Dynamical signatures of symmetry protected topology following symmetry breaking

We investigate topological signatures in the short-time nonequilibrium dynamics of symmetry protected topological (SPT) systems starting from initial states which break a protecting symmetry. Naively one might expect that topology loses meaning when a protecting symmetry is broken. Defying this intuition, we illustrate, in an interacting Su-Schrieffer-Heeger (SSH) model, how this combination of symmetry breaking and quench dynamics can give rise to both single-particle and many-body signatures of topology. From the dynamics of the symmetry broken state, we find that we are able to dynamically probe the equilibrium topological phase diagram of a symmetry respecting projection of the post-quench Hamiltonian. In the ensemble dynamics we demonstrate how spontaneous symmetry breaking (SSB) of a protecting symmetry can result in a quantized many-body topological “invariant” which is not pinned under unitary time evolution. We dub this “dynamical many-body topology” (DMBT). We show numerically that both the pure state and ensemble signatures are remarkably robust, and argue that these nonequilibrium signatures should be quite generic in SPT systems, regardless of protecting symmetries or spatial dimension.

Figure 1

Left panel: Sketch of a slice of the equilibrium phase diagram for interacting SSH model (1), where $d=0.4$ and $\tau =1.0$, as detailed in [46]. The phase diagram hosts four phases; the topological insulating (TI) and band insulating (BI) phases, and their order-obstructed analogs OOTI and OOBI, in which the topology is obstructed by a charge density wave. The phase diagram is qualitatively unchanged for different values of $d$ and $\tau$, so long as $\tau \ne 0$. Quantized charge pumping is possible only in the TI and OOTI phases. Right panel: Qualitative sketch of Zak phase $\mathcal{Z}\left(t\right)$ for quenches from an order-obstructed initial state (with ${\widehat{H}}_0$ signified by the black diamond in the left panel). Upper (lower) right: Quenches into trivial (topological) phase, with ${\widehat{H}}_1$ represented by red (blue) diamond in the left panel. Dashed vertical lines in the right panel depict time of first “cusp” in Zak phase, which can be used to determine the ground state topology of quench Hamiltonian ${\widehat{H}}_1$.

Figure 2

Illustrative examples of trivial (left) and topological (right) cusps in the Zak phase during quenches from symmetry broken initial state (11). $t_{\text{cusp}}$ denotes the time of the first cusp according to (33). Main panels on left and right are computed for systems of $L=100$ sites. The insets show that $t_{\text{cusp}}$ converges rapidly with system size. For both left and right quenches, ${\tau }_1=1.0$, and ${\mathrm{\Delta }}_1=V_1=0$. Left: ${\widehat{H}}_1\in \mathrm{BI}$ with $J_1=1.5$, $d_1=0.4$. Right: ${\widehat{H}}_1\in \mathrm{TI}$ with $J_1=0.2$, $d_1=0.8$.

Figure 3

Illustration of the formation and deformation of topological and trivial cusps in the symmetry broken Resta polarization $|P^{\text{PS}}\left(t\right)|$ near equilibrium phase boundary, in units of $q{a}_0/2$. Here $J_c=d_1=0.4$. The vertical dashed line on each axis represents the first genuine cusp, according to (34). Data shown for systems of $L=12$ sites with initial Hamiltonian given by $J_0=0.1$, $d_0=0.22$, ${\tau }_0=0.3$, $V_0=5.0$, and quench Hamiltonian given by $d_1=0.4$, ${\tau }_1=1.0$, $V_1=0.0$.

Figure 4

Typical example of quench protocol for recovering the ground state topology of ${\widehat{H}}_1^{\prime }$ via cusps in the Resta polarization according to (34). The black dashed line signifies the equilibrium phase boundary ($J=d$), and the yellow region encompassing it denotes the region of uncertainty given small system size ($L=10$). Initial state was generated as the ground state of ${\widehat{H}}_0$ with ${\mathrm{\Delta }}_0=100{\tau }_0$, and all other parameters set to zero. In the post-quench Hamiltonian ${\widehat{H}}_1$, with ${\tau }_1=1$ setting the scale, both sublattice imbalance ${\mathrm{\Delta }}_1=0.1$ and interaction $V_1=0.2$ are included. All quenches are stopped at $t_{\text{max}}=20$. A threshold of $\eta =0.1$ is used.

Figure 5

Quenches from ${\widehat{H}}_0\in$ OOBI to (left panel) ${\widehat{H}}_1\in$ TI and (right panel) ${\widehat{H}}_1\in$ BI. In both panels, red denotes symmetry broken state (i.e., $P^{\text{PS}}$) and blue denotes ZTE. In both cases, jumps in $P^{\text{ZTE}}$ are directly associated with zero crossings of $\mathrm{Re}\left[{\gamma }^{\text{PS}}\right]$. $P$ is in units of $q{a}_0/2$. Data shown for $L=12$ sites with $V_1={\mathrm{\Delta }}_1=0$ and ${\tau }_1=1$. For the left panel, $J_1=0.06$ and $d_1=1.0$, while for the right panel $J_1=1.3$ and $d_1=0.4$. The times of the first and second jumps in $P^{\text{ZTE}}$, as used in the stability analysis below, are labeled $t_1$ and $t_2$ in the lower panels. In the left panel, $t_2$, which is very large, is not shown. Inset: Zoom in on first jump.

