Leggett's Bound for Amorphous Solids

Leggett’s bound for amorphous solids Giulio Biroli, Bryan Clark, Laura Foini, Francesco Zamponi

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Giulio Biroli, Bryan Clark, Laura Foini, Francesco Zamponi. Leggett’s bound for amorphous solids. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2011, 83, pp.094530. 10.1103/PhysRevB.83.094530. cea-02524971

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:1011.2863v2 [cond-mat.dis-nn] 31 Mar 2011 vdne fvr lwdnmc,tehlmr fglassy of hallmark showed the [10] dynamics, in slow experiments very the of solids, fact evidences disordered In very glasses. produce used possibly likely protocols “super- Helium the experimental solidify this the to Actually, labeled al. phase. et glass” equilibrium Boninsegni the into solid. freezes ordered sweeps eventually was Carlo Monte system phase of the this number large before 60%; a as for high last to as observed frac- density a with superfluid and of phase liquid, tion a the to of similar observation structurally the is reported which They stable. is phase a u owr yBnnen ta.[] h possibility the [6]: al. et a Boninsegni of by forward put was not still is solids quantum between in clear. fre- relationship superfluidity and the and theoret- studies, temperature disorder of experimental series of long and a function ical of a spite in as Consequently, quency. inertia mo- the induce of modify to that not ment sample instead is the but disorder in fraction states of non-equilibrium superfluid role the the enhance that to experiments. suggest the [9] in studies role Other important an play must disorder o temperature low a nadto,terl fHe 20%. of of order role the the on extremely addition, density, an In superfluid to in of corresponds fraction change that high a inertia fast of with that moment samples showed the theoretical 8] disordered [7, earlier produce super- Reppy Following quenches and of Ritner formation samples. [4–6], the analyses question solid foster important in can the fluidity disorder others, whether many of among raised, 3] unhdfo h qiiru iudpaea high at phase liquid equilibrium the from quenched ueia iuaino eim4a eaieyhg den- high relatively at ( 4 sity (PIMC) Helium Carlo of Monte simulation Integral numerical Path a performed authors eeteprmnso oi He solid on experiments Recent eew att ou noepriua rpslthat proposal particular one on focus to want we Here ukln-ie eatbegasphase glass metastable long-lived bulk ρ ∼ 0 1 . 3 ntttd hsqeTóiu Ih) E,adCR R 23 URA CNRS and CEA, Théorique (IPhT), Physique de Institut 03 PES NSUR84,ascé` ’PCPrs0,2 u Lh Rue 24 06, Paris associée à l’UPMC 8549, UMR CNRS LPTENS, n oprsn oohrrslso h ue-ls hs ar phase super-glass i the being on results of other instead pa to equilibrium, 4 comparisons from Helium and far in evolving fi fractions samples We superfluid to large Over transition. of glassiness. observations freezing intermediate tal of the sign of any sup without density quickly, quen of Moreove the rapidly terms at 4 ratio. in Helium temperature, difference Lindemann distinguishable much of same simulations not the Carlo is by there characterized bound, crystal upper the of inflcutosaon lsia eut.Teeruhest rough mann These different thi of results. accomplish variety classical to a around order in fluctuations In solids sian Leggett. amorphous by for derived generated bound upper an A ˚ eivsiaetecntanso h uefli rcinof fraction superfluid the on constraints the investigate We 2 rneo etrfrTertclSine rneo Unive Princeton Science, Theoretical for Center Princeton .INTRODUCTION I. − 3 ,weetesse a eyquickly very was system the where ), T 4 IS n NN ein iTise i ooe 6,I-34136 265, Bonomea via Trieste, di Sezione INFN, and SISSA 0 = iloBiroli, Giulio . ,a hc h C solid HCP the which at K, 2 3 muiis[]sgetthat suggest [3] impurities egt’ on o mrhu solids amorphous for bound Leggett’s 4 yKmadCa [1– Chan and Kim by 1 ra Clark, Bryan fHe of 4 These . 2 T ar Foini, Laura to uefliiyi mrhu oisadt xli h nu- the results. explain of experimental to understanding and and merical microscopic solids better amorphous a in superfluidity reach to between investigation, order relation further in deserves the superfluidity Clearly, interacting in and [13]. disorder fraction of phase condensate model superglass small realistic) the extremely In- yet an found density. (and phase bosons superglass particle simplified the same a of the in investigation at theoretical crystals a comparable of deed, density one superfluid glasses the of dense to fraction that a above, have discussed con- should findings local expect, the might the one to on Thus, trary mostly exchange, particle. depends a to and of related process condensate neighborhood local is small a very Superfluidity is or which 12]. zero show [11, in- to fractions which why thought case, is is crystalline enhance stead question the should to profile open compared density still superfluidity amorphous and an in natural freezing The behavior. uaeifrain ntegastasto nquantum in transition ac- glass obtaining the on [15], informations simulations curate (PIMD) Molec- Integral Dynamics Path has with ular glasses compared in and dynamics developed of been Theory version Mode-Coupling quantum dynam- a the recently, PIMC of More whose the equilibrates. system via at slowly a 4 ics looked for Helium has it quenched a calculating [6] of directly al. density superfluid et of nu- Boninsegni fraction Previous involv- of problematic. properties work is merical calculating system and quantum of glassy because problem, dynamics ing numerically sign real-time accessible the the not of contrast, is equations In systems Newton’s physical quantum solving [14]. the by simulate motion system easily of the can reasonably phase one of is metastable because dynamics it a glass, system, of a classical properties or a get phase In to equilibrium straightforward true Helium. the phase is solid respect which glass with of phase, metastable the crystal be that the to to fact expected always the is any) from (if comes problem this h andffiut ntenmrclivsiainof investigation numerical the in difficulty main The l u eut ugs htteexperimen- the that suggest results our all ,4 3, mtssgetta,a es ttelevel the at least at that, suggest imates hdfo h iudpaet eylower very to phase liquid the from ched tbegaspae te scenarios Other phase. glass stable a n tce fe ai unhcorrespond quench rapid a after rticles lodiscussed. also e n rnec Zamponi Francesco and dta h ytmcytlie very crystallizes system the that nd r nldn yivsiaigGaus- investigating by including ers st,Pictn J054 USA 08544, NJ Princeton, rsity, ,w efr ahItga Monte Integral Path perform we r, naopossldfloigfrom following solid amorphous an ,w s siptdniyprofiles density input as use we s, 6 -19 i-u-vte France Gif-sur-Yvette, F-91191 06, rudt ewe ls n a and glass a between erfluidity mn,705Prs France. Paris, 75005 omond, ret,Italy Trieste, 3 2 hard spheres. However, in this study exchange effects density ρs by: were neglected and therefore superfluidity could not be 2 investigated. Therefore, for the moment path integral ρs 1 ∂ Emin(ω) = lim 2 simulations are not conclusive. ρ ω→0 I0 ∂ω Here we approach the problem in a different way. In 2 one of the first works on supersolidity, Leggett showed where ρ is the particle density and I0 = NmR the how one can derive an upper bound for the fraction of su- classical moment of inertia. From this expression it is perfluid density of a generic many-body system in which clear that upper bounds on the superfluid density can be translational invariance is broken, by means of a varia- obtained by using variational wavefunctions that in the tional computation [16]. The output of Leggett’s compu- ω 0 limit tend to the wavefunction for a non-rotating → tation is a formula that needs as only input the average container. Leggett used a variational wavefunction of density profile of the solid. This formula has been ap- the form Ψ(~r1, , ~rN )=Ψ0(~r1, , ~rN ) exp[i ϕ(~ri)], ··· ··· i plied to Helium crystals, and the aim of this work is to where Ψ0 is the ground state wavefunction for the non- P use it to study the amorphous solid. At present, there is rotating case and φ = i ϕ(~ri) a sum of phases satisfying not yet any reliable first principle computation or experi- the condition ϕ(θ)= ϕ(θ+2π) 2πmR2ω/~ [16, 17]. The P − mental measurements of the density profile of amorphous bound can be improved by including two-body correla- Helium 4. We endeavor to generate robust estimates of tions [18]. Defining it using a number of different techniques, in particular by investigating a model of zero-point Gaussian fluctuations ρ(~r)= d~r d~r Ψ (~r , , ~r ) 2 δ(~r ~r ), (1) 1 ··· N | 0 1 ··· N | − i around classical configurations, and PIMC simulations Z i without exchange (which should be closer to the classical X dynamics). Checking whether these techniques all give which is the density profile in the ground state, one finds roughly similar orders for the bound is a way to assess that the variational estimation of Emin(ω) reads: the robustness of our result. In the following, we will ~2 denote the fraction of superfluid density by “superfluid E (ω)= E + d~r[ ϕ(~r)]2ρ(~r), (2) fraction” and we always refer to Leggett’s upper bound min 0 2m ∇ to this quantity, unless otherwise specified. Z The rest of this paper is organized as follows. In sec- where E0 is the ground state energy in the non-rotating tion II, we discuss how to adapt Leggett’s bound to an case. amorphous solid. In section III A, we compute the bound Because of the assumption that the thickness of the for a profile made of Gaussian fluctuations around a clas- cylinder is much smaller than the radius, one can simplify sical configuration, and compare the results for an amor- the problem even further by “unrolling” the annulus and phous and an ordered solid, while in section III B we dis- consider the system inside a parallelepiped of length L = cuss previous numerical computations [6]. In section IV 2πR in the x direction. In this geometry the phase ϕ has we try to obtain more precise information by comparing to satisfy the boundary condition ϕ(0,y,z)= ϕ(L,y,z) ~ − a classical simulation of a glass-forming system with a v0L where v0 = mRω/ . The minimization of (2) with PIMC numerical simulation of Helium. In section V, we respect to ϕ leads to the equation for ϕ(~r): show that under some approximations one can obtain a ~ [ρ(~r) ϕ(~r)]=0 (3) formula for the bound that can – at least in principle – ∇ · ∇ be computed from neutron or X-ray scattering data. and results in an upper bound on the superfluid density:

