Susceptibility of entanglement entropy: a universal indicator of quantum criticality

TL;DR

This work introduces the susceptibility of entanglement entropy as an information-geometric indicator of quantum criticality, demonstrating that its finite-size behavior in the XY chain and TFIM reveals universal scaling around critical points. By deriving a simple closed-form expression for the susceptibility from the one-body reduced density matrix, the authors show that turning points converge to the thermodynamic critical points with model-specific power laws: $|\gamma_c^{\infty}-\gamma_c^{N}| \sim \dfrac{\pi}{2\sqrt{2}\,N}$ for XY and $|h_c^{\infty}-h_c^{N}| \sim N^{-3/2}$ for TFIM, while the maximum susceptibility scales as $(\log N)^2$ in TFIM and saturates in XY. The approach is model-agnostic, relies only on reduced density matrices, and is extendable to dynamics and possibly non-integrable systems, offering a practical, angle-free diagnostic for quantum criticality with potential applications in quantum control and technology. The analytic results are complemented by numerical simulations and perturbative/elliptic-function analyses, highlighting the utility of information-geometric tools in capturing critical behavior.

Abstract

A measure of how sensitive the entanglement entropy is in a quantum system, has been proposed and its information geometric origin is discussed. It has been demonstrated for two exactly solvable spin systems, that thermodynamic criticality is directly \textit{indicated} by finite size scaling of the global maxima and turning points of the susceptibility of entanglement entropy through numerical analysis - obtaining power laws. Analytically we have proved those power laws for $| \ λ_c(N)-λ_c^{\infty}|$ as $N\to \infty$ in the cases of finite 1D transverse field ising model (TFIM) ($λ=h$) and XY chain ($λ=γ$). The integer power law appearing for XY model has been verified using perturbation theory in $\mathcal{O}(\frac{1}{N})$ and the fractional power law appearing in the case of TFIM, is verified by an exact approach involving Chebyshev polynomials, hypergeometric functions and complete elliptic integrals. Furthermore a set of potential applications of this quantity under quantum dynamics and also for non-integrable systems, are briefly discussed. The simplicity of this setup for understanding quantum criticality is emphasized as it takes in only the reduced density matrix of appropriate rank.

Figures (6)

• Figure 1: Examples of transverse magnetization for different system sizes
• Figure 2: Susceptibility of entanglement entropy 1. $\Sigma_{N}^{\text{XY}}(\gamma)$, 2. $\Sigma_{N}^{\text{TFIM}}(h)$ in a few small 1. XY and 2. TFIM chains
• Figure 3: Susceptibility of Entanglement-Entropy $\Sigma_{\gamma \gamma}^{(h=0)}$ for XY model
• Figure 4: 1. $| \gamma^{\infty}_c - \gamma^{ N}_c|$ in log-log scale, 2. Maximum susceptibility, 3.Numerics vs analytics
• Figure 5: Susceptibility of entanglement entropy $\Sigma_{N}^{\text{TFIM}}(h)$ for transverse field Ising model