Diagnosis of mixed-state topological phases in strongly correlated systems via disorder parameters

TL;DR

The paper tackles diagnosing mixed-state topology in strongly correlated fermionic systems by introducing a disorder-parameter framework based on Z(\theta) = Tr[\hat{\rho} e^{i\theta \hat{Q}}] and its generating function F(\theta). The authors derive a numerically stable method to compute the second derivative F^{(2)}(\pi) via determinant quantum Monte Carlo, even in regimes with sign problems, using an effective non-Hermitian Hamiltonian and HS decoupling. They show that in topological phases, a topological scaling indicator \mathcal{F} exhibits a characteristic linear dependence on the system size along the open direction, driven by gapless edge modes, while trivial phases do not display this scaling. The KMH model maps a interaction-driven QSH to a trivial insulator, and the HH model reveals robust QAH behavior at finite temperature despite sign problems, illustrating the method’s ability to access otherwise challenging regions. Overall, the work provides a practical, scalable tool to numerically explore topological phenomena in interacting mixed states, enabling studies beyond the reach of conventional approaches.

Abstract

Characterizing topological phases for strongly interacting fermions in the mixed-state regime remains a major challenge. Here we introduce a general and numerically efficient framework to diagnose mixed-state topological phases in strongly interacting systems via the disorder parameter (DP) of the U(1) charge operator. Specifically, from the finite-size scaling of the second derivative of the DP generating function, we introduce the topological scaling indicator, which exhibits a characteristic linear scaling with the system's linear dimension for topological phases, a signature that vanishes upon transition into a topologically trivial phase. Crucially, we develop an efficient determinant Quantum Monte Carlo algorithm that facilitates the evaluation of this indicator in interacting systems. We apply our approach to two paradigmatic models: for the Kane-Mele-Hubbard model, we successfully map the interaction-driven transition from a quantum spin Hall insulator to a trivial Mott insulator. Furthermore, our method circumvents the limitations imposed by the severe sign problem in the Haldane-Hubbard model, enabling robust identification of the quantum anomalous Hall phase at accessible temperatures. This work provides a powerful and accessible tool for the numerical exploration of topological phenomena in interacting mixed states, opening a pathway to study systems previously inaccessible due to computational obstacles.

Figures (8)

• Figure 1: (a) $F(\theta)$ calculated from the 1D SSH model ($\beta=10.0$ and $L=128$). In the topological phase ($\delta t=-0.1$), a distinct cusp emerges at $\theta=\pi$, which is absent in the trivial phase ($\delta t=0.1$). (b) Response of $F(\pi)$ to a threaded magnetic flux $2\pi\tau$ in the Haldane model. A sharp cusp develops exclusively in the topological phase ($M < 3\sqrt{3}\lambda$). The parameters are $t_1 = 1.0$, $\lambda = 0.1$, $L_x= L_y = 36$ and $\beta = 10.0$. (c,d) The topological scaling indicator $\mathcal{F}$ scales linearly with $\Delta L_x$ in the Haldane model ($M=0, \lambda=0.1$). (c) The slope of $\mathcal{F}$ is proportional to $1/L_y$ at fixed $\beta=10$. (d) The results of $\mathcal{F}$ for fixed $L_y=12$ under different $\beta$.
• Figure 2: (a) Distinct linear scaling of $\mathcal{F}$ with $\Delta L_x$ for different $L_y$, calculated in the topological phase ($U = 2.5$) of KMH model at $\beta = 5.0$. (b) Linear scaling of $\mathcal{F}$ with $\Delta L_x$ for fixed $L_y = 8$ at different inverse temperatures $\beta$. The reference system used in defining $\mathcal{F}$ has $L_x^0 = 7$.
• Figure 3: The results of TSI $\mathcal{F}$ across the quantum phase transition in the KMH model. The inverse temperature is fixed at $\beta=8$. (a) Deep in the topological phase ($U = 3.6$), $\mathcal{F}$ exhibits the expected linear relation with $\Delta L_x$. the reference system size is $L_x^0 = 7$. (b) Approaching the phase transition ($U = 4.5$), the linear scaling remains robust, confirming the persistence of the topological phase even as correlations strengthen. The reference size is increased to $L_x^0 = 9$. (c)In the trivial AFM insulating phase ($U=6.0$), the linear scaling breaks down entirely and the TSI is suppressed towards zero, correctly identifying the non-topological nature of the state.
• Figure 4: The results of TSI in the HH model, with the inverse temperature fixed at $\beta = 3.6$ where the sign problem is still moderate. (a) $\mathcal{F}$ exhibits a clear linear relation with $\Delta L_x$, with the slope inversely proportional to $L_y$, indicating a topologically nontrivial phase. (b) The linear relation disappears, and $\mathcal{F}$ nearly vanishes, suggesting a transition into a topologically trivial insulating phase.
• Figure S1: (a) $F(\theta)$ in the trivial ($\delta t=0.25$) and topological ($\delta t=-0.1$) regimes, showing a cusp only in the latter. (b) $F(\theta)$ in the topological regime ($\delta t=-0.1$) for various system sizes $L$. (c) $F^{(1)}(\theta)$ in the topological regime ($\delta t=-0.1$) for different $L$, exhibiting a divergence at $\theta=\pi$ as $L$ increases. The inverse temperature is set to $\beta = 1.0$ throughout.