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Home , Solid, State of matter, Superfluidity, Supersolid

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Physics Department Faculty Publication Series

2005 State of Nikolai Prokof 'ev University of Massachusetts - Amherst, [email protected]

Boris Svistunov University of Massachusetts - Amherst, [email protected]

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Recommended Citation Prokof'ev, Nikolai and Svistunov, Boris, "Supersolid " (2005). Physics Review Letters. 1175. Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1175

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For more information, please contact [email protected]. arXiv:cond-mat/0409472v2 [cond-mat.stat-mech] 23 Nov 2004 h oi tt fmte sisltn 4 rvst ea be to proves [4] insulating is matter of state the a as 4 interstitial and/or essary vacancies implies it happen. what to and this points for lattice conditions necessary of com- the in number being are atoms the possible of with is number mensurate the superfluidity with whether structures question the swer SF [10]. in vacancies result of may absence own, the its in on interpretation exchange, interatomic misleading that the caused apparently, which, ae nteecag nerlbetween integral exchange the on based o h pe on ntespruddensity, superfluid the on bound upper formula the rigorous for a derived and wavefunction groundstate icse nRf 9.Asprt su ste“extremely the is issue estimate separate A order-of-magnitude was tentative” connectivity [9]. the Ref. to in relation discussed their and defects tal essprud(F rpris n a osdrtheir consider may establishing One as properties. pos- thus (SF) con- and superfluid Bose-Einstein atoms) sess interstitial a (or feature vacancies may of Early densate Chester and that matter. [3] showed of Lifshitz [4] and renewed state Andreev prompted (SFS) by work supersolid has theoretical the which in result (NCMI) interest inertia breakthrough of a moment is non-classical a have samples siltreprmnsa rgntn rmsal(about small from torsion originating the ∼ in as decoupling experiments mass oscillator interpret to then natural etlwr frrve,se .. 5)idctsta va- that indicates in [5]) interstitials e.g., and see, cancies review, (for work mental ikbtenteS epneadthe a and established response Leggett SF the is ab- [9]. between idea the paper link this Mistakenly, Leggett’s in 8]. to even [7, attributed superfluidity defects zero-point to of idea quan- sence an may in facts processes exchange these tum strong with that forward deal put To was vacancies. solid of of variations presence the at looked 4 [6] 0 study below recent negligible is concentration their e hse’ fia pclto”ta ihu them without that speculation” “final Chester’s He. ebtentocpctrpae n i o eelany reveal not did and plates two between He eo er-xmn egt’ okadso that show and work Leggett’s re-examine we Below an- to is follow to discussion the of point central The eetdsoeyb i n hn[,2 htsolid that 2] [1, Chan and Kim by discovery Recent oee,teoewemn uko rvosexperi- previous of bulk overwhelming the However, % ocnrto fzr-on aace [1]. vacancies zero-point of concentration 1%) odto o F nbsncssessmlrto similar systems bosonic in SFS for condition ASnmes 74.w 78.s 05.30.-d 67.80.-s, 67.40.-w, numbers: PACS and interfaces. interpretation, supersolid superfluid of bulk its question theorem, edsusrcn xeietb i n hn[, Chan and Kim b by which experiment solids recent pa commensurate integral discuss in an We possible as not both, is or superfluidity atoms, interstitial or vacancies, epoeta h eesr odto o oi ob loas a also be to solid a for condition necessary the that prove We sufficient eateto hsc,Uiest fMsahsts Amher Massachusetts, of University Physics, of Department 4 usa eerhCne KrhtvIsiue,138 Mos 123182 Institute”, “Kurchatov Center Research Russian ecytl are crystals He odtosfrSS twas It SFS. for conditions nteSproi tt fMatter of State Supersolid the On ioa rkfe n oi Svistunov Boris and Prokof’ev Nikolay connectivity . ρ 2 3 s eaos[9], atoms He K activated ≤ h most The . ρ 3 s × Crys- . fthe of 10 nec- 4 and not He − 4 ag re TDR)i eurdfrS 1,1,17]. Ψ 16, connected [15, SF that for requirement required is The just long- (TODLRO) is off-diagonal order topological and superfluidity, that range saying for of key way is another connec- [9]) (called Leggett property by say, This tivity fixed. coordinates, kept are several coordinates or one x when small scopically oevr nsprud Ψ superfluids in Moreover, t fpritn urnsi ape ihtecylindrical the with stabil- [9]. samples thus geometry in annulus and currents circulation persistent of of ity quantization the to e ftoietclbsn omn on (molecu- bound a forming groundstate, identical lar) two of tem h hsc fdsree n frustrated and on based to disordered in- data forced of their supersolid physics are of we the bulk explanation results alternative a an Chan for excluding & look Kim By the of parameters, [13]. terpretation system changing by e.g. which broken symme- interstitials immediately exact and is vacancies an zero-point require between they try be because to ground- Nature, measure” SFS in “zero that have found then solids commensurate is The in question states central to [12]). the due bulk to that the answer from to/from constant lattice the adding/removing coming of and change separate the the to from to crucial contribution is the pinned suppress (pinning of ar- vacancies compressibility an and SF, forward solids relates put which we uncertainty [11] –-number Thirdly, relation the SF. on to based lead gument provides not their also on do processes It exchange own explicitly. why showing seen picture is simple state a va- SF of the presence in the theorem cancies which the in of language proof path-integral alternative using an present We theorem. ooi ytmhsn eo,Ψ zeros, no has system bosonic igo ag circumference, large of ring a l ϕ rdc ftetepaewv o h etro mass of center the for wave plane the coordinate, the of product sntsnl-aud eas if because single-valued, not is h igw get we ring the avl,tefis oaigsaeo h oeueon the of state rotating first the Naively, . 1 1 oilsrt h on,cnie n-iesoa sys- one-dimensional a consider point, the illustrate To ssonb ena 1] h rudtt fthe of groundstate the [14], Feynman by shown As x , = 2 to h rudsae saconsequence, a As state. ground the of rt x . . . , ffra lentv xlnto nterms in explanation alternative an offer e 427 i ekcniuu rnlto symmetry. translation continuous reak 2 πR/L 2 20) ntecneto this of context the in (2004)] 225 , m R r ae rudtesse hl other while system the around taken are , ϕ prudi ohv zero-point have to is uperfluid ( = 0 ϕ ( t A003and 01003 MA st, ϕ | 1 r 0 r 1 − → ( 1 | − r + 1 r − cow 2 r ϕ | 2 .Ti xrsin however, expression, This ). 