Can a crystal be the ground state of a Bose system?
aa r X i v : . [ c ond - m a t . o t h e r] O c t Can a crystal be the ground state of a Bose system?
Maksim D. Tomchenko
Bogolyubov Institute for Theoretical Physics14b, Metrolohichna Str., Kyiv 03143, Ukraine
It is usually assumed that the Bose crystal at T = 0 corresponds to the genuine groundstate of a Bose system, i.e., it is described by the wave function without nodes. By meansof a simple analysis based on the general principles, we prove that the ground state of aperiodic Bose system corresponds to a liquid or gas, but not to a crystal. One can expect thatit is true also for a system with zero boundary conditions, because the boundaries should notaffect the bulk properties. Hence, a zero-temperature natural crystal should correspond to anexcited state of a Bose system. The wave functions Ψ of a zero-temperature Bose crystal areproposed for zero and periodic boundary conditions. Such Ψ describe highly excited states ofthe system that correspond to a local minimum of energy (absolute minimum corresponds toa liquid). Those properties yield the possibility of existence of superfluid liquid H , Ne, Ar,and other inert elements. We propose possible experimental ways of obtaining them. In nature, liquids usually crystallize at the cooling. This leads to the natural commonlyaccepted assumption that the lowest state of a dense Bose system corresponds to a crystal.However, we will see in what follows that this is apparently not the case. The question aboutthe structure of the ground state is of primary importance. In a strange way, it has beenlittle investigated in the literature. Below, we will try to clarify this question mathematically(Sect. 2) and consider the possible experimental consequences (Sect. 3). In this regard, wemention the book by K. Mendelssohn [1], that provides an excellent review of the history ofthe development of low-temperature physics till 1965.
In the literature, two types of solutions were proposed for the wave function (WF) Ψ c ofthe ground state (GS) of a Bose crystal: condensate type and condensateless one. Bothcorrespond to WF without nodes. Thus, it is assumed that the crystal at T = 0 corresponds o the genuine GS of the system. Based on the general principles, we will show in whatfollows that the genuine GS of a periodic Bose system of any density corresponds to a liquidor gas, but not to a crystal. We consider a system of spinless bosons that is uniform on largescales and does not undergo the action of external fields.The condensateless ansatz reads (see works [2, 3, 4, 5, 6] and reviews [7, 8, 9])Ψ c ≈ e S X P c N Y j =1 ϕ ( r j − R j ) , (1)where r j and R j are the coordinates of atoms and lattice sites, respectively, N is the numberof atoms in the system, and P c means all possible permutations of coordinates r j . Thefunction S is usually written in the Bijl–Jastrow approximation [10, 11]: S = 12 X i = j S ( r i − r j ) . (2)The exact formula for S is as follows [12, 13]: S = 12! j = j X j j S ( r j − r j ) + 13! j = j ; j = j X j j j S ( r j − r j , r j − r j ) + . . . + 1 N ! j = j ; j = j ; ... ; j N − = j N X j j ...j N S N ( r j − r j , r j − r j , . . . , r j N − − r j N ) . (3)Here, the sum including S j describes the j -particle correlations. In ansatz (1), the crystallattice is postulated, and it is assumed that the atoms execute small oscillations near thesites. The function ϕ ( r ) from (1) in the approximation of small oscillations is ϕ ( r ) = e − α r / [2, 3, 4, 5, 6, 7]. The simple analysis shows that, for such solution, no condensate of atoms ispresent [8, 14, 15].The condensate ansatz reads [16, 17]Ψ c ≈ e S e − N P j =1 θ ( r j ) , (4)where function θ ( r ) is periodic with periods of the crystal. This solution is of the wave typeand is characterized by a condensate with WF Ψ c ( r ) ≃ e − θ ( r ) . The crystal-like solutions witha condensate were considered in other approaches as well [18, 19, 20, 21, 22, 23, 24].In order to verify the bulk structure of solutions, we can use any boundary conditions(BCs). Let us test the crystal solutions (1), (4) for periodic BCs. The periodic system istranslationally invariant, which yields two consequences. (i) The properties of a system on aring must not change at a rotation of the ring. This holds provided that, at a displacementof the system as a whole by the radius-vector δ r → , WF of the system is multiplied by aconstant: Ψ( r + δ r , . . . , r N + δ r ) = (1 + i p δ r )Ψ( r , . . . , r N ) = e i p δ r Ψ( r , . . . , r N ) . (5) ii) Since Ψ( r + δ r , . . . , r N + δ r ) = δ r X j ∂∂ r j ! Ψ( r , . . . , r N ) , (6)relation (5) yields ˆ P Ψ ≡ − i ~ X j ∂∂ r j Ψ = ~ p Ψ . (7)Therefore, any solution Ψ( r , . . . , r N ) of the Schr¨odinger equation with periodic BCs shouldbe an eigenfunction of the momentum operator ˆ P , i.e., it should satisfy conditions (5) and(7). This is well known from quantum mechanics.The most widely used ansatz is WF (1), where the coordinates of sites R j are fixed and areindependent of the coordinates of atoms r j . However, such ansatz does not satisfy conditions(5) and (7). Indeed, for ϕ ( r ) = e − α r / , we have:ˆ P Ψ c = i ~ α Ψ c X j ( r j − R j ) = ~ p Ψ c . (8)With regard for the anharmonic corrections to ϕ ( r ) , the formula for ˆ P Ψ c is complicated, butthe conclusion does not change. More complicated modifications of WF (1) were proposed in[25, 26] (visual interpretation of the approach in [26] was given in [27]). For them, relation (7)does not hold as well. In particular, for a one-dimensional (1D) periodic system, let us maketranslation r j → r j + δr in the function Ψ T ( R ) = R Ξ( R, S ) dS [26] with the replacement s j → s j + δr . Then the limits of integration change. Ψ T will be invariable, if the values ofΞ( R, s , . . . , s N ) are the same at s j = δr and s j = L + δr for any j (one, two, . . . , N ; L issystem size). But this is not satisfied for Ξ( R, S ) from [26]. That is, Ψ T ( R ) changes at atranslation and does not correspond to zero momentum.Solution (1) for periodic BCs is impossible also because of the fact that, for a pe-riodic Bose system, the concentration is an exact constant: n ( r ) = const [24, 28, 29].This surprising property was first noticed, apparently, in [28]. It is related to the trans-lation invariance and can be easily proved (for T = 0, see the calculation of the den-sity matrix in the coordinate representation in [29] and in the operator approach in [24];for T > , this can be proved analogously to the analysis in [29], using the formula n ( r ) = F ( r , r ) = const · R d r . . . d r N P j e − E j /k B T | Ψ j ( r , r , . . . , r N ) | and property (5)). Theconstancy of the density means that, in a periodic system, the crystalline ordering is hidden.It must manifest itself in oscillations (with the period of a crystal) of the two-particle densitymatrix F ( r , r | r , r ), rather than in the density. But solution (1) corresponds exactly tothe oscillating density : n ( r + b ) = n ( r ), where b x , b y , b z are the sizes of crystal cell. Let usshow it. Since S in (2) and (3) correspond to a constant density, we set S = 0 in (1). Thenwe get n ( r ) = ˜ C · P j e − α ( r − R j ) = const . On the other hand, n ( r + b ) = n ( r ), since thetranslation of the crystal by one step is equivalent to the renumbering of sites, which doesnot change the sum. However, the oscillating density is impossible for periodic BCs, sincethe relation n ( r ) = const must hold. or the condensate ansatz (4), we obtainˆ P Ψ c = Ψ c i ~ X j ∂θ ( r j ) ∂ r j . (9)This equals ~ p Ψ c , if θ ( r j ) = − i pr j /N + const . The ground state must correspond to zeromomentum ~ p . We get p = 0 at θ ( r j ) = const . Then ansatz (4) is reduced to the solutionΨ = C · e S (10)with S (3). However, it is a solution for WF of the ground state of a uniform system (liquidor gas) [11, 12, 13, 30].The key point is that, for the periodic system, WF of the ground