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Numerical evaluation of Rényi entropy with path integrals

Miha Srdinšek

Published Sep 26, 2023• 0 citations
Publisher

Abstract

Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.

DOI
10.70675/701196eezd03cz4671z8d45z6492afe498d7

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Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.
Despite being a well-established operational approach to quantify entanglement, Rényi entropy calculations have been plagued by their computational complexity.We introduce here a theoretical framework based on an optimal thermodynamic integration scheme, where the Rényi entropy can be efficiently evaluated using regularizing paths.This approach avoids slowly convergent fluctuating contributions and leads to low-variance estimates.In this way, large system sizes and high levels of entanglement in model or first-principles Hamiltonians are within our reach.We demonstrate this approach in the one-dimensional quantum Ising model and perform an evaluation of entanglement entropy in the formic acid dimer, by discovering that its two shared protons are entangled even above room temperature.
The interplay of quantum and classical fluctuations in the vicinity of a quantum critical point (QCP) gives rise to various regimes or phases with distinct quantum character. In this work, we show that the R\'enyi entropy is a precious tool to characterize the phase diagram of critical systems not only around the QCP but also away from it, thanks to its capability to detect the emergence of local moments at finite temperature. For an efficient evaluation of the R\'enyi entropy, we introduce an algorithm based on a path-integral Langevin dynamics combined with a previously proposed thermodynamic integration method built on regularized paths. We apply this framework to study the critical behavior of a linear chain of anharmonic oscillators, a particular realization of the ${\ensuremath{\phi}}^{4}$ model. We fully resolved its phase diagram, as a function of both temperature and interaction strength. At finite temperature, we find a sequence of three regimes---para, disordered, and antiferro---met as the interaction is increased. The R\'enyi entropy divergence coincides with the crossover between the para and disordered regimes, which shows no temperature dependence. The occurrence of the antiferro regime, on the other hand, is temperature dependent. The two crossover lines merge in proximity of the QCP, at zero temperature, where the R\'enyi entropy is sharply peaked. Via its subsystem-size scaling, we confirm that the transition belongs to the two-dimensional Ising universality class. This phenomenology is expected to happen in all ${\ensuremath{\phi}}^{4}$-like systems, as well as in the elusive water-ice transition across phases VII, VIII, and X.
Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. \nThrough measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. \nWe describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. \nThis description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. \n \nUsing the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. \nThe scaling near the critical point also shows signatures identifying the universality class of the model. \nImproved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. \nWe explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. \nThis same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. \nAll of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.
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.
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.
Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose an alternative and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one- and two-dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. We analyze the computational cost and provide analytical and numerical evidences for it to be polynomial in system size. This work also suggests a unifying framework for calculating various types of entropy quantities including entanglement R\'enyi entropy and entanglement R\'enyi negativity.