arXiv:2603.22338 · Physical review. B./Physical review. B
Path Integral Monte Carlo on a Sphere
Riccardo Fantoni
cond-mat.quant-gascond-mat.stat-mechcond-mat.str-elphysics.comp-phquant-phSubmitted Mar 21, 2026• Announced Mar 24, 2026• 0 citations• Revised Aug 21, 2026 (v2)
We solve numerically exactly a simple toy model to quantum general relativity or more properly to path integral on a curved space. We consider the thermal equilibrium of a quantum many body problem on the sphere, the surface of constant positive curvature. We use path integral Monte Carlo to measure the kinetic energy, the internal energy and the static structure of a bosons, fermions and anyons fluid at low temperatures on the sphere. For bosons we also measure the superfluid fraction and compare its behavior at the critical temperature with the universal jump predicted by Nelson and Kosterlitz in flat space in the thermodynamic limit at the superfluid phase transition. For fermions and anyons it is necessary to use the restricted path integral recipe in order to overcome the sign problem. Even if this recipe is exact for the non interacting fluid it reduces to just an approximation for an interacting system. And we make the example of the electron gas at low temperature. Snapshots of the many body path configuration during the evolution of the computer experiment show that the ``speed'' of the single particle path near the poles slows down as a consequence of the ``hairy ball theorem'' of Poincaré. The influence of curvature on the thermodynamic and structural properties of the many body fluid is also studied.
We solve numerically exactly a simple toy model to quantum general relativity or more properly to path integral on a curved space. We consider the thermal equilibrium of a quantum many body problem on the sphere, the surface of constant positive curvature. We use path integral Monte Carlo to measure the kinetic energy, the internal energy and the static structure of a bosons, fermions and anyons fluid at low temperatures on the sphere. For bosons we also measure the superfluid fraction and compare its behavior at the critical temperature with the universal jump predicted by Nelson and Kosterlitz in flat space in the thermodynamic limit at the superfluid phase transition. For fermions and anyons it is necessary to use the restricted path integral recipe in order to overcome the sign problem. Even if this recipe is exact for the non interacting fluid it reduces to just an approximation for an interacting system. And we make the example of the electron gas at low temperature. Snapshots of the many body path configuration during the evolution of the computer experiment show that the speed'' of the single particle path near the poles slows down as a consequence of the hairy ball theorem'' of Poincaré. The influence of curvature on the thermodynamic and structural properties of the many body fluid is also studied.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
We study a bosonic gas of hard spheres by using the exact zero-temperature path-integral ground-state (PIGS) Monte Carlo method and the equations of superfluid hydrodynamics. The PIGS method is implemented to calculate for the bulk system the energy per particle and the condensate fraction through a large range of the gas parameter na3 (with n the number density and a the s-wave scattering length), going from the dilute gas into the solid phase. The Maxwell construction is then adopted to determine the freezing at $n{a}^{3}=0.264\ifmmode\pm\else\textpm\fi{}0.003$ and the melting at $n{a}^{3}=0.290\ifmmode\pm\else\textpm\fi{}0.003$. In the liquid phase, where the condensate fraction is finite, the equations of superfluid hydrodynamics, based on the PIGS equation of state, are used to find other relevant quantities as a function of the gas parameter: the chemical potential, the pressure, and the sound velocity. In addition, within Feynman's approximation, from the PIGS static structure factor we determine the full excitation spectrum, which displays a maxon-roton behavior when the gas parameter is close to the freezing value. Finally, the equations of superfluid hydrodynamics with the PIGS equation of state are solved for the bosonic system under axially symmetric harmonic confinement obtaining its collective breathing modes.
It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.
We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.
By using exact path-integral Monte Carlo methods we calculate the equation of state of an interacting Bose gas as a function of temperature both below and above the superfluid transition. The universal character of the equation of state for dilute systems and low temperatures is investigated by modeling the interatomic interactions using different repulsive potentials corresponding to the same s-wave scattering length. The results obtained for the energy and the pressure are compared to the virial expansion for temperatures larger than the critical temperature. At very low temperatures we find agreement with the ground-state energy calculated using the diffusion Monte Carlo method.
A novel path integral Monte Carlo (PIMC) approach for correlated many-particle systems with arbitrary pair interaction in continuous space at low temperatures is presented. It is based on a representation of the N -particle density operator in a basis of (anti-)symmetrized N -particle states (configurations of occupation numbers). The path integral is transformed into a sum over trajectories with the same topology and, finally, the limit of M →∞ , where M is the number of high-temperature factors, is analytically performed. This yields exact expressions for the thermodynamic quantities and allows to perform efficient simulations for fermions at low temperature and weak to moderate coupling. Our method is expected to be applicable to dense quantum plasmas in the regime of strong degeneracy where conventional PIMC fails due to the fermion sign problem. c