state of a crystal mustbe an eigenfunction of the operator of momentum and should correspond to zero momentum : ~ p = 0. The case p = 0 is unphysical. Indeed, if the crystal would contain a quasiparticle,then the momentum ~ p = 0 would be associated with this quasiparticle. But the groundstate contains no quasiparticles, and the crystal as a whole does not move. Therefore, theseparated direction that is set by nonzero momentum cannot be associated with a physicalproperty. In view of this, we have p = 0.This can be shown mathematically. Let GS be described by the real positive nodeless WFΨ corresponding to the momentum ~ p = 0 and the energy E : ˆ H Ψ = E Ψ . This WF canbe written in the form Ψ = C · e ψ p + S , (11)where ψ p and Ψ are eigenfunctions of the operator of momentum with the same eigenvalue ~ p . In the simplest case, ψ p = const · ρ − p , where ρ − p = √ N P Nj =1 e i pr j is the collectivevariable (the general form ψ p is more complicated, see [31, 32]). We choose the coordinateorigin so that x ∈ [ − L x / , L x / y ∈ [ − L y / , L y / z ∈ [ − L z / , L z / { x, y, z } at the inversion r → − r transits into itself. On the other hand, as r → − r , we have ψ p → ψ − p . This follows from (7) and the form of ψ p . Hamiltonian (seeformula (13) below) remains invariant. Therefore, the stateΨ = C · e ψ − p + S (12)is also a solution of the Schr¨odinger equation with the energy E . This means that the groundstate corresponding to nonzero momentum is degenerate. Since WFs (11) and (12) correspondto different values of the observable quantity (momentum), they must be orthogonal. Theinversion r → − r transforms Ψ into Ψ . On the other hand, the space Ω transits intoitself at such inversion. Therefore, if the function Ψ takes values in the interval [ A, B ] for r ∈ Ω, then the function Ψ takes values only from [ A, B ] as well. Since Ψ is positiveeverywhere, Ψ must be positive everywhere as well. Such Ψ and Ψ cannot be orthogonal.Hence, the assumption about that GS is characterized by nonzero momentum results in thecontradiction. Therefore, the true GS can correspond only to zero momentum. S with nonzero momentum leads also to another difficulty: apparently, it is impossibleto find such ψ p that Ψ is real.In principle, GS can correspond to zero momentum and be degenerate. This possibilityis discussed in Appendix 1. It was argued in book by R. Feynman that GS of a system ofinteracting bosons should be nondegenerate [33]. Therefore, we will not consider this case.Thus, we have shown that the ground state of a periodic system of interacting bosonscorresponds to zero momentum. Since functions (1) and (4) describe the states without adefinite momentum, WF of the ground state of a crystal must be different .The structure of WF of the ground state, Ψ , of a Bose system with periodic BCs canbe easily determined. The condition p = 0 and formulae (5) and (7) imply that Ψ shouldnot vary at a translation. Therefore, it can depend only on the difference of coordinates.The general form of such nodeless function is given by formulae (10) and (3). This can beproved strictly (see Appendix 1). Thus, the genuine ground state of a periodic Bose systemcorresponds always to a liquid or gas.
This is the main result of the present work. Thetheorem of nodes implies that any excited state of the system corresponds to WF with nodes[34]. This means that any state of a crystal with periodic BCs corresponds to WF with nodes.A zero-temperature state of a crystal we will call the ground state of a crystal.We note that the structure of Ψ (10) of a liquid is usually obtained from the requirementthat Ψ should be invariable at a translation [ p = 0 in (5)] [12, 35]. However, the translationinvariance of a system admits p = 0 in (5). Therefore, the condition p = 0 is primary, andthe translation invariance of Ψ is a consequence of this requirement.The natural crystals are in contact with the other medium: air, earth, etc. Such crystalcan be approximately modeled with the help of the zero BCs: Ψ = 0 on the crystal surface.In this case, our arguments are not valid, since the system with the zero BCs has no definitemomentum. For such crystals, we have two possibilities: (i) GS of a crystal with the zeroBCs corresponds to the genuine GS and is described by WF without nodes or (ii) the genuineGS of the system corresponds to a liquid, and GS of a crystal corresponds to an excited stateof the system and is described by WF with nodes (as for periodic BCs). Possibility (i) meansthat the boundaries affect strongly the structure of solutions. Both possibilities are nontrivialand indicate that the modern theory of Bose crystals is not perfect (because it usually usesperiodic BCs and considers that WF of the ground state of a crystal has no nodes).It can be seen from a qualitative reasoning that the energy of a system should depend onBCs negligibly. Consider a 1D crystal consisting of N atoms under the zero BCs. Such acrystal is characterized by the oscillating density ρ ( x + b ) = ρ ( x ). Let us join the ends of thiscrystal. Thus, we introduced the periodic BCs. Since the interatomic potential U ( | r − r | )is usually significantly different from zero only at | r − r | < ∼ N -th ones. For the remaining atoms, the transition to the periodic BCs does not make ny changes, and, therefore, their state should not change. In reality, the picture is morecomplex. At the transition from the zero BCs to the periodic ones, the translation symmetryarises. Therefore, the profile of the density ρ ( x ) is transformed by jump from the oscillatingone to a constant (see the analysis above). In addition, the collection of harmonics changesas well (from k j = πj/L we pass to k j = ± πj/L , j = 1 , , . . . , ∞ ). Despite these properties,the energy E of a zero-temperature crystal can depend on BCs only very weakly. Indeed,we have E = Z d r · · · d r N Ψ c ∗ " − ~ m N X j =1 △ j + X j On the basis of the above analysis, we assume that, for any BCs, the inequality E c ( ρ, N ) > E l ( ρ, N ) (19)holds. Here, E c and E l are the energies of GS of a Bose crystal and a Bose liquid, respectively, N is the number of atoms, and ρ = mn is the density. In (19), E c and E l are compared atthe same ρ . However, the phase transitions occur in experiments at the same pressure P .Here, two cases are possible: E c ( P, N ) > E l ( P, N ) (20)or E c ( P, N ) < E l ( P, N ) . (21)For He, inequality (20) is satisfied (at the pressure of crystallization P ≈ atm , see Ap-pendix 2). The liquid satisfying condition (20) must be stable, at low P and T , againstcrystallization. If (21) is satisfied, the liquid corresponds to a metastable state, but theduration of the transition into the stable crystalline state may be long.Inequality (19) testifies to the existence of a large number of quantum states correspondingto a liquid and possessing the energies less than the energy of GS of a crystal. We will callsuch states “under-crystal liquid” (“underliquid” for short). Since this region of states islarge, one can expect that at least part of it is observable. It is also clear that, at sufficientlylow temperatures, the underliquid have to be superfluid. The creation of such superfluidswill mean that, in addition to the vessels with He II, physical laboratories will possess thevessels with other superfluids. Let us try to ascertain how the underliquid can be produced.For all known liquids, except for He, the ( P, T ) diagram of states is separated into theregions corresponding to a gas, a liquid, and a crystal and has the triple point (see Fig. 2).The ( P, T ) diagram of He has no triple point. In this diagram, the gas contacts only withthe liquid. Each of the transitions (gas–liquid, liquid–crystal, and gas–crystal) is operated bythree equations describing the equilibrium between phase 1 and phase 2 [46]: P = P ≡ P , T = T ≡ T , and P ( v − v ) + T [ s ( P, T ) − s ( P, T )] = E ( P, T ) − E ( P, T ) , (22)where E j is the internal energy