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arXiv:1807.08123 · Physical Review A

Diffusion Monte Carlo methods for spin-orbit-coupled ultracold Bose gases

Juan Sánchez-Baena, Jordi Boronat, Ferran Mazzanti

Published Nov 27, 2018• 14 citations
arXivHTMLPDF

Abstract

We present two diffusion Monte Carlo (DMC) algorithms for systems of ultracold quantum gases featuring synthetic spin-orbit interactions. The first one is a spin-integrated DMC method which provides fixed-phase energy estimates. The second one is a discrete spin generalisation of the T-moves spin-orbit DMC [Melton et al., J. Chem. Phys. 144, 244113 (2016)], which provides an upper bound to the fixed-phase energy. The former is a more accurate method but it is restricted to spin-independent two-body interactions. We report a comparison between both algorithms for different systems. As a check of the efficiency of both methods, we compare the DMC energies with results obtained with other numerical methods, finding agreement between both estimations.

DOI
10.1103/physreva.98.053632

Related papers

Correlations in spin-orbit coupled ultracold quantum gases

Juan Sánchez-Baena

The main goal of this thesis is the computation of ground state properties of quantum many-body systems under Spin Orbit coupling (SOC) interactions out of the ultradilute regime. We present two approaches to fulfill this goal: the development and application of quantum Monte Carlo methods and the realization of beyond mean field calculations following the Bogoliubov-de Gennes formalism. Regarding the first approach, we show how to modify the standard Diffusion Monte Carlo (DMC) algorithm such that it can sample SOC interactions. We develop the formalism of two DMC methods suited to the simulation of SOC systems: the Discrete Spin T-moves DMC (DTDMC) and the Spin-Integrated DMC (SIDMC). Since the ground state wave function of a SOC system is complex, these methods work under the fixed phase approximation, meaning that they can not provide exact ground state estimates. DTDMC corresponds to an adaptation to discrete spin variables of a preexisting SOC DMC algorithm. This method requires the definition of an effective Hamiltonian to cure a sign problem in the propagator, which worsens the quality of its estimations. On the other hand, SIDMC, a completely original method developed in this thesis, is able to bypass the definition of the effective Hamiltonian by propagating the spin-integrated wave function of the system in imaginary time. As a consequence, SIDMC yields physical estimates closer to their ground state values. However, the SIDMC method is unable to sample spin-dependent two-body interactions. Through the use of the DTDMC algorithm, we elaborate the extension to the correlated regime of the phase diagram of a many-body Raman SOC system with spin-dependent two-body interactions. The results show that correlations favor the exotic stripe phase, which features density modulations, meaning that the stripe region of the phase diagram is enlarged as the level of correlations is increased. We also report results for the pair-distribution function, the static structure factor, the one-body density matrix and the superfluidity of the system. We show that the superfluid fraction across the stripes is non-zero, meaning that the stripes are superfluid, a consistent result with previous mean field calculations. Regarding the second prescription to account for correlations in a SOC system, we follow the Bogoliubov-de Gennes formalism in order to compute the first order correction by correlations to the mean field result. We compute the Lee-Huang-Yang (LHY) energy correction for a Raman SOC system in the stripe phase for the first time. As an application, we are able to determine the role played by quantum fluctuations in a Raman SOC system in the stripe phase that is unstable at the mean field level due to attractive interactions. Such system is currently under experimental development. The results show that quantum fluctuations stabilize the collapse predicted by mean field theory, giving rise to a stabilized gas, or a stabilized liquid, depending on the values of the parameters of the Hamiltonian. Moreover, the finite system supports the existence of self-bound droplets as its ground state. These droplets show density modulations, induced by SOC. Therefore, they represent a novel state of matter in the field of ultracold atoms that combines the self-bound character of liquids, density modulations reminiscent of solids, and superfluidity. We also provide a phenomenological analytical functional of the density in order to ease the evaluation of the LHY energy. Finally, we also report a brief analysis of a quantum many-body system that features Spin Orbital Angular Coupling (SOAC). We perform DTDMC calculations in order to evaluate the impact of correlations in the system. We show that the mean field and DTDMC predictions for the energy and spin polarization agree in the range of parameters considered, meaning that correlations are not very relevant. L'objectiu principal d'aquesta Tesi és el càlcul de propietats de l'estat fonamental de sistemes quàntics de molts cossos amb interaccions d'espí-òrbita fora del règim ultra-diluït. Es presenten dues prescripcions per dur a terme aquest objectiu: l’elaboració i utilització de mètodes de Monte Carlo i la realització de càlculs més enllà de l'aproximació de camp mig amb la tècnica de Bogoliubov-de Gennes. Pel que fa a la primera opció, es mostra com adaptar el mètode de Monte Carlo de Difusió (DMC) estàndard per tal de samplejar adequadament interaccions espí-òrbita. Es desenvolupa el formalisme de propagadors per dos mètodes diferents capaços d'assolir aquest objectiu. Aquest dos mètodes són el Discrete Spin T-moves DMC (DTDMC) i el Spin-Integrated DMC (SIDMC). El primer, l'algorisme DTDMC, es correspon a una adaptació d'un mètode prèviament existent a variables d'espí discretes. Aquest mètode requereix de la definició d'un Hamiltonià efectiu per curar un problema de signe, la qual cosa redueix la qualitat de les seves estimacions. D'altra banda, el segon mètode (SIDMC) és un algorisme completament original desenvolupat en aquest treball. Aquest mètode és capaç d'ignorar la definició d'aquest Hamiltonià efectiu per mitjà de la propagació en temps imaginari de la funció d'ona integrada a l'espai d'espins. Com a conseqüència, la qualitat de les estimacions obtingudes amb el SIDMC és superior a aquelles obtingudes amb el DTDMC. No obstant, el SIDMC és incapaç de samplejar interaccions a dos cossos que depenguin de l'espí. Fent servir el mètode DTDMC, s’elabora el l'extensió del diagrama de fases d'un sistema amb espí-òrbita Raman al règim correlacionat. Els resultats mostren que les correlacions afavoreixen l'exòtica fase amb modulacions de densitat, la fase stripe, que abasta una extensió del diagrama més gran quant major són les correlacions. A més, es reporten resultats per la distribució de parelles, el factor d'estructura estàtic, la matriu densitat a un cos, i es caracteritza quantitativament la superfluïdesa d'aquesta fase, mostrant una fracció superfluida no nul·la a la direcció transversal respecte a les modulacions de densitat. Pel que fa a la segona prescripció esmentada prèviament, s'ha seguit el formalisme de Bogoliubov-de Gennes per realitzar càlculs analítico-numèrics que tinguin en compte les correlacions a primer ordre respecte a un càlcul de camp mig per un sistema amb espí-òrbita Raman. Això ha permès obtenir la correcció de Lee-Huang-Yang (LHY) a l'energia de camp mig per a la fase stripe d'aquest sistema per primer cop. Fent servir aquest resultat, s'ha pogut determinar el paper que juguen les fluctuacions quàntiques a un sistema a la fase stripe que és inestable a nivell de camp mig a causa de la presencia d'interaccions entre components d'espí suficientment atractives. Actualment, aquests sistemes es troben en desenvolupament experimental. Els resultats mostren que les fluctuacions quàntiques estabilitzen el sistema, i dónen lloc a una fase líquida o gas en funció del valor dels paràmetres de l'Hamiltonià. A més, el sistema finit suporta l'existència de gotes líquides auto-lligades com a estat fonamental. Aquestes gotes mostren modulacions de densitat induïdes per la interacció espí-òrbita. Per tant, representen un estat de la matèria novedós al camp dels gasos quàntics ultra-freds, ja que presenten una combinació de propietats de líquid, de sòlid i superfluïdesa. A fi de facilitar els càlculs de la correcció de LHY, es reporta un funcional fenomenològic que reprodueix amb precisió els resultats provinents de càlculs complerts. Finalment, també es reporta una breu anàlisi d'un sistema quàntic de molts cossos amb acoblament espí-òrbita de tipus angular. Es reporta un estudi amb DTDMC de l'impacte de les correlacions a un sistema d'aquest tipus.