1 r ) h orc ouini to is correct The . G / 2 | ,tmstebudstate: bound the times 2, ,wt oaiainlength localization with ), osntbcm macro- become not does r L 1 G ( or ≫ x 1 G r x , l 2 switna a as written is , esingle-valued be 2 stknaround taken is 4 x , . . . , einterfaces. He N ) 0. 6= 2 replace ϕ0 withϕ ˜0, which has a zero at |r1 − r2|≈ L/2, point fluctuations of atoms away from lattice points will i.e. in the region where ϕ0 is exponentially small; at dis- be erased from the picture. The procedure continues un- tances |r1 − r2| ≪ L the two functions are almost identi- til we have erased all pairs with sizes much smaller than L cal, |ϕ˜0| ≈ ϕ0. Note, that the energetic cost of creating but much larger than all microscopic scales. We say that a zero in this case is exponentially small and vanishes in the crystal state has no vacancies and interstitials if the the limit of infinite system size. final coarse-grained configuration is empty. On the other The same considerations apply to the disconnected hand, if the coarse-grained configuration still contains crystal state consisting of N bosons when ΨG decays to lattice points, or particles, or both, at arbitrary large the macroscopically small value when any finite number distances, we say that it has crystal defects in it. The of coordinates are taken away from their typical posi- decimation procedure explains how vacancies are possi- tions in the crystal (other coordinates are kept fixed) and ble in commensurate solids, and perfectly agrees with the moved around the system. The first rotating state in the conventional view of classical crystals at finite tempera- system with periodic boundary conditions can be written ture. For the commensurate solid with connected ΨG we i2πR/L N may start with the perfect-lattice configuration of parti- as φ1 = e ϕ˜0 with R = Pi=1 ri/N andϕ ˜0 having zeros in regions where the modulus of ϕ0 is exponentially cle coordinates and its empty coarse-grained picture, and suppressed and thus extra zeros do not cost finite (system then move x1 arbitrary distance away to produce an im- size independent) . Clearly, the phase gradient cir- age of the vacancy and interstitial. This will not result culation of φ1 is ∼ 1/N, and this system will not show in the exponential suppression of the configuration prob- the NCMI which is based on the impossibility of setting ability (in fact, such configurations will dominate in the system in rotation with a macroscopically small veloc- normalization integral). ity or circulation. This should be compared with the In the absence of exact interstitial/vacancy symme- first rotating state of the single- superfluid system, try the concentration of vacancies, nv, and interstitials, N (SF ) i2π Pi=1 ri/L nint, in the supersolid will be different, e.g. nv > nint, φ = e ϕ0 with ϕ0 > 0 and phase gradi- 1 since broken continuous symmetry allows production of ent circulation 2π. Now, making zeros in connected ϕ0 is so costly energetically that the lowest energy state corre- excess vacancies by making small changes in the lattice sponds to the relatively high of rotation. constant. [This mechanism is is not available in discrete For definiteness, we consider below only single-atomic su- models, and then nv = nint is possible.] In the transla- perfluids, but all considerations are readily generalized tion invariant system at T=0 one expects then ρs = Anv, to the m-atomic case. We have essentially reproduced just like in any other interacting bosonic system. above the Leggett’s argument that SF wavefunctions are Our second consideration is based on the path-integral necessarily connected. formulation of quantum statistics [14] in terms of many- body trajectories, {x (τ)}, in imaginary time τ ∈ [0,β] By definition, |Ψ (x , x ,...,x )|2 is the probability i G 1 2 N with periodic boundary conditions {x (β)} = {x (0)}. density to observe particles at the specified positions. We i i The most important superfluid characteristic of parti- fix all coordinates except one, x , and observe that in 1 cle trajectories, or world lines, is their winding num- connected wavefunctions |Ψ (x )|2 remains finite when G 1 bers, M α, α =1, 2,...,d. We assume periodic boundary x is taken arbitrary far from the initial position. For- 1 conditions in space in all d-dimensions with linear sizes mally, this property is identical to statistical properties Lα = L. To determine M α imagine a cross-section go- of atomic configurations in classical crystals at finite tem- ing through point R perpendicular to the direction α perature and was used by Chester to introduce vacancies and count how many times particles cross it from left to in the . This final correspondence was not right, kα , and from right to left, kα . By definition, wind- elaborated in Ref. [9]. − + ing numbers are M α = kα −kα . They are independent of How do we “visualize” vacancies/interstitials in the + − the cross-section location R because trajectories are con- espe- state of with strong exchange, tinuous and periodic in imaginary time. The superfluid cially when the number of atoms coincides with the num- density is then given by [19] ber of lattice points? It appears that the common per- ception is that such solids do not have vacancies and αγ 2−d α γ ρs =2mTL h M M i , (1) interstitials, by definition, or else they are indistinguish- able from the standard zero-point fluctuations. Imagine where m is the particle mass. In d = 3, the superfluid a solid sample pinned by an (arbitrarily weak) external density is finite in the thermodynamic limit, L → ∞, potential preventing it from moving as a whole. There T → 0, and T/L → 0, if the probability of having world is no problem in identifying lattice points using the av- lines with non-zero winding numbers in the ground state erage particle density profile ρ(r) which is periodic in is close to unity. space. Consider now some typical spatial configuration We now demonstrate that crystal states without zero- of particle positions along with the lattice points and point vacancies are described by world-line configurations start the coarse-graining procedure of “erasing” the clos- with M = 0, i.e. they are not superfluid. We start est particle-lattice point pairs in the spirit of the spa- with the picture of a perfect crystal with particles tightly tial renormalization group treatment. As we progress localized around equilibrium lattice points. In this state towards mesoscopic length-scales all short-ranged zero- the trajectories are nearly straight lines with M = 0 as 3

