Nonunitary Preparation of Nontrivial States from Trivial Regimes in Two-Dimensional Topological Insulators

Abstract

While remarkable progress has been achieved in engineering nontrivial Hamiltonians across a wide range of physical platforms, preparing their corresponding nontrivial ground states remains a major experimental challenge. The commonly used strategy for state preparation relies on adiabatic protocols. However, when a trivial initial state is unitarily driven toward nontrivial regimes, the dynamics must cross gap-closing critical points, rendering the process intrinsically nonadiabatic, and the state remains topologically trivial. Here, we present a nonunitary method for dynamically preparing nontrivial states in two-dimensional topological insulators. By introducing dephasing noise into a slowly driven unitary evolution, we demonstrate that the topological number of the resulting dephased states can coincide with that of the target nontrivial Hamiltonian. This nearly adiabatic nonunitary state-preparation protocol provides a powerful alternative to conventional adiabatic approaches for accessing topological states.

Figures (4)

• Figure 1: Nonunitary preparation of topologically nontrivial states. (a) Topological phase diagram of the Qi-Wu-Zhang model. White, blue, and red regions correspond to topological phases with lower-band Chern numbers $n_{\mathrm{C}}=0$, $+1$, and $-1$, respectively. Black dashed lines indicate the topological phase boundaries associated with energy-gap closings at $\mathbf{k}_c\in\{0,\pi\}$. The Hamiltonian parameter is ramped from an initially trivial regime (star symbol) to a final nontrivial regime (diamond symbol) along the trajectory $m(t)=m_i+(m_f-m_i)(1-e^{-vt})$. Parameters are $m_i=-4$, $m_f=-1$, and $v=0.2$. (b) Chern number of the coherently evolved state (bar chart) compared with that of the instantaneous Hamiltonian (purple dotted line). (c) Uhlmann number of the instantaneously dephased state (bar chart) compared with that of the instantaneous Hamiltonian (purple dotted line). The dephasing rate is set to $\gamma_{\mathbf{k}}=2$. (d–f) Uhlmann geometric phase of the coherently evolved final state, the dephased final state, and the ground state of the final nontrivial Hamiltonian at $t=25$, respectively, as functions of $k_x$. The topological number of the evolving state in (b, c) is plotted at discrete times. The momentum space $k_x\in[-\pi,\pi]$ in (d-f) is discretized into 101 equally spaced points.
• Figure 2: Topological phase transition in parameter ramp. Top panels (a-c) show the amplitude spectra of the holonomy eigenvalues $\lambda_{\pm}(k_x)$ for the evolving dephased state at $t=5$, $5.68$, and $25$, respectively. Bottom panels (d-f) display the corresponding phases of the holonomy eigenvalues $\mu_{\pm}(k_x)$ as a function of $k_x$. Red and blue dots represent the complex eigenvalues $z_{\pm}$ associated with the larger and smaller amplitudes, respectively. Parameters are $m_i=-4$, $m_f=-1$, $v=0.2$ and $\gamma_{\mathbf{k}}=2.0$ for all panels.
• Figure 3: Noise-induced topological phase transition. Top panels (a-c) show the amplitude spectra of the holonomy eigenvalues $\lambda_{\pm}(k_x)$ of the dephased final state at $t=25$ for three dephasing rates $\gamma_{\mathbf{k}}=0.01$, $0.024$, and $0.05$, respectively. Bottom panels (d-f) display the corresponding phases of the holonomy eigenvalues $\mu_{\pm}(k_x)$ as a function of $k_x$ for the same dephasing rates. Parameters are $m_i=-4$, $m_f=-1$, and $v=0.2$ for all panels.
• Figure 4: Ramp-velocity-driven topological phase transition. (a-c) Upper band occupation of the dephased final state at $t=25$, $[1+r_{k_{x},0}(t)]/2$, as a function of $k_x$ for three parameter ramp velocities $v=4.0$, $2.5$, and $1.0$, respectively. (d-f) Amplitude spectra of the holonomy eigenvalues $\lambda_{\pm}(k_x)$ of the dephased final state for the same parameter ramp velocities. (g-i) Phases of the holonomy eigenvalues $\mu_{\pm}(k_x)$ of the dephased final state for the same parameter ramp velocities. Parameters are $m_i=-4$, $m_f=-1$, and $\gamma_{\mathbf{k}}=2$ for all panels.