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Ultracold atoms at unitarity within quantum Monte Carlo methods

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Variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the properties of the zero-temperature fermionic gas at unitarity are reported. Our study differs from earlier ones mainly in that we have constructed more accurate trial wave functions and used a larger system size, we have studied the dependence of the energy on the particle density and well width, and we have achieved much smaller statistical error bars. The correct value of the universal ratio of the energy of the interacting to that of the noninteracting gas, $\ensuremath{\xi}$, is still a matter of debate. We find DMC values of $\ensuremath{\xi}$ of $0.4244(1)$ with 66 particles and $0.4339(1)$ with 128 particles. The spherically averaged pair-correlation functions, momentum densities, and one-body density matrices are very similar in VMC and DMC, which suggests that our results for these quantities are very accurate. We find, however, some differences between the VMC and DMC results for the two-body density matrices and condensate fractions, which indicates that these quantities are more sensitive to the quality of the trial wave function. Our best estimate of the condensate fraction of 0.51 is smaller than the values from earlier quantum Monte Carlo calculations.

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Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.

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The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.

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Population size bias in diffusion Monte Carlo

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The size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.

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Balázs Hetényi, Eran Rabani, B. J. Berne

We propose a new method to calculate ground state position dependent observables in quantum many-body systems. The method, which we call the path-integral diffusion Monte Carlo (PI-DMC) method, is essentially a combination of path-integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. The distribution resulting from a DMC simulation is further propagated in imaginary time by PIMC sampling. Tests of the new method for simple cases such as the harmonic oscillator, a double well, and a set of ten coupled harmonic oscillators show that the results for several observables converge rapidly to the exact result.

