Topological quantization of energy transport in micro- and nano-mechanical lattices

Abstract

Topological effects typically discussed in the context of quantum physics are emerging as one of the central paradigms of physics. Here, we demonstrate the role of topology in energy transport through dimerized micro- and nano-mechanical lattices in the classical regime, i.e., essentially “masses and springs”. We show that the thermal conductance factorizes into topological and nontopological components. The former takes on three discrete values and arises due to the appearance of edge modes that prevent good contact between the heat reservoirs and the bulk, giving a length-independent reduction of the conductance. In essence, energy input at the boundary mostly stays there, an effect robust against disorder and nonlinearity. These results bridge two seemingly disconnected disciplines of physics, namely topology and thermal transport, and suggest ways to engineer thermal contacts, opening a direction to explore the ramifications of topological properties on nanoscale technology.

I. INTRODUCTION

Topology gives rise to fascinating phenomena and can lead to the emergence of many exotic states of matter^1–4, from condensed matter^5,6 to cold atoms⁷ to quantum computation^8,9. An example of a lattice with a nontrivial topology is the Su-Schrieffer-Heeger (SSH) model of electrons hopping in polyacetylene¹⁰, which is the focus of many cold-atom studies, e.g., for measuring the Zak phase¹¹ and demonstrating topological Thouless pumping^12,13. While there are works focusing on topological effects in the classical regime^14–25, few connect topology and energy transport²⁶.

We present a mechanical system that manifests topological effects in energy transport and has relevance to many nanoscale scenarios^27,28. This system is the mechanical counterpart to the SSH model in Fig. 1(a) where alternating nearest neighbor coupling strengths “dimerize” the lattice. When both ends terminate on weak bonds, the whole lattice pairs into dimers. Terminating on a strong bond, though, leaves the end sites unpaired, resulting in the formation of an edge mode. Hence, depending on the topology – e.g., swapping the nearest neighbor couplings constants, which does not change the bulk – there will be zero, one (on either the left or right), or two edge modes.

FIG. 1.

(a) Illustration of dimerization in a mechanically alternating lattice. Intracell (K₁) and intercell (K₂) nearest-neighbor couplings are shown by green and black lines, respectively. Depending on the value of the winding number, Eq. (3), and the parity of the lattice length, N, there can be zero, one, or two edge modes, Eq. (4). For example, when N is even and K₁ > K₂ then $\mathcal{W}=0$ (K₁ < K₂ with $\mathcal{W}=1$) and no edge mode (two edge modes) are present as the strong couplings pair all the sites (all but the two sites at the boundaries) as shown in the left (right) lattice. (b) Illustration of energy transport in an envisioned micro- or nano-mechanical topological lattice. The large and small spacings between beams of length L alternate the nearest neighbor couplings (varying widths can alternate other parameters). Other device characteristics (thickness t, undercut d, device materials) can be used to tune the parameters in Eq. (1). A modulated laser of wavelength λ can equilibrate the end beam of the lattice at some elevated temperature T_L, while the other end is either damped into equilibrium with its surroundings at T_R or its local temperature is measured optically or electronically via its oscillations. A difference in these two temperatures will drive an energy current J. The presence of the edge modes will create an interfacial resistance, as the localized modes tend to decouple the bulk lattice from the boundaries, thereby reducing the ability of energy to flow away from the edge. (c) Three representative normal modes, plotted as the polarization vector squared, $u_{qn}^2$, versus beam position, n, along the lattice. The two edge modes (solid red and dashed blue lines) are localized around the left and right edges, respectively, while delocalized modes are spread across the entire lattice (dot-dashed green line shows $\sqrt{N}{u}_{qn}^2$ for one delocalized mode). The parameters are K₂ = 2K₁, m₂ = 3m₁/4, and w₂ = w₁. The inset shows the frequencies of all modes enumerated in ascending order of frequency. The edge modes reside in the gap between the two bands of delocalized modes.

II. RESULTS

The mechanical lattice we examine has vibrational spectrum equivalent to the energy spectrum of the SSH model when all parameters (except the alternating nearest-neighbor coupling) are uniform. Otherwise it is identical to the spectrum of a slice of the time-dependent Rice-Mele model, which has a quantized Chern number on the extended 2D plane⁴. The lattice has the Hamiltonian $H={\sum }_n\frac{m_n}{2}\left({\dot{x}}_n^2+{\omega }_n^2{x}_n^2\right)+{\sum }_n\frac{k_n}{2}{(x_n-x_{n+1})}^2$ with masses m_n, onsite frequencies w_n, and nearest-neighbor couplings K_n for site n with coordinate x_n. These parameters, [m_n,w_n,K_n], are [m₁,w₁,K₁] or [m₂,w₂,K₂] for odd or even n, respectively. After a lattice Fourier transform, we get the Bloch Hamiltonian

