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arXiv:1310.8332 · Physical Review B

Particle entanglement in continuum many-body systems via quantum Monte Carlo

C. M. Herdman, Pierre–Nicholas Roy, Roger G. Melko, Adrian Del Maestro

Published Apr 4, 2014• 34 citations
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Abstract

Entanglement of spatial bipartitions, used to explore lattice models in condensed matter physics, may be insufficient to fully describe itinerant quantum many-body systems in the continuum. We introduce a procedure to measure the R\'enyi entanglement entropies on a particle bipartition, with general applicability to continuum Hamiltonians via path integral Monte Carlo methods. Via direct simulations of interacting bosons in one spatial dimension, we confirm a logarithmic scaling of the single-particle entanglement entropy with the number of particles in the system. The coefficient of this logarithmic scaling increases with interaction strength, saturating to unity in the strongly interacting limit. Additionally, we show that the single-particle entanglement entropy is bounded by the condensate fraction, suggesting a practical route towards its measurement in future experiments.

DOI
10.1103/physrevb.89.140501

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Entanglement of spatial bipartitions, used to explore lattice models in condensed matter physics, may be insufficient to fully describe itinerant quantum many-body systems in the continuum. We introduce a procedure to measure the R'enyi entanglement entropies on a particle bipartition, with general applicability to continuum Hamiltonians via path integral Monte Carlo methods. Via direct simulations of interacting bosons in one spatial dimension, we confirm a logarithmic scaling of the single-particle entanglement entropy with the number of particles in the system. The coefficient of this logarithmic scaling increases with interaction strength, saturating to unity in the strongly interacting limit. Additionally, we show that the single-particle entanglement entropy is bounded by the condensate fraction, suggesting a practical route towards its measurement in future experiments.
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We consider the R\'enyi entanglement entropy of bosonic Tomonaga-Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher-order corrections involve a microscopic length scale, the leading-order scaling depends only on this sole dimensionless parameter which characterizes the low-energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Tomonaga-Luttinger liquid theory for the Lieb-Liniger model of $\ensuremath{\delta}$-function interacting bosons in the one-dimensional spatial continuum.
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.
We present a general scheme for the calculation of the R\'enyi entropy of a subsystem in quantum many-body models that can be efficiently simulated via quantum Monte Carlo. When the simulation is performed at very low temperature, the above approach delivers the entanglement R\'enyi entropy of the subsystem, and it allows us to explore the crossover to the thermal R\'enyi entropy as the temperature is increased. We implement this scheme explicitly within the stochastic series expansion as well as within path-integral Monte Carlo, and apply it to quantum spin and quantum rotor models. In the case of quantum spins, we show that relevant models in two dimensions with reduced symmetry (XXXXXX model or hard-core bosons, transverse-field Ising model at the quantum critical point) exhibit an area law for the scaling of the entanglement entropy.
Abstract We develop a nonequilibrium increment method in quantum Monte Carlo simulations to obtain the Rényi entanglement entropy of various quantum many-body systems with high efficiency and precision. To demonstrate its power, we show the results on a few important yet difficult (2 + 1)dquantum lattice models, ranging from the Heisenberg quantum antiferromagnet with spontaneous symmetry breaking, the quantum critical point with O(3) conformal field theory (CFT) to the toric code Z2{{\mathbb{Z}}}_{2}Z2​ Z2 topological ordered state and the Kagome Z2{{\mathbb{Z}}}_{2}Z2​ Z2 quantum spin liquid model with frustration and multi-spin interactions. In all these cases, our method either reveals the precise CFT data from the logarithmic correction or extracts the quantum dimension in topological order, from the dominant area law in finite-size scaling, with very large system sizes, controlled errorbars, and minimal computational costs. Our method, therefore, establishes a controlled and practical computation paradigm to obtain the difficult yet important universal properties in highly entangled quantum matter.