Scaling of the entanglement spectrum near quantum phase transitions

The entanglement spectrum describing quantum correlations in many-body systems has been recently recognized as a key tool to characterize different quantum phases, including topological ones. Here we derive its analytically scaling properties in the vicinity of some integrable quantum phase transitions and extend our studies also to nonintegrable quantum phase transitions in one-dimensional spin models numerically. Our analysis shows that, in all studied cases, the scaling of the difference between the two largest nondegenerate Schmidt eigenvalues yields with good accuracy critical points and mass scaling exponents.

Figure 1

Schmidt gap closing in the vicinity of the critical point for the transverse Ising model. Comparison between the analytical prediction from CFT (15) (solid gray line), the power-law scaling (23) (blue dashed line), and the numerical values (crosses) obtained by exact diagonalization. We consider a chain of 12 000 sites.

Figure 2

Schmidt gap closing in the vicinity of the critical point for the longitudinal Ising model. Comparison between the analytical prediction from CFT (19) (solid gray lines), the power-law scaling (23) (blue dashed line), and the numerical values (crosses) obtained by solving the model with DMRG. We consider a chain of 1536 sites. Notice that the fit is restricted to the six largest values.

Figure 3

Phase diagram of the spin-1 Hamiltonian (24) as a function of $\theta$ and $D$ computed by DMRG. The details about the phases can be found in Ref. 31. We also put an arrow and a corresponding number for the four critical points that we study more extensively in the text. (Figure from Ref. 31).

Figure 4

The Schmidt gap, the string order parameter, and the staggered magnetization for the Hamiltonian (24) on the $\theta =0$ line, shown as a function of $D/J$. The shaded area represents the scaling regions close to the critical points where our numerical analysis is less reliable. (Figure from Ref. 31.)

Figure 5

Schmidt spectrum close to the dimer-Haldane phase transition for $D=0$ as a function of $\theta$ and for $L=384$. The critical point is at $\theta /\pi =-1/4$ and it is indicated by a vertical dashed line. The discrepancy between the location of the critical point and the zone where the statistics of the entanglement spectrum changes is a finite size effect. We indicate the degeneracies with small clusters of dots in the two phases. Lines joining the numerical data are only a guide to the eye.

Figure 6

Finite size scaling close to the Haldane-dimer transition. From the left size, the first and second panels are the FSS results for the Schmidt gap $\Delta \lambda$, while the third and fourth panels refer to the FSS of the dimer order parameter.