Entanglement-limited linear response in fermionic systems

TL;DR

This paper establishes a general link between entanglement-entropy scaling and linear response in particle-conserving fermionic ground states by showing that static particle-number fluctuations within a finite region scale with the same universal function $F(L)$ that governs $\mathcal{S}_{vN}$. It then demonstrates that time-dependent fluctuations and linear-response functions inherit this entanglement-driven scaling, leading to counterintuitive subextensive responses such as energy absorption and particle-number fluctuations scaling with a region's boundary rather than its volume in gapped systems. The authors verify these entanglement-limited scalings in free-fermion models across area-law, critical, and volume-law regimes, using both general arguments and explicit calculations of the dynamical structure factor $\overline S(\omega_0)$, including area-law insulators, 2D metallic phases, and lower-dimensional subsystems embedded in higher-dimensional hosts. The work provides a principled connection between entanglement and measurable linear-response observables, suggesting new avenues to probe many-body entanglement in experiments and to constrain dynamical behavior in interacting systems and at finite temperature.

Abstract

We propose a general connection between entanglement-entropy scaling laws and the linear response functions of particle-conserving fermionic systems in their ground state. Specifically, we show that the response to perturbations coupled to the particle number within a finite region exhibits the same size scaling as the entanglement entropy of that region. We explicitly verify this scaling in free-fermion systems that display area-law, volume-law, and critical forms of entanglement. The resulting entanglement-governed scaling of response functions leads to unexpected physical consequences. For instance, contrary to conventional expectations, the energy absorption rate and particle-number fluctuations in gapped systems scale with the boundary of the perturbed region rather than with its volume. Our work thus establishes a direct link between linear-response properties and many-body entanglement.

Figures (6)

• Figure 1: Conserved lattice fermions exposed to a perturbation which couples to the particles on subsystem $A$ (red region) $\hat{N}_A=\sum_{i\in A}\hat{n}_i$ with linear dimension $L$. The leading order scaling in $L$ of various physical responses follow the same form as the scaling of the entanglement entropy of subsystem $A$.
• Figure 2: (Color online) Behavior of the dynamical noise in the QWZ model with $m=1.0$. For all system sizes the subsystem is chosen as $l_x = N_x/2$ and $l_y = N_y/2$. The coarse-graining width is fixed for $N_x = N_y = 12$ as $\Delta \omega_0 = 2 \delta \omega$, where $\delta \omega = \max_{\bf k}| \omega_{{\bf k}+\delta {\bf k}} - \omega_{\bf k} |$ with ${\bf k}$ are discrete momentum values (due to finite system lengths) in a mesh grid given by Eq. \ref{['eq:discrete-k-2D']}, and momentum step $\delta {\bf k} = (2\pi/N_y)\hat{\bf y}$. Since, for the set of parameters considered here, the valence and conduction bands span $\omega_{\pm} \in (\pm 1, \pm 3)$, the dynamical structure factor is ideally nonzero only for $\omega_0 \in (2,6)$, corresponding to possible transitions from valence to conduction states, when the frequency window $\Delta \omega$ is very small. For a finite $\Delta \omega$, however, we observe a slight broadening of the range over which $\bar{S}(\omega_0)$ remains finite.
• Figure 3: (color online) Noise power in a 2D metallic lattice model with single–band dispersion $\omega_{\mathbf{k}} = - t \bigl[\cos(k_x) + \cos(k_y)\bigr] + \mu$, where $t = 1.0$ and $\mu = 0.5$. For all system sizes the subsystem is chosen as $l_x = N_x/2$ and $l_y = N_y/2$. The frequency-smoothing width is fixed for $N_x = N_y = 14$ as $\Delta \omega_0 = 2 \delta \omega$, where $\delta \omega = \max_{\mathbf{k}} | \omega_{\mathbf{k}+\delta \mathbf{k}} - \omega_{\mathbf{k}} |$. The discrete momentum values $\mathbf{k}$ are defined on the same mesh introduced in Eq. \ref{['eq:discrete-k-2D']}, and the momentum step in the $y$–direction is $\delta \mathbf{k} = (2\pi/N_y)\hat{\mathbf{y}}$.
• Figure 4: (color online) Behavior of the noise power in a 1D one-band model. The dispersion relation is given by $\omega_{k} = -t\cos(k) + \mu$, where we use the parameters $t = 1.0$ and $\mu = 0.5$. The subsystem length is fixed at $l = N/2$. The frequency-swapping window $\Delta\omega_0 = 2\delta\omega$ is obtained following the same procedure as in Fig. \ref{['Fig-2D1B']}, but applied to the 1D case and for a fixed system size $N = 40$.
• Figure 5: (color online) Behavior of the noise power in a 1D subsystem ($l_x = N_x$ and $l_y = 1$) of a two-dimensional single-band model on a square lattice. The frequency window $\Delta \omega_0 = 2\delta\omega$ is fixed based on the value of $\delta\omega$ obtained for a system of size $N_x = N_y = 14$ ($\delta\omega$ is obtained in the same way as for the results shown in Fig. \ref{['Fig-2D1B']}). Panel (a) shows the unscaled noise power, while panel (b) presents the same data scaled as $\bar{S}(\omega_0)/N_x$. The 1D subsystem embedded in the 2D system clearly exhibits a one-dimensional volume-law behavior, evidenced by the collapse of all curves in the scaled results of panel (b), as well as the linear dependence on the subsystem length $l_x$ shown in the inset of panel (a).