Tuning the quantumness of simple Bose systems: A universal phase diagram Youssef Koraa,1 , Massimo Boninsegnia , Dam Thanh Sonb,1 , and Shiwei Zhangc,d aDepartment of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada; bKadanoff Center for Theoretical Physics, The University of Chicago, Chicago, IL 60637; cCenter for Computational Quantum Physics, Flatiron Institute, New York, NY 10010; and dDepartment of Physics, The College of William and Mary, Williamsburg, VA 23187 Contributed by Dam Thanh Son, September 17, 2020 (sent for review August 20, 2020; reviewed by Joseph Carlson and Boris Svistunov) We present a comprehensive theoretical study of the phase dia- for fruitful comparison with experiment. Of particular interest gram of a system of many Bose particles interacting with a is the understanding of how quantum-mechanical effects alter two-body central potential of the so-called Lennard-Jones form. the qualitative behavior of the system predicted classically. For First-principles path-integral computations are carried out, pro- systems obeying Fermi statistics, it is not yet possible to system- viding essentially exact numerical results on the thermodynamic atically reach the accuracy necessary for reliable predictions of properties. The theoretical model used here provides a realistic new reactions, new structures, or new phases of matter; indeed, and remarkably general framework for describing simple Bose this remains a grand challenge. However, if the constituent systems ranging from crystals to normal fluids to superfluids and particles obey Bose statistics, it is now possible in principle gases. The interplay between particle interactions on the one to obtain exact numerical estimates of thermodynamic aver- hand and quantum indistinguishability and delocalization on the ages of relevant physical observables, for many relevant physical other hand is characterized by a single quantumness parameter, systems. which can be tuned to engineer and explore different regimes. A broad class of condensed-matter systems is well character- Taking advantage of the rare combination of the versatility of ized by pairwise, central interactions among constituent particles the many-body Hamiltonian and the possibility for exact compu- (e.g., atoms), featuring 1) a strong repulsion at short interparti- tations, we systematically investigate the phases of the systems cle separations (from Pauli exclusion principle, acting to prevent as a function of pressure (P) and temperature (T), as well as the electrons from different atomic or molecular clouds from over- quantumness parameter. We show how the topology of the phase lapping spatially) and 2) a weak attractive tail at long distances, PHYSICS diagram evolves from the known case of 4He, as the system is arising from mutually induced electric dipole moments. A widely made more (and less) quantum, and compare our predictions with used approximate model to describe such an interaction is the available results from mean-field theory. Possible realization and Lennard-Jones (LJ) potential observation of the phases and physical regimes predicted here are discussed in various experimental systems, including hypothetical σ 12 σ 6 VLJ (r) = 4 − , [1] muonic matter. r r statistical physics j quantum many-body physics j quantum fluids and where is the depth of the attractive well, σ is the characteris- solids j Bose–Einstein condensation j superfluidity tic range of the interaction, and r is the separation between the two particles. Despite its simplicity, the LJ potential effectively ne of the major themes of modern physics is the prediction accounts for the physical behavior of a large number of simple Oof macroscopic properties and phases of thermodynamic liquids. assemblies of atoms and molecules directly from first princi- A chief example of this type of condensed-matter system is ples. A famous quote by Weisskopf from 1977 captures the helium. Helium is unique among all substances, in that it does aspiration and also underscores the challenge: “Assume that a not solidify at low temperature, under the pressure of its own group of intelligent theoretical physicists have lived in closed buildings from birth that they never had occasion to see nat- Significance ural structures:::What would they be able to predict from a fundamental knowledge of quantum mechanics? They would Predicting the properties of a quantum-mechanical system predict the existence of atoms, of molecules, of solid crys- of many interacting particles is a major goal of modern sci- tals, both metals and insulators, of gases, but most likely not ence and an outstanding challenge. We consider a compact the existence of liquids.” (ref. 1, p. 202). Although Weisskopf and versatile model which captures the essential features of focused on liquids, his remark highlighted the broader diffi- a broad class of systems made of particles obeying Bose– culty in treating interparticle interactions and emergent phe- Einstein statistics and which allows one to systematically dial nomena. Interestingly, factoring in