Measurement protocol for detecting correlated topological insulators in synthetic quantum systems

Abstract

Two-dimensional topological insulators, characterized by symmetry-protected anomalous boundary modes, have been generalized to the strongly correlated regime for both bosonic and fermionic systems. As correlated topological insulators (TI) approach experimental realization in quantum simulators, conventional probes, such as transport measurements, are not easily applicable to these synthetic platforms. In this study, we focus on two examples of correlated TI: a bosonic TI protected by $\mathbb{Z}_2\times U(1)$ symmetry and the fermionic quantum spin Hall insulator protected by time-reversal symmetry. We propose a unified, readily implementable protocol based on measuring the disorder parameter $\langle\exp(\mathrm{i}θ\hat{Q}_A)\rangle$ for a large subregion $A$, with $\hat{Q}_A$ the total charge operator within $A$. Our key finding is that this quantity exhibits non-analytical dependence on $θ$ for correlated TI, a signature robust against decoherence. We establish this diagnostic through both numerical simulations and analytical derivations. This protocol is well-suited for implementation on near-term quantum simulation platforms, providing a direct route to experimentally confirm correlated TI.

Figures (4)

• Figure 1: Detection scheme for disorder parameters in quantum simulators. The protocol involves performing $N_M$ projective measurements of the many-body wavefunction $\lvert\psi\rangle$ in the particle number basis for a sub-region $A$. This yields the distribution $q(n)$, counting the occurrences of particle number $n$ within $A$. The scaled disorder parameter $f(\theta)\equiv-{\left\lvert\partial A\right\rvert^{-1}}\cdot \ln\left\lvert U_A(\theta)\right\rvert$ is then computed from $q(n)$ via post data-processing. A characteristic non-smooth behavior of $f(\theta)$ serves as a direct signature of a nontrivial correlated TI.
• Figure 2: Numerical results for the disorder parameter. (a,d) Bosonic correlated TI. (b,e) Fermionic QSH insulator. (c,f) Trivial fermionic insulator with time reversal symmetry. Top row (a-c): scaled disorder parameter $f(\theta)\equiv -\left\lvert\partial A\right\rvert^{-1} \ln\langle{U_A(\theta)}\rangle$. Bottom row (d-f): its derivative $f'(\theta)$. Different subsystem sizes $\left\lvert\partial A\right\rvert$ are indicated in the legends. The discontinuity behavior of $f'(\theta)$ at $\theta=\pi$ in (d,e) provides a direct signature of correlated TI.
• Figure 3: Bulk fixed-point wavefunction and boundary theory for the $U(1)\times \mathbb{Z}_2$ bosonic correlated TI. (a) The fixed-point wavefunction is an equal-weight superposition of all allowed configurations. (b) The decoration pattern of a typical configuration. Here, the orange/teal point refers to a positive/negative charged boson. (c) A typical boundary configuration. Here, the boundary $U(1)$ symmetry action is non-onsite. (d) An extended boundary Hilbert space, where each site holds two Ising spins. The symmetry action is onsite in this extended Hilbert space.
• Figure 4: Calculating disorder parameter with RDM defined at $\partial A$. After the isometry $U$, the RDM $\rho_A$ is mapped to $\rho_{\partial A}$ defined on the entanglement space. The disorder parameter $\langle U_A(\theta)\rangle$ can be computed as the expectation value of $W_{\partial A}(\theta)$ in the 1D mixed state $\rho_{\partial A}$.