per atom for the system staying in the j -th phase, v j and s j are the volume and entropy (per atom) of the j -th phase. Equation (22) is equivalent to theequality of the chemical potentials of phases 1 and 2: µ ( P, T ) = µ ( P, T ).The possible position of the underliquid region on the ( P, T ) diagram is shown by starsin Fig. 2. The upper and lower boundaries of the underliquid region are set by condition(22). The lower boundary corresponds to the transition underliquid–gas. The possible upper ,0 0,5 1,0 1,5-20-15-10-505101520 l g ( P / P ) T/T Fig. 2: [Color online] ( P, T ) diagram of states for inert elements (H , Ne, Ar, etc., except for He) withthe assumed region of the under-crystal liquid (bounded by stars ⋆ ⋆ ⋆ ); lg ≡ log . Lines of the transitionsgas–crystal ( ◦ ◦ ◦ , Eq. (33) with ξ = 9 . NNN ), and liquid–crystal ( (cid:4)(cid:4)(cid:4) ) are shown. P and T are the pressure and temperature at the triple point. Dotted line (Eq. (35) with ξ = 9 . f = 7) marks thecontinuation of the line gas–liquid to the region of low T . This curve lies somewhat higher than the curve ◦ ◦ ◦ , but the difference is visually indistinguishable (maximum distance between the curves along the verticalis equal to 0 . 13 and corresponds to T /T ≈ . NNN , (cid:4)(cid:4)(cid:4) , and the right vertical boundary ofthe region of the under-crystal liquid are drawn by eye. boundary corresponds to the transition underliquid–crystal. In the limits of these boundaries,the liquid can be stable or metastable, as was noted above. The right boundary P ul − c ( T ) isshown in Fig. 2 approximately. It corresponds not to a phase transition, but to the condition E l ( P, T ) = E c ( P ) (that is possible, if (20) is satisfied). The equation for the lower boundarywill be obtained in what follows. It is easy to estimate the location of the upper boundary P ul − c ( T ). At T = 0 , relation (22) yields P ul − c ( T = 0) = E c ( P ) − E l ( P ) v l − v c . (23)For the realistic values v l − v c = 0 . v l = 0 . . and E c ( P ) − E l ( P ) = 10 Kk B , we find P ul − c ( T = 0) ≈ atm (here, k B is the Boltzmann constant). The upper boundary exists,if P ul − c ( T = 0) > 0. At P ul − c ( T = 0) < E l ( P, T ) = E c ( P )) at P ≈ k B T ≈ ε/ 3, where ε is the energy from the Lennard-Jonespotential, holds for the inert elements. Assume that E l ( P = 0) − E c ( P = 0) ∼ − . ε ∼− . k B T , similarly to helium-4. At low T, we may consider only the phonon contributionto the energy. Then E l ( T ) = E l + π (cid:16) k B T ~ c s (cid:17) k B Tn [47], where c s is the first sound velocity.The relations E l ( P = 0 , T ) = E c ( P = 0) and E c ( P = 0) − E l ( P = 0) ≃ . k B T yield T T ≃ · . nπ (cid:18) ~ c s k B T (cid:19) . (24) sing the parameters of the triple point for neon ( T = 24 . K [48], c s = 628 m/s , ρ =1 . g/cm [49]), we get T ≈ . T . For liquid argon at the triple point, we have T = 83 . K [48], c s = 870 m/s , ρ = 1 . g/cm [50]. From whence with the help of (24), we obtain T ≈ . T . We expect that these estimates of T are valid by the order of magnitude.The basic question is: How can we “get to” the region of underliquid in experiments?(The underliquid state of He has already been obtained: as it is easy to guess, this is HeII.) On top and to the right from the region of underliquid, the crystalline states are placed.The region of underliquid corresponds to very low temperatures: T < ∼ . T . The crystallinestates at such T were experimentally studied for many substances, but the underliquid stateswere not found. According to (20) and (21), the crystal with T and P from the region ofunderliquid should be metastable or stable, respectively. In the metastable state, such crystalslive, apparently, very long (otherwise, the underliquid would be found experimentally longago). Therefore, we believe that it is impossible to produce an underliquid from a crystal (bydecreasing T or P ).The underliquid can be, apparently, obtained by strong supercooling of a liquid whoseinitial temperature is higher than the melting one. To avoid the crystallization, it is necessaryto purify a liquid from impurities and to use a vessel with smooth walls. A shortcoming ofthe method consists in the necessity of a strong supercooling, which requires the high degreeof smoothness of walls and of purity of a liquid.Most likely, it is easier to get in the underliquid region by isothermal compression of agas at T ≪ T . According to Fig. 2, at the isothermal increase in the pressure of a gaswith T ≪ T , we cross firstly the curve gas–crystal and then the curve gas–liquid. Therefore,the gas must turn into a crystal (not in the underliquid), which corresponds to experiments.Nevertheless, we will show below that the underliquid can be obtained in such a way. Forthis purpose, it is necessary to perform the transition at the temperature T ≪ T and tocreate the conditions preventing the crystallization (see below).To substantiate this point, we consider the transitions gas–crystal (g-c) and gas–liquid(g-l) in more details. First, one needs to get the dependences P ( T ) setting the curves g-cand g-l. As is known, along the line P ( T ) of the phase transition the Clapeyron–Clausiusrelation should hold: dPdT = s ( P, T ) − s ( P, T ) v − v . (25)Let index 1 correspond to a gas, and index 2 to a liquid or crystal. The data on the pressureof saturated vapors for He II [51] show that, at T < ∼ T λ , the temperature of a vapor ismuch larger than the temperature of the Bose condensation. Therefore, the vapor can beconsidered as an ideal gas. Assume that the vapors of other inert elements at T ≤ T canalso be considered as an ideal gases. The entropy of a one-atom ideal gas consisting of atoms ith zero spin and zero orbital moment (all inert elements, except for H ) is known [46, 52]: s g = 52 k B + k B ln (cid:20) ( k B T ) / P (cid:16) m π ~ (cid:17) / (cid:21) . (26)Note that the first term in this formula is given in the literature with different coefficients:3 / / / / / s of a liquid or crystal at T ≤ T is determined mainly by the phononcontribution ( ∼ T ), which is much less than the entropy s = s g of a saturated vapor (weremark that the Debye temperature for inert elements is comparable with T ). The entropyof a Bose liquid at T → s l = 2 π k B n (cid:18) k B T ~ c s (cid:19) . (27)For He atoms at T = 1 K and at the pressure of saturated vapors P ≈ . · − atm [51], weget s g /s l ≃ P = 0 . atm [48] and s g /s l ≈ . K to T , the relation s l , s c ≪ s g holds. In this case, v l , v c ≪ v g . Therefore, in the zero approximation, the curvesgas-crystal and gas-liquid at 0 ≤ T ≤ T are given by the formula dPdT = s g v g = P s g k B T . (28)Denote ˜ T = T /T , ˜ P = P/P . Then formula (26) can be written as s g /k B = 52 ln ˜ T − ln ˜ P + s , (29)where s is the value of s g /k B at the triple point. Equation (28) takes the form d ˜ Pd ˜ T = ˜ P ˜ T (cid:18) 52 ln ˜ T − ln ˜ P + s (cid:19) . (30)Now, denote y = ln ˜ P and x = ln ˜ T . Then Eq. (30) becomes dydx = 52 x − y + s . (31)We need to find a solution satisfying the boundary condition y = x = 0 (for the triple point).The solution can be sought as a series y = a x + a x + . . . + a j x j + . . . . After the simpletransformations, we get y = ξ + 5 x − ξe − x , ξ = s − . (32)From (32) we obtain P ( T ) for a saturated vapor at 0 < T ≤ T :˜ P = e ξ ˜ T / e − ξ/ ˜ T . (33) t is a solution in the zero approximation. It holds for both curves g-l and g-c. We do notknow whether this solution was obtained previously. It seems that formula (33) is a rathergood approximation. He has no triple point. If we set formally T = 1 K , then Eq. (33) agrees well withthe experimental pressure P ( T ) of saturated vapors of He at T ≤ K [51]. If we take T = 3 K , then Eq. (33) describes experiments only qualitatively (perhaps because formula(28) becomes a poor approximation for (25)). For Ne, Ar, Kr, and Xe, the dependence P ( T )for the sublimation curve was measured for temperatures T ≃ (3 / ÷ T (by the data before1976 [53]). In particular, the experimental dependence P ( T ) for neon at T = 16 − K isdescribed by the fitting formula lg ˜ P ≈ . − . 