1 II. LEGGETT’S BOUND ρ = d~rρ(~r) ϕ(~r) 2 . (4) s V v2 |∇ | 0 ZV

Leggett showed in his pioneering work on supersolidity Note that if ϕv0 (~r) is a solution of (3) with bound- that the wavefunction of the ground state of a system of ′ ary conditions ϕ(0,y,z) = ϕ(L,y,z) v0L, then ϕv0 = ′ − bosonic particles inside a rotating cylindrical container (v0/v0)ϕv0 is a solution with boundary conditions cor- ′ can be obtained by finding the ground state for the non- responding to v0. Hence, Eq. (4) does not depend on rotating system but with new boundary conditions [16]. v0 and we can choose v0 = 1 without loss of generality. Using cylindrical polar coordinates and assuming that Furthermore, while in the geometry described above the the thickness of the cylinder is much smaller than the wavefunction should satisfy hard wall conditions at the radius R, the new boundary conditions correspond to boundary of the box in the y and z directions, we will imposing that the wave function gets an extra phase fac- simplify the problem by considering periodic boundary 2 tor exp( 2πimR ω/~) when the angle θi of any particle conditions in the y and z directions [19]. − i is shifted by 2π. Here m is the particle mass and ω In order to find a solution of Eq. (3) satisfying the the radial velocity. From the ω dependence of the en- correct boundary condition is useful to rewrite ϕ as ergy of the ground state, Emin(ω), obtained with these new boundary conditions one can compute the superfluid ϕ(~r)= ~v ~r + δϕ(~r), (5) 0 · 3 where δϕ(~r) is defined inside the volume V and satisfies 1 periodic boundary conditions, and ~v0 is a unit vector. In the original problem ~v0 =x ˆ, but since we reformu- lated the problem in a periodic cubic box, the direction 0.8 of ~v0 can be varied without affecting the result, in the limit V . Since δϕ(~r) is periodic, we can write the → ∞ equations in Fourier space (see Appendix A for details): 0.6 ρ ρ s/ ~q ~v0ρ~q = (~q ~p)ρ~q−~p iδϕ~p , (6) · · 0.4 X~p=6 ~0 and from the solution for iδϕ~q one can obtain the Leggett bound [17], that reads in Fourier space: 0.2 N=20 glass N=100 glass ρ 1 fcc s =1 (~v ~q)iδϕ ρ . (7) 2 0 ~q −~q 0 ρ − ρv0 · 0.1 0.21/3 1/2 0.3 0.4 X~q=6 ~0 ρ A

