Detecting topology through dynamics in interacting fermionic wires

Probing the topological invariants of interacting systems stands as a grand and open challenge. Here we describe a dynamical method to characterize 1D chiral models, based on the direct observation of time-evolving bulk excitations. We present analytical and state-of-the-art numerical calculations on various flavors of interacting Su-Schrieffer-Heeger (SSH) chains, demonstrating how measuring the mean chiral displacement allows us to distinguish between topological insulator, trivial insulator, and symmetry-broken phases. Finally, we provide a readily-feasible experimental blueprint for a model displaying these three phases and we describe how to detect those.

Figure 1

(a) Sketch showing three unit cells of a spinful SSH chain with sublattices $A$ and $B$, spin states $\uparrow$ and $\downarrow$, intra- and intercell tunnelings $J$ and $J^{\prime }$, and interactions $U_{\parallel }$ and $U_{\perp }$. (b) Dynamics of a local excitation: Adding one extra fermion at ($t=0,x=0$) to a half-filled single-component Fermi sea generates an evolution with a characteristic conelike spreading pattern (model: noninteracting SSH chain with $L=64$ unit cells, $J/J^{\prime }=1.2$, and open boundary conditions). (c) Mean chiral displacement for $J/J^{\prime }=0.8$ (red, $\gamma =2$) and 1.2 (blue, $\gamma =0$). In noninteracting systems, the MCD is quantized to the winding number. Minute oscillations appear (and relax quickly) upon scattering with the chain boundaries, regardless of the presence or absence of edge states.

Figure 2

MPS analysis of the MCD of the short-ranged Hamiltonian ${\mathcal{H}}_{\mathrm{sr}}$ in a system with $L=32$ unit cells and open boundary conditions. The phase diagram contains trivial (TRI) and topological insulator (TOI) phases, separated by a transition (dashed line) independent of $U_{\perp }$. (a) Time-averaged disconnected part ${\overline{\mathrm{\Delta }\mathcal{C}}}_d$ of the MCD. (b) Time-averaged full correlator $\overline{\mathrm{\Delta }\mathcal{C}}$. (c),(d) Detailed (i.e., unaveraged) time traces of $\mathrm{\Delta }\mathcal{C}\left(t\right)$ at parameters marked by colored symbols in panel (b) are shown in panels (c) and (d). Both here and in Fig. 3, the time averages in panels (a) and (b) are performed over the time duration $T=10/J^{\prime }\ll L/J^{\prime }$, which ensures that the excitation propagates through the bulk only, without reaching the edges of the chain.

Figure 3

MPS analysis of the long-ranged Hamiltonian ${\mathcal{H}}_{\mathrm{lr}}$ for $L=32$ unit cells and open boundary conditions. Aside from trivial (TRI) and topological insulator (TOI) regions, the phase diagram includes an extra symmetry-breaking phase (SB). (a) The disconnected part ${\overline{\mathrm{\Delta }\mathcal{C}}}_d\left(T\right)$ of the MCD. (b) Time average $\overline{\mathrm{\Delta }\mathcal{C}}\left(T\right)$ of the full correlator. The hatched region with zigzag lines denotes $\overline{\mathrm{\Delta }\mathcal{C}}\propto T^2$. Details of the dynamics of $\mathrm{\Delta }\mathcal{C}$ at parameters marked by colored symbols in panel (b) are shown in panels (c) and (d). Although the time-averaged MCD for a finite system does not distinguish well between SB and TOI, the behavior of $\mathrm{\Delta }\mathcal{C}$ is markedly different in all three regions: It oscillates around the initial value in the TRI, around a finite (but not quantized) value in the TOI, and it diverges quadratically in the SB phase. Close to phase transitions, a finite-size scaling lifts the ambiguity between the different cases (see Fig. 10).

Figure 4

Proposal to engineer the long-range Hamiltonian ${\mathcal{H}}_{\mathrm{lr}}={\mathcal{H}}_0+{\mathcal{H}}_{\parallel }$ with two-component ultracold fermions in a tilted optical lattice. A Zeeman term splits the degeneracy between the $\uparrow$ and $\downarrow$ states. A linear tilt is used to hinder particles from naturally hopping between lattice sites. The desired hopping processes are reintroduced by (i) RF transitions between the $\uparrow /\downarrow$ states (green) and (ii) two-photon Raman transitions (yellow). To create a hole excitation, a blast beam tightly focused on $x=0$ transfers one particle to an internal state which is uncoupled to the Hamiltonian.

