Quantum complexity across thermal phase transition in the transverse field Ising chain with long-range couplings

TL;DR

This work reveals that quantum-information-based markers—specifically the Schmidt gap of the purified thermal state and non-stabilizerness measured via stabilizer Rényi entropies—exhibit pronounced signatures at the classical thermal transition of the one-dimensional long-range transverse-field Ising model. Using TDVP on locally purified tensor networks, the authors map finite-temperature states to pure states and perform finite-size scaling to extract critical parameters, showing robustness in the Schmidt-gap signal and a distinct peak in quantum magic near criticality. The results highlight the emergence of quantum complexity as a hallmark of thermal critical behavior, with stronger finite-size effects in the long-range regime ($\alpha=0.8$) compared to the short-range-like case ($\alpha=1.8$). They also discuss methodological caveats related to purification bias and sampling, and propose directions for refining magic calculations and extending the framework to broader models and dynamical scenarios.

Abstract

We investigate the behavior of the Schmidt gap, the von Neumann entanglement entropy, and the non-stabiliserness in proximity to the classical phase transition of the one-dimensional long-range transverse-field Ising model (LRTFIM). Leveraging the time-dependent variational principle (TDVP) within a tensor-network formulation, we simulate thermal states through their purified tensor-network representations. Our results show that these observables, typically regarded as hallmarks of quantum criticality, exhibit pronounced and coherent signatures even at a classical thermal transition, highlighting the emergence of quantum complexity as the system nears thermal criticality.

Figures (14)

• Figure 1: Evolution of the eigen value gap $\Delta$, the corresponding susceptibility $\chi$, and the von Neumann entanglement entropy $S$ of central bipartition, as functions of $\beta$ for $h = 0.0$, system sizes $N=16,32,64,200,250,300,350,400$ and (a) $\alpha = 1.8$; (b)$\alpha = 0.8$. The vertical dashed lines in all the plots are our estimate of $\beta_c$ from FSS analysis.
• Figure 2: Comparison of the $\beta$-dependence of $\Delta$, $\chi$ and $S$, for fixed $N=400$: (a) $h=0.0,~0.3$ for $\alpha=1.8$; (b) $h=0.0,~0.3$ for $\alpha=0.8$; (c) $\alpha=1.8,~0.8$ for $h=0.0$.
• Figure 3: Finite-size scaling of $\Delta$ for $h=0.0$ and for (a) $\alpha =1.8$, (b) $\alpha =0.8$. The data collapse scaling fit was performed within the dashed lines.
• Figure 4: Evolution of the nonstabilizerness density, $m_n$ as a function of $\beta$ . Panel (a) and (b) shows $m_1$ versus $\beta$: - In (a) $\alpha=1.8$; the blue and yellow lines correspond to $h=0.0$ and $h=0.3$ respectively. - in (b) $h=0.0$: the blue and yellow lines correspond to $\alpha=1.8,~0.8$ respectively. Panels (c) and (d) are the same as (a) and (b) but show the variation of $m_2$ instead of $m_1$. All plots are for $N=32$ .
• Figure 5: Evolution of the nonstabilizerness density, $m_n$ as a function of $\beta$ for $N=16$ (Blue lines) and $N=32$ (Orange lines), and $N=64$ (Green line), in the absence of an external field. (a) and (b) shows $m_1$ versus $\beta$ for $\alpha=1.8,0.8$ respectively. The black dashed line in both plots indicates $\beta_c$ that we estimate from FSS .