The Journal of Chemical Physics•Apr 1, 1999•21 citations
We present two diffusion Monte Carlo (DMC) algorithms for systems of ultracold quantum gases featuring synthetic spin-orbit interactions. The first one is a spin-integrated DMC method which provides fixed-phase energy estimates. The second one is a discrete spin generalisation of the T-moves spin-orbit DMC [Melton et al., J. Chem. Phys. 144, 244113 (2016)], which provides an upper bound to the fixed-phase energy. The former is a more accurate method but it is restricted to spin-independent two-body interactions. We report a comparison between both algorithms for different systems. As a check of the efficiency of both methods, we compare the DMC energies with results obtained with other numerical methods, finding agreement between both estimations.
The main goal of this thesis is the computation of ground state properties of quantum many-body systems under Spin Orbit coupling (SOC) interactions out of the ultradilute regime. We present two approaches to fulfill this goal: the development and application of quantum Monte Carlo methods and the realization of beyond mean field calculations following the Bogoliubov-de Gennes formalism. Regarding the first approach, we show how to modify the standard Diffusion Monte Carlo (DMC) algorithm such that it can sample SOC interactions. We develop the formalism of two DMC methods suited to the simulation of SOC systems: the Discrete Spin T-moves DMC (DTDMC) and the Spin-Integrated DMC (SIDMC). Since the ground state wave function of a SOC system is complex, these methods work under the fixed phase approximation, meaning that they can not provide exact ground state estimates. DTDMC corresponds to an adaptation to discrete spin variables of a preexisting SOC DMC algorithm. This method requires the definition of an effective Hamiltonian to cure a sign problem in the propagator, which worsens the quality of its estimations. On the other hand, SIDMC, a completely original method developed in this thesis, is able to bypass the definition of the effective Hamiltonian by propagating the spin-integrated wave function of the system in imaginary time. As a consequence, SIDMC yields physical estimates closer to their ground state values. However, the SIDMC method is unable to sample spin-dependent two-body interactions. Through the use of the DTDMC algorithm, we elaborate the extension to the correlated regime of the phase diagram of a many-body Raman SOC system with spin-dependent two-body interactions. The results show that correlations favor the exotic stripe phase, which features density modulations, meaning that the stripe region of the phase diagram is enlarged as the level of correlations is increased. We also report results for the pair-distribution function, the static structure factor, the one-body density matrix and the superfluidity of the system. We show that the superfluid fraction across the stripes is non-zero, meaning that the stripes are superfluid, a consistent result with previous mean field calculations. Regarding the second prescription to account for correlations in a SOC system, we follow the Bogoliubov-de Gennes formalism in order to compute the first order correction by correlations to the mean field result. We compute the Lee-Huang-Yang (LHY) energy correction for a Raman SOC system in the stripe phase for the first time. As an application, we are able to determine the role played by quantum fluctuations in a Raman SOC system in the stripe phase that is unstable at the mean field level due to attractive interactions. Such system is currently under experimental development. The results show that quantum fluctuations stabilize the collapse predicted by mean field theory, giving rise to a stabilized gas, or a stabilized liquid, depending on the values of the parameters of the Hamiltonian. Moreover, the finite system supports the existence of self-bound droplets as its ground state. These droplets show density modulations, induced by SOC. Therefore, they represent a novel state of matter in the field of ultracold atoms that combines the self-bound character of liquids, density modulations reminiscent of solids, and superfluidity. We also provide a phenomenological analytical functional of the density in order to ease the evaluation of the LHY energy. Finally, we also report a brief analysis of a quantum many-body system that features Spin Orbital Angular Coupling (SOAC). We perform DTDMC calculations in order to evaluate the impact of correlations in the system. We show that the mean field and DTDMC predictions for the energy and spin polarization agree in the range of parameters considered, meaning that correlations are not very relevant. L'objectiu principal d'aquesta Tesi és el càlcul de propietats de l'estat fonamental de sistemes quàntics de molts cossos amb interaccions d'espí-òrbita fora del règim ultra-diluït. Es presenten dues prescripcions per dur a terme aquest objectiu: l’elaboració i utilització de mètodes de Monte Carlo i la realització de càlculs més enllà de l'aproximació de camp mig amb la tècnica de Bogoliubov-de Gennes. Pel que fa a la primera opció, es mostra com adaptar el mètode de Monte Carlo de Difusió (DMC) estàndard per tal de samplejar adequadament interaccions