A Universal Phase Diagram

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Downloaded by guest on September 28, 2021 www.pnas.org/cgi/doi/10.1073/pnas.2017646117 a Kora Youssef A diagram systems: phase Bose universal simple of quantumness the Tuning solids physics statistical matter. hypothetical including muonic systems, are experimental here various predicted in regimes discussed physical and and phases realization the Possible of theory. observation mean-field with predictions from our results compare of available and quantum, case less) (and known more made the from phase evolves the of diagram topology the how show (P systems We parameter. pressure quantumness the of of function phases a of the compu- as investigate versatility exact systematically for the possibility we of the tations, regimes. and combination different Hamiltonian rare explore many-body the and the engineer of to parameter, advantage quantumness tuned Taking single be a can one by which the characterized the is on on hand delocalization interactions and other indistinguishability particle quantum and between and hand superfluids interplay Bose to The fluids simple normal gases. to describing realistic crystals for from a ranging framework provides systems general here remarkably used model and thermodynamic the a theoretical on The with pro- results properties. numerical out, interacting exact carried essentially particles are form. viding Lennard-Jones Bose computations so-called path-integral many the First-principles of of potential dia- system central Svistunov) phase Boris two-body the a and Carlson of Joseph of study by reviewed theoretical gram 2020; comprehensive 20, a August review present for We (sent 2020 17, September Son, Thanh Dam by Contributed 23187 VA Williamsburg, Mary, and 60637; IL Chicago, lctdt eslbe”(e.2 .74,w r o nthe in now sufficient are precision with we systems, 714), many-body com- of to p. mechanics too number quantum 2, large of much laws a exact (ref. fundamental equations the “the apply soluble,” to to that position be leads Dirac to laws by plicated these e.g., etc. of expressed, science, application fears, materials chemistry, early physics, Despite of subareas encompass- now many computation, ing and the simulations expanded computer greatly of have role algorithms computational of opment those including systems, liquids. simple of ther- many and interactions of structure equilibrium properties atomic assess- of modynamic of predictions make this models to one changed simple allow have using computer Simulations certainly of ment. phe- almost possibility diffi- emergent would the broader and in simulations factoring interactions the Interestingly, interparticle Weisskopf highlighted nomena. not Although treating remark crys- 202). likely in his p. most solid culty liquids, 1, but of (ref. on gases, liquids.” molecules, focused of of insulators, of existence and atoms, the metals would of both They existence tals, nat- mechanics? the see quantum to predict of occasion knowledge had closed fundamental never in . . they structures. lived that the ural have birth captures a physicists from that 1977 theoretical buildings “Assume intelligent from challenge: princi- of the Weisskopf group first underscores by also from quote and directly aspiration famous molecules A and ples. atoms of assemblies O eateto hsc,Uiest fAbra dotn BTG21 Canada; 2E1, T6G AB Edmonton, Alberta, of University Physics, of Department h ai nraeo oencmuigpwraddevel- and power computing modern of increase rapid The fmcocpcpoete n hsso thermodynamic of phases prediction and the is properties physics modern macroscopic of of themes major the of ne | oeEnti condensation Bose–Einstein a,1 | c unu aybd physics many-body quantum etrfrCmuainlQatmPyis ltrnIsiue e ok Y100 and 10010; NY York, New Institute, Flatiron Physics, Quantum Computational for Center htwudte eal opeitfo a from predict to able be they would What asm Boninsegni Massimo , n eprtr (T temperature and ) | superfluidity | unu ud and fluids quantum 4 a e stesse is system the as He, a hn Son Thanh Dam , ,a ela the as well as ), b,1 b 1 ulse ne the under Published interest.y competing no Massachusetts declare of authors University The B.S., and Laboratory; National Amherst. 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PHYSICS 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 predictions. 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. The 3 He, a fermion). For comparison, for a fluid of parahydrogen density of the system is therefore ρ = N /V . Particles interact via molecules, i.e., spin-zero bosons of mass one-half of that of a the LJ potential. As mentioned in the Introduction, we take the 4He atom, = 34.16 K and σ = 2.96 A,˚ yielding Λ = 0.08. In characteristic length σ as our unit of length and the well depth stark contrast to helium, fluid parahydrogen crystallizes at a tem- as that of energy. The dimensionless quantum-mechanical many- perature T = 13.8 K, well above that at which Bose–Einstein body Hamiltonian reads as condensation might take place. Although quantum effects are observable (6) near melting, there is no evidence of a superfluid N N 1 X 2 X 1 1 Hˆ = − Λ ∇ + 4 − , [4] phase, even in reduced dimensions, where quantum effects are 2 i r 12 r 6 amplified (7). i i