Given the density profile, the linear equation (6) for iδϕ~q can be solved by truncating the sum over momenta at FIG. 1: Leggett upper bound for ρs/ρ, for a Gaussian profile of width A1/2 around an amorphous jammed configuration a given cutoff, ~q < qmax, so that the problem reduces | | and in a FCC lattice, as a function of the adimensional pa- to solving a finite set of linear equations, which can be 1 3 1 2 rameter ℓ = ρ / A / (the Lindemann ratio). done by matrix inversion. We accomplish this via a LU decomposition [20]. An important remark is that the truncation preserves the variational nature of the computation. Indeed, it hard spheres. Although this certainly is not a realis- can be seen as setting δϕ = 0 for ~q q , which tic model of density profiles for He4, it allows us to ad- ~q | | ≥ max amounts to a particular choice of the variational function dress the role of disorder on ρs. Furthermore, a mapping δϕ(~r) and hence still gives an upper bound on the true from quantum systems at zero temperature and classi- superfluid fraction. cal Brownian systems allows one to find quantum many Another important remark is that the bound derived particle models whose ground state wave-function can be above applies only, strictly speaking, to the true ground mapped exactly on (the square of) the probability distri- state of the system. In the following however, we are in- bution of classical hard spheres systems [13]. Thus, the terested in applying it to the glass state, which is at best results of this section apply directly to those models. a long-lived metastable state, the crystal being always Classical hard spheres are known to be characterized the true ground state. Still, it is clear from the deriva- by a high density crystal FCC phase. However, if com- tion that if the life time τ of the state is very long, such pressed fast enough, or due to a small polydispersity, the that for any experimentally accessible frequency one has hard spheres freeze in an amorphous glassy state. A typ- ωτ 1, then the system does not have time to escape ical density profile of a very quickly compressed glassy from≫ the metastable state during the experiment and the state can be obtained by the Lubachevski-Stillinger com- bound should apply without modification. pression algorithm [21] (we used the implementation of [22]), which is know to be very efficient in producing amorphous jammed configurations. The output of the III. SUPERFLUID FRACTION OF algorithm are the positions = R1, , RN of the AMORPHOUS SOLIDS particles in a random close packedR state{ ··· (at infinite} pressure). The algorithm is deterministic, but different final A. Hard sphere systems configurations are obtained by starting the compression from random initial configurations of points. The com- In order to understand whether disorder in the density pression runs were performed at very fast rates (we fixed profile can lead to an increase of the superfluid density, the parameter γ = 0.1, see [22, 23] for details) in order we shall compare the result of the bound for an amor- to avoid crystallization. phous glassy profile and the corresponding crystal. The Furthermore, we will assume that the density profile of only input for our study are the density profiles of the a typical glassy configuration at finite pressure is the sum amorphous and crystal state. Unfortunately, the former of Gaussians centered around the amorphous sites, which is not available for He4 in realistic conditions. As a con- are the output of the previous algorithm. For classical sequence, we decided for a first study to focus on a more systems, this assumption has been tested numerically for simple and academic case that can still provide insights FCC crystals [24], and has been often used in density on the role of disorder. We consider the amorphous and functional computations of both ordered [25] and amor- crystalline density profiles that one obtains for classical phous structures [23, 26], giving accurate results. For 4 quantum systems, the Gaussian model has been shown To conclude this section, we observe that the above re- to be accurate enough, at least for the purpose of com- sults allow to obtain a quantitative upper bound for the puting the Leggett’s upper bound [27–29]. superfluid fraction of a system whose wavefunction is ex- For a given configuration , the density profile we use actly the Jastrow wavefunction corresponding to classical is defined as R hard spheres. The quantum glassy phase of this system has been discussed in [13]. In both the crystal and glassy phases, the values of A1/2 for classical hard spheres do ρ(~r )= γA( ~r R~ i )= d~sγA( ~r ~s ) δ(~s R~ i) , |R | − | V | − | − not exceed 0.1 (in units of the sphere diameter) [23–25], i Z i X X (8) and the same is true for ℓ, since the density is very close 2 3/2 where γA(~x) = exp( ~x /(2A))/(2πA) is a normal- to 1 (in the same units) in both solid phases. Using the −| | results of Fig. 1, we obtain an upper bound ρ /ρ < 0.1%, ized Gaussian of width A, and ~r R~ i is the distance on s the periodic box, i.e. it is the distance| − | between ~r and its which is consistent with the extremely small values∼ of the condensate fraction found in [13]. closest image of R~ i. The corresponding Fourier transform reads (neglecting terms of order exp( L2/A)): − 2 1 ~ B. Superfluid fraction of amorphous solid Helium 4 ρ ( )= e−Aq /2 ei~q·Ri . (9) ~q R V i X In this section, we attempt an application of our results 4 In solving Eqs. (6) and (7) we considered amorphous to the more interesting case of disordered solid He , based configurations of N = 20 and N = 100 particles. All on the observation above, that an estimate of the Linde- mann ratio ℓ = ρ1/3A1/2, together with the results of the calculations were done with the cut-off set at qmax = 20π/L. We checked that the result does not depend on Fig. 1, should provide a reasonable estimate of Leggett’s the specific amorphous configuration used by considering bound. different amorphous configurations α, α = 1, , ; At the end of Ref.[29] it is stated that, by fitting this is expected since the superfluidR density is a··· macro-N the Path Integral Monte Carlo density profile of HCP 4 √ scopic quantity. The reported results are therefore av- solid He , one obtains a value A = 0.1274 d at ρ = ˚−3 √ ˚−3 eraged over 10 independent configurations. More details 0.0353 A and A = 0.1486 d at ρ = 0.029 A . on the numerics can be found in Appendix A. Here d is the nearest-neighbor distance for the HCP lattice. The number density of the HCP lattice satis- The results are plotted in Figure 1. One can notice fies the relation ρd3 = √2, hence d = 21/6/ρ1/3 and that, apart from the smallest values of the dimension- √ 1/3 1/6√ less parameter, the two curves corresponding to 20 and ℓ = Aρ =2 A/d. In the same reference it is also 100 particle configurations perfectly agree. The discrep- stated that the upper bound computed by using the fitted ancy in the region of small ℓ = ρ1/3A1/2 is due to the Gaussian density profile coincides with the one obtained approximation brought by the introduction of a cut-off, by using the true PIMC density profile, and corresponds respectively to ρs/ρ = 0.06 and 0.22. These values are and vanishes in the limit qmax 1/√A. In order to understand to what≫ extent the disorder in- reported in table I. fluences the value of the superfluid density, we compare We now make the following assumptions: the superfluid fraction found in the amorphous system 1. At least for the purpose of computing Leggett’s up- to the values obtained through the same calculations in per bound, the true density profile can be fitted to the case of a crystal [17, 27–29]. Figure 1 reports the re- a Gaussian profile. This is true for the crystal [29] sults for the average superfluid fraction of the amorphous and we assume that it remains true for an amor- solid just described and those corresponding to the FCC phous solid. lattice (which is the thermodynamically stable one for hard spheres) for the R~ i, according to the same Gaus- 2. The parameter ℓ for the amorphous solid is smaller sian model (in the latter case our results are consistent than that of the crystal at the same density. This with previous ones [17, 27–29]). The difference between can be understood by observing that crystalline the two is very small, suggesting two conclusions. configurations are better packed than amorphous configurations, therefore leaving more room (“free 1. Disorder does not influence much the superfluid volume”) for fluctuations. It is true for Jastrow behavior of the system for comparable values of wavefunctions [13] (i.e. classical system) and we ρ1/3A1/2, at least at the level of this variational do not find any reason why quantum fluctuations calculation. should dramatically affect this property.

