Universal Behavior in Entanglement Entropy Reveals Quantum Criticality and Underlying Symmetry Breaking

Abstract

Entanglement plays a key role in quantum physics, but how much information it can extract from many-body systems is still an open question, particularly regarding quantum criticalities and emergent symmetries. In this work, we systematically study the entanglement entropy (EE) and derivative entanglement entropy (DEE) near quantum phase transitions in various quantum many-body systems. A one-parameter scaling relation between the DEE and system size at the critical point has been derived for the first time, which successfully obtains the critical exponent via data collapse. Furthermore, we find that the EE peaks at the (emergent) symmetry enhanced first-order transition, reflecting higher symmetry breaking. This work provides a new paradigm for quantum many-body research from the perspective of EE and DEE.

Figures (7)

• Figure 1: Quantum spin models on a square lattice with periodic boundary condition. The lattice size is $L_{x}\times L_{y}$, and the subsystem is a $(L_{x}/2)\times L_{y}$ cylinder region, which is separated from the rest part by the dashed lines. (a) Spin-1/2 bilayer Ising-Heisenberg model. $-J_{z}< 0$ denotes the intralayer FM Ising interaction, while $J>0$ is the interlayer AF Heisenberg interaction. (b) Spin-1/2 dimerized Heisenberg model. The AF interaction strengths on the thin and the thick bonds are $J_{1}$ and $J_{2}$, respectively. The anisotropy parameter $\Delta$ is the same on all bonds.
• Figure 2: (a) The second Rényi EE and (b) its derivative, and (c) the scaling function $\tilde{S}_{0}'(x)$ obtained from the data collapse analysis of the DEE near the Ising transition of the bilayer Ising-Heisenberg model. In the calculations, $J=3.045$ is fixed, while $J_{z}$ is tuned near the QCP $J_{z,c}=1$. (d--f) Data of the dimerized Heisenberg model with the anisotropy parameter $\Delta=0.9$. $J_{2}=2.1035$ is fixed, while $J_{1}$ is tuned around the O(2) QCP $J_{1,c}=1$. (g--i) Data of the dimerized isotropic Heisenberg model. $J_{2}=1$ is fixed, while $J_{1}$ is tuned around the O(3) QCP $J_{1,c}=0.52337$.
• Figure 3: The second Rényi EE divided by the boundary length $S^{(2)}/\ell$ of (a) the anisotropic Heisenberg model and (b) the checkerboard $J$-$Q$ model near the symmetry-enhanced first-order transitions.
• Figure S1: The checkerboard $J$-$Q$ model on a $L_{x}\times L_{y}$ square lattice with periodic boundary condition in both directions. The subsystem $A$ is a $(L_{x}/2)\times L_{y}$ cylinder region with the boundary length $\ell=2L_{y}$. The AF Heisenberg interaction ($J$-term) acts on all n.n. bonds, while the four-spin interaction ($Q$-term) acts on n.n. bond pairs that form a shaded plaquette.
• Figure S2: Failed data collapse analysis of the second Rényi EE data near the (2+1)D O($N$) ($1\leq N\leq 3$) QCPs based on the proposed scaling form Eq. (\ref{['eq:ee']}) Huang2024a. The quantum spin models with these QCPs are described in the main text. The vertical axis is the rescaled EE $S^{(2)}(\ell,g)\ell^{-1}$, while the horizontal axis is the dimensionless scaling variable $x=(g-g_{c})\ell^{1/\nu}$ calculated with the QCPs and the critical exponents well-established in the literature Wu2023bMatsumoto2001aZhu2021bGuida1998Deng2003Zhang2017: (a) $J_{z,c}=1$ and $\nu=0.630$ for the Ising transition, (b) $J_{1,c}=1$ and $\nu=0.671$ for the O(2) transition, and (c) $J_{1,c}=0.52337$ and $\nu=0.707$ for the O(3) transition. Were the scaling form Eq. (\ref{['eq:ee']}) valid, the rescaled data for different lattice sizes would collapse onto a single curve, which is unfortunately not the case.