2 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.2017646117 Kora et al. 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PHYSICS the cubic term. This gives D/σ4 = 57 ± 8, which is again consistent with the estimates made from few-body calculations in refs. 35 and 36. The finite-temperature behavior of all three physical regimes is also shown in Fig. 3. The crystalline phase melts into a nonsuperfluid liquid upon increase of the temperature. This is not sur- prising and underscores the importance of quantum-mechanical exchanges, which underlie superfluidity, in the melting of the Bose solid (4). Melting occurs at a temperature which decreases on increasing the value of Λ. In the liquid, we computed three different temperatures: 1) the liquid–gas critical temperature, 2) the superfluid transition temperature at the ground-state equilibrium density, and 3) the Bose–Einstein condensation temper- 2/3 ature of the noninteracting system TBEC ≈ 3.3125 Λ ρ , also at the ground-state equilibrium density. The interplay between the three temperatures is plotted in Fig. 3 and is discussed in more detail in Intermediate Λ Regime. Finally, we have the superfluid gas regime, in which the system behaves very similarly to a dilute Bose gas. Fig. 3. Ground-state and liquid–gas critical temperature of the system as a In Low Λ Regime, Intermediate Λ Regime, and High Λ Regime, function of Λ. The liquid–gas critical temperature (TLG, diamonds) is deter- we provide more detailed descriptions of the different physical mined by the procedure discussed in Methodology and illustrated for X = 3 regimes in Fig. 3, moving from smaller to larger values of Λ. in Fig. 1. Also shown are the superfluid (Tλ, circles) and Bose–Einstein (TBEC, P T squares) transition temperatures of homogeneous fluids, as well as the melt- Detailed - phase diagrams are computed at representative Λ values to probe the different phases and the topology of the ing temperatures (TM, stars) of crystals. Tλ, TBEC, and TM are computed by holding the density fixed at the ground-state equilibrium value. When not phase transitions. It is important to reiterate that our results are shown, statistical uncertainties are smaller than the size of the symbols. all using the simple LJ atom–atom interaction. Despite its gener- Lines are guides to the eye. ality, there will be situations, for example low-density diatomic gases or very high-pressure states, where new phases emerge which are not captured by our Hamiltonian. dominate the potential energy, causing the system to become less bound and suppressing the liquid phase, causing TLG to Low Λ Regime. At values of Λ corresponding to X > 4.8, quan- go down. tum mechanical exchanges are suppressed, and the ground state We systematically investigate these trends in Results. is primarily the result of minimizing the potential energy, i.e., a crystal. Nevertheless, as shown in ref. 4, exchanges play a crucial Results role in the determination of the melting temperature. (It is worth Overview. The results of our extensive QMC computations are noting that the isotopes 8He and 6He have both been realized summarized in Fig. 3. In this subsection, we discuss the main in the laboratory, with single-nuclei half-lives of 0.12 and 0.8 s, ground-state features of this diagram and give a brief overview of respectively.) the different physical regimes at zero temperature, before mov- The pressure–temperature phase diagram at Λ = 0.09075, ing on to describe finite-temperature characteristics. The various corresponding to X = 8, is shown in Fig. 5. transition temperatures in Fig. 3 were computed at specific val- The melting temperature of the equilibrium crystalline phase ues of Λ. The corresponding equivalent helium mass X values goes down as Λ grows, as shown in Fig. 3. In particular, we are also shown in Fig. 3. We indicate with arrows the locations the muonic counterparts of some molecules. The different shades in Fig. 3 represent the different ground states of the system, depending on the value of Λ. Three dis- tinct physical regimes can be identified. At low values of Λ (high values of the nuclear mass), the ground state is a crystal. At a value of Λ ≈ 0.15, the system quantum melts into a superfluid that remains self-bound. As Λ is further increased, the binding is weakened. This behavior is illustrated in Fig. 4, which shows the equilibrium density going down as Λ grows, to finally hit zero upon reaching another critical value Λc , whereupon the system undergoes quantum unbinding. In the regime Λ > Λc , the ground state is a superfluid gas. From the many-body equation-of-state results, we obtain an estimate of Λc ≈ 0.46 as shown in Fig. 4, which corresponds to X ≈ 1.6. This result agrees with an earlier prediction made in ref. 35 using the zeros of the two-body scattering length (34), confirming the argument based on few-body considerations. In the series expansion of the effective potential in terms of a classical field, the three-body term has the opposite sign with respect to the two-body term, as we approach Λc from below. We can Fig. 4. The ground-state equilibrium density of the system as a function of also compute the three-body scattering hypervolume D, related 2 the inverse de Boer parameter. The intercept at ρeq = 0 shows the minimum to the three-body coefficient λ3/3 by ~ D/6m = λ3/3. Our esti- nuclear mass that remains self-bound at zero temperature; the red square is mate is obtained by fitting the energy as a function of density an exact result obtained from the two-body scattering length (34, 35). (Inset) at a value (chosen to be Λ = 0.44) close to Λc with a third- Examples of the equation of state at Λ = 0.1815, 0.242, 0.363, respectively, degree polynomial and extracting the value of the coefficient of from bottom to top.

