Quantum Dynamical Characterization and Simulation of Topological Phases With High-Order Band Inversion Surfaces

How to characterize topological quantum phases is a fundamental issue in the broad field of topological matter. From a dimension reduction approach, we propose the concept of high-order band inversion surfaces (BISs), which enable the optimal schemes to characterize equilibrium topological phases by far-from-equilibrium quantum dynamics, and further report the experimental simulation. We show that characterization of a $d$-dimensional ($d\mathrm{D}$) topological phase can be reduced to lower-dimensional topological invariants in the high-order BISs, of which the $n\mathrm{th}$-order BIS is a $(d-n)\mathrm{D}$ interface in momentum space. In quenching the system from trivial phase to topological regime, we unveil a high-order dynamical bulk-surface correspondence that the quantum dynamics exhibits nontrivial topological pattern in arbitrary $n\mathrm{th}$-order BISs, which universally corresponds to and so characterizes the equilibrium topological phase of the postquench Hamiltonian. This high-order dynamical bulk-surface correspondence provides new and optimal dynamical schemes with fundamental advantages to simulate and detect topological states, in which through the highest-order BISs that are of zero dimension, the detection of topological phase relies on only minimal measurements. We experimentally build up a quantum simulator with spin qubits to investigate a three-dimensional chiral topological insulator through emulating each momentum one by one and measure the high-order dynamical bulk-surface correspondence, with the advantages of topological characterization via highest-order BISs being demonstrated.

Popular Summary

Topological quantum phases are currently a mainstream of research in condensed-matter physics, of which how to characterize the topological phases is a fundamental issue. One way to identify topological quantum states and measure topological invariants is through the celebrated bulk-boundary correspondence (BBC), in which the topological number of the bulk links to the number of the robust gapless states in the real-space boundary.

This work extends the BBC in real space and in equilibrium to the high-order dynamical bulk-surface correspondence (DBSC) in momentum space and in nonequilibrium. In doing this, a new basic concept, i.e., high-order band inversion surfaces (BISs), is proposed. Topological phases can be characterized and simulated by the minimal information of the high-order BISs, providing a class of new and optimal dynamical schemes to explore and detect topological phases. This work provides insight into the exploration of topological phases and may have a significant impact on broad studies of quantum simulation of topological quantum phases. In particular, it is challenging to achieve a complete quantum simulation of both, bulk and boundary topological physics, in typical quantum simulators. For example, in ultracold atoms, it is convenient to measure the bulk topology in the momentum space but hard to simulate the real-space boundary.

Being the momentum counterpart of the real-space BBC, the high-order DBSC result provides optimal dynamical schemes to completely simulate both bulk and boundary physics of topological phases, which may largely expand the study of quantum simulation of topological physics. A quantum simulator is experimentally built using spin qubits, allowing the study of high-order DBSC in a three-dimensional chiral topological insulator. An optimal quantum simulation of the topological phase via highest-order BISs is demonstrated.

Figure 1

Sketch of high-order BISs. A $d\mathrm{D}$ BZ goes through $d$-dimension reductions and finally becomes a zero-dimensional (0D) BIS. Black lines and the number in square brackets correspond to the dimension of the BZ or the BIS. Two insets with (a) green and (b) orange backgrounds illustrate two different kinds of dimension reduction processes of the BIS from three dimensions to zero dimension.

Figure 2

(a1) TASP ${\overline{⟨{\sigma }_x⟩}}_x$ in the BZ after a quench from a large $m_x$ to $0$. The $1\mathrm{BIS}$ is marked by red dashed lines ($k_x=0$ and $\pi$). (a2) DSTs $g_y^{\mathrm{first}}$, $g_z^{\mathrm{first}}$, and ${\mathbf{g}}^{\mathrm{first}}$ along the $k_y$ direction on the $1\mathrm{BIS}$. (b1) ${\overline{⟨{\sigma }_y⟩}}_y$ on the $1\mathrm{BIS}$ after a quench from a large $m_y$ to $0$. The $2\mathrm{BIS}$ is marked by four black points $\left(k_x,k_y\right)=\{\left(0,0\right),\left(0,\pi \right),\left(\pi ,0\right),\left(\pi ,\pi \right)\}$. (b2) ${\overline{⟨{\sigma }_z⟩}}_y$ on the $1\mathrm{BIS}$ after a quench from a large $m_y$ to $0$. Black arrows represent the DSTs ${\mathbf{g}}^{\mathrm{second}}$ on the $2\mathrm{BIS}$. Here (and in the following figures) red and blue arrows correspond to the dimension reduction (DR) of BIS and the calculation of DSTs, respectively.

Figure 3

(a1) TASP ${\overline{⟨{\sigma }_z⟩}}_z$ in the BZ after a quench from a large $m_x$ to $0$. The $1\mathrm{BIS}$ is marked by a red dashed closed loop. (a2) DSTs $g_x^{\mathrm{first}}$, $g_y^{\mathrm{first}}$, and ${\mathbf{g}}^{\mathrm{first}}$ on the $1\mathrm{BIS}$. (b1) ${\overline{⟨{\sigma }_x⟩}}_x$ and (b2) ${\overline{⟨{\sigma }_y⟩}}_x$ on the $1\mathrm{BIS}$ after a quench from a large $m_x$ to $0$. Black points and arrows represent the $2\mathrm{BIS}$ and DSTs ${\mathbf{g}}^{\mathrm{second}}$, respectively.

