Theory for supersolid $^4$He

0.22 MB·

by  Xi Dai

Theory for supersolid 4He
Xi Dai1, Michael Ma2, and Fu-Chun Zhang1,2,3
1Department of Physics, University of Hong Kong, Hong Kong
2Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA
3Department of Physics, Zhejiang University, Hangzhou, China
5 (Dated: February 2, 2008)
0
0 Although both vacancies and interstitial have relatively high activation energies in the normal
2 solid, we propose that a lower energy bound state of a vacancy and an interstitial may facilitate
vacancy condensation to give supersolidity in 4He . We use a phenomenological two-band boson
n
latticemodeltodemonstratethisnewmechanismanddiscuss thepossiblerelevancetotherecently
a
J observedsuperfluid-like,non-classicalrotationalinertialexperimentsofKimandChaninsolid4He.
Someof our results may also be applicable to trapped bosons in optical lattices.
7
1
PACSnumbers: 05.30.-d,03.75.Hh,67.40.-w

]
n
o Recently Kim and Chan have reported observation of The essence of our theory is that while, consistent with
c superfluid-like, non-classical rotational inertial (NCRI) all known experiments, the normal solid state is the DF
- behavior in solid 4He, both when embedded in Vycor state, the supersolid state results from condensation of
r
p glass[1]andinbulk4He [2]. Theirexperimentshavere- vacancies and/or interstitials about a defect rich back-
u vivedgreatinterestinsupersoliditywithbothcrystalline ground of excitons. This physics is shown quantitatively
s
and superfluid orderings in helium. The possibility of using a phenomenological two-band lattice boson model
.
at a supersolid phase in 4He was theoretically proposed by to represent the defects in solid 4He. Using mean field
m Chester [3], Leggett [4], Saslow [5] and by Adreeve and theory (MFT), we show that superfluidity in solid he-
Lifshitz[6]in1970’s. AdreeveandLifshitzproposedBose lium can exist in parameter regimes qualitatively con-
-
d condensationof vacancies as the mechanismfor superso- sistent with all the known experiments and microscopic
n lidity. Chesterspeculatedthatsupersoliditycannotexist calculations. We will arguethat the key results willhold
o
without vacancies and/or interstitials [3], a claim made beyondMFT,andindeedarerenderedmorerobustbyin-
c
more rigorous recently by Prokofev and Svistunov [7]. clusionofquantumfluctuations. Becauseofthe”vacuum
[
Experiments and more sophