Precise computation of universal corner entanglement entropy at 2+1 dimension: From Ising to Gaussian quantum critical points

Abstract

Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.

Figures (17)

• Figure 1: Model, phase diagram and the universal corner EE from Ising to Gaussian QCPs. (a) Phase transition on $h-K$ plane. The light pink line represents the continuous phase transition, while the blue line indicates the first-order phase transition. The dots indicate the points where we performed the crossing-point analysis to determine the locations of the phase transitions. The PQMC determined Gaussian fixed point is identified as the yellow star. The top left inset in (a) illustrates the entanglement regions. In a $L \times L$ square lattice, the entangled region $A_1$ has the dimension $L \times L/2$ (orange background) and it has a smooth boundary [Eq. \ref{['eq:Second order REE of smooth boundary']}]. The region $A_2$ has a $L/2 \times L/2$ (blue background) and it has four 90$^\circ$ corners log-contribution [Eq. \ref{['eq:Second order REE of corner boundary']}]. The lattice is periodic along both directions (denoted by the dotted lines), and the entanglement regions $A_1$ and $A_2$ have the same boundary length. The bottom right inset in (a) illustrates the action of $K$-term in the Hamiltonian of Eq. \ref{['eq:eq1']}. (b) Scaling of the subtracted entanglement entropy $S_s$. The dependence of $S_s$ on $\ln(L)$ at the phase transition boundary for selected values of $K = 0$, 5, 10, $K_c$, 160 and 320 are shown, where $K_c = 16.02(6)$ is the Gaussian QCP. (c) The universal log-coefficient $s$, extracted from the data in (b), as a function of $K$. (d), (e), and (f) show the extrapolation for $K=0$, $K=5$, and $K=10$ at their corresponding (2+1)d Ising QCPs. The computed corner log-coefficients converge to a universal average value of $s=0.020(1)$ as $L_{\min}$ increases. (g) corresponds to the Gaussian fixed point at $K_c$, where the corner log-coefficient converges to the universal value of $s=0.025(1)$. (h) is for $K=320$ at its first-order phase transition point; here, the computed corner log-coefficients vanishes as $s=0.000(1)$.
• Figure 2: Finite-size scaling analysis of the phase transitions driven by transverse field $h$ at different values of parameter $K$. (a-c) $K = 5$: (a) Binder cumulant $U_L$ as a function of $h$ for different system sizes up to $L = 64$. The crossing point of the curves determines the critical field $h_c^{K=5}=23.9(5)$. (b) Data collapse of $U_L$ using the finite-size scaling relation $U_L \sim (h/h_c - 1)L^{1/\nu}$ with the 3D Ising universality critical exponent $\nu = 0.63$. The high quality of the collapse confirms a continuous phase transition. (c) Heat map of the $R$ statistic for the data collapse, evaluated over a range of $\nu$ and $h_c$ values. The optimal region of high $R$ aligns with $\nu = 0.63$, providing further evidence that the continuous phase transition belongs to the 3D Ising universality. (d-f) $K = 16.02$: (d) Binder cumulant $U_L$ versus $h$ for different $L$ up to 64. The intersection point identifies $h_c^{K=16.02}=67.57(4)$. (e) Data collapse of $U_L$ using the Gaussian (free) universality critical exponent $\nu = 0.5$, indicating a continuous phase transition belonging to this class. (f) Heat map based on the $R$ criterion for the finite-size scaling relation, exploring different values of $\nu$ and $h_c$. The bright region confirming the best data collapse includes $\nu = 0.5$, consistent with the Gaussian universality. (g-i) $K = 320$: (g) Binder cumulant $U_L$ as a function of $h$ for different $L$ up to 16. The emergence of negative $U_L$ values for $L=12$ and $L=16$ at certain values of $h$ is a signature of a first-order phase transition. (h) Attempted data collapse of $U_L$ using the 3D Ising critical exponent ($\nu = 0.63$) fails. (i) Attempted data collapse using the Gaussian critical exponent ($\nu = 0.5$) also fails. The poor quality of both collapse attempts provides additional evidence for a first-order phase transition at this parameter.
• Figure 3: Universal corner term in a free theory. In 2d square lattice with the entanglement area having four 90$^\circ$ corners, one can compute the universal term $s$ in Eq. \ref{['eq:eq7']} and extrapolate the value with the inverse linear size of the region $1/L_{\min}$, the extrapolated value is $s=0.02567$. The red dashed line is the reference value from literature UniversalCasini2007UniversalHelmes2016.
• Figure 4: Illustration of the bubble basis and its operator evolution. a) and b) show the action of $H_J$, c) and d) show the action of $H_h$ and e) - h) show the action of $H_K$ on the bubble basis. The $H_K$ acts on a square of four spins, but for illustration purposes, it is compressed and shown as a one-dimensional bar here. After acting the operators on the basis, they would give different constants as described in Eq. \ref{['eq:weight']} and these constants are also shown next to the bubbles.
• Figure 5: Propagation of the operators. This graph gives a pictorial representation of Eq. \ref{['eq:timeslice']}. It shows a configuration with $m=2$ and the operators $H_{a_1}$ - $H_{a_4}$ are $H_J$, $H_h$, $H_k$ and $H_j$ respectively. The actions of different operators on the bubble basis are illustrated in Fig. \ref{['fig:fig1']}, and the constants are neglected here.