Probing non-equilibrium physics through the two-body Bell correlator

TL;DR

This work shows that a two-body Bell-CHSH correlator can efficiently diagnose dynamical quantum phase transitions in a long-range XY chain under sudden quenches. By deriving the Bell parameter $\mathcal{B}$ from the two-site reduced state and expressing it through the correlation matrix, the authors reveal that the long-time saturated value $\mathcal{B}_s$ sharply differentiates quenches that stay within a phase from those that cross a phase boundary, with a robust threshold $\mathcal{B}_c$ that follows Gaussian or tri-Gaussian laws depending on the quench type. The approach remains effective across varied LR exponents $\alpha$ and anisotropies $\gamma$, and it outperforms bipartite entanglement and most classical correlators in signaling DQPTs, while being experimentally accessible without full state tomography. These findings establish the Bell correlator as a practical, local diagnostic tool for non-equilibrium criticality in many-body quantum systems. The results have relevance for quantum simulators with LR interactions, where rapid, local measurements can reveal global phase structure during dynamics.

Abstract

Identifying equilibrium criticalities and phases from the dynamics of a system, known as a dynamical quantum phase transition (DQPT), is a challenging task when relying solely on local observables. We exhibit that the experimentally accessible two-body Bell operator, originally designed to detect nonlocal correlations in quantum states, serves as an effective witness of DQPTs in a long-range (LR) XY spin chain subjected to a magnetic field, where the interaction strength decays as a power law. Following a sudden quench of the system parameters, the Bell operator between nearest-neighbor spins exhibits a distinct drop at the critical boundaries. In this study, we consider two quenching protocols, namely sudden quenches of the magnetic field strength and the interaction fall-off rate. This pronounced behavior defines a threshold, distinguishing intra-phase from inter-phase quenches, remaining valid regardless of the strength of long-range interactions, anisotropy, and system sizes. Comparative analyses further demonstrate that conventional classical and quantum correlators, including entanglement, fail to capture this transition during dynamics.

Figures (10)

• Figure 1: (Color online) Temporal behavior of Bell CHSH correlator under field quench. The behavior of $\mathcal{B}(t)$ (ordinate) against time (abscissa) for different pairs of initial and quenched magnetic field strengths, ($h_i, h_f$), of the nearest-neighbor transverse Ising models ($\gamma =1, \alpha =10$). Solid lines (dark and light) represent when the initial and final fields belong to the same phase while dashed lines (dark and light) are for inter-field quenches. We notice that intra-field quenches produce a higher saturated $B(t)$ as compared to the inter-field quenches. Based on these observations, we introduce the physical quantity, $\mathcal{B}_s(t)$ in Eq. (\ref{['eq:Bsat']}) to probe DQPT. All axes are dimensionless.
• Figure 2: (Color online) Comparison between steady-state and time-averaged Bell correlators across dynamical phases of the nearest-neighbor transverse Ising model. (a) Map plot of the steady-state value, $\mathcal{B}_{s}$, over the parameter space defined by $h_i$ (abscissa) and $h_f$ (ordinate) with ($h_i, h_f$). Regions where $h_i$ and $h_f$ lie in different phases exhibit reduced $\mathcal{B}_{s}$, while quenches within the same phase yield elevated values. (b) Map plot for the time-averaged Bell correlator, $\mathcal{B}_{avg}$, exhibiting similar behavior to $\mathcal{B}_{s}$. All axes are dimensionless.
• Figure 3: (Color online) Contour corresponding to the benchmarking value $\mathcal{B}_c$ in the parameter space $(h_i,h_f)$ of quantum XY-model with $\gamma = 0.2$. The four panels correspond to the map plot of $\mathcal{B}_{s}$ for the long-range $XY$ models characterized by the fall-off rates, $\alpha$ -- (a) $\alpha = 0.9$, (b) $\alpha =1.5$, (c) $\alpha =3.5$, and (d) the nearest-neighbor limit ($\alpha =10$) by varying $h_i$ (horizontal axis) and $h_f$ (vertical axis). The contours (solid lines) defined by the benchmarking value $\mathcal{B}_c$ delineates the region associated with same-phase detection, and the enclosed area serves as a figure of merit for the efficiency of the proposed quantifier. All axes are dimensionless.
• Figure 4: (Color online) Critical benchmarking threshold as DQPT quantifier. (a) Variation of $\mathcal{B}_c$ (solid line, ordinate) with $\alpha$ (abscissa) for $\gamma = 0.2, 0.4, 0.6, 0.8$ and $1.0$ during magnetic field quenches, $(h_i, h_f)$. The dashed lines are the Gaussian fit, given in Eq. (\ref{['eq:fieldqfit']}). (Inset) Dependence of the inverse of variance of the fitted Gaussians on $\gamma$ upto a scale. (b) $\mathcal{B}_c$ (ordinate) vs $h$ (abscissa) within the range $[-0.75,0.41]$ for $\gamma = 0.4, 0.8$ and $1.0$ during coupling quenches ($\alpha_i, \alpha_f$). In this case, the fitted dash curve is modeled after a tri-Gaussian function (see Eq. (\ref{['eq:tri_gaussian_fit']})). In both the cases, dark to light shades represent the decrease of $\gamma$. All the axes are dimensionless.
• Figure 5: (Color online) Dependence of the sensitivity of the proposed quantifier on the anisotropy parameter $\gamma$. The four panels correspond to the map plot of $\mathcal{B}_{s}$ for the $XY$ model characterized by the field quench with $\alpha =10$ ((a) and (b)) and the coupling quenches, having a fixed magnetic field, $h =-0.5$((c) and (d)). We compare the sharpness of the quantifier for different values of the anisotropy parameter -- $\gamma = 1.0$ ((a), (c)) and $0.2$ ((b), (d)). All axes are dimensionless.