Fine-grained entanglement loss along renormalization group flows

Abstract

We explore entanglement loss along renormalization group trajectories as a basic quantum information property underlying their irreversibility. This analysis is carried out for the quantum Ising chain as a transverse magnetic field is changed. We consider the ground-state entanglement between a large block of spins and the rest of the chain. Entanglement loss is seen to follow from a rigid reordering, satisfying the majorization relation, of the eigenvalues of the reduced density matrix for the spin block. More generally, our results indicate that it may be possible to prove the irreversibility along RG trajectories from the properties of the vacuum only, without need to study the whole hamiltonian.

Figures (3)

• Figure 1: Entropy $S_L(\lambda)$ of the quantum Ising chain as a function of the magnetic field $\lambda$ for $L = 100$. The lowest branch has $\epsilon > 0$.
• Figure 2: Scaling of the Ising chain entropy $S_L(\lambda)$ as $\lambda$ approaches its critical value $\lambda^* = 1$ ($\lambda\in(0.99,1.01)$), for various values of $L$ up to $L = 700$ (solid black triangles). Both left and right asymptotes scale as $S_L(\lambda)\sim -(1/6)\log_2 |1-\lambda|$.
• Figure 3: Entropy $S_L(\gamma,\lambda)$ for the XY spin chain as a function of anisotropy $\gamma$ and magnetic field $\lambda$ for $L = 100$. The solid red line in the projection (phase diagram) is the critical XX-model (XX*), and the solid blue line is the critical XY-model (XY*). QCP is the quantum critical point of the Ising chain.