2. The dependence of ρs on the density proﬁle is Based on these assumptions, the true Leggett’s bound mainly through the Lindemann ratio ℓ = ρ1/3A1/2. for the amorphous system should be smaller than the This conjecture allows us to obtain an estimate of same bound for the crystal at the same density. This can the Leggett upper bound for ρs in more realistic be estimated using the values of ℓ reported in [29] and cases as we will do in the next section. reading the corresponding superﬂuid fraction from Fig. 1 5

HCP, Leggett’s bound (Ref.[29]) Glass, Leggett’s bound (this work) Glass, QMC (Ref. [6]) −3 ρ (A˚ ) ℓ ρs/ρ ρs/ρ ρs/ρ 0.029 0.167 0.22 0.282 0.6 0.0353 0.143 0.06 0.127 0.07

TABLE I: Leggett’s bound for He4 in the HCP crystal state [29] and glassy state. Quantum Monte Carlo results for the glass are also reported [6]. or using the results obtained in [29] for the HCP crystal. We quenched a dense (ρ = 1.2) system of N = 216 These values are reported in table I and are similar. particles from very high temperature (T = 2) to very We compare the upper bound obtained in this way with low temperature (T =0.05) deep in the glass phase (the the values of ρs obtained numerically by Boninsegni et glass transition temperature being around T = 0.435 al. via PIMC [6]. Interestingly, we find that the bound is at this density [14]). We run the simulation for a to- very close to the PIMC numerical result, and in particu- tal time τ = 15000 and we printed configurations every lar at the smallest density the bound is violated by the ∆t = 5 which is of the order of the decorrelation time in PIMC result. This can be due either to the very rough the glass (estimated from the decay of the self scattering approximations involved in our computation, or to the functions). From each configuration we deduced fact that the glass is not a really long-lived metastable 1 state at this very low density. The latter possibility, i.e. ρ (t)= ei~q·~rj (t) , (10) ~q V that the system is rapidly evolving out of equilibrium, j would invalidate the derivation of Leggett’s bound but X it would also raise problematic questions regarding the where ~rj (t) is the position of particle j at time t, and the measurement of ρs using the Ceperley formula, which is corresponding instantaneous value of the static structure strictly valid if thermodynamic equilibrium is achieved factor S (t)= V ρ (t) 2/ρ. ~q | ~q | and in the limit of small frequency. In Fig. 2 we plotted ρ~q(t) and the structure factor S~q(t) as a function of MD time after the quench. The vectors ~q =2π/L(nx,ny,nz) and the corresponding integers are IV. DOES A STABLE GLASS STATE EXISTS given in the caption. We see that after a short tran- FOR HELIUM 4? sient, the density profiles fluctuate around a non-zero value which is quite stable, except for some rare “crack” events where the density changes abruptly. These are In order to study the stability of the glass phase in He- probably due to groups of particles that switch back and lium 4, we performed Path Integral Monte Carlo simula- forth between two different locally stable configurations. tions, that we discuss in this section. Before discussing This system is indeed extremely dense and at very low the more complex quantum simulation, we present some T , therefore its dynamics is basically that of harmonic classical simulations in order to deal with a well con- vibrations around local minima of the potential (except trolled situation, where the presence of a glass transition for the rare cracks). The largest instantaneous value of has been firmly established. S~q(t) corresponds to the (2, 1, 6) curve in Fig. 2 for all t > 1000; therefore, all values− are smaller than 20 at all times, showing that there are no Bragg peaks. This is 1. What should we expect from a glass-forming system? A what we expect to see in a glass. In this case, we can classical simulation easily deduce the average values of ρ~q for a given glassy configurations by taking the average of ρ~q(t) over a time We performed standard Molecular Dynamics (MD) interval where there are no crack events. From these, simulations of the Kob-Andersen binary mixture [14], we could compute the Leggett bound as previously dis- which is known to be a good glass former and does not cussed. show any sign of crystallization even after very long MD runs at low temperature. The latter is a mixture of two types of particles (A and B), interacting through different 2. Absence of a stable glass phase from a Path Integral Lennard-Jones potentials, with the parameters specified Monte Carlo simulation in [14]. In the rest of this section we use reduced Lennard- Jones units, namely we use σAA and εAA as units of Motivated by results of [6] we tried to compute the length and energy, and m as unit of mass. Consequently, superfluid fraction based directly on Path Integral Monte 2 mσAA/εAA is the unit of time (the latter convention Carlo data. Unfortunately, PIMC does not give access to is slightly different from the one of [14]). Note that to the real time dynamics of the system, but following [6] pcompare with Helium one should keep in mind that for we studied the Monte Carlo dynamics, in the hope that that system σ 2.56 A˚ and ε 10.2 K. this is a reasonable proxy for the real time dynamics. ∼ ∼ 6