39 lg ˜ T + 5 . T − . / ˜ T [53]. The approximatesolution (33) with ξ for neon ( ξ = 9 . 69) gives the values of lg ˜ P less approximately by 10%.Solution (33) was found, by neglecting the corrections s and v in (25). At ˜ T ≪ T , but remain small even at ˜ T = 1. Inorder to estimate the influence of corrections on the solution, we take the entropy s /k B =4 ˜ T − ( f + 1) ˜ T f with f > s /k B proper (for comparison, s l /k B = 1 . 58 ˜ T for neon at ˜ T ≪ 1; while estimating s l , we take c s ( T ≤ T ) = c s ( T ) and ρ ( T ≤ T ) = ρ ( T )). The second term effectively describes thecorrection v in the denominator. In this case, we get the solution y = ξ + 5 x − ξe − x + e x − e fx , (34)˜ P = e ξ ˜ T / e − ξ/ ˜ T e ˜ T − ˜ T f . (35)In Fig. 2, this solution is shown as the curve g-l and solution (33) as the curve g-c. Ofcourse, such correspondence between the formulae and the curves is only qualitative: theabove analysis evidences that the curves g-l and g-c should be close. For Fig. 2, we useparameter ξ = 9 . 69 corresponding to neon. In this case, the parameter f = 7 is chosen sothat curve (35) lies above curve (33), and the slope of curve (35) at T → T is less than thatof curve (33). As a result, curves (33) and (35) are similar to experimental curves g-c andg-l, respectively. The exact curves g-l and g-c can significantly differ from those presentedin Fig. 2, because the corrections s and v were taken into account in a rough model form.However, this analysis is sufficient to show that the solutions of such type correctly describeexperimental curves g-l and g-c. The second conclusion is that though the corrections s and v separate the curves g-l and g-c, this difference should be small. For example, at ˜ T = 0 . P gl /P gc = e ˜ T − ˜ T f ≈ . f = 7.It is significant that the slopes of the experimental P ( T ) curves g-l (at T > T ) and g-c(at T < T ) near the triple point are very close: in particular, for inert elements [48]. Thisagrees with our conclusion that these curves should be close at T ≤ T .If the ratio P gl /P gc = ζ is close to 1, the phase transitions gas–crystal and gas–liquid are“switched-on” almost simultaneously. At the compression, the system transits in a liquid or rystal depending on that which nuclei are generated faster: microdrops or microcrystals. InAppendix 3, where the formation of nuclei is considered, we will show that it is necessaryto increase the pressure of a gas up to P ≈ P gc ζ φφ in order that the microdrops will begenerated faster, than microcrystals ( φ depends on the substance; the characteristic valueis φ ≃ . P gc ( T ) lies below the curve P gl ( T ) (seeAppendix 3). Let the gas be compressed at the pressure P ≈ P gc ζ φφ ∼ P gc ζ ∼ . P gc (for ζ = 1 . E l ( T, P ) < E c ( P ), then such underliquidwill not crystallize. Our estimates are crude. Therefore, the exact formulae can give a muchlarger ratio P/P gc . However, we expect that ζ φφ < ∼ 2, i.e., the pressure P should be increasedby at most several times as compared with P gc , in order that the spontaneous (bulk or surface)condensation of a gas into a liquid will start.According to the analysis in Appendix 3, in order to prevent the crystallization of a gasand to “switch-on” the bulk spontaneous mechanism of formation of nuclei, one needs topurify a gas from suspended solid impurities and to prevent the formation of crystallinenuclei on the walls. To achieve the latter, one can take a vessel with smooth walls (though,it is impossible to obtain an ideally smooth walls), and the molecules of walls should weaklyinteract with the molecules of a gas (or the crystalline ordering of walls should significantlydiffer from that of crystal nuclei forming from a gas). If the walls is covered from inside bya microscopically thin film of He II, then the surface of walls should be liquid and smooth.Therefore, the formation of crystal nuclei on the walls becomes difficult. In addition, theinteraction of helium atoms with molecules of the majority of gases is weak, which mustprevent the adsorption of molecules of a gas on the walls and the formation of surface nuclei.It is noted in books [54, 55] that, at the compression of a gas at a temperature T < T , themetastable liquid is sometimes formed and then crystallizes. These properties are evidenceof the validity of the inequality E c ( P ) < E l ( P ) (21). However, our analysis shows that, forsome substances, the inequality E c ( P ) > E l ( P ) (20) should hold. In this case, the liquidformed at the compression of a gas should be stable and should not crystallize.Interestingly, the transition crystal–underliquid can occur at a negative pressure. By(23), we have P ul − c ( T = 0) < v l − v c > E c ( P ) − E l ( P ) < v l − v c < E c ( P ) − E l ( P ) > 0. We may expect that P ul − c ( T = 0) ∼ − (100 ÷ atm . In thiscase, the state of underliquid can apparently be obtained by creating a negative pressure ina crystal. The idea of the creation of a liquid from a crystal by applying a negative pressurewas advanced by J. Frenkel [55, 56]. Since the existence of a liquid at T = 0 was consideredimpossible, J. Frenkel proposed to make the experiment at some T > 0. However, we sawabove that the liquid state can exist at T = 0. Therefore, the experiment can be carried on t any low temperature.The above analysis shows that the form of the ( P, T )-diagram at low P and T shoulddepend on how we got in this region. If we have got into it from above (by cooling a crystal),we obtain an ordinary diagram with the triple point. The result will be the same, if wewill try to get into the region of underliquid from the bottom (from the region of a gas) orfrom the right (from the region of a liquid) in the absence of the conditions preventing theformation of crystal nuclei. But if we try to enter from the bottom (or from the right) in thepresence of such conditions, then we should obtain the state of underliquid. In other words,the phase diagram in the region to the left from the melting curve and above the sublimationcurve has two levels (or two “layers”): liquid-type and crystal-type ones. Such liquid stateswere obtained previously by supercooling a liquid. In this case, it was considered in theliterature that the liquid state at T = 0 is impossible. Therefore, the supercooling of a liquiddown to T ≈ T ≈ , Ne, Ar, etc.),except for helium. We propose to carry out three following experiments . (1) To supercoolisobarically liquid H , Ne, and Ar down to temperatures that are by several times less thanthe temperature of the Bose condensation of the ideal gas T c = . s +1) / ~ k B m n / [46] (here, s is the spin of a particle; for liquid inert elements, except for hydrogen and helium, T c < K ).In this case, we should obtain a superfluid underliquid. (2) To compress isothermally a dilutegas (H , Ne, Ar) at T ∼ . T up to a pressure that exceeds by several times the saturatedvapor pressure for the same T . The underliquid should also be created. In both experiments,it is necessary to create the conditions hampering the crystallization (see the discussion aboveand Appendix 3), and condition (20) must be satisfied for the stability of a liquid. In thesecond experiment, a less supersaturation is required. Therefore, the requirements to thepurification and to walls can be apparently less strict. (3) To create a negative pressure P ul − c < ∼ − atm in a crystal made of an inert element at T < ∼ . T . One can expect thatsome of such crystals will melt into an underliquid.If the underliquid is metastable (inequality (21)) and has a small life-time, then suchunderliquid state can be unobservable. However, He is stable (inequality (20)). Therefore, itis natural to expect that, among inert elements, there are several other ones with the stableunderliquid state. In view of this, it is