Thermodynamic singularities in the entanglement entropy at a two-dimensional quantum critical point

We study the bipartite entanglement entropy of the 2D transverse-field Ising model in the thermodynamic limit. Series expansions are developed for the Renyi entropy around both the small-field and large-field limits, allowing the separate calculation of the entanglement associated with lines and corners at the boundary between subsystems. Series extrapolations are used to extract power laws and logarithmic singularities as the quantum critical point is approached, giving access to new universal quantities. In 1D, we find excellent agreement with exact results as well as quantum Monte Carlo simulations. In 2D, we find compelling evidence that the entanglement at a corner is significantly different from a free boson field theory. These results demonstrate the power of the series expansion method for calculating entanglement entropy in interacting systems, a fact that will be particularly useful in searches for exotic quantum criticality in models with and without the sign problem.

Figure 1

The infinite square plane is partitioned into four quadrants $a$, $b$, $c$, and $d$. The region $A$ could be a half-plane such as $a\cup b$ or a quadrant such as $b$ or $c$, while the rest of the square lattice forms the region $B$. For the former partition, several low-order clusters that cross the boundary between $A$ and $B$ are also shown.

Figure 2

Entanglement entropy of the transverse-field Ising chain, for two edges, obtained by series expansions. For comparison QMC data on finite systems are also shown. In the ordered phase $\log 2$ has been subtracted from the QMC data to correspond to the fact that series expansions are done around a single ordered ground state.

Figure 3

“Area law” term $s_2$ and corner term $c_2$ of the entanglement entropy of the 2D transverse-field Ising model obtained by series expansions in the variables $h/J$ and $J/h$. In each case several approximants with critical point biased at $h/J=3.044$ are shown.