(1,7,0) 60 25 (0,0,8) (0,0,6) (2,1,-6) (5,0,4) 50 (5,4,2) 20

40 15 Sq(t) Sq(t) 30

10 20

5 10

0 0 0 5000 t 10000 15000 0 50000 t 100000

0.2 0.03 ρ (0,0,8) Re q(t) (0,0,6) ρ Re q(t) 0.02 0.1 0.01

0 0

-0.01 -0.1 ρ Im q(t) ρ -0.02 Im q(t)

-0.2 -0.03 0 5000 t 10000 15000 0 50000 t 100000

20 50

40 15

S(q) 30 10 S(q)

5 10

0 0 0 5 q 10 15 0 1 2q 3 4

FIG. 2: Evolution of the density proﬁle after a quench from FIG. 3: Evolution of the density proﬁle after a quench from high to low temperature for a classical glass forming sys- high to low temperature for a quantum Helium 4 system, tem, using Molecular Dynamics. (Top) Instantaneous value using Path Integral Monte Carlo. Time here represents the of S~q (t) for three representative values of ~q (the corresponding number of Monte Carlo sweeps. The panels are the same as (nx, ny, nz) are indicated in the caption). (Middle) Instanta- in Fig. 2, except that the average of S~q(t) in the lower panel neous values of ρ~q(t) for a representative value of ~q. (Bottom) has been taken for t> 75000, and the angular average is not The time average of S~q(t) over the whole simulation, as a func- reported because of the strong anisotropy of the result. All tion of q (in reduced LJ units). Scatter points are values for quantities are plotted using A˚ as units of length. a given ~q, the full black line is the angular average over all vectors with the same modulus. 7