Figure 5

The interacting Creutz model. We represent hopping processes by arrows indicating the direction of movement with corresponding colored amplitude. Interactions are represented by colored boxes.

Figure 6

The interacting rotated Creutz model at $\chi =\pi$. We represent hopping processes by arrows indicating the direction of movement with corresponding colored amplitude. Interactions are represented by colored boxes.

Figure 7

The tilted lattice of Fig. 6 at the fine-tuned point $m=0$ and $g=t=\chi /\pi$.

Figure 8

Panel (a) displays the density bulk propagation of a single-particle excitation above the vacuum for a spinless SSH chain at $J/J^{\prime }=0.7$ for $L=128$ unit cells (i.e., the topological phase with winding number $\gamma =1$) and open boundary conditions. The grid lines display times at which the excitation would scatter with the boundary of smaller system sizes, which results in deviations from the fully translationally invariant solution of the MCD presented in (b). The black line is the analytical solution of Eq. (B10), and colored lines represent numerical solutions on chains with different lengths. Panel (c) depicts the density bulk propagation of a single-particle excitation above a half-filled Fermi sea with the same system parameters as in panel (a). The nontrivial interference pattern visible in the vacuum case disappears in this case. Panel (d) displays the MCD, which becomes quantized in the thermodynamic limit in agreement with Eq. (B11) (shown as a black line). For finite systems, minute oscillations appear when the excitation scatters with the boundary.

Figure 9

Entanglement growth for a local excitation created at $x_0=L/2$ for different bond dimensions and bipartitions. The system consists of $L=32$ unit cells for a single SSH wire at parameters $J=J^{\prime }$ without interactions $U_{\parallel }=U_{\perp }=0$ (the noninteracting quantum critical point) displayed in panels (a),(b) and with $H_{\mathrm{lr}}$ interactions $U_{\parallel }=2$, $U_{\perp }=0$ shown in (c),(d). (a),(c) The bipartition is chosen such that $\ell =L/2$ and the excitation is created at the interface between $A|B$. Panels (b),(d) demonstrate that increasing the distance between the boundary and the initial location of the excitation (as indicated by the numbers $1,...,5$ within the panels) causes an approximately linear delay before the rapid growth of the entanglement.

Figure 10

Panel (a) displays the conelike spreading of the density of a propagating excitation in the topological (TOI) phase of ${\mathcal{H}}_{\mathrm{lr}}$ at $J/J^{\prime }=0.0469$ and $U_{\parallel }/J^{\prime }=2.75$ [the yellow triangle in Figs. 3 and 3], computed with bond dimension $M=256$ on a chain of length $L=128$. In panel (b) we display corresponding MCD, computed for chains of lengths $L\in \{32,64,128\}$ (yellow, blue, red). The vertical dotted lines highlight the maximum observation time of bulk propagation for each size. The graphs shown in (c),(d) illustrate instead the situation in the symmetry-broken (SB) phase ${\mathcal{H}}_{\mathrm{lr}}$ at $J/J^{\prime }=0.0469$ and $U_{\parallel }/J^{\prime }=5.5$ [gray downward triangle in Figs. 3 and 3]. The injection of the excitation induces a spontaneous symmetry breaking, which leads to the formation of the domain-wall visible in panel (c). As a consequence, the MCD displays the characteristic quadratic divergence in time shown in panel (d).

Figure 11

(a) Truncation error in the reduced density matrix of the static ground state approximation for ${\mathcal{H}}_{\mathrm{lr}}$ for bond dimension $M=64$. The maximum error is $\mathrm{\Delta }\rho =1.6\times {10}^{-9}$, showing that the MPS calculation is essentially exact for ${\mathcal{H}}_{\mathrm{lr}}$. (b) The same analysis performed on ${\mathcal{H}}_{\mathrm{sr}}$ shows larger truncation errors, prompting for a check of results vs bond dimension. (c) Time traces of the MCD for ${\mathcal{H}}_{\mathrm{sr}}$ with $J/J^{\prime }=0.6$ and $U_{\perp }=6$ for different bond dimensions. Relatively small bond dimensions ensure to converge within the time window we investigated ($0<t<8/J^{\prime }$). (d) The time-averaged $\overline{\mathrm{\Delta }\mathcal{C}}$ converges therefore very quickly.