desirable to execute three proposed experiments withall inert elements (except for He).The inert elements were investigated mainly at T > ∼ T . The number of experiments at T ≪ T is much less. In the last ones the crystals, being in equilibrium with their vapors, werestudied at T ≥ K [53]. We assume that the state of underliquid was not obtained earlierbecause the conditions hampering the crystallization were not created. Therefore, a gas or iquid turned into a crystal, rather than into an underliquid. Moreover, the underliquid statewas considered impossible and was not sought.We note also that, at the strong supercooling, the viscosity of some liquids increasessharply, and they transit into a glass-like state. The atoms of inert gases are sphericallysymmetric (except for hydrogen which forms molecules H ) and, therefore, should not turninto a glass at the supercooling, as we understand it. According to the above analysis, at T < ∼ . T c the liquids consisting of inert elements have to be similar to He II, i.e., they shouldcontain a condensate of atoms with zero momentum and should be superfluid.Undoubtedly, inequality (19) should be correct. Therefore, the region of underliquid mustexist, and the task is to enter this region in experiments.In Appendix 2, we consider the third principle of thermodynamics, properties of He, andnumerical solutions for crystals. Our analysis shows that the genuine ground state of a Bose system should correspond to aliquid or gas. In this case, the lowest states of a liquid and a crystal must satisfy theinequality E c ( P ) > E l ( P ) (20) or E c ( P ) < E l ( P ) (21). If inequality (21) holds, the stablestate of the system at T → He). The underliquid state, that does not crystallize at the coolingand is superfluid at very low temperatures, should exist for such substances. This is the mainprediction of the present work. We assume that the underliquid can be created in experimentsby compressing a gas at a low temperature or by strong supercooling an ordinary liquid (inboth cases, it is necessary to create the conditions preventing the crystallization).According to the above analysis, a Bose crystal is a standing wave in the probability field.Most likely, this property is a general principle valid not only for Bose systems. Therefore,it is possible that the underliquid state and the superfluidity are inherent not only in inertelements, but even in water.If inequality (19) is true under any boundary conditions, it will change our comprehensionof the nature of crystals and lead to the discovery of new physical phenomena. For example,the space apparatus “New Horizons” found in 2015 that the Sputnik Planitia surface on Plutoconsists of solid nitrogen and is similar to a mosaic made of hexagons and pentagons. Thisvalley has no craters, though they are present on the remaining Pluto’s surface. It is possiblethat a liquid water ocean exists under the surface [57, 58, 59]. That is surprising becausethe Pluto’s surface temperature is about 40 K . However, we have established above that theliquids of low viscosity can exist even at T = 0 K . This can help one to understand some Such idea was proposed previously in [44], but that work is immature and contains errors; see, in particular, the Introductionin [36] nomalous properties of cosmic objects.We hope that the above-proposed experiments to create the underliquid state will becarried out.The present work is partially supported by the National Academy of Sciences of Ukraine(project No. 0116U003191). Ψ corresponding to zero momentum We now find the general form of the ground-state WF. As was shown in Sect. 2, such WFmust correspond to zero momentum. Consider the functions ψ f = c , (36) ψ f k = c ρ − k , (37) ψ f k k = c (cid:18) ρ − k ρ − k − ρ − k − k √ N (cid:19) , (38) ψ f k k k = c (cid:20) ρ − k ρ − k ρ − k − √ N ( ρ − k ρ − k − k + ρ − k ρ − k − k + ρ − k ρ − k − k ) ++ 2 N ρ − k − k − k (cid:21) , (39) . . . , (40) ψ f k ... k N = c N ρ − k · · · ρ − k N + c N X P ( k j ) ρ − k · · · ρ − k N − ρ − k N − − k N ++ . . . + c NN ρ − k − ... − k N , (41)where c ij are constants, and P P ( k j ) is the sum over all permutations of the vectors k j . Theseare the wave functions of a periodic system of N free spinless bosons. Here, ψ f corresponds tothe ground state; ψ f k describes the state, where one boson has a momentum ~ k , and N − ψ f k ... k N describes the state in which each of thebosons has some nonzero momentum ~ k j . These functions are solutions of the Schr¨odingerequation with the given BCs, and, therefore, form the complete orthonormalized collectionof basis functions. Any Bose-symmetric WF of the variables r , . . . , r N for the Schr¨odingerproblem with interatomic interaction and periodic BCs can be expanded in this basis. Thisis the ground for the theory of quantum liquids constructed in [31, 35]. Hence, any WFΨ( r , . . . , r N ), being an eigenfunction of the momentum operator of the system of N identicalbosons and corresponding to the momentum ~ p , can be presented in the form of a sumΨ p = a ψ f p + k + k = p X k k a ( k , k ) ψ f k k + . . . + k + ... + k N = p X k ... k N a N ( k , . . . , k N ) ψ f k ... k N . (42) sing Eqs. (36)–(41), this expansion can be written asΨ p = b ( p ) ρ − p + q + p =0 X q =0 b ( q ; p )2! N / ρ q ρ − q − p + q + q + p =0 X q , q =0 b ( q , q ; p )3! N ρ q ρ q ρ − q − q − p ++ . . . + q + ... + q N − + p =0 X q ,..., q N − =0 b N ( q , . . . , q N − ; p ) N ! N ( N − / ρ q · · · ρ q N − ρ − q − ... − q N − − p . (43)Here, the wave vectors k l , q l , p l , p are quantized by the law (for 3D) q =2 π ( j x /L x , j y /L y , j z /L z ), where j x , j y , j z are integers, and L x , L y , L z are the system sizes.If GS is nondegenerate, then the wave function of GS is always positive and correspondsto zero momentum. Therefore, it can be presented in the form Ψ = C · e S , where S is Ψ p (43) with p = 0. In this case, the constant b ( ) ρ ≡ √ N b ( ) is taken into account in C .However, such S is the Fourier-transform of S (3). This proves that if GS is nondegenerateand ln Ψ can be expanded in a Fourier series, then ln Ψ = S + const , where S has theliquid-type form (3). If GS is degenerate and corresponds to several WFs, then one of themmay apparently be set by formulae (10), (3), and the remaining WFs should be given byformula (43) with p = 0. To our knowledge, no solutions with the degenerate GS were foundfor a uniform many-particle Bose system in the absence of an external field.Thus, function (10), (3) specifies the general form of WF of the nondegenerate GS of aperiodic Bose system.It was assumed [7, 60] that, in order to get a crystal-type solution, one needs to intro-duce a small bare external periodic potential into the Hamiltonian. This is equivalent to the spontaneous appearance of the crystal-type not translationally invariant solution (1) froma translationally invariant liquid-type solution. N. Bogoliubov explains [60] that the con-stant mean density (liquid-type solution) arises due to the averaging over many crystallineconfigurations shifted in the space relative to one another. Therefore, the separation of oneconfiguration should lead to a periodic density, which is specific to a crystal. This reasoningrequires that such a configuration exists . In other words, the spontaneous choice of a solu-tion is possible only if the complete set of solutions for the initial unperturbed Hamiltoniancontains this solution. However, it was shown above that genuine GS for a boundary-valueproblem with periodic BCs corresponds always to a translationally invariant solution. Thatis, the crystal-type solution without nodes is absent among the solutions of the boundaryproblem. Therefore, under periodic BCs the spontaneous breaking of a translation symmetrycannot lead to a nodeless crystalline solution. We now consider several more questions. Why do the liquids in nature crystallize at thecooling, if the crystal corresponds to a