The representation of quantum systems in PIMC in- Note that in the quantum case, at variance with the clas- volves certain important extensions beyond the classical sical case, these two quantities are not directly related. representation of point particles. To begin with, particles At each PIMC sweep we recorded the values of the above are represented by paths (or polymers) in space. These quantities, which we then averaged over 50 PIMC sweeps paths manifest the zero point motion inherent in the in order to eliminate part of the fluctuations. quantum mechanical system. For distinguishable parti- The results for a representative run of the above proce- cles, this is the only difference. For particles with statis- dure are reported in Fig. 3. Unfortunately, the dynamics tics (bosons), these paths then can permute onto each of this system looks quite different from the formation of other forming larger paths or cycles. a glass from a quenched liquid. First of all, the structure We initially focus on studying a quenched quantum factor becomes quite large for some values of ~q, therefore system of Helium particles but require that they act like suggesting the presence of large crystallites in the sam- distinguishable particles. There are a number of poten- ple. Indeed, the largest value of the structure factor cor- tial advantages of this approach. To begin with, one responds to the (5, 0, 4) curve in Fig. 3 at large times and may hope that distinguishable particles are more likely to the (5, 4, 2) curve in Fig. 3 at short times. We see that to retain the relationship between real dynamics and while at short times the values of S~q(t) are smaller than the Monte Carlo dynamics. Secondly, the simulation of 10, at larger times they grow up to 50, which clearly indi- distinguishable particles is faster and more easily paral- cates the presence of large crystallites in the sample (note lelized over many processors allowing for longer simula- in addition that these values have been averaged over 50 tions. PIMC sweeps and also over imaginary time). Moreover, We used the Aziz potential as a model for Helium [30], the ρ~q(t) (reported for a representative value of ~q in the and in this section we always use Angstroms as units of middle panel of Fig. 3) are not fluctuating around some length and Kelvins as units of temperature. The pair stable value; they display a sluggish evolution that does product action is used as the approximation for the high not allow us to identify a region of times where the system temperature density matrix and an imaginary time step is close to some metastable density profile that does not of δτ = 0.025 K is used. We equilibrated a system of evolve in time. What we can learn from this is that the N = 216 particles in the liquid phase at a density of quenching from a (exchange-free) liquid to a (exchange- −1 0.029 A˚ and a temperature of T = 2 K. The system free) low temperature liquid froze to a (possibly very is then instantaneously quenched to T = 0.166 K. This broken) crystal relatively quickly without showing any is accomplished by taking a snapshot of the paths from intermediate signs of glassiness. Note however that this T = 2 K and then, for each time slice of the old path, behavior was not observed in all runs: some runs did not placing 12 time slices for the new lower temperature path; display signs of crystallization for times up to 200000 this is similar to what was done by Boninsegni et al. [6]. PIMC sweeps. Still the dynamics was sluggish∼ enough We then run the PIMC from this quenched configura- to prevent the identification of a stable glass phase. We tion. These paths are obviously highly artificial because also tried turning off some moves (the displace moves) in the distances between many adjacent time slices are zero. order to slow down the relaxation to the crystal, but the Over a very short period at the beginning of the quenched system still seems to freeze just as quickly. run, though, this artificial aspect of the path quickly re- In conclusions, we were not able to find a long-lived laxes leaving the paths in a configuration that mirrors metastable glassy state in our quantum simulations. This the higher temperature formation. is probably due to the fact that monodisperse systems al- In the following we refer to t as the PIMC “time” (num- 1 ways crystallize quite fast. This is well known in the clas- ber of PIMC sweeps ), while τ is the imaginary time. At sical case and seems to also hold true when quantum zero each “time” t, the PIMC code returns a configuration τ point motion is introduced (at least in this specific ex- ~rj (t), the latter being the imaginary time trajectory of ample). This leaves the discrepancy between our findings particle j as function of the imaginary time τ. We can and those of [6] to be explained. One possibility is that define the instantaneous density as exchange, that we neglected, may be critically important β for exhibiting the glassy behavior of Helium 4: it could 1 i~q·~rτ (t) ρ~q(t)= dτ e j , (11) be that the path integral at the low temperatures we are βV j 0 focusing on is dominated by exchange paths, whereas the X Z paths that make the glass unstable are mainly without and the instantaneous structure factor exchange; indeed we find them with our PIMC. In this β 1 i~q·[~rτ (t)−~rτ (t)] case, the instability of the glass would be a much rarer S~q(t)= dτ e j k . (12) process once one takes into account exchange paths. In βN 0 Xj,k Z particular, since crystals have a very low or zero superfluid fraction, we know that their corresponding path integral is dominated by paths without exchange. In conse- quence, eliminating the exchange could also make crystal 1 We define a sweep as attempting a displace move on (an expected) 10% of the particles and attempting bisection moves on nucleation easier since it makes it a less rare process. (not necessarily unique) 0.4N/(T δτ) time slices. An additional possibility is that the glassy behavior is 8 sensitive to the specific details of the simulation (type of 1 Monte Carlo moves, length of the paths, etc.). We leave a more detailed investigation of this point for future study. 0.8

V. TOWARDS A METHOD FOR EXPERIMENTALLY ASSESSING THE LEGGETT 0.6 BOUND ρ ρ s/ As we discussed previously, the problem in applying 0.4 our analysis to realistic system is that the amorphous 4 density profile of He cannot be easily measured experi- N=20 exact mentally. Below, we endeavor to connect the bound on 0.2 N=100 exact N=20 approx ρs to the so-called non-ergodic factor gq, which in princi- N=100 approx ple could be measured in experiments, e.g. by neutrons 0 or X-ray scattering. It is defined as 0.1 0.21/3 1/2 0.3 0.4 e ρ A 2 ρ 1 α α gq = ρ~q ρ−~q = ρ~qρ−~q , (13) 1/3 1/2 N FIG. 4: Result for ρs/ρ as a function of ℓ = ρ A , where N α X R~ i are the center of the spheres in an amorphous jammed where the overbare denotes the statistical average over configuration of N spheres with periodic boundary conditions. the amorphous states sampled statistically by the sys- We report the exact computation according to Eq. (6) and the tem. These are indexed by α =1, , , and under the approximate result Eq. (20). ··· αN Gaussian approximation each profile ρ~q is obtained from Eq. (9) by plugging the reference positions correspond- for ~p, ~q = 0, we get ing to each different amorphous configuration α. The 6 statistical average is performed with the weightsR α that F (~q, ~p)= ρ~q−~p iδϕ~pρ−~q = ρδ~q,~p iδϕ~qρ−~q ρδ~q,~p F (~q) . correspond to the frequency with which they appear in ≡ (17) an experiment, or equivalently their Boltzmann weight. Substituting the last expression in (14), we obtain First, let us focus on ρs, which is the average of the α superfluid density ρs corresponding to each amorphous ρ(~q ~v )g F (~q)= · 0 q . (18) state. Since the superfluid density is a macroscopic quan- Nq2 tity we expect (and we have checked numerically, see Ap- e pendix A) a self-averaging behavior, i.e. the fluctuations Averaging (7) over α, we get α of ρs are negligible. However, as usual for disordered 2 systems, the computations are easier for ρs. Multiplying ρs 1 1 (~v0 ~q) α =1 2 (~v0 ~q)F (~q)=1 2· 2 gq . Eq. (6) by ρ−~q and averaging over α we obtain ρ − ρv0 · − N v0 q X~q=6 ~0 X~q=6 ~0 (19) ρ2 e (~q ~v0) gq = (~q ~p)F (~q, ~p) , (14) In the thermodynamic limit, the sum can be replaced by · N · an integral, and performing the angular integration we X~p=6 ~0 e obtain: where we define, for ~p, ~q = 0 (that are the only cases 6 ρ 2 ∞ dq q2 involved in the equation above) s =1 g . (20) ρ − 3 (2π)2ρ q Z0 1 α α α F (~q, ~p)= ρ~q−~p iδϕ~p ρ−~q = ρ~q−~p iδϕ~pρ−~q . (15) The same result can be obtained by meanse of a large A α N X expansion of the system of equations, which however is poorly convergent and cannot be used in a systematic Clearly iϕ is strongly correlated to ρ , being the solution ~q ~q way, see Appendix B. of (6). In order to simplify the problem we assume that As before we need to introduce a cut-off in the sum on these variable are Gaussian distributed. Using Wick’s ~q in (9) and calculate numerically the non-ergodic factor theorem, one has gq by averaging the density over the same configurations α considered above. We set the cutoff according to the F (~q, ~p)= ρ~q−~p iδϕ~p ρ−~q + ρ~q−~p iδϕ~pρ−~q R (16) spherical constraint ~q qmax . We increased qmax until e | |≤ + ρ~q−~piδϕ~p ρ−~q + ρ~q−~pρ−~q iδϕ~p . qmax = 20π/L, when the convergence in gq was reached. For the purpose of computing the non-ergodic factor and Note that, due to translation invariance of the averages 2 then the approximate bound, as given in Eq. (19), we ρ e over α, one has ρ~q = ρδ~q,~0 and ρ~qρ−~p = N gqδ~q,~p. Hence, averaged over 100 different configurations. In this case,

