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arXiv:2412.02236

Susceptibility of entanglement entropy: a universal indicator of quantum criticality

Sarkar, Pritam

Published Dec 3, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2412.02236

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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 ∣ λc(N)−λc∞∣| \ λ_c(N)-λ_c^{\infty}|∣ λc​(N)−λc∞​∣ as N→∞N\to \inftyN→∞ in the cases of finite 1D transverse field ising model (TFIM) (λ=hλ=hλ=h) and XY chain (λ=γλ=γλ=γ). The integer power law appearing for XY model has been verified using perturbation theory in O(1N)\mathcal{O}(\frac{1}{N})O(N1​) 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.
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The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).
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Critical phenomena, characterized by abrupt changes in the macroscopic properties of a system, exhibit a profound degree of universality, meaning that vastly different physical systems can display identical behavior near their critical points. This universality is typically understood through the lens of scaling laws and the Renormalization Group theory, which classify critical points into distinct universality classes. This paper proposes and explores the "Entropic Law of Critical Universality," an overarching principle positing that not only do thermodynamic quantities scale universally, but the fundamental information-theoretic measures of a system, particularly entropy in its various forms (thermodynamic entropy, entanglement entropy, Renyi entropy), also conform to universal scaling behaviors that uniquely characterize these universality classes. We review how entanglement entropy exhibits universal logarithmic corrections at quantum critical points, linked to the central charge of underlying conformal field theories. We also discuss how specific entropy behaves non-monotonically near quantum chromodynamics critical points and how entropic measures can identify phase transitions and their classes in various condensed matter systems, including graphene and spin models. The paper synthesizes theoretical frameworks and empirical evidence to demonstrate that entropy serves as a powerful and often more direct probe of the universal properties underlying critical phenomena, offering a comprehensive perspective on the shared information landscape of diverse critical systems.
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