Calculating the entanglement spectrum in quantum Monte Carlo with application to ab initio Hamiltonians

Several algorithms have been proposed to calculate the spatial entanglement spectrum from high order Rényi entropies and maximum entropy techniques. We present an alternative approach for computing the entanglement spectrum with quantum Monte Carlo for both continuum and lattice Hamiltonians. This method provides direct access to the matrix elements of the spatially reduced density matrix and we determine an estimator that can be used in variational Monte Carlo as well as other Monte Carlo methods. The algorithm is based on using a generalization of the swap operator, which can be extended to calculate a general class of density matrices that can include combinations of spin, space, particle, and even momentum degrees of freedom. We demonstrate the method by applying it to the ${\mathrm{H}}_2$ and ${\mathrm{N}}_2$ molecules and describe how the spatial entanglement spectrum encodes a covalent bond that includes all the many body correlations.

Figure 1

The entanglement spectrum of the full configuration interaction ground state wave function of ${\text{H}}_2$ (note the inverted vertical scale). The red dashed line located at 0.25 corresponds to the value of the Hartree-Fock entanglement eigenvalues. The four sectors (columns) represent the number of particles and spins in region A. From left to right the sectors are zero, spin up, spin down, and both electrons. The statistical uncertainty in the smallest eigenvalues is $3\times {10}^{-5}$.

Figure 2

The five orbitals with the largest eigenvalues of the effective entanglement Hamiltonian for ${\text{H}}_2$. The orbitals, unlike normal single particle orbitals, exist only in the half space of region A. Orbitals 4 and 5 are related by a ${90}^{\circ }$ rotation along the bonding axis.

Figure 3

The 64 largest values of the entanglement spectrum of a multideterminant ground state wave function of ${\text{N}}_2$. The red dashed line marks the 64 largest Hartree-Fock entanglement eigenvalues (0.0156 25). The $x$ axis serves as an index for 16 different sectors which are distinguished by particle number and spin polarization in region A. Sectors 1–8 are symmetric with respect to sectors 9–16 within error bars.