e 9 in fact, one does not face the computational problem of phase in imaginary time simulations with and without inverting the linear system (6) and thus a larger statistics exchange. can easily be taken. The results of the computations are It is worth to note that we neglected the role of small shown in Figure 4. We plotted the superfluid fraction concentration of He3 impurities (of the order of few ppm) obtained through the exact procedure (7) and the ap- that has received a lot of attention in experiments [3]. proximated one (20), both for the configurations with 20 The reason is that we focused on a bulk glass phase of and 100 particles. The agreement between the approx- He4, whose density profile should be largely independent imated curve and the exact one is good for large value of such a small concentration of He3 impurities. It could of ℓ while they start to differ when the localization pa- be, however, that He3 impurities affect the dynamical rameter decreases, for values of the bound around 0.7. stability of the glass phase. Based on the experience Unfortunately for the interesting values of ℓ the approxi- on classical systems, it is likely that in presence of a mated calculation gives wrong results. However, we find large concentration of impurities crystallization will be it useful, since it allows to estimate the typical scale of avoided [14] and a long-lived quantum glass phase [15, 31] ℓ at which the bound starts decreasing fast from 1 to 0 will be stable. In this case, it should be very easy to mea- and we hope that it will be possible to improve it in the sure the density profile and compute the Leggett bound future, in order to be able to apply it to realistic cases. using the procedure detailed above. However, it has been estimated that a concentration of at least 0.1% of impurities is needed to stabilize the glass [32]. Therefore, VI. CONCLUSIONS the typical concentration of He3 ( ppm) should not be enough to produce a sensible effect,∼ unless some unex- The aim of this paper was to study Leggett’s upper pected phenomenon related to the quantum mechanical bound for amorphous quantum solids. We showed that nature of the systems (e.g. exchange, as already dis- for quantum systems described by a hard sphere Jastrow cussed) becomes relevant. wavefunction, the superfluid fraction must be smaller that 0.1%, which is consistent with a previous investi- Acknowledgments gation that found extremely small condensate fractions for this system [13]. Moreover, the hard sphere result suggests that crystal and glass phase characterized by We wish to thank S. Baroni, M. Boninsegni, G. Car- the same Lindemann ratio should have similar Leggett’s leo, S. Moroni and L. Reatto for very useful discussions. upper bounds for the superfluid fraction. FZ wishes to thank the Princeton Center for Theoreti- On this basis, we attempted to apply our results to cal Science for hospitality during part of this work. This 4 research was supported in part by the National Science glassy He [6]. We found that the upper bound for ρs is in general very close to the numerical results of Ref. [6], Foundation under Grant No. NSF PHY05-51164. Part of and at density ρ = 0.029 A˚−3 it is below. One possi- the numerical calculations have been performed on the ble origin of this discrepancy could be that at such low cluster “Titane” of CEA-Saclay under the grant GENCI density the life time of the metastable glassy state is 6418 (2010). too short, and the system is intrinsically out of equilibrium; in that situation Leggett’s bound is inapplicable, Appendix A: Details on the numerical procedure since it assumes that the reference wave-function corresponds to a truly metastable state. Indeed we generically found from Path Integral Monte Carlo calculations that We define the Fourier transforms in the cubic box of 3 (at least if exchange is neglected) the system crystallizes side L and volume V = L as follows: very fast after the quench, which is consistent with a very 1 i~q·~r −i~q·~r short lifetime of the metastable glass. ρ~q = d~rρ(~r)e , ρ(~r)= ρ~qe , (A1) V V Overall, our findings suggest two possible scenarios Z X~q (not necessarly antithetic). (1) An amorphous stable where ~q = 2π (n ,n ,n ), and each of the integers n Z, glass has a superfluid fraction, not only a Leggett’s upper L x y z i and similarly ∈ bound, very similar to a defect-free crystal with the same

Lindemann ratio. Since we know from experiments and 1 i~q·~r simulations that this superﬂuid fraction is very small, δϕ~q = d~rδϕ(~r)e . (A2) V V or possibly zero, we are bound to conclude that the Z glassy supersolid phase found in experiments do not cor- Note that δϕ~0 is an irrelevant constant phase in the vari- respond to a truly stable glass: the system is instead ational wavefunction so we set it to zero. Finally, rapidly evolving out of equilibrium and, somehow, this ~v ~q = ~0 , enhances superﬂuidity. (2) Exchange promotes glassiness ~v = 0 (A3) ~q ~ and whereas a stable glass phase cannot exist, because it ( i~qϕ~q ~q = 0 . has a very short life-time, a superglass can. This could − 6 be partially tested by comparing the stability of the glass which leads immediately to Eq. (6). 10

