Topological Invariants for Quantum Quench Dynamics from Unitary Evolution

Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.

Figure 1

Topological $\pi$ defects (solid red dots) as hedgehogs in the momentum-time space for quantum quenches. They are Weyl-like degeneracy points in the phase band of loop unitary $U_l$ at $\varphi =\pi$. The arrow denotes the phase-band spin vector $\widehat{\mathit{m}}$ near the defect. (a),(b) Quench from a trivial initial state $|{\xi }_0⟩=(0,1{)}^T$ to $H_{q=1}$ and $H_{q=2}$ defined in model (7). (c),(d) Quench from the ground state of $H_{q=1}$ to $h={\sigma }_z$ and $H_{q=2}$, respectively. The topological charge of the hedgehog is $+1$, $+2$, $-1$, $+1$ from (a) to (d), and equal to the Chern number change ${\mathcal{C}}_f-{\mathcal{C}}_i$ in each case. $R=0$, $M=1.6$. (e) Schematics of the DQPT curve $\mathrm{\Gamma }$ (magenta line). It is the intersection of the $t=\pi /2$ plane and the equal-$\varphi$ surface of $\varphi =\pi /2$ (green), which encloses the $\pi$ defect. Along $\mathrm{\Gamma }$, vectors ${\widehat{\mathit{n}}}_0$, $\widehat{\mathit{h}}$ and $\widehat{\mathit{m}}$ are perpendicular to each other.

Figure 2

(a)–(c) Winding of the phase-band spin vector $\widehat{\mathit{m}}$ (arrows) along the DQPT curve $\mathrm{\Gamma }$ for quench from initial state $|{\xi }_0⟩=(0,1{)}^T$ to $H_q$ with $q=1$, 2, 3 (left to right). The corresponding winding number as defined in Eq. (8) is $\nu =1$, 2, 3. (d)–(f) Hopf links in momentum-time space for the same quench to $H_q$ with $q=1$, 2, 3 (left to right). Shown are preimages of two time-evolved states $\pm \mathit{\xi }=(1,0,0)$. $R=0.2$, $M=1.6$.

Figure 3

Top view of (a) a knot and (b) a link formed by two curves ${\mathrm{\Gamma }}_{\pm }$ traced by vector $\pm \widehat{\mathit{m}}$ along $\mathrm{\Gamma }$ for quenches to critical $H_q$ with Dirac points. (Insets) Topological charges on the $t=\pi /2$ plane. They cross $\mathrm{\Gamma }$ exactly at the Dirac points where the two curves switch, ${\mathrm{\Gamma }}_{+}\leftrightarrow {\mathrm{\Gamma }}_{-}$. (a) A single curve ties into a knot with crossing number $c_K=5$, $q=3$, $R=0.512$. (b) Two closed curves form a torus link with linking number $c_L=1$, $q=2$, $R=0.64$. $M=1.6$.