Quantum critical universality and singular corner entanglement entropy of bilayer Heisenberg-Ising model

We consider a bilayer quantum spin model with anisotropic intralayer exchange couplings. By varying the anisotropy, the quantum critical phenomena changes from XY to Heisenberg to Ising universality class, with two modes, three modes, and one mode, respectively, becoming gapless simultaneously. We use series-expansion methods to calculate the second and third Renyi entanglement entropies when the system is bipartitioned into two parts. Leading area-law terms and subleading entropies associated with corners are separately calculated. We find clear evidence that the logarithmic singularity associated with the corners is universal in each class. Its coefficient along the Ising critical line is in excellent agreement with those obtained previously for the transverse-field Ising model. Our results provide strong evidence for the idea that the universal terms in the entanglement entropy provide an approximate measure of the low energy degrees of freedom in the system.

Figure 1

A bilayer of quantum spins can be divided into half bilayers or quarter bilayers by a suitable choice of partitions. These can, in turn, be used to define the area-law entanglement entropy (or the entanglement entropy per unit length of the boundary) and the corner entanglement entropy.

Figure 2

The phase diagram of the model with the singlet phase at small $\alpha$ and the ordered XY and Ising phases at large $\alpha$. The dashed line represents the system with full Heisenberg symmetry. The critical point $B$ has $O\left(3\right)$ universality class and separates the $O\left(2\right)$ universality class along $AB$ from the Ising universality class along $BC$.

Figure 3

Critical exponents $\gamma$ and $\nu$ are shown for different values of the anisotropy parameter $\lambda$. Our results are consistent with constant values for the exponents in the XY and Ising regimes, with a larger exponent at the special Heisenberg symmetry case. The results from $\epsilon$ expansions [37] are shown as solid lines for $\gamma$ and dashed lines for $2\nu$ (on the right for Ising, on the left for XY, and in the middle for the classical Heisenberg universality class).

Figure 4

Plots of the area law or entanglement entropy per unit boundary length for the second Renyi entropy $S_2$. Quantum Monte Carlo data from Helmes et al. are shown by solid circles. For $\lambda =1$ two approximants are shown for the $s_2/{\lambda }^2$ series: a [4, 4] Padé approximant and a [$T=0$, $M=3$, $L=5$] biased first-order integrated differential approximant [35].

Figure 5

Plots of the area law or entanglement entropy per unit boundary length for the third Renyi entropy $S_3$.

Figure 6

Coefficient $-a_n$ of the logarithmic singularity associated with a corner in the boundary. These results show that the Ising and XY universality classes have constant coefficients. At Heisenberg symmetry the coefficient is largest. Our results are consistent with a rough proportionality for the log coefficients with the number of gapless modes in the system.