espí-òrbita. Es desenvolupa el formalisme de propagadors per dos mètodes diferents capaços d'assolir aquest objectiu. Aquest dos mètodes són el Discrete Spin T-moves DMC (DTDMC) i el Spin-Integrated DMC (SIDMC). El primer, l'algorisme DTDMC, es correspon a una adaptació d'un mètode prèviament existent a variables d'espí discretes. Aquest mètode requereix de la definició d'un Hamiltonià efectiu per curar un problema de signe, la qual cosa redueix la qualitat de les seves estimacions. D'altra banda, el segon mètode (SIDMC) és un algorisme completament original desenvolupat en aquest treball. Aquest mètode és capaç d'ignorar la definició d'aquest Hamiltonià efectiu per mitjà de la propagació en temps imaginari de la funció d'ona integrada a l'espai d'espins. Com a conseqüència, la qualitat de les estimacions obtingudes amb el SIDMC és superior a aquelles obtingudes amb el DTDMC. No obstant, el SIDMC és incapaç de samplejar interaccions a dos cossos que depenguin de l'espí. Fent servir el mètode DTDMC, s’elabora el l'extensió del diagrama de fases d'un sistema amb espí-òrbita Raman al règim correlacionat. Els resultats mostren que les correlacions afavoreixen l'exòtica fase amb modulacions de densitat, la fase stripe, que abasta una extensió del diagrama més gran quant major són les correlacions. A més, es reporten resultats per la distribució de parelles, el factor d'estructura estàtic, la matriu densitat a un cos, i es caracteritza quantitativament la superfluïdesa d'aquesta fase, mostrant una fracció superfluida no nul·la a la direcció transversal respecte a les modulacions de densitat. Pel que fa a la segona prescripció esmentada prèviament, s'ha seguit el formalisme de Bogoliubov-de Gennes per realitzar càlculs analítico-numèrics que tinguin en compte les correlacions a primer ordre respecte a un càlcul de camp mig per un sistema amb espí-òrbita Raman. Això ha permès obtenir la correcció de Lee-Huang-Yang (LHY) a l'energia de camp mig per a la fase stripe d'aquest sistema per primer cop. Fent servir aquest resultat, s'ha pogut determinar el paper que juguen les fluctuacions quàntiques a un sistema a la fase stripe que és inestable a nivell de camp mig a causa de la presencia d'interaccions entre components d'espí suficientment atractives. Actualment, aquests sistemes es troben en desenvolupament experimental. Els resultats mostren que les fluctuacions quàntiques estabilitzen el sistema, i dónen lloc a una fase líquida o gas en funció del valor dels paràmetres de l'Hamiltonià. A més, el sistema finit suporta l'existència de gotes líquides auto-lligades com a estat fonamental. Aquestes gotes mostren modulacions de densitat induïdes per la interacció espí-òrbita. Per tant, representen un estat de la matèria novedós al camp dels gasos quàntics ultra-freds, ja que presenten una combinació de propietats de líquid, de sòlid i superfluïdesa. A fi de facilitar els càlculs de la correcció de LHY, es reporta un funcional fenomenològic que reprodueix amb precisió els resultats provinents de càlculs complerts. Finalment, també es reporta una breu anàlisi d'un sistema quàntic de molts cossos amb acoblament espí-òrbita de tipus angular. Es reporta un estudi amb DTDMC de l'impacte de les correlacions a un sistema d'aquest tipus.
Variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the properties of the zero-temperature fermionic gas at unitarity are reported. Our study differs from earlier ones mainly in that we have constructed more accurate trial wave functions and used a larger system size, we have studied the dependence of the energy on the particle density and well width, and we have achieved much smaller statistical error bars. The correct value of the universal ratio of the energy of the interacting to that of the noninteracting gas, $\ensuremath{\xi}$, is still a matter of debate. We find DMC values of $\ensuremath{\xi}$ of 0.4244(1)0.4244(1)0.4244(1) with 66 particles and 0.4339(1)0.4339(1)0.4339(1) with 128 particles. The spherically averaged pair-correlation functions, momentum densities, and one-body density matrices are very similar in VMC and DMC, which suggests that our results for these quantities are very accurate. We find, however, some differences between the VMC and DMC results for the two-body density matrices and condensate fractions, which indicates that these quantities are more sensitive to the quality of the trial wave function. Our best estimate of the condensate fraction of 0.51 is smaller than the values from earlier quantum Monte Carlo calculations.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.
The size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.
We propose a new method to calculate ground state position dependent observables in quantum many-body systems. The method, which we call the path-integral diffusion Monte Carlo (PI-DMC) method, is essentially a combination of path-integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. The distribution resulting from a DMC simulation is further propagated in imaginary time by PIMC sampling. Tests of the new method for simple cases such as the harmonic oscillator, a double well, and a set of ten coupled harmonic oscillators show that the results for several observables converge rapidly to the exact result.