m+\sum_{j=1,2}(S_{j}+S_{j}^{\dagger})\big{]}.$
(4)

The particle-hole symmetry is implemented by the anti-unitary operation
$\Theta_{\rm PH}=(\sigma_{1}\otimes I){\mathcal{K}}$ and its phase diagram
consists of topological phases with Chern number $C=-1$ for $m\in(-4,0)$ and
$C=+1$ for $m\in(0,4)$, as well as of a trivial phase $C=0$ for
$m\notin[-4,4]$.

Figure 2: H-shaped acoustic resonators and their positive and negative
couplings (a) First resonant mode has frequency $f\approx 5000$Hz and consists
of out-of-phase pressure field $|p\rangle$ that is, to a high degree, uniform
along the arms of the H-shape. The red and green dots will be used for
reference, to communicate the connections in the more schematic
representations in Fig. 3. (b) An acoustic bridge between a pair of
resonators, which fits the mode-coupling Hamiltonian
$f^{2}|p_{1}\rangle+f^{2}|p_{2}\rangle+\beta(|p_{1}\rangle\langle
p_{2}|+|p_{2}\rangle\langle p_{1}|)$ with a positive coupling coefficient
$\beta$. (c) An acoustic bridge leading to a negative $\beta$ but identical
coupling strength $|\beta|$.

The mapping (1) leads to a crystal with four resonators in the primitive cell
and the connections shown in Figs. 1(b-d). Each resonator needs a total of
eight connections of equal strength with its neighboring resonators.
Implementing these connections was one of the main design challenges. A search
among various options led us to the H-shaped resonator shown in Fig. 2, which
is quite unique. Its first resonating mode, shown in Fig. 2(a), displays a
uniform pressure field along the long arms. Positive and negative couplings
between these resonant modes carried by two resonators can be established by
the short bridges shown in Figs. 2(b,c). The main advantage of the H-shaped
resonators and the use of the first resonant mode is that the bridges can be
attached to four ends, hence reducing the load per end, and can be moved
between the arms of the H-shape sharing the same color, without affecting the
strength and sign of the coupling. This feature ultimately enabled us to
implement the eight needed connections of equal strengths and desired signs.
The details of the primitive cells and of the specific connections are
supplied in Fig. 3. The mass term in Eq. (4) is implemented by a mismatch
$\Delta H=H_{1}-H_{2}$ between the heights of resonators located inside the
same primitive cell (see Fig. 3(a)). The geometric parameters are as follows:
The lattice constant of the square lattice in xy-plane is $a=12.73$mm. The
diameter of the long arms of the H-shaped resonator is $d=4$mm and, in all
experiments except for those reported in Fig. 1(e), their heights were chosen
as $H_{1}=22$mm for R1, R3 and $H_{2}=21.5$mm for R2, R4. All coupling bridges
have a length $20$mm and diameter $0.6$mm.

Figure 3: Unit cell design of the topological acoustic crystal (a) The domain
of the acoustic wave equation used in the COMSOL simulations, corresponding to
the unit cell. It consists of four H-shape resonators placed in two rows and
two columns. (b) Schematic of the unit cell showing only the labels (red/green
dots) of the arms. (c) COMSOL simulation of the bulk band structure along the
high symmetry lines of the first Brillouin zone of a square lattice, revealing
a topological bandgap and almost perfect doubly degenerate bands that are
mirror symmetric relative to the mid-gap. (d,e) Schematic and realistic
representations of the theoretical couplings specified in Fig. 1(c,d). Note
that some of the couplings are at the top and and some at the bottom of the
H-shaped resonators.

We now demonstrate that the acoustic crystal indeed delivers the expected
physics from the topological D class. Fig. 1(e) reports COMSOL simulations of
the resonant spectrum of the crystal in a ribbon geometry as function of the
mass parameter $\Delta H$. When periodic boundary conditions are imposed in
the transversal direction, a clean gap opens and closes in the bulk spectrum
in full agreement with the phase diagram of the model mentioned earlier.
Furthermore, when hard boundary conditions are used, edge spectrum shown with
red markers in Fig. 1(e) fills entirely the bulk gaps in the topological sides
of the phase diagram, while being completely absent in the non-topological
side. Fig. 1(f) reports the resonant spectrum of the crystal in the same
ribbon geometry and for the specific value $\Delta H=0.5$mm, which is in the
middle of a topological phase. This time, the spectrum is rendered as function
of the quasi-momentum parallel to the ribbon. The connecting bridges
intersecting the transversal boundaries of the ribbon have been removed
entirely as, for example, visible in panels (g-h) of same figure. The expected
chiral edge bands of opposite slopes and localized on opposite boundaries of
the ribbon are indeed present. As clearly visible in Fig. 1(f), these chiral
bands are doubly degenerate, which is the proof for the enforcement of the
symmetry $U$ mentioned earlier. Another remarkable feature of the reported
spectrum is the almost perfect mirror symmetry with respect to the mid-gap
line, which demonstrates the expected particle-hope symmetry for both the bulk
and edge states. Thus, the COMSOL simulations reproduce all the expected
features derived from $\rho(H)$.