We performed the calculations for different values of In this case the values of the bound were more sensitive the Lindemann parameter ℓ = ρ1/3A1/2, increasing the to the particular realization, so we took averages over 30 number of vectors ~q according to the spherical constraint configurations. For every value of the localization pa- ~q qmax, until a reasonable convergence in the value rameter, the superfluid fractions that we found were on |of| the ≤ bound (7) was achieved, at least for large val- average smaller, as reported in Figure 5. ues of A. From Eq. (9) one sees that for large ~q the | | 1 corresponding component ρ~q is suppressed through the 2 factor e−Aq /2. Thus, one needs to truncate the sum over ~q at qmax 1/√A, as higher terms will not con- 0.8 tribute. Unfortunately,∼ for small A, this cut-off is too heavy in terms of computational time and we should use a lower one. Still, considering small configurations and 0.6 sufficiently large values of A, which nevertheless span the ρ /ρ physical region of interest, we could reach a good conver- s gence or keep the error under control. Note additionally 0.4 that by increasing the number of vectors ~q in (9), the value found for the superfluid fraction monotonically de- N=20 exact creases, as expected because of the variational property 0.2 N=20 approx already discussed. This permits to preserve the nature of Analytic approx upper bound for Eq. (7), despite the cut-off approxima- 0 tion. Overall, we found that the better compromise was 0 0.21/3 1/2 0.4 ρ A to set qmax = 20π/L. In order to check the independence of the bound on the flow direction, we also compared the results obtained FIG. 5: Result for ρs/ρ as a function of the localization pa- 1/3 1/2 ~ 3 with the velocity v0 along the (1, 0, 0) direction to those rameter ρ A , where Ri are N random points in [0, L] along (1, 1, 1) and we observed a negligible difference with periodic boundary conditions. which is expected to vanish in the thermodynamic limit, because amorphous solids are statistically homogeneous on large scales. Appendix B: Large A expansion We have also checked that the bound for the superfluid density almost does not fluctuate by considering different amorphous configurations α, α =1, , , as it is ex- For large A, we expect that the density becomes uni- R ··· N pected since the superfluid density is a macroscopic quan- form. Hence, ρ~ ρ, and ρ~q 0 for ~q = ~0. We can use 0 → → 6 tity. We computed the corresponding superfluid fraction this to expand iδϕ~q systematically in powers of ρ~q. We ρα and the average ρ = ρα/ for 10 different config- rewrite Eq. (6) as s s α s N urations. The variance of ρs is very small. In this paper P α we presented results averaged over 10 realizations of , 2 R ~q ~v ρ = q ρiδϕ + (~q ~p)ρ − iδϕ . (B1) a larger statistics do not lead to appreciable differences. · 0 ~q ~q · ~q ~p ~p Finally, as a check of our codes, we repeated all the ~pX=6 ~0,~q calculations on configurations of 20 particles occupying uncorrelated uniformly random positions in the box, i.e. (1) (2) ~ We write δϕ~q = δϕ~q + δϕ~q + where the different where Ri are uniform and independent random vari- k ··· ables in [0,L]3. In this case it is easy to show that terms are of order (ρ~q) . At first order g = exp( Aq2). Hence Eq. (20) becomes q − ∞ (1) ~q ~v0 ρ 2 2 1 iδϕ = · ρ~q , (B2) e s 2 −Aq ~q q2ρ =1 2 dq q e =1 3/2 3/2 . ρ − 3(2π) ρ 0 − 24π ρA Z (A4)

at second order

(2) 1 (1) (~q ~p)(~p ~v0) iδϕ = (~q ~p)ρ − iδϕ = · · ρ − ρ , (B3) ~q −q2ρ · ~q ~p ~p − p2q2ρ2 ~q ~p ~p ~pX=6 ~0,~q ~pX=6 ~0,~q at third order ′ ′ (3) 1 (2) (~q ~p)(~p ~p )(~p ~v0) ′ ′ iϕ~q = 2 (~q ~p)ρ~q−~piϕ~p = · 2 ·2 ′2 3 · ρ~q−~pρ~p−~p ρ~p (B4) −~q ρ · ′ q p p ρ ~pX=6 ~0,~q ~pX=6 ~0,~q ~pX=6 ~0,~p 11 from which we can guess the order k:

(k) k−1 (~q ~p1)(~p1 ~p2) (~pk−1 ~v0) iϕ = ( 1) · · ··· · ρ − 1 ρ 1− 2 ρ −2− −1 ρ −1 (B5) ~q − q2p2 p2 ρk ~q ~p ~p ~p ··· ~pk ~pk ~pk ~ ~ ~ 1 k−1 ~p1=6 0,~q; ~p2=6 0,~pX1; ··· ~pk−1=6 0,~pk−2 ··· and so on. Plugging this in Eq. (7) we get

2 ρs (~v0 ~q) (~v0 ~q)(~q ~p)(~p ~v0) =1 2 ·2 2 ρ~qρ−~q + · 2 2· 2 3 · ρ~q−~pρ~pρ−~q ρ − ρ v0 q q p v0 ρ X~q=6 ~0 X~q=6 ~0 ~pX=6 ~0,~q ′ ′ (B6) (~v0 ~q)(~q ~p)(~p ~p )(~p ~v0) ′ ′ · 2· 2 ′2 ·2 4 · ρ~q−~pρ~p−~p ρ~p ρ−~q + . − ′ q p p v0 ρ ··· X~q=6 ~0 ~pX=6 ~0,~q ~pX=6 ~0,~p While this expansion seems a simple strategy of solution of Eq. (6), it is very poorly convergent and in practice it is not very helpful.

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