highly excited state of a system? Mathematically, this s related to the fulfillment of condition (22) for the liquid–crystal transition. The physicalexplanation consists in the following: at the cooling of a liquid down to some temperature, themicrocrystals arising as fluctuations become stable. And the visual reason is that the systemfalls into the local energy minimum corresponding to a crystal (see Fig. 1). As a result,the liquid crystallizes, and we obtain a crystal with some number of quasiparticles. In thiscase, the cooling of the crystal means a decrease in the number of quasiparticles introducedrelative to GS of a crystal.Let us consider the third principle of thermodynamics. Some researchers believe thatnamely a crystal (rather than a liquid) corresponds to the genuine GS of a system, sincethe crystal is more ordered in the r -space and, therefore, should be characterized by a lowerentropy. However, according to quantum statistics, the entropy is determined by propertiesof a system in the space of quantum states (not in the r -space). It is given by the formula S = k B ln( N ( E )) [52], where N ( E ) is the number of states with energy close to E . Towhat is N ( E ) equal for the GS of a crystal? Inequality (19) implies that many liquid stateswith energy close to E c must exist. If we take them into account, we get N ( E c ) ≫ S = 0. Therefore, it is necessary to introduce the natural postulate: in the calculations ofthermodynamic quantities, one needs to take only states of the phase under consideration(gas, liquid, or crystal) into account in the partition function. Then for each phase at T = 0 , we obtain N ( E ) = 1 and S = 0, i.e., the Nernst theorem is satisfied. He has particular properties. According to experiments, liquid He (He II) at P ≈ atm and T < ∼ . K solidifies and transforms into a hcp crystal [7]. In this case, for liquid and solid He, we have, respectively, E l ≈ − . K [61, 62] and E c ≈ − . K [63]. That is, the GSenergy of a crystal by 0 . K higher , than E of a liquid. In this case, liquid and solid heliumhave densities of 0 . g/cm [61, 64] and 0 . g/cm [63, 64], respectively. To verify thebasic inequality (19), we need to compare E of liquid and solid helium at the same ρ . We candetermine E of liquid helium at ρ = 0 . g/cm by the known formula in [61, 62]. In thiscase, one needs to know P ( ρ ) of He II at ρ = 0 . . g/cm . However, such data are notavailable, since He II does not exist at such densities (it solidifies). It is significant that GSof liquid helium at the crystallization pressure ( P ≈ atm ) has a lower energy, than GS ofa crystal. Nevertheless, liquid helium crystallizes. The reason is known: He possesses largezero oscillations. Therefore, at low pressures, the crystal is unstable. As a result, the systemat low pressures and temperatures is in the state of underliquid. This is He II. As the pressureincreases, the ratio of the amplitude of zero oscillations to the lattice period decreases [7, 65].At P > ∼ atm , the crystal embryos become stable and liquid He crystallizes [66, 67]. In thiscase, the formation of microcrystals and the external pressure make the liquid state unstable:the external pressure compresses the system, performs the work, and increases the energyof the system up to E of a crystal. This results in the formation of a crystal. By sucha scenario, the ground state of He at P > atm corresponds to a liquid, but this state annot be obtained. However, the crystallization pressure should rise above 25 atm providedwe prevent the formation of crystal nuclei (one needs to purify helium from impurities anduse the smooth walls with a microstructure different from the structure of a helium crystal).We do not know whether attempts to obtain high-density liquid helium in this way weremade before. If He II with ρ = 0 . . g/cm could be obtained, it would be possibleto verify inequality (19) for ρ = 0 . g/cm .In two last decades, the crystal-type solutions were studied mainly numerically, bythe Monte-Carlo method [9, 68] and by the multi-configuration time-dependent Hartree(MCTDH) method [69]. The MCTDH method is apparently exact, but it is realizable onlyfor a small number of particles ( N < ∼ several particles per a cell of the trap. In thiscase, GS can indeed have a quasicrystalline form. However, if the number of atoms in acell of the trap is large, GS should correspond to a liquid. We think that, with the help ofthe MCTDH method, one can to verify inequality (19). To make this, one needs to find thegenuine GS for N > ∼ 10 interacting atoms without a trap . In a number of works, the inequality E c ( N, ρ ) < E l ( N, ρ ) was obtained by the Monte-Carlo method for an untrapped Bose systemwith large N , various BCs, with and without the dipole-dipole interaction (see referencesin reviews [9, 68]). However, according to the analysis in Sect. 2, this result is incorrect,at least for periodic BCs. This means that though the Monte-Carlo method is efficient formany problems [75], sometimes it gives a false solution. In our opinion, one of the possiblereasons is that the distance between the crystalline and liquid minima is apparently small: | E c ( N, ρ ) − E l ( N, ρ ) | ∼ . | E c ( N, ρ ) | . Therefore, the Monte-Carlo method, by probing thesolutions near a trial function, can find a local minimum instead of the absolute one. Thismethod can be tested on a 1D system of point bosons: it is known from the exact solutionsthat GS of such system corresponds to a liquid at any coupling constant, under the periodic[37, 38] and zero [39, 40] BCs. Will the Monte-Carlo method with the trial function (1) leadto such a solution? The theory of formation of nuclei of a new phase is not completed (especially, the theoryof crystallization), but its general contours are apparently clear (see review [76] and books[54, 55, 77, 78, 79, 80, 81, 82, 83, 84, 85]). The nuclei of the other phase can be created onthe walls of a vessel and in bulk. We will consider only the simpler bulk case.The bulk condensation of a gas into a crystal or liquid occurs under the avalanche-likeincrease in the number of nuclei of the new phase. Such growth is possible, if P or T iffers from the value P = P ∞ or T = T ∞ corresponding to the condition of equilibrium(22). Consider a gas at low P = P ∞ and low T = T ∞ . Let us compress it isothermallyso that the pressure increases up to some P > P ∞ . In such gas, the embryos of the liquidphase (microdrops) and the crystalline one (microcrystals) should randomly appear. In asupersaturated gas (vapor) at the pressure P r ≥ P ∞ , a droplet is in equilibrium with a gas,if its radius r satisfies the Kelvin formula [52, 55, 78]: P r ( T ) = P ∞ ( T ) exp (cid:18) α lg nk B T r (cid:19) , (44)where n is the concentration of atoms in a droplet, P ∞ is the saturated vapor pressure, α lg is the coefficient of surface tension of a liquid on the boundary with a gas. Let the pressure P r correspond to the radius r = r , according to (44). Then the droplets of radius r < r must evaporate. The condensation of atoms of a gas on a droplet decreases the pressure in agas, which makes it possible for the droplets of radius r > r to exist. As a result, the meanradius of droplets must increase with the time, until the whole gas transforms into one largedrop [52]. The crystal embryos in a gas can be described analogously. We will get formula(44), where the parameters of a microdrop should be replaced by those of a microcrystal.According to a more detailed theory, the process of formation of embryos is as follows[55, 76, 78, 81, 82, 83, 84]. The fluctuations in a gas result in the spontaneous formationof microscopic embryos of a liquid (microdrops) and a crystal (microcrystals) in a gas. Theembryo can randomly capture atoms of the gas, which will lead to the growth of this embryo.The reverse process is possible as well. As a result, some (non-stationary, generally speaking)distribution of embryos over sizes should be