Fig. 1(g) illustrates the spatial profile of an eigen-mode with eigen-
frequency inside the topological bulk gap, for a geometry with clean edges. As
expected, the mode is localized at the edge of the crystal and surrounds
entirely the crystal, hence supplying a wave channels for sound propagation.
Fig. 1(h) demonstrates that this channel is robust against the imperfections
of the boundaries, as expected from a strong topological crystal.

The edge channels reported in Figs. 1(g,h) are doubly degenerate and they
support clockwise and anti-clockwise propagating modes, which cannot back-
scatter due to the $U$ symmetry mentioned earlier. A generic source of sound
will excite both modes, hence the sound propagation will not be uni-
directional. This brings us to the issue of projecting onto one of the
symmetry sectors $\Pi_{\pm}$. As we already mentioned, this can be done by
using specialized sources of sound that excite only modes with desired
symmetry. While the design of these sources is known [1], they can be
complicated and require fine tuning. In Fig. 1(i,j), we demonstrate a simple
principle to filter the clockwise or the anti-clockwise propagating modes by
using a generic source of sound and a patch of absorbing boundary placed at
one side or the other of the source. This together with the ability of the
crystal to support pairs of topological edge channels can prove useful in
various applications where on-demand clockwise or anti-clockwise signal
propagation is needed.

Looking forward into the near future, we plan to further improve the design
and to ultimately demonstrate the concept with a real laboratory model. This
will then enable the exploration and confirmation of the unique
characteristics of class D topological systems. For example, topological phase
transitions revealed in the phase diagram from Fig. 1(e) can be investigated
in the presence of disorder and the expected localization-delocalization phase
transition can be finally confirmed experimentally. Furthermore, the critical
behavior of the physical observables should be in line with the Wigner surmize
at $\beta=2$ [15], hence quite distinct from anything reported experimentally
so far. We point out the relatively recent success [24], where the critical
behavior for class AIII in dimension one was finally observed with atomic
chains.

Another interesting direction enabled by our successful design of a crystal
from class D is the observation of majorana-like modes stabilized by
$\pi$-fluxes. The explicit tight-binding models were already supplied in [16]
and, by applying the map $\rho$ from Eq. (1), we can generate explicit
acoustic crystals that simulate the majorana states. As demonstrated in [16],
the braiding and fusion of $\pi$-fluxes is non-trivial and worth exploring
even at the one-particle level covered by the linear wave equation regime
employed here. We mention that, in one dimension, the work on majorana-like
modes implemented with classical degrees of freedom is vigorously underway
[25, 26, 27, 28, 29], with the recent works [28, 29] demonstrating
experimental control at a level that makes the classical systems feasible for
information storage and processing. In 2 dimensions, implementing the
$\pi$-fluxes will require non-uniform perturbations of the couplings and this
will require a major effort.

Looking further into the future, we are confident that the H-shaped resonators
can be successfully employed to assemble acoustic crystals that simulate the
remaining classes in the classification table. As we already mentioned, some
of the classes will require eight resonators per unit cell [1], but each
resonator is coupled with its neighbors in a fashion very similar to what was
already implemented in this work, specifically, eight equal-strength
connections per resonator. The H-shape can be iterated to generate geometries
that expose more legs where robust and effective couplings can be established.
This will also answer the more general current challenge in meta-materials
science of implementing complex unit cells and connectivities.

###### Acknowledgements.

All authors acknowledge support from the National Science Foundation through
the grant CMMI-2131760. E. P. acknowledges additional support from the
National Science Foundation through the grant DMR-1823800.