Figure 4

(a1) $1\mathrm{BIS}$ with ${\overline{⟨{\gamma }_0⟩}}_0=0$ in the BZ. (a2) The DSTs $g_1^{\mathrm{first}}$, $g_2^{\mathrm{first}}$, $g_3^{\mathrm{first}}$, and ${\mathbf{g}}^{\mathrm{first}}$ on the $1\mathrm{BIS}$. (b1) TASP ${\overline{⟨{\gamma }_1⟩}}_1$ at the plane of $k_x=0$. The $2\mathrm{BIS}$ with ${\overline{⟨{\gamma }_1⟩}}_1=0$ is marked by a red dashed closed loop ($A$-$B$-$C$-$D$-$A$). (b2) DSTs $g_2^{\mathrm{second}}$, $g_3^{\mathrm{second}}$, and ${\mathbf{g}}^{\mathrm{second}}$ on the $2\mathrm{BIS}$. (c1) The TASP ${\overline{⟨{\gamma }_2⟩}}_2$ on the $2\mathrm{BIS}$. The $3\mathrm{BIS}$ is marked by two black points $B$ and $D$. (c2) TASP ${\overline{⟨{\gamma }_3⟩}}_2$ on the $2\mathrm{BIS}$. Black arrows correspond to the DST ${\mathbf{g}}^{\mathrm{third}}$ on the $3\mathrm{BIS}$. Red and blue arrows correspond to the dimension reduction (DR) of BISs and the calculation of DSTs, respectively.

Figure 5

(a) BISs with ${\overline{⟨{\gamma }_0⟩}}_1=0$. (b) The $1\mathrm{BIS}$ formed by two planes $k_x=0$ and $-\pi$. (c) The $2\mathrm{BIS}$ formed by four lines $(k_x,k_y)=\{(0,0),(0,-\pi ),(-\pi ,0),(-\pi ,-\pi )\}$. (d) The $3\mathrm{BIS}$ formed by eight points with $k_{x,y,z}=0,-\pi$. Blue arrows represent the DST $g_0^{\mathrm{third}}(\mathbf{k})$.

Figure 6

(a) Energy levels of the NVC, the four in the green frame are employed for the experiment and coupled by a microwave pulse ${\omega }_0$. (b) Upper panel: quantum circuit for the dynamical simulation. The operator $e^{-iHt}$ denotes the evolution by $H_{\mathrm{RWA}}$, while the net effect together with two nuclear spin operations $R_{\pm Y}^{\theta }$ renders an evolution under $H_{\mathrm{eff}}$. Lower panel: operations of quench and readout steps in different directions. (c)–(f) Experimental measurements of TASPs ${\overline{⟨{\gamma }_1⟩}}_1$ on the $1\mathrm{BIS}$, ${\overline{⟨{\gamma }_2⟩}}_2$ on the $2\mathrm{BIS}$, and ${\overline{⟨{\gamma }_3⟩}}_2$ near the $3\mathrm{BISs}$ for $m_0=1.4t_0$ and $t_{\mathrm{SO}}=0.2t_0$. $\varphi$ marked in (d) corresponds to the momentum on the $2\mathrm{BIS}$. Solid red lines in (e) and (f) are theoretical results for comparison. In the inset of (f), blue arrows represent the DST $g_3^{\mathrm{third}}(\mathbf{k})$ on the $3\mathrm{BIS}$.

Figure 7

(a) Sketch of the step function ${\alpha }_{\mathbf{k}}$ in the BZ. (b) The functions $\sin {\alpha }_{\mathbf{k}}$ (red) and $\cos {\alpha }_{\mathbf{k}}$ (blue) along the dashed line in (a). (c) Schematic diagram of the dimension reduction. The number in the square bracket is the dimensionality of the corresponding BIS.

Figure 8

(a) Quantum circuit for postquench dynamics on the left-hand side. $R_{\pm Y}^{\theta }$ represents rotation of nuclear spin about $\pm y$ axis by an angle $\theta$. $e^{-iHt}$ corresponds to evolution of the system under the rotating-wave approximation for a time duration of $t_{\mathrm{eff}}$. The net effect is equivalent to the evolution under $H_{3\mathrm{D}}$ for a duration of $t$ on the right-hand side. (b)–(c) Sequences to measure populations. The operation labeled ${\gamma }_i$ (pink) is a deep quench along $i$ direction, which requires unitary operations to transform the state $|00>$ to an eigenstate of the prequench Hamiltonian. $e^{-i{H}_{3\mathrm{D}}t}$ (purple) corresponds to the postquench dynamical evolution, whose detailed quantum circuit is plotted in (a). The operation ${\gamma }_j$ (azure) corresponds to measurement of the ${\gamma }_j$ component, which requires an operation to transform ${\gamma }_j$ to the $z$ basis. ${\pi }_{\mathrm{MW}}$ and ${\pi }_{\mathrm{rf}}$, respectively, represent the microwave and ratio-frequency pulses, which are applied to rotate spins. PLM (green) corresponds to a photoluminescence measurement. Idle corresponds to a waiting time equal to the total time of the ${\gamma }_i$, $H_{3\mathrm{D}}$ and ${\gamma }_j$ steps.

Figure 9

(a),(b), (c),(d), and (e),(f), respectively, show measurement data of ${⟨{\gamma }_1⟩}_1$, ${⟨{\gamma }_2⟩}_2$, and ${⟨{\gamma }_3⟩}_2$ on the $3\mathrm{BIS}$. (a), (c), and (e) plot average photon number for different sequences as shown in Fig. 8 at different time. Error bars are estimated considering photon shot noise. In (b), (d), and (f) populations and spin polarizations are obtained from results in (a), (c), and (e), respectively. Solid green lines in (b), (d), and (f) are calculated from the theoretical model. We set $m_0=1.4t_0$, $t_{\mathrm{SO}}=0.2t_0$, and the momentum point on the $3\mathrm{BIS}$ corresponds to $\varphi =\pi$ in Fig. 6.