Indicators of conformal field theory: Entanglement entropy and multiple-point correlators

The entanglement entropy (EE) of quantum systems is often used as a test of low-energy descriptions by conformal field theory (CFT). Here we point out that this is not a reliable indicator, as the EE often shows the same behavior even when a CFT description is not correct (as long as the system is asymptotically scale-invariant). We use constraints on the scaling dimension given by the CFT with SU(2) symmetry to provide alternative tests with two- and four-point correlation functions, showing examples for quantum spin models in 1+1 dimensions. In the case of a critical amplitude-product state expressed in the valence-bond basis (where the amplitudes decay as a power law of the bond length and the wave function is the product of all bond amplitudes), we show that even though the EE exhibits the expected CFT behavior, there is no CFT description of this state. We provide numerical tests of the behavior predicted by CFT for the correlation functions in the critical transverse-field Ising chain and the $J\text{−}Q$ spin chain, where the conformal structure is well understood. That behavior is not reproduced in the amplitude-product state.

Figure 1

Two- and three-point functions for the classical critical 2D Ising model. The exponents for $\langle \sigma \sigma \rangle$, $\langle \epsilon \epsilon \rangle$, and $\langle \sigma \sigma \epsilon \rangle$ are found to be $0.252\left(4\right)$, $2.01\left(3\right)$, and $1.263\left(5\right)$, respectively, where the numbers within parenthesis indicate the statistical error (one standard deviation) of the preceding digit. The curves show fits to Eq. (18) using $b=2$.

Figure 2

Power-law behavior of correlation functions in the 1D TFIM chain at criticality. The exponents for the spin correlation, energy density correlation, and three-point correlation are $0.2498\left(2\right)$, $1.9\left(4\right)$, and $1.29\left(1\right)$, respectively.

Figure 3

Scaled spin correlation function, Eq. (19), for the critical TFIM on chains of different lengths ($x/L=0.5$ is the largest separation on a periodic chain). Here we have used ${\mathrm{\Delta }}_{\sigma }=1/8$ as that is the scaling dimension of the Ising spin operator [9].

Figure 4

The spin and dimer two-point correlation functions of the critical $J\text{−}Q$ chain scaled with the conformal distance according to Eq. (19).

Figure 5

Four-point correlators of the critical $J\text{−}Q$ chain along with a fit to the form expected for scaling dimension 2, shown here for two different combinations of cross-ratios and system sizes. The prefactor for the two fits is almost the same.

Figure 6

Phase diagram of the APS with a critical curve separating the Néel phase and the VBS phase; this matches well the phase diagram previously reported in Ref. [20].

Figure 7

Flow of the Binder cumulant of the APS with $\kappa$ for different system sizes at $\lambda =3$. We see here the crossing points for three sets of $(L,2L)$ (example of extrapolation to critical point shown in inset of Fig. 8) and also that $B_c$ becomes negative for large sizes after the crossing point.

Figure 8

EE of the 1D critical APS at $\lambda =1$ for a periodic chain of $L=64$ sites. A fit to the form predicted by Eq. (23), along with finite size corrections [25] causing the visible even-odd oscillations, gave $c=1.08\left(4\right)$. Error bars are of the order of the symbol size. The inset shows the extrapolation of the Binder cumulant crossings points (examples of which are seen in Fig. 7), which we have used to extract the critical $\kappa$ for the infinite system. Here ${\kappa }_{\mathrm{crit}}$ is 1.519(5).

Figure 9

EE for the 1D critical APS versus the parameter $\lambda$, extracted as illustrated in Fig. 8.

Figure 10

Dimension of the spin operator in the 1D critical APS graphed vs. the short-bond parameter $\lambda$. The values were deduced using two- and four-point functions calculated on the critical curve.

Figure 11

Scaled spin correlation functions, defined in Eq. (19), plotted against the separation as a fraction of the system size for the critical APS at $\lambda =1$ (where ${\kappa }_{\mathrm{crit}}=1.519$). The inset shows the ratio $R(0.2,0.5)$ (from Eq. (24)), of the correlations at two distances.