(a) (b) (c)

FIG. 1: Particle worldlines in different crystals at low . The time axis is vertical. Dashed lines show the equilibrium lattice points. (a) Nearly classical crystal at low temperature; M = 0. (b) Insulating quantum crystal with large zero-point fluctuations and frequent particle exchange processes; M = 0. (c) Particle worldlines with non-zero winding number. shown in Fig. 1(a). When tunnelling exchange processes are added into the picture the world lines are no longer in coarse-graining scale one-to-one correspondence with the lattice points. The nature of the exchange process, however, is such that τ when one particle leaves its equilibrium crystal point R1 r and goes to point R2, the other particle goes from R2 to R1 (for pairwise exchange). Thus, the net current of world lines through any cross-section remains strictly zero. The same conclusion follows from the consideration of multiparticle exchange events [18], see Fig 1(b). Consider now a world-line configuration with non-zero winding number, Fig. 1(c). At any moment of imagi- nary time we consider the spatial configuration of par- ticle positions and lattice points and apply the coarse- FIG. 2: Evolution of the coarse-grained picture for the tra- graining procedure discussed above. Again, all short- jectory with non-zero winding number similar to Fig. 1(c). lived and short-ranged exchange process and zero-point Filled and open circles show particle and lattice site positions fluctuations will be erased once we pass several atomic correspondingly. Arrows indicate the direction of the particle distances, and in the insulating state the “movie” of the number current. coarse-grained configuration evolution in time will show an empty “screen” for Figs. 1(a,b). If M 6= 0, as in