formed. In this case, the embryos of sizes largerthan the critical one (Eq. (44)) must unboundedly grow. Such embryos are usually callednuclei. If the system is supplied with a gas in the amount compensating the loss due to theformation of nuclei, we get a stationary distribution of nuclei and the continuous transforma-tion of small nuclei into large ones. If such pumping of a gas is absent, then in the usual case(isothermal formation of nuclei in a closed system with permanent supersaturation) the non-stationary process eventually becomes stationary [83, 84, 85]. Therefore, we may consider theprocess to be stationary. The kinetic analysis shows that, in this case, the rate J of homoge-neous (i.e., without exterior impurities) formation of nuclei is [54, 55, 78, 80, 81, 82, 83, 84, 85] J = n g Be − WkbT , (45)where n g is the gas concentration, B is the kinetic factor (which can depend on P and T ), W > W is maximum at the given P and T ). Condition (44) yields the radius of such a nucleus as afunction of P = P r at T = T ∞ = const . Frequently, the dependence of the critical radius r on T at P = P ∞ = const is studied. Then [55, 81] r = 2 α lg T ∞ nq ( T ∞ − T ) , (46) here n is the concentration of atoms in a nucleus, q = T [ s g ( P, T ) − s l,c ( P, T )] is the latentheat of the phase transition per atom. As is seen, the higher the supercooling of a vapor, theless the nucleus radius.It is difficult to calculate the value of B in (45). Different models give different values.Within the classical approach (high T and large nuclei), J. Gibbs found W for a critical liquidnucleus [77], W l = ςα lg / , (47)and for a critical crystal nucleus, W c = X j ς j α j / . (48)Here, ς = 4 πr is the droplet surface area, j is the number of a crystal face, ς j is the area ofthe j -th face of a crystal, and α j is the coefficient of surface tension for the j -th face of thecrystal which contacts with the gas. It is useful to write formula (48) in the form [54] W c = ς ¯ α cg / , (49)where ¯ α cg is the average coefficient of surface tension of the crystal on the boundary with thegas, and ς is the area of a sphere with the volume equal to that of a crystalline nucleus. At T = T ∞ and P = P ∞ , the radius of a critical nucleus is r = ∞ . Therefore, J turns to zero,which corresponds to the equilibrium of phases.Apparently, the underliquid can be obtained easier by means of the isothermal compressionof a gas, than by its isobaric cooling. We now consider only the first way. The modern theorycannot exactly conclude whether the compressed gas will turn into a liquid or a crystal. Thisis not surprising, because the process of transition of one phase into another one is complexand depends on many factors.First, we note that W is less at the condensation of a gas on the surface, than at thecondensation in bulk [76, 78, 79, 82, 84]. In particular, the work of formation of a dome-shaped critical nucleus (liquid or crystalline) of radius r on a solid wall is [78, 79, 82, 85] W D = ( ςα/ θ ][1 − cos θ ] , (50)where ς = 4 πr , θ is the angle between the nucleus surface and the wall, α is the surfacetension of the nucleus that is in contact with a gas. At the complete nonwetting ( θ = π ), thevalue of W D = ςα/ W (47) or (49). If the wetting ispresent ( θ < π ), then W D < W l , W c , and a nucleus can be easier formed on the wall, than inbulk. Of course, crystallization is a complex process that is not reduced to the formation ofdome-like nuclei. However, formula (50) shows that a crystal nucleus can be easier formed onthe surface, than in bulk [54, 78, 81, 82, 84]. Therefore, if the gas contains solid impurities (orthe walls of a vessel contain some inhomogeneities able to become the centers of condensation), hen the surface condensation, rather than the bulk one, is realized. In practice, the impuritiesand inhomogeneities of walls are usually present. Moreover, W Dl < W l even for the ideallysmooth wall provided θ < π . Therefore, the condensation of a gas usually occurs on the wallsor on impurity particles.According to experiments, at T < T a gas condenses into a crystal. This is because thegas-crystal curve lies below the gas-liquid one (see Fig. 2). Microscopically, this means that W Dc corresponding to the formation of a two-dimensional critical crystal nucleus is less thanthe work W Dl of the formation of an analogous liquid nucleus. The reason for this is that thecrystalline structure of a substrate usually decreases W Dc and thus stimulates the formationof namely crystalline nuclei. In particular, the condensation of a gas into a crystal becomesmore intense, if a substrate on which the condensation occurs is a crystal of a close structure[76, 78, 80, 82, 84], because, in this case, W Dc decreases.In practice, the formation of crystal nuclei can be prevented if the gas is well purifiedfrom impurity particles and the vessel with very smooth walls is used. In addition, themicrostructure of walls of a vessel should be significantly different from the microstructureof a crystal, into which the gas can condense. Under these conditions, the condensation of agas into a liquid (on the walls or in bulk) should be dominant.Assume that the bulk homogeneous mechanism of spontaneous formation of nuclei isrealized. In this case, crystalline and liquid nuclei will arise. The rate of each of theseprocesses is given by formula (45), where W is determined by formulae (47) or (49). It isclear that J c ≪ J l at T → λ ≡ W c /W l > . (51)In this case, the condensation of a gas into droplets is more probable. Let us find theconditions under which relation (51) is satisfied. Formulae (44), (47), and (49) yield λ = ¯ α cg α lg n l n c [ln ( P/P lg ∞ ] [ln ( P/P cg ∞ ] , (52)where n l and n c are the concentrations of atoms in a microdrop and a microcrystal, respec-tively, at the same pressure P . We set P lg ∞ = ζ · P cg ∞ , ¯ α cg = (1 + η ) α lg , n c = (1 + ϑ ) n l ,and (1 + η ) (1 + ϑ ) − = (1 + φ ) . Here, P lg ∞ and P cg ∞ are the equilibrium pressures on thegas-liquid and gas-crystal curves, respectively. As a rule, | η | , | ϑ | ≪ 1. Therefore, | φ | ≪ P/P cg ∞ > ζ φφ . (53)That is, at T → P exceeding P cg ∞ by ζ φφ times.The quantity ¯ α cg can be estimated in the following way. By the rate of formation of crystalnuclei in a liquid, we can find ¯ α cl : usually, ¯ α cl ≈ (0 . ÷ . α gl (for temperatures close to he melting one; see Table III.1 in [54]). It is natural to assume that ¯ α cg = α gl + ˜ c ¯ α cl , where˜ c ≃ − 1, if the density of a crystal is less than that of a liquid, and ˜ c ≃ ϑ ≃ . α cg ≈ α gl + ¯ α cl , i.e., η ≃ . 15. However, for some substances (e.g., ice) η and ϑ are significantly different and can be negative. For the characteristic values η = 0 . 15 and ϑ = 0 . , we get φ ≈ . 1, and (53) gives P/P cg ∞ > ζ . For the inert elements, the triple pointcorresponds to P ∼ atm . Therefore, at T ≪ T we have P cg ∞ ≪ atm . According to theanalysis in Sect. 3, at T ≪ T the value of ζ is close to 1. Therefore, the pressure P > ζ P cg ∞ at which the gas should condense into droplets is quite achievable.It was asserted in some works [78, 84] that, for the vapor–crystal and vapor–liquid tran-sitions, one needs to set B = B ′ e Cq/ ( k b T ) in formula (45). Here, the constant C depends onthe mechanism ( | C | ≃ q is the latent heat of sublimation or evaporation, and B ′ mayslightly depend on T. Above, we neglected the factor e Cq/ ( k b T ) . This is justified, if the phasetransition occurs at a not too high supersaturation (in this case, the critical radius r is large,and so, W ≫ | C | q ).For the surface mechanism of formation of nuclei, the formulae are significantly morecomplicated, especially for crystalline nuclei. In the last case, the work W c depends also onthe relationship of the crystalline structures of a nucleus and the substrate [76, 78, 80, 82, 84].We did not make estimates for this case. Most likely, the ratio P/P cg ∞ is not too different from(53). Therefore, if