Ground state entanglement in quantum spin chains

TL;DR

This work demonstrates that ground-state entanglement in 1D spin chains exhibits universal scaling at quantum critical points, governed by conformal symmetry and characterized by the block entropy S_L. By analyzing XY and XXZ models through exact methods and Bethe Ansatz, the authors connect S_L to the central charge of the underlying conformal field theory and reveal how entanglement saturates away from criticality while diverging logarithmically at criticality. They further relate entanglement ordering to majorization and discuss implications for the renormalization group and the efficiency of numerical techniques like DMRG, including extensions to higher dimensions where area laws constrain computational approaches. The results provide a unified picture linking quantum information measures, critical phenomena, and field-theoretic concepts such as the c-theorem. This synthesis highlights the central role of entanglement in governing quantum many-body behavior and its practical consequences for simulating complex systems.

Abstract

A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.

Figures (16)

• Figure 1: The entropy $S_L$ corresponds to the von Neumann entropy of the reduced density matrix $\rho_L$ for a block of $L$ adjacent spins, and measures the entanglement between the block and the rest of the chain. State $\rho_L$ is obtained from the ground state $\hbox{$| \Psi_g \rangle$}$ of the N spin chain by tracing out all $N-L$ spins outside the block.
• Figure 2: Bounds for the entropy $S_L$ for some pure states in a system with N=26 spins. Triangles correspond to the linear upper bound (\ref{['eq:bound']}), and applies to translationally invariant states (and, more generally, to arbitrary $N$-qubit states). Stars are the logarithmic upper bound for a symmetric state under permutations. The diamonds are the values of the entropy for a GHZ state.
• Figure 3: Energy of the system for different values of the parameters $\lambda$ and $\gamma$ as a function of $\phi$. The thick plot corresponds to the XX model without magnetic field, the dashed one to a system with $\lambda=\gamma=0.5$, the flat one is the Ising limit without magnetic field and the dot-dashed one and the thick dashed plot correspond to the isotropic and Ising model with $\lambda=1$, respectively.
• Figure 4: Some critical regions in the parameter space ($\gamma,\lambda$) for the XY model. The Ising model, $\gamma=1$, has a critical point at $\lambda=1$. The XX model, $\gamma=0$, is critical in the interval $\lambda\in [0,1]$. The whole line $\lambda=1$ is also critical. A complete analysis of the critical regions in this model was done by Barouch and McCoy in Bar
• Figure 5: This road map describes the steps followed in order to obtain the entropy $S_L$ of $L$ contiguous spins from an infinite $XY$ chain. We diagonalize the Hamiltonian $H_{XY}$ by rewriting it first in terms of Majorana operators $\check{a}$ and then in terms of Majorana operators $\check{b}$. The ground state $\hbox{$| \Psi_g \rangle$}$ is characterized by a correlation matrix $\Gamma^B$ for operators $\check{b}$, $\Gamma^A$ for operators $\check{a}$. Correlation matrix $\Gamma^A_L$ describes the reduced density matrix $\rho_L$ for a block of $L$ spins. $S_L$ is finally obtained from $\Gamma_L^C$, the block-diagonal form of $\Gamma_L^A$.