$${\mathbf{H}}_q=h_0\mathbf{I}+h_z{\sigma }_z+{\overline{\mathbf{H}}}_q,$$ (1)

where σ_z is the z Pauli matrix, I is the 2 × 2 identity matrix, and h_z and h₀ are given in the Supplemental Material (SM)²⁹, and

$${\overline{\mathbf{H}}}_q=-\frac{1}{m}\left[\begin{array}{cc}0& f^{*}(q)\\ f(q)& 0\end{array}\right],$$ (2)

with $m=\sqrt{m_1{m}_2}$ and f(q) = K₁ + K₂e^iq. A natural realization of this model is in one-dimensional micro- and nano-electromechanical systems (MEMS-NEMS), which provide a versatile platform for dynamical phenomena and devices³⁰. As we will discuss, a combination of laserinduced heating and optical/electronic readout can topologically characterize energy transport in MEMS-NEMS, as shown in Fig. 1(b). However, since this model is one of the most elementary examples of a physical system – classically coupled “masses and springs” – there exists many alternative realizations.

The topological nature of the lattice can be seen by considering ${\overline{\mathbf{H}}}_q=R_x{\sigma }_x+R_y{\sigma }_y$, where σ_x,y are the Pauli matrices. The curve (R_x = K₁ + K₂ cos(q), R_y = −K₂ sin(q)) may or may not wrap around the origin in the complex plane as q goes from 0 to 2π. Counting how many times the curve encircles the origin gives the winding number,

$$\mathcal{W}=\{\begin{array}{ll}1,\hfill & K_1<K_2\hfill \\ 0,\hfill & K_1>K_2\hfill \end{array}.$$ (3)

This number is an important topological property of a 1D system’s band structure^1,5. The Zak phase³¹ is the 1D Berry phase and is 2π times the winding number. When the winding number is nonzero, the lattice is topologically non-trivial and edge modes appear, decaying exponentially from the edges with a decay length ξ = ln(K₁/K₂). Without loss of generality, we use the convention that if only one edge mode is present, it is on the left. The number of left (N_L) and right (N_R) edge modes is thus

$$N_L=\mathcal{W},N_R=\frac{1-e^{i\pi (N+\mathcal{W})}}{2}.$$ (4)

Figure 1(c) shows the decaying amplitude squared of two edge modes. The σ_z term in Eq. (1), while not present in the SSH model, does not destroy the edge modes as one can verify explicitly (see the SM²⁹). Moreover, the edge modes persist in the presence of nonlinearity (see Fig. 2).

FIG. 2.

Band structure and edge modes for (a,b) the harmonic lattice (that leads to Eq. (1)) versus the ratio of couplings and (c,d) a nonlinear, Fermi-Pasta-Ulam (FPU) like^32–35 generalization versus temperature. (a,b) When K₁ < K₂ in an even site lattice, there are two edge modes (purple and wine lines) that reside in the gap between the two phonon bands (outlined with blue and red for the upper and lower bands, respectively). As K₁ increases the spatial extent of the edge states grows until they merge with the bulk states at K₁ = K₂. This process is shown for alternating couplings only (i.e., m₁/m₂ = 1; w₁/w₂ = 1) and for the masses alternating (i.e., m₁/m₂ = 2; w₁/w₂ = 1). (c,d) The edge modes also exist in nonlinear lattices and persist even as the nonlinearity increases with temperature T (at high enough temperatures, the nonlinearity merges the edge modes with the bulk). The persistence of the edge modes is relevant to MEMS/NEMS, which are often operated in nonlinear regimes^30,36. Prior studies also show that classical nonlinear systems can exhibit topological excitations¹⁵.

These modes are more than just a physical curiosity, however. Figure 1(b) illustrates a lattice of interacting cantilevers with one end at a temperature T_L. When the lattice is locally excited at the boundary, e.g., via a laser as shown in Fig. 1(b), or via an electromagnetic coupling, the resulting energy current will depend on the presence of the edge modes, whether this energy flow is due to a single transient excitation or in a steadystate. The conductance, κ ≡ J/ΔT, where ΔT is the temperature difference between two reservoirs, captures this effect. For the bulk, the intrinsic conductance, κ₀, is given by its average phonon group velocity³⁷

$${\kappa }_0=\frac{k_B}{2\pi }{{\displaystyle \int }}_{\mathrm{\Omega }}dq{v}_q=\frac{k_B\mathrm{\Omega }}{2\pi },$$ (5)

where we use Ω to indicate both the bands and the total bandwidth. This is the maximum rate at which a harmonic lattice can transport heat between two equilibrium reservoirs at different temperatures. Since it depends only on the bulk band structure, Ω, it is independent of winding number, i.e., swapping the order of K₁ and K₂ – or changing the parity of the lattice – will not affect it. Reaching this conductance in practice, however, requires that all phonon modes are su ffciently in contact with the reservoirs so that they are supplied ample thermal energy³⁷. In the presence of topological edge modes this limit is never reached, and the thermal conductance is always lower than κ₀, regardless of the system length. This is rather surprising, considering the fact that there are at most two edge modes, whereas the number of modes grows linearly with the system size.