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PHYSICS fluid and gas phases at low pressure are correctly captured by mean-field and analytic theory (15). Discussion and Conclusions We performed extensive, numerically exact many-body computations of simple Bose systems interacting through the Lennard-Jones potential and investigated their physical properties throughout a wide range of the “quantumness” parameter Λ. As a function of Λ, we studied the evolution of the phase diagram and provided detailed predictions at several values of Λ representative of the different physical regimes. One goal of our study was to establish the kind of phases and phase diagram topology that one can encounter in this very broad class of systems. Only insulating crystal and (super)fluid phases are present; no “supersolid” is observed, consistent with a wealth of theoretical predictions pointing to the absence of a supersolid phase in a system in which the dominant interac- Fig. 7. The pressure–temperature phase diagram of 2He (Λ = 0.363). The tion is pairwise and spherically symmetric and features a “hard main graph shows a zoom-in of the low-pressure region, while Inset gives core” repulsion at short distances (43, 44). No coexistence of a more global view. Lines are to guide the eye. Solid lines correspond to two superfluid phases is observed either, which is also consis- first-order phase transition and dashed lines to second order. tent with the thermodynamics of the liquid–gas transition and our current understanding of the relation between superfluidity and Bose–Einstein condensation in gases. is consistent with the prediction made by the variational theory in Given the generality of the LJ interaction, mapping out in ref. 20, in which the authors contend that the solidification pres- detail the thermodynamic phase diagram can guide in the design sure of a bosonic 3He is greater than that of the Fermi system by and interpretation of experiments aimed at observing additional at least a factor of 2. phases of matter, as more experimental avenues continue to 2 open up. Experimental realization of the systems studied here High Λ Regime. On further increasing Λ, one counters He, which is certainly not limited to helium. Among all naturally occurring is located at the region where T exceeds TLG. While the system λ substances, significant quantum effects are observed in parahy- remains self-bound at zero temperature, it boils before losing drogen and can also be expected in two unstable isotopes of its superfluidity upon increasing temperature. This is in contrast 4 helium which possess an even number of nucleons (i.e., they are with the case in He which, as the temperature is raised, loses bosons). Higher values of Λ may be achieved in a laboratory set- superfluidity long before it boils. 2 ting by preparing systems of ultracold atoms, via exotic matter, The pressure–temperature phase diagram for He is shown in 3 or in excitonic systems. Fig. 7, which is simpler compared to that of He. At low tem- In addition to providing a universal phase diagram for peratures and pressures, a first-order boundary separates the this class of simple Bose system, we hope that our investi- superfluid phase and the gas phase. Beyond TLG, the phases gation also serves as an example of the progress in making are separated instead by a second-order line, which continues to definitive and comprehensive predictions on interacting quan- grow as a function of pressure. In Fig. 7, Inset we present a more tum many-body systems. Such examples are still uncommon, complete diagram that includes higher pressures. The behavior 3 but are certainly becoming increasingly possible, owing to the at high pressure is similar to that of He, where the second-order development of reliable and robust computational methods line doubles back and intersects the solid–liquid boundary with a and more cross-fertilization between them and with analytical negative slope. approaches. As one continues raising the value of Λ, the first-order line separating superfluid and normal phases progressively recedes Data Availability. There are no data underlying this work. toward the origin, until the system no longer features a first-order phase transition. The first-order portion vanishes precisely when ACKNOWLEDGMENTS. This work was supported by the Natural Sciences and Λ = Λc , where quantum unbinding takes place in the ground Engineering Research Council of Canada, a Simons Investigator grant (to Λ > Λ D.T.S.), and the Simons Collaboration on Ultra-Quantum Matter, which is a state, as we discussed in Fig. 4. For c there is only a second- grant from the Simons Foundation (651440 to D.T.S.). Computing support of order line separating the superfluid phase and the normal gas Compute Canada and of the Flatiron Institute are gratefully acknowledged. phase. These features of the phase boundary between the super- The Flatiron Institute is a division of the Simons Foundation.

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