## References

  * [1] Y. Barlas, E. Prodan, Topological classification table implemented with classical passive metamaterials, Physical Review B 98, 094310 (2018).
  * [2] F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ”parity anomaly”, Physical review letters 61, 18 (1988).
  * [3] C. L. Kane, E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
  * [4] C. L. Kane, E. J. Mele, Z(2) topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005).
  * [5] B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757-1761 (2006).
  * [6] M. König, et al., Quantum spin hall insulator state in HgTe quantum wells, Science 318, 766-770 (2007).
  * [7] J. E. Moore, L. Balents, Topological invariants of time-reversal-invariant band structures, Phys. Rev. B 75, 121306 (2007).
  * [8] L. Fu, C. L. Kane, Topological insulators in three dimensions, Phys. Rev. B 76, 045302 (2007).
  * [9] L. Fu, C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76, 045302 (2007).
  * [10] A. P. Schnyder, S. Ryu, A. Furusaki, A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  * [11] X.-L. Qi, T. L. Hughes, S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  * [12] A. Kitaev, Periodic table for topological insulators and superconductors, in Advances in Theoretical Physics: Landau Memorial Conference, edited by V. Lebedev and M. Feigel’man, AIP Conf. Proc. No. 1134 (AIP, New York, 2009), p. 22.
  * [13] S. Ryu, A. P. Schnyder, A. Furusaki, A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  * [14] E. Prodan, Disordered topological insulators: A non-commutative geometry perspective, J. Phys. A: Math & Theor 44, 113001 (2011).
  * [15] A. Altland, M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142-1161 (1997).
  * [16] Y. Liu, Y. Liu, E. Prodan, Braiding flux-tubes in topological auantum and classical lattice models from class-D, Annals of Physics 414, 168089 (2020).
  * [17] F. D. M. Haldane, S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett. 100, 013904 (2008).
  * [18] E. Prodan, C. Prodan, Topological phonon modes and their role in dynamic instability of microtubules, Phys. Rev. Lett. 103, 248101 (2009).
  * [19] L. Nash, D. Kleckner, A. Read, V. Vitelli, A. Turner, W. Irvine, Topological mechanics of gyroscopic metamaterials, Proceedings of the National Academy of Sciences 112, 47 (2015).
  * [20] R. Susstrunk, S. Huber, Observation of phononic helical edge states in a mechanical topological insulator, Science 349, 47 (2015).
  * [21] Chiral tubes are acoustic bridges with lengths comparable with the acoustic wavelength. Proper choice of this length supplies a (frequency dependent!) phase delay between the ends of the tube and, hence, between the coupled resonators.
  * [22] Note that Ref. [20] generated a spin-Chern insulator rather than a topological insulator from Class AII. The two have different critical phase transition theories, hence a spin-Chern number cannot be consider as a representative of class AII.
  * [23] Mode-coupling theory assumes that the pressure field of a collective resonant acoustic mode over a lattice of coupled resonators is identical with the one of a decoupled resonator when the pressure field is probed inside that resonator. This theory is accurate when the connections have a small footprint and they do not influence each other.
  * [24] E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massignan, T. L. Hughes, B. Gadway, Observation of the topological Anderson insulator in disordered atomic wires, Science 362, 929-933 (2018).
  * [25] C-W Chen, N. Lera, R. Chaunsali, D. Torrent, J. V. Alvarez, J. Yang, P. San-Jose, J. Christensen, Mechanical analogue of a Majorana bound state, Advanced Materials 31, 1904386 (2019).
  * [26] Y. Barlas, E. Prodan, Topological braiding of non-abelian midgap defects in classical metamaterials, Phys. Rev. Lett. 124, 146801 (2020).
  * [27] Z. Guo, J. Jiang, H. Jiang, J. Ren, H. Chen, Observation of topological bound states in a double Su-Schrieffer-Heeger chain composed of split ring resonators, Phys. Rev. Res. 3, 013122 (2021).
  * [28] K. Qian, D. J. Apigo, K. Padavić, K. H. Ahn, S. Vishveshwara, C. Prodan, Observation of phase controllable majorana-like bound states in metamaterial-based Kitaev chain analogues, arXiv:2201.12377 (2022).
  * [29] F. Allein, R. Chaunsali, A. Anastasiadis, I. Frankel, N. Boechler, F. K. Diakonos, G. Theocharis, Duality of topological edge states in a mechanical Kitaev chain, arXiv:2203.10311 (2022).