Topological Protection Can Arise from Thermal Fluctuations and Interactions

Topological quantum and classical materials can exhibit robust properties that are protected against disorder, for example, for noninteracting particles and linear waves. Here, we demonstrate how to construct topologically protected states that arise from the combination of strong interactions and thermal fluctuations inherent to soft materials or miniaturized mechanical structures. Specifically, we consider fluctuating lines under tension (e.g., polymer or vortex lines), subject to a class of spatially modulated substrate potentials. At equilibrium, the lines acquire a collective tilt proportional to an integer topological invariant called the Chern number. This quantized tilt is robust against substrate disorder, as verified by classical Langevin dynamics simulations. This robustness arises because excitations in this system of thermally fluctuating lines are gapped by virtue of interline interactions. We establish the topological underpinning of this pattern via a mapping that we develop between the interacting-lines system and a hitherto unexplored generalization of Thouless pumping to imaginary time. Our work points to a new class of classical topological phenomena in which the topological signature manifests itself in a structural property observed at finite temperature rather than a transport measurement.

Figure 1

Directed lines and doubly periodic substrate potentials. (a) Schematic of single directed line in a potential described by Eq. (1) with $(p,q)=(1,2)$, $y$-axis period $\lambda$, and $x$-axis period $a$. (b) Substrate potential (blue curves) and the theoretical density distribution [from Eq. (2), yellow curves] for a single chain at three $y$ positions indicated by the dotted lines. (c) The compound potential $V(x,y)$ (blue) combines two components, static (gray) with wavelength $a/q=a/2$ and sliding (red) with wavelength $a/p=a$. (d) Illustration of a Thouless pump for a potential with $(p,q)=(1,2)$, corresponding to $\mathcal{C}=1$. Under a filling density of one electron per lattice constant, each electron is exponentially localized to a unique unit cell. The drift of one such localized wave function over an adiabatic cycle is shown schematically; it is exactly quantized to $\mathcal{C}$ steps of lattice size $a$ over each period $\lambda$ of the potential variation along the $y$ direction. The tunneling of probability weight between adjacent potential minima during the adiabatic evolution, indicated by the dashed arrow, is crucial for the shift. (e) Same as (d) for a potential with $(p,q)=(2,3)$ for which $\mathcal{C}=-1$.

Figure 2

Topological tilt of directed line conformations. (a) Snapshot of a molecular dynamics simulation [27] of ten noncrossing directed lines experiencing the substrate potential from Fig. 1 with $\mathcal{C}=1$ under commensurate filling (one chain per unit cell of the potential along the $x$ direction). (b) Equilibrium monomer density distribution, and numerically computed scattering intensity profile (inset, see Supplemental Material [27]). (c) Probability Density Function (PDF) of monomer $x$ position for a subset of monomers along the length of the chains. Data from different chains are aggregated by first shifting the $p$th chain by an amount $pa$ along $x$, where $p\in \{0,\dots ,9\}$ indexes the chains in order from left to right. (d) Center of mass computed from the probability distributions in (c). The inset shows the shift over one period, $x_{\mathrm{c}.\mathrm{m}.}(y)-x_{\mathrm{c}.\mathrm{m}.}(y-\lambda )$, which agrees with the prediction of $\mathcal{C}=1$ away from the line ends. (e)–(h) Same as (a)–(d) for the potential from (b) with $\mathcal{C}=-1$. The lines display an anomalous tilt to the left, even though the potential slides to the right with increasing $y$.

Figure 3

The line tilt is robust against disorder. (a) Example of a substrate potential with $\mathcal{C}=1$ from Fig. 2, with random disorder added. (b) Equilibrium monomer density distribution for potential in (a) under commensurate filling. (c) Aggregated PDF of monomer $x$ positions. Although the density profiles of individual chains show deviations, the aggregated profile maintains the quantized tilt. (d) Tilt as measured in simulations for increasing disorder added to the substrate interaction for the potentials studied in Fig. 2. Each point represents an average over ten realizations of random disorder; the error bars represent estimated standard deviations. Triangles and squares correspond to underlying periodic potentials with Chern numbers $\mathcal{C}=1$ and $-1$, respectively, with $n_d$ additional modes with random amplitude and phase added on. The quantized tilt is preserved until the disorder strength ${\sigma }_d$ becomes comparable to the excitation gap $E_g$.