Figure 6

Stability analysis for quenches from (left panel) ${\widehat{H}}_0\in$ OOTI and (right panel) ${\widehat{H}}_0\in$ OOBI, plotted for varied quench intersite hopping anisotropy ${\tau }_1$. Here $t_1$ and $t_2$ are the times of the first and second jumps in $P^{\text{ZTE}}\left(t\right)$, and the dimensionless quantity $w_{12}=log(t_2/t_1)$ tells us about the stability of the many-body dynamical phase. Upper panel illustrates the dependence on $J_1$ and ${\tau }_1$, with $t_2\approx t_1$ implying stability for quenches within the same phase, and stability manifesting in $t_2\gg t_1$ for quenches across equilibrium phase boundaries. Lower panel visualizes the positions of local minima in $w_{12}$ as a function of $J_1$ for various ${\tau }_1$, as extracted from the data in the upper panel. The presence of minima at $J_1\approx d_1$ for a wide range of ${\tau }_1$ suggests universality in the quench dynamics. Data shown for $L=10$ sites with $V_1=0$ and $d_1=1$ to set the relative scale.

Figure 7

Demonstration of robustness of successful pure state protocols for dynamical recovery of equilibrium topology from quench dynamics of (a) Zak phase (33) and (b) Resta polarization (34). In both cases, the system starts as a $\mathrm{\Delta }\to \infty$ eigenstate, and evolves according to quench Hamiltonian ${\widehat{H}}_1$, with $V_1=0$. ${\mathrm{\Delta }}_1^{\text{max}}$ represents the largest ${\mathrm{\Delta }}_1$ such that the protocol is successful. (a) Data shown for $L=32$, obtained from tight-binding calculations. For ${\mathrm{\Delta }}_1>{\mathrm{\Delta }}_1^{\text{max}}$, the cusps are trivial. Values only shown for $J_1>d_1$, because when $J_1<d_1$, the system is trivial. Addition of any sublattice imbalance, which explicitly breaks the protecting chiral symmetry, results in another trivial Hamiltonian with cusps at ${\mathcal{Z}}^{\text{PS}}=0$, so there is no ${\mathrm{\Delta }}_1^{\text{max}}$. A tolerance of ${\eta }_{\mathcal{Z}}$ was used. (b) Resta polarization for system of size $L=8$ computed in ED. Unlike the Zak phase, when ${\mathrm{\Delta }}_1>{\mathrm{\Delta }}_1^{\text{max}}$, $P^{\text{PS}}$ exhibits no cusps at all. As a result, it makes sense to define ${\mathrm{\Delta }}_1^{\text{max}}$ for $J_1>d_1$ as well as $J_1<d_1$. A tolerance of ${\eta }_{\mathcal{P}}$ was used. In both (a) and (b), robustness increases with distance from the phase boundary (equivalently depth of quench). This general trend holds regardless of ${\tau }_1$. Additionally, ${\tau }_1>d_1$ bolsters the TI, whereas ${\tau }_1<d_1$ bolsters the BI phase. This mirrors the results shown in Fig. 6 in the main text. All quenches are stopped at $t_{\text{max}}=20$.

Figure 8

The pure state Zak phase at time $\tilde{t}$ for quench Hamiltonians with variable intersite hopping. The first nonequilibrium pure state criterion [Eq. (A8)] diagnoses all Hamiltonians with $\mathcal{Z}\left(\tilde{t}\right)>\frac{1}{2}$ (above the horizontal gray dashed line) as topological. The true equilibrium topological condition $J_1<d_1$ (to the left of the vertical dashed-dotted line) as topological. While these two conditions align for ${\tau }_1=d_1$, there is a mismatch for ${\tau }_1\ne d_1$. Data shown for $L=24$ sites with ${\mathrm{\Delta }}_1=V_1=0$.

Figure 9

Dependence of $t_1$ on various system parameters for quenches which cross an equilibrium topological phase boundary. (a) and (b) Quenches from OOBI $\to$ TI, whereas (c) shows quenches from OOTI $\to$ BI. (a) Influence of non-SSH-like intersite hopping on first critical time. $t_1$ clearly decreases for deeper quenches (smaller $J_1$). This trend holds for all values of ${\tau }_1$. As ${\tau }_1\to 0$, topology becomes ill-defined, which is reflected in the drastic increase in $t_1$, likely decreasing the stability of the dynamical topological phase. (b) Influence of quench interaction $V_1$ on $t_1$. While the exact time of the jump in $P^{\text{ZTE}}$ changes with $V_1$, the behavior is qualitatively the same as for $V_1=0$, demonstrating robustness of the dynamical topological signatures. (c) Influence of initial interaction $V_0$ on $t_1$. Larger $V_0$ (relative to other parameters in ${\widehat{H}}_0$) brings the initial state closer to the trivial product state. The initial symmetry broken Resta polarization $P^{\text{PS}}$ thus starts (at $t=0$) closer to $|P^{\text{PS}}|=q{a}_0/4$, and has a shorter distance to travel in Hilbert space before undergoing a jump in $P^{\text{ZTE}}$. Systems of size $L=64$ sites are considered in all cases. For all quenches, $d_0=0.22$, ${\tau }_0=0.3$, and $d_1=1.0$. For (a) and (b), $J_0=0.75$ and $V_0=5.0$, while $V_1=0$ for (a) and ${\tau }_1=0.6$ for (b). In (c) we have set $J_0=0.1$, $V_0=5$, and ${\tau }_1=0.9$ for each quench.