the possibility of computer up the effect of quantum entanglement in the presence of simulations would almost certainly have changed this assess- particle interaction by tuning a single parameter. We are able ment. Simulations using simple models of atomic interactions to obtain exact numerical results for the phase diagrams of allow one to make predictions of equilibrium structure and ther- these systems. Possible directions for experimental realization modynamic properties of many simple systems, including those of the predictions are discussed. of liquids. The rapid increase of modern computing power and devel- Author contributions: Y.K., M.B., D.T.S., and S.Z. designed research, performed research, opment of computational algorithms have greatly expanded the contributed new reagents/analytic tools, analyzed data, and wrote the paper.y role of computer simulations and computation, now encompass- Reviewers: J.C., Los Alamos National Laboratory; and B.S., University of Massachusetts ing many subareas of physics, chemistry, materials science, etc. Amherst. y Despite early fears, expressed, e.g., by Dirac that “the exact The authors declare no competing interest.y application of these laws leads to equations much too com- Published under the PNAS license.y plicated to be soluble,” (ref. 2, p. 714), we are now in the 1 To whom correspondence may be addressed. Email: [email protected] or dtson@ position to apply the fundamental laws of quantum mechanics to uchicago.edu.y a large number of many-body systems, with precision sufficient www.pnas.org/cgi/doi/10.1073/pnas.2017646117 PNAS Latest Articles j 1 of 7 Downloaded by guest on September 28, 2021 vapor. Its most common isotope, 4He, undergoes a transition Z mµ me Meq = 1 + M , [3] to a superfluid phase at a temperature of 2.17 K. Both the fact A mN mµ that no crystallization occurs and the superfluid transition are 4 understood as consequences of Bose statistics (3, 4), which He where mµ and me are the masses of the muon and the electron, atoms (composite particles of zero total spin) obey. At higher respectively. The replacement of electrons by muons causes 1) temperature, 4He shows a behavior typical of other fluids; e.g., a shrinkage of the range (σ) of the interparticle potential by a it has a liquid–gas critical point at temperature about 5.19 K and factor of mµ=me (∼200) and 2) an increase in the well depth pressure 227 kPa. () by the same factor, resulting in a 200-fold increase of Λ— The question immediately arises of how general some of the sufficient to bring even systems made of heavier elements, e.g., properties of 4He are among Bose systems featuring the same Ne, whose condensed phase displays essentially classical physical kind of interaction, or how they might evolve with the mass of the behavior, into the highly quantum regime. particles and the interaction parameters, or whether new phases In this work, we perform a comprehensive study of the univer- might appear. sal phase diagram of LJ Bose systems. We use state-of-the-art A theoretical description of a system of interacting bosons quantum Monte Carlo (QMC) methods to compute numerically based on the LJ potential constitutes a simple but remarkably exact thermodynamic averages of relevant physical observables general framework in which such questions can be addressed. at finite temperatures. Given the presence of both strong inter- On taking (σ) as our unit of energy (length), the Hamiltonian actions and large quantum effects in these systems, systematically is fully parametrized by the dimensionless parameter∗ accurate many-body computations are crucial for reliable predic- tions. We map out the complete thermodynamic phase diagram 2 as a function of pressure and temperature, varying the parame- ~ ter Λ to explore a variety of physical regimes ranging from almost Λ = 2 , [2] mσ entirely classical to the ultraquantum. The remainder of this paper is organized as follows: in Theoret- whose magnitude expresses the relative importance of the kinetic ical Framework we describe the model of the system and briefly and potential energies. The larger the value of Λ is, the more summarize the methodology we utilized. In Results, we present significant the quantum effects in the dynamics of the particles and discuss our results in several subsections separated by the and the higher the temperature to which they can be expected different regimes of Λ, and we finally outline our conclusions in to persist. Conversely, in the Λ ! 0 limit the potential energy Discussion and Conclusions. dominates, and the behavior of the system is largely classical. To make this argument more quantitative, we note that for Theoretical Framework 4 He, ≡ He = 10:22 K and σ ≡ σHe = 2:556 A,˚ i.e., Λ = 0:18, Model. We consider an ensemble of N identical particles of mass which is the second highest value among naturally occurring sub- m obeying Bose statistics, enclosed in a cubic box of volume V , stances (the highest being 0.24 for the lighter helium isotope, with periodic boundary conditions in the three directions.