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The Journal of Chemical Physics

A path integral ground state replica trick approach for the computation of entanglement entropy of dipolar linear rotors

Tapas Kumar Sahoo, Dmitri Iouchtchenko, C. M. Herdman, Pierre–Nicholas Roy

Published May 14, 2020• 15 citations
Publisher

Abstract

We calculate the second Rényi entanglement entropy for systems of interacting linear rotors in their ground state as a measure of entanglement for continuous rotational degrees of freedom. The entropy is defined in relation to the purity of a subsystem in a bipartite quantum system, and to compute it, we compare two sampling ensembles based on the path integral ground state (PIGS) formalism. This scheme centers on the replica trick and is aided by the ratio trick, both developed in this context by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)]. We study a system composed of linear quantum rotors on a lattice in one dimension, interacting via an anisotropic dipole-dipole potential. The ground state second Rényi entropies estimated by PIGS are benchmarked against those from the density matrix renormalization group for various interaction strengths and system sizes. We find that the entropy grows with an increase in interaction strength, and for large enough systems, it appears to plateau near log(2). We posit that the limiting case of many strongly interacting rotors behaves akin to a lattice of two-level particles in a cat state, in which one naturally finds an entanglement entropy of log(2).

DOI
10.1063/5.0004602

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We calculate the second Rényi entanglement entropy for systems of interacting linear rotors in their ground state as a measure of entanglement for continuous rotational degrees of freedom. The entropy is defined in relation to the purity of a subsystem in a bipartite quantum system, and to compute it, we compare two sampling ensembles based on the path integral ground state (PIGS) formalism. This scheme centers on the replica trick and is aided by the ratio trick, both developed in this context by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)]. We study a system composed of linear quantum rotors on a lattice in one dimension, interacting via an anisotropic dipole-dipole potential. The ground state second Rényi entropies estimated by PIGS are benchmarked against those from the density matrix renormalization group for various interaction strengths and system sizes. We find that the entropy grows with an increase in interaction strength, and for large enough systems, it appears to plateau near log(2). We posit that the limiting case of many strongly interacting rotors behaves akin to a lattice of two-level particles in a cat state, in which one naturally finds an entanglement entropy of log(2).
We consider the entanglement between two spatial subregions in the Lieb-Liniger model of bosons in one spatial dimension interacting via a contact interaction. Using ground-state path integral quantum Monte Carlo we numerically compute the R\'enyi entropy of the reduced density matrix of the subsystem as a measure of entanglement. Our numerical algorithm is based on a replica method previously introduced by the authors, which we extend to efficiently study the entanglement of spatial subsystems of itinerant bosons. We confirm a logarithmic scaling of the R\'enyi entropy with subsystem size that is expected from conformal field theory, and compute the nonuniversal subleading constant for interaction strengths ranging over two orders of magnitude. In the strongly interacting limit, we find agreement with the known free fermion result.
In recent years, experimental techniques have enabled the creation of ultracold optical lattices of molecules and endofullerene peapod nanomolecular assemblies. It was previously suggested that the rotor model resulting from the placement of dipolar linear rotors in one-dimensional lattices at low temperature has a transition between ordered and disordered phases. We use the density matrix renormalization group (DMRG) to compute ground states of chains of up to 100 rotors and provide further evidence of the phase transition in the form of a diverging entanglement entropy. We also propose two methods and present some first steps toward rotational spectra of such molecular assemblies using DMRG. The present work showcases the power of DMRG in this new context of interacting molecular rotors and opens the door to the study of fundamental questions regarding criticality in systems with continuous degrees of freedom.
We provide a perspective on the calculation of ground state entanglement properties of many-body bosonic systems with continuous degrees of freedom using path integral molecular dynamics sampling. To date, such ground state calculations have been performed using quantum Monte Carlo sampling and we show here that the Langevin Equation path integral ground state (LePIGS) dynamics sampling approach can be practical. A key challenge is the computation of reduced density matrices required for the estimation of the so-called Rényi entanglement entropy. We provide proof of concept results using a novel path integral molecular dynamics implementation of the permutation operator. This implementation allows us to efficiently estimate particle entanglement entropy in arbitrary continuum systems, among other quantities.
The second Rényi entropy has been used as a measure of entanglement in various model systems, including those on a lattice and in the continuum. The present work focuses on extending the existing ideas to measurement of entanglement in physically relevant systems, such as molecular clusters. We show that using the simple estimator with the regular Path Integral Ground State (PIGS) distribution is not effective, but a superior estimator exists so long as one has access to other configuration sectors. To this end, we implement the ability to explore different sectors in the Molecular Modelling Toolkit (MMTK) and use it to obtain the entanglement entropy for a test system of coupled harmonic oscillators. \n \nThe Semiclassical Initial Value Representation (SC-IVR) method for real-time dynamics using the Herman-Kluk propagator is known to be an effective semiclassical method. In the present work, we combine this approximate real-time propagator with exact and approximate ground state wavefunctions in order to find ground state survival amplitudes. The necessary integrals are first performed numerically on a grid (which is feasible for only low-dimensional systems) and then stochastically using MMTK (which has applicability to high-dimensional systems). The stochastic approach is used to compare two estimators, and it is again demonstrated that better results are obtained in a specialized configuration sector.
We introduce a quantum Monte Carlo algorithm to measure the Rényi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path-integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability, and interactions. We present proof-of-principle calculations and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large-scale many-body systems of interacting bosons.
A ground state path integral quantum Monte Carlo algorithm is introduced that allows for the study of entanglement in lattice bosons at zero temperature. The Rényi entanglement entropy between spatial subregions is explored across the phase diagram of the one dimensional Bose-Hubbard model for systems consisting of up to L=256 L = 256 sites at unit-filling without any restrictions on site occupancy, far beyond the reach of exact diagonalization. The favorable scaling of the algorithm is demonstrated through a further measurement of the Rényi entanglement entropy at the two dimensional superfluid-insulator critical point for large system sizes, confirming the existence of the expected entanglement boundary law in the ground state. The Rényi estimator is extended to measure the symmetry resolved entanglement that is operationally accessible as a resource for experimentally relevant lattice gases with fixed total particle number.