the microstructures of the wall and crystal nuclei are strongly differentand the wall is very smooth, we may expect that at the pressure P > (2 ÷ P cg ∞ the surfaceformation of liquid nuclei is more probable, than the surface formation of crystal nuclei. Inthis case, the gas should condense into a liquid when compressed. Moreover, if atoms of a gasinteract weakly with atoms of the walls, then the bulk formation of nuclei (drops or crystals)should be more intense, as compared with the formation of nuclei on the walls.Our analysis is rather crude, but the main conclusions are apparently qualitatively right.Thus, the experiment on gas compression should be carried out with different walls of vesselsat several different temperatures T ≪ T . The condensation of a gas into a liquid have tobe more probable than the crystallization, provided that (i) the microstructures of the walland crystal nuclei are significantly different (or atoms of the gas interact weakly with atomsof the wall and relation (53) holds) and (ii) the gas is purified from impurities. The weakinteraction of atoms of a gas with walls can apparently be obtained if the walls are coveredby a microlayer of helium-II. Perhaps, this is the simplest way to obtain the underliquid. References [1] K. Mendelssohn, The Quest for Absolute Zero: the Meaning of Low Temperature Physics (McGraw-Hill, New York, 1966). 2] N. Bernardes, H. Primakoff, Phys. Rev. , 968 (1960).[3] E.M. Saunders, Phys. Rev. , 1724 (1962).[4] L.H. Nosanow, G.L. Show, Phys. Rev. , 546 (1962).[5] K.A. Brueckner, J. Frohberg, Progr. Theoret. Phys. (Kyoto), Suppl. 383 (1965).[6] L.H. Nosanow, Phys. Rev. , 120 (1966).[7] R.A. Guyer, Solid State Phys. , 413 (1970).[8] A.G. Leggett, Quantum Liquids (Oxford Univ. Press, New York, 2006), Chap. 8.[9] C. Cazorla, J. Boronat, Rev. Mod. Phys. , 035003 (2017).[10] A. Bijl, Physica , 869 (1940).[11] R. Jastrow, Phys. Rev. , 1479 (1955).[12] C.-W. Woo, Phys. Rev. A , 2312 (1972).[13] E. Feenberg, Ann. Phys. , 128 (1974).[14] L. Penrose, O. Onsager, Phys. Rev. , 576 (1956).[15] N. Prokof’ev, Adv. in Phys. , 381 (2007).[16] D.M. Ceperley, G.V. Chester, M.H. Kalos, Phys. Rev. D , 3208 (1976).[17] D. Ceperley, G.V. Chester, M.H. Kalos, Phys. Rev. B , 1070 (1978).[18] E.P. Gross, Ann. Phys. , 57 (1958).[19] E.P. Gross, Phys. Rev. Lett. , 599 (1960).[20] A. De Luca, L.M. Ricciardi, H. Umezawa, Physica , 61 (1968).[21] A. Coniglio, M. Marinaro, B. Preziosi, Nuovo Cimento B , 25 (1969).[22] D.A. Kirzhnits, Yu.A. Nepomnyashchi˘i, Sov. Phys. JETP , 1191 (1971).[23] Yu.A. Nepomnyashchii, Theor. Math. Phys. , 928 (1971).[24] M. Tomchenko, J. Low Temp. Phys. , 100 (2020).[25] T.R. Koehler, Phys. Rev. Lett. , 89 (1966).[26] S. Vitiello, K. Runge, M.H. Kalos, Phys. Rev. Lett. , 1970 (1988).[27] K.E. Schmidt, D.M. Ceperley, Monte Carlo techniques for quantum fluids, solids anddroplets , in Monte Carlo Methods in Condensed Matter Physics , ed. by K. Binder,Topics in Applied Physics, vol. 71, pp. 205–248 (Springer, Heidelberg, 1992). 28] J. Sato, R. Kanamoto, E. Kaminishi, T. Deguchi, arXiv:1204.3960 [cond-mat.quant-gas].[29] K. Sacha, J. Zakrzewski, Rep. Progr. Phys. , 016401 (2018).[30] N.N. Bogoliubov, D.N. Zubarev, Sov. Phys. JETP , 83 (1956).[31] I.A. Vakarchuk, I.R. Yukhnovskii, Theor. Math. Phys. , 73 (1980).https://doi.org/10.1007/BF01019263[32] M. Tomchenko, accepted in J. Low Temp. Phys. ; arXiv:1905.03712 [cond-mat.quant-gas].[33] R.P. Feynman, Statistical Mechanics: A Set of Lectures (W. A. Benjamin, Mas-sachusetts, 1972).[34] R. Courant, D. Hilbert, Methods of Mathematical Physics , Vol. 1 (Interscience, NewYork, 1949), Chap. 6.[35] I.A. Vakarchuk, I.R. Yukhnovskii, Theor. Math. Phys. , 626 (1979).https://doi.org/10.1007/BF01019246[36] M.D. Tomchenko, Ukr. J. Phys. , 250 (2019).[37] M. Girardeau, J. Math. Phys. (N.Y.) , 516 (1960).[38] E.H. Lieb, W. Liniger, Phys. Rev. , 1605 (1963).[39] M. Gaudin, Phys. Rev. A , 386 (1971).[40] M. Tomchenko, J. Phys. A: Math. Theor. , 365003 (2015).[41] M.D. Tomchenko, Dopov. Nac. Akad. Nauk Ukr. No. 12, 49 (2019)https://doi.org/10.15407/dopovidi2019.12.049[42] M.A. Cazalilla, J. Phys. B: At. Mol. Opt. Phys. , S1 (2004).[43] M.D. Tomchenko, Ukr. J. Phys. , 720 (2005).[44] M. Tomchenko, arXiv:1201.2623 [cond-mat.other].[45] C. Cazorla, J. Boronat, J. Phys. Cond. Mat. , 015223 (2008).[46] L.D. Landau, E.M. Lifshitz, Statistical Physics , Part 1 (Pergamon Press, Oxford, 1980).[47] I.M. Khalatnikov, An Introduction to the Theory of Superfluidity (Perseus Pub., Cam-bridge, 2000).[48] V.M. Glazov, V.B. Lazarev, V.V. Zharov, Phase Diagrams of Simple Substances (Nauka,Moscow, 1980) [in Russian]. 49] D.G. Naugle, J. Chem. Phys. , 5730 (1972).[50] A. van Itterbeek, W. Grevendonk, W. van Dael, G. Forrez, Physica , 1255 (1959).[51] B.N. Esel’son, V.N. Grigor’ev, V.G. Ivantsov, E.Ya. Rudavskii, D.N. Sanikadze, I.A.Serbin, Properties of Liquid and Solid Helium (Izd. Standartov, Moscow, 1978) [inRussian].[52] K. Huang, Statistical Mechanics (Wiley, New York, 1963).[53] V.A. Rabinovich, A.A. Vasserman, V.I. Nedostup, L.S. Veksler, Thermophysical Proper-ties of Neon, Argon, Krypton, and Xenon (Izd. Standartov, Moscow, 1976) [in Russian].[54] R.F. Strickland-Constable, Kinetics and Mechanism of Crystallization (Academic Press,London, 1968).[55] J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955).[56] J. Frenkel, Acta Physocochimica URSS , 913 (1935).[57] N.P. Hammond, A.C. Barr, E.M. Parmentier, Geophys. Res. Lett. , 6775 (2016).[58] D.P. Hamilton, S.A. Stern, J.M. Moore, L.A. Young, Nature , 97 (2016).[59] C.J. Bierson, F. Nimmo, S.A. Stern, Nature Geoscience , 468 (2020).[60] N.N. Bogoliubov, Lectures on Quantum Statistics. Quasi-Averages (Gordon and Breach,New York, 1970).[61] B.M. Abraham, Y. Eckstein, J.B. Ketterson, M. Kuchnir, P.R. Roach, Phys. Rev. A ,250 (1970).[62] P.R. Roach, J.B. Ketterson, C.-W. Woo, Phys. Rev. A , 543 (1970).[63] D.O. Edwards, R.C. Pandorf, Phys. Rev. , A816 (1965).[64] E.R. Grilly, J. Low Temp. Phys. , 33 (1973).[65] J. Pomeranchuk, Zhur. Eksp. Theor. Fiz. , 919 (1950).[66] W.H. Keesom, Helium (Elsevier, Amsterdam, 1942).[67] K. Mendelssohn, Liquid helium , in Low Temperature Physics II , ed. by S. Fl¨ugge(Springer, Berlin, 1956), pp. 370–461.[68] D.M. Ceperley, Rev. Mod. Phys. , 279 (1995).[69] Multidimensional Quantum Dynamics: MCTDH Theory and Applications , ed. by H.-D. Meyer, F. Gatti, G.A. Worth (Wiley-VCH, Weinheim, 2009).[70] F. Deuretzbacher, J.C. Cremon, S.M. Reimann, Phys. Rev. A , 063616 (2010). 71] S. Z¨ollner, G.M. Bruun, C.J. Pethick, S.M. Reimann, Phys. Rev. Lett. , 035301(2011).[72] S. Z¨ollner, Phys. Rev. A , 063619 (2011).[73] B. Chatterjee, A.U.J. Lode, Phys. Rev. A , 053624 (2018).[74] S. Bera, B. Chakrabarti, A. Gammal, M.C. Tsatsos, M.L. Lekala, B. Chatterjee,C. L´evˆeque, A.U.J. Lode, arXiv:1806.02539 [cond-mat.quant-gas].[75] W.M.C. Foulkes, L. Mitas, R.J. Needs, G. Rajagopal, Rev. Mod. Phys. , 33 (2001).[76] N. Fuks, Usp. Fiz. Nauk , 496 (1935) [in Russian]. 10.3367/UFNr.0015.193504c.0496[77] J.W. Gibbs, The Collected Works , v. 1 Thermodynamics (Longmans, New York, 1928).[78] M. Volmer, Kinetik der Phasenbuildung (Steinkopff, Dresden, 1939).[79] V.D. Kuznetsov, Crystals and Crystallization (GITTL, Moscow, 1954) [in Russian].[80] V.I. Danilov, Structure and Crystallization of a Liquid (AN UkrSSR, Kiev, 1956) [inRussian].[81] B. Chalmers, Principles of Solidification (Wiley, New York, 1964).[82] M.C. Flemings, Solidification Processing (McGraw-Hill, New York, 1974).[83] V.P. Skripov, E.N. Sinitsyn, P.A. Pavlov, G.V. Ermakov, G.N. Muratov, N.V. Bulanov,V.G. Baidakov, Thermophysical Properties of Liquids in a Metastable State (Atomizdat,Moscow, 1980) [in Russian].[84] D. Kashchiev, Nucleation (Butterworth-Heinemann, Oxford, 2000).[85] K.F. Kelton, A.L. Greer, Nucleation in Condensed Matter , Pergamon Materials Series,vol. 15 (Elsevier, Holland, 2010)., Pergamon Materials Series,vol. 15 (Elsevier, Holland, 2010).