Universal Entanglement Growth along Imaginary Time in Quantum Critical Systems

TL;DR

The paper reveals a universal imaginary-time non-equilibrium scaling for entanglement in (2+1)D quantum critical systems, showing that the corner contribution to the second Rényi entropy $\Delta S_2$ scales as $\Delta S_2 \sim s_c \ln \tau$ in the short-time regime ($\tau \ll L$, with dynamic exponent $z=1$). It introduces a Projected Quantum Monte Carlo framework and two linked algorithms—an incremental sampling method and Subtracted Corner Entanglement Entropy (SCEE)—to extract universal coefficients efficiently from early relaxation dynamics. Validation in both free Dirac fermions and the interacting Gross-Neveu-Yukawa (GNY) QCP yields precise $s_c$ values and demonstrates data collapse across regimes, highlighting enhanced entanglement due to fermion-boson coupling. This work provides a practical route to entanglement spectroscopy of quantum critical matter and suggests non-equilibrium protocols can efficiently reveal universal CFT data on classical and quantum platforms, with potential applications to ground-state preparation and quantum simulations.

Abstract

Characterizing universal entanglement features in higher-dimensional quantum matter is a central goal of quantum information science and condensed matter physics. While the subleading corner terms in two-dimensional quantum systems encapsulate essential universal information of the underlying conformal field theory, our understanding of these features remains remarkably limited compared to their one-dimensional counterparts. We address this challenge by investigating the entanglement dynamics of fermionic systems along the imaginary-time evolution. We uncover a pioneering non-equilibrium scaling law where the corner entanglement entropy grows linearly with the logarithm of imaginary time, dictated solely by the universality class of the quantum critical point. Through unbiased Quantum Monte Carlo simulations, we verify this scaling in the interacting Gross-Neveu-Yukawa model, demonstrating that universal data can be accurately recovered from the early stages of relaxation. Our findings significantly circumvent the computational bottlenecks inherent in reaching full equilibrium convergence. This work establishes a direct link between the fundamental theory of non-equilibrium critical phenomena and the high-precision determination of universal entanglement properties on both classical and quantum platforms, paving the way for probing the rich entanglement structure of quantum critical systems.

Figures (7)

• Figure 1: (a) Lattices and EE subsystem cuts considered in this Letter. The full system has periodic boundary conditions in both horizontal and vertical directions. The orange region (Region $A$) has dimensions $\frac{L}{3}\times \frac{2L}{3}$ and features two $\frac{\pi}{3}$ and two $\frac{2\pi}{3}$ corners. The blue region (Region $B$) has dimensions $\frac{L}{3}\times L$ and possesses a smooth boundary. Crucially, both regions share the same boundary length. (b) Schematic illustration of the corner entanglement growth in imaginary-time evolution. The springs represent the degree of corner contribution to entanglement between the subsystem and the environment. In the non-equilibrium regime, the corner contribution to the entanglement entropy increases linearly with the logarithm of the imaginary time $\tau$.
• Figure 2: Non-equilibrium relaxation and scaling of the corner entanglement entropy $\Delta S_2$ in the non-interacting ($U=0$) system.(a) Imaginary-time evolution of $\Delta S_2$ from a product state toward the free Dirac fermion ground state. (b) Scaling analysis of the equilibrium corner entanglement $\Delta S_2$ with system size $L$, demonstrating the logarithmic dependence of $\Delta S_2$ on $L$. We determine the equilibrium coefficient $s_c^{\mathrm{eq}}$ by fitting $\Delta S_2(L)$ to the form $s_c \ln L +\text{const.}$ over fitting windows with varying lower bounds $L_{\text{min}}$ and a fixed maximum $L_{\text{max}}=90$. Inset: Extrapolation of the fitting results of $s_c^{\mathrm{eq}}(L_{\text{min}})$ to $L_{\text{min}}^{-1}\rightarrow 0$ using a polynomial fit, yielding $s_c^{\mathrm{eq}} = 0.3116(13)$. (c)$\Delta S_2$ plotted against system size $L$ for different fixed values of imaginary time $\tau$. In the non-equilibrium relaxation regime ($\tau \ll L$), $\Delta S_2$ exhibits a "size-independent plateau" determined solely by $\tau$. (d) The universal size-independent scaling of $\Delta S_2$ plotted as a function of imaginary time $\tau$. Inset: We extract the coefficient by fitting $\Delta S_2(\tau)$ to the logarithmic form $s_c \ln \tau + \text{const.}$ in fitting windows with varying minima $\tau_{\text{min}}$ and a fixed $\tau_{\text{max}}=6.0$. Extrapolating to $\tau_{\text{min}}^{-1}\rightarrow 0$ yields $s_c^{\mathrm{neq}} = 0.3111(15)$.
• Figure 3: Non-equilibrium dynamics of $\Delta S_2$ at the GNY QCP ($U = 3.8$). All simulations start from an AFM product state. (a)$\Delta S_2$ versus system size $L$ for different $\tau$. In the non-equilibrium regime ($\tau \ll L$), $\Delta S_2$ exhibits a size-independent plateau. (b) Logarithm plotting of $\Delta S_2$ versus $\tau$. The data for different $L$ collapse and are fitted to $\Delta S_2 = s_c \ln \tau + \text{const.}$, yielding the non-equilibrium coefficient $s_c^{\text{neq}} = 0.345(7)$. (c) Data collapse validating the universal scaling form in Eq. (\ref{['scalingeq1']}). Plotting $\Delta S_2 - s_c \ln L$ versus the scaling variable $\tau/L$ collapses all data onto a single master curve. The dashed line with the slope of $0.345$ is plotted for comparison with the rescaled curve.
• Figure 4: Non-equilibrium dynamics of $\Delta S_2$ at the AFM phase ($U = 7$). (a) $\Delta S_2$ versus $L$ for different $\tau$. In the non-equilibrium regime ($\tau \ll L$), $\Delta S_2$ exhibits a size-independent plateau. (b) Logarithm plotting of $\Delta S_2$ versus $\tau$. The data for different $L$ collapse and are fitted to $\Delta S_2 = s_c \ln \tau + \text{const.}$, yielding the non-equilibrium coefficient $s_c^{\text{neq}} = 0.070(12)$.
• Figure S1: (a) Sampling distribution of $\log(\det \mathbf{g}^A / \det \mathbf{g}^B)$, which follows a normal distribution $\mathcal{N}(\mu, \sigma^2)$. (b) Scaling of the statistical mean $\mu$ (left) and standard deviation $\sigma$ (right) with system size $L$. The solid lines represent fits to the scaling relations $\mu \approx -0.02L^{1.995}$ and $\sigma \approx 0.165L^{0.981}$.