Fig. 1(c), there is no way for the renormalization proce- , and ∆E = EN+1 − EN−1. Incompressible states dure to erase all particles at all moments in time since have finite ∆E. On another hand, in a macroscopic sys- topologically winding numbers correspond to particle tra- tem ∆E can be obtained by considering the energy in- jectories moving continuously in the same spatial direc- crease by creating an interstitial-vacancy pair with ar- tion and thus creating imbalance between particles and bitrary large separation between them (in pinned solids lattice points at arbitrary large distances. After all other the notion of vacancy or interstitial is rigorously defined particles are associated with lattice points and erased, within the coarse-graining procedure as the presence or the winding trajectory describes an interstitial-vacancy absence of perfect registry between particles and unit pair which separates over the distance of order L and cells). Crystals without zero-point defects are gapped eventually makes a closed loop around the system, see with respect to the interstitial-vacancy production (oth- Fig. 2. For the statistics of such loops to give non- erwise these defects would be an essential part of the α γ vanishing h M M i in the thermodynamic limit, they groundstate) and thus are incompressible (if pinned) and have to be typical in a given lattice structure. In other vice versa. One step further, this implies that superfluid- words, zero-point vacancies and interstitials are an inte- ity and pinned compressibility come together and either gral part of the groundstate. one (including long-wave acoustic properties with addi- Our last consideration is based on the relation between tional mode) can be used for the detection of the the pinned compressibility, κ, and zero-point vacancies. SFS state experimentally. This conclusion is in line with Compressibility can be calculated through the the famous uncertainty relation [11] between the phase of states with one extra particle, EN+1, and one extra of the superfluid order parameter, φ, and particle num- vacancy, EN−1, as κ = 1/V ∆E, where V is the system ber, ∆φ∆N ≥ 1/2. Because of this relation, one may not 4 introduce a well defined phase field for the incompress- crystallographic indices provides a broad distribution of ible state of matter which tends to completely suppress transition . particle number fluctuations. One of the experimental mysteries is extreme sensi- We have little doubt that large activation energies tivity to the addition of 3He at the level of for vacancies and interstitials in 4He measured down to 3 nim ∼ 100ppm. To minimize kinetic energy, He ∼ 1 K temperatures [5] will not radically change to near atoms are likely to end up at frustrated interfaces, and zero at lower temperatures, and that helium is a commen- then at the edges where different interfaces meet. This surate solid at T = 0. Since it has no symmetry between may increase 3He edge vs bulk concentration by a fac- the vacancies and interstitials, we conclude that there (edge) tor as large as (D/d)2 and produce n ∼ 1. This are no zero-point vacancies in bulk solid 4He. By ex- im will have a profound effect on the edge-connected inter- cluding superflow through the crystal bulk we are forced face superfluidity (edges then act as a disordered two- to look more closely at the superfluid properties of dis- dimensional network of Josephson junctions). ordered helium-substrate layers and frustrated interfaces between helium micro-. We are not aware of any systematic study what are the 4 There are indications of strong disorder in experimen- properties of interfaces between the He micro-crystals 4 tal system of Ref. [1]. The dependence of the super- and helium He crystals on disordered substrates at el- fluid density on reduced temperature parameter t = evated . Model simulations of domain wall boundaries in the checkerboard solid (obtained for in- (Tc −T )/Tc has little to do with the expected bulk super- 0.671 teracting lattice bosons at half-integer filling factor) show fluidity t dependence. Instead, ρs appears to vanish that they remain superfluid deep into the bulk solid phase at Tc with zero derivative. Such a behavior can be mod- elled by a broad distribution of transition temperatures [20]. In the outlined picture three predictions are certain: in the heterogeneous sample. This observation correlates (i) the superfluid fraction must strongly depend on the 3 with the gradual decrease of the decoupled mass with process, (ii) the amount of He sufficient 2 the increase of the torsion oscillator amplitude by orders to suppress superfluidity scales as nim ∝ ρs, and (iii) of magnitude. Let us assume that a sample consists of transition temperatures on interfaces do not depend on micro-crystallites of linear size D with superfluid inter- D and ρs(T = 0). faces of typical thickness d between them. The superlfuid We are grateful to M. Chan, R. Guyer, R. Hallock, fraction may be estimated then as ρs/ρ ∼ d/D. To have Yu. Kagan, A. Leggett, J. Machta, and W. Mullin for ∼ 1% of the superfluid mass coming from interfaces with stimulating discussions. The research was supported by d ∼ 10 A˚ one will need sizes about a frac- NSF under grant No. PHY-0426881 and by NASA under tion of a micron. The variety of interfaces with different grant NAG32870.

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