We note that the ability of a specific mode q to conduct heat will depend on its contact with the external reservoirs and its intrinsic conductance (determined by its group velocity). In the setup of Fig. 1(b), the strength of the contact of a specific mode q with the reservoirs is given by $\gamma u_{q1}^2$ and $\gamma u_{qN}^2$ for the left and right, respectively. The coupling (i.e., damping rate) γ is the strength of contact of the reservoirs to the cantilever beam at the lattice boundary. The polarization vector of the mode on the boundaries, $u_{qn}^2$ with n = 1 or N, attenuates the coupling of the mode q to the reservoirs. When the lattice weakly contacts the reservoirs – in order to minimally perturb the boundaries – the conductance for mode q, κ_q, is due to two contributions in series (see the SM²⁹)

$$\frac{k_B}{{\kappa }_q}=\frac{1}{\gamma u_{q1}^2}+\frac{1}{\gamma u_{qN}^2},$$ (6)

where the first term is from the left interface and the second from the right interface. To describe the behavior for arbitrary γ, the bulk contribution – N/v_q, the intrinsic ability of the mode to transfer heat – and an overdamping contribution proportional to would need to be included in Eq. (6). Many of the results below hold up to moderate values of γ, as explained in the SM²⁹.

The edge modes have an exponentially vanishing amplitude, $u_{qn}^2\approx 0$, for either n = 1 or N, which yields k_q ≈ 0 for q ∈ $\mathcal{E}$, where $\mathcal{E}$ is the set of edge modes. The total conductance will then be

$$\kappa =\frac{1}{2\pi }{{\displaystyle \int }}_{\mathrm{\Omega }}dq\underset{N\to \infty }{\lim }N{\kappa }_q,$$ (7)

where the integral is over only the phonon bands and thus the edge state contribution – which would be a separate sum – is absent. This equation has a similar form to Eq. (5) but κ_q contains the non-ideal contact to the external heat source and sink.

We proceed by giving a heuristic derivation of the effect of topology, and a rigorous derivation is in the SM²⁹. Considering all normal modes of a lattice, one has simple “sum rules” for the boundary amplitudes, ${\sum }_q{u}_{q1}^2=1/m$ and ${\sum }_q{u}_{qN}^2=1/m$ for the case when m₁ = m_N = m, that reflect the (mass) scaling and orthogonal transformations that yield the normal modes. In the absence of edge modes, the bulk modes have a contact strength $u_{qn}^2\propto \gamma /(mN)$ for n = 1 and N. Using this value for $u_{qn}^2$, the non-topological interfacial conductance for an even length lattice is

$$\overline{\kappa }=\frac{k_B\gamma }{2m},$$ (8)

which is limited by the coupling of the external reservoirs to the lattice, i.e., the heat injected is the bottleneck to current flow³⁷ (a similar situation occurs in electronic transport^38–40).

In the presence of edge modes – states localized at the boundaries – the total coupling of the bulk to the reservoirs is reduced: ${\sum }_{q\in \mathrm{\Omega }}{u}_{qn}^2=1/m-{\sum }_{q\in \mathcal{E}}{u}_{qn}^2$.The bulk modes therefore have a contact strength ∝ $\gamma \left(1-m{\sum }_{q\in \mathcal{E}}{u}_{qn}^2\right)/(mN)$ . The amplitude squared of an edge mode on a boundary of its origin is (1-e^−2ξ)/m, which follows from the normalization of an exponentially decaying state (see the SM²⁹). The bulk modes therefore have contact $\gamma u_{qn}^2\propto \gamma \exp (-2\xi )/(mN)\text{for}n=1$ and N. The conductance in the presence of edge modes is then

$$\kappa =\frac{k_B\gamma e^{-2\xi }}{2m}=e^{-2\xi }\overline{\kappa }.$$ (9)

Thus, there is a topologically induced component of the conductance that manifests itself as a prefactor e^−2𝜉.

The case of odd or even N can support N_L = 0,1 states on the left and and N_R = 0,1 states on the right, according to the winding number, Eq. (4). Generalizing Eq. (9) to arbitrary length lattices, and also inhomogeneous mass and on-site frequency cases, the conductance of the lattice is

$$\kappa =\Xi \overline{\kappa },$$

where $\overline{\kappa }$ is the nontopological component of the conductance (for N odd, $\overline{\kappa }=k_B\gamma /2m_1$; for N even, $\overline{\kappa }$ is the conductance in the absence of edge modes, i.e., K₁ and K₂ swapped, see the SM²⁹) and Ξ gives the three discrete topological levels

$$\Xi =\frac{2}{e^{2N_L\xi }+e^{2N_R\xi }}.$$

The quantity Ξ is thus a function of winding number, as N_L(R) depend on it through Eq. (4). Out of the different configurations (using the parameters that give the same bulk properties but ordering them in different ways), Eq. (II) will give only three possible values, corresponding to the presence of 0,1 or 2 edge modes. All trivial mass effects (at the boundaries) are in $\overline{\kappa }$.

III. DISCUSSION

The effect of topological edge modes on thermal conductance is demonstrated in Figure 3, which discusses a uniform lattice with alternating K₁ and K₂ only (solid lines), a lattice with m₁ = 2m₂ and w₁ = w₂ (dashed lines) and a lattice with m₁/m₂ = K₁/K₂, w₁ = w₂ (dash-dotted lines). For each bulk lattice, we show the case with zero edge modes (green), two edge modes (purple), and one edge mode (red and blue for a left and right edge mode, respectively).

FIG. 3.

Topological quantization of the conductance. (a) The conductance versus K₁/K₂ of infinite length dimerized lattices (in the setup shown in Fig. 1(b)). We show only a subset of the curves for clarity, see the SM²⁹ for details. (b) When taking the ratio of the conductance to the non-topological component, $\overline{\kappa }$, all the curves in (a) collapse onto three distinct levels. Green, purple, and red-blue lines show the cases with zero, two, and one edge mode(s) (left or right), respectively, generically suppressing the conductance. (c) The topological conductance is robust to nonlinearity and disorder. The yellow circles and wine squares show the normalized conductance of two disordered lattices (with both an odd and even number of sites), and the magenta diamonds show the normalized conductances of the FPU lattice. Empty (full) symbols are for one (two) edge modes. Since the nonlinearity also introduces an overall shift in the parameters, the data points are plotted versus an effective K₁/K₂, as described in the SM²⁹. In all cases, the results agree with the theoretical result, Eqs. (II) and (II).

Figure 3(a) plots the thermal conductance κ versus the ratio K₁/K₂. The conductance can take on essentially any value (by changing masses and on-site frequencies, one can fill in the whole plot). However, when taking simple ratios, $\kappa /\overline{\kappa }$, of the thermal conductance, a simple quantization emerges, as shown in Fig. 3(b). These ratios take on just three values given by Ξ. Generically, the introduction of edge modes suppresses the conductance, as it reduces the contact between the energy sources/sinks and the bulk states. Non-topological effects (i.e., the changing bulk state structure as K₁/K₂ increases) can also significantly influence the conductance for certain sets of parameters⁴¹.

Since the suppression of the thermal conductance is a topological effect³³, it will not depend on the specific details of the system and reservoirs and is anticipated to be robust against various modifications to the lattice (so long as the topology is maintained). We demonstrate this robustness in Fig. 3(c) where we plot the normalized conductance versus K₁/K₂ for three additional lattices; the FPU-β lattice of Fig. 2(c) and two disordered, dimerized lattices. For all these cases, the conductance follows Eq. (II), showing the universality of the topology-induced reduction of thermal conductance. If the Langevin reservoirs are replaced by uniform harmonic lattices with constant coupling 0 < K < min(K₁,K₂) representing trivial topology, the edge modes and their influence on thermal conductance should still survive at the boundary due to a change of topology.

We further note that Eq. (6) is a general result. It entails, therefore, that even³⁴ non-topological localized modes (e.g., due to a light mass at the boundary) can suppress the thermal conductance. However, nontopological modes will not display the quantized conductance of Eq. (II) and shown in Fig. 3b,c. As well, there are many channels for heat/energy transport. In the setup envisioned in Fig. 1, heat will also be carried by vibrations of the underlying crystal lattice. Therefore, it is necessary to use a low thermal conductivity material so that vibrations of the cantilevers are the dominant channel for energy transport.

IV. CONCLUSION

Just as thermal transport can serve as a probe of nonlinear structural transitions^42,43, these results show that signatures of nontrivial topologies appear in classical (or quantum) energy transport in conventional physical systems, such as MEMS/NEMS or at crystal-polymer interfaces. In particular, a combination of laser-induced heating and optical/electronic readout will allow for the topological characterization of energy transport in micro- or nano-mechanical lattices and control of heat flow⁴⁴. The emergence of edge states may help design, e.g., thermoelectric devices, where the lattice thermal conductance needs to be suppressed independently of the electronic conductance. Moreover, energy flow and thermal properties are critical to the operation of nanotechnologies, where they can limit and even define the functionality of devices^44–46. The results presented here thus generate exciting prospects for observing topological properties in conventional physical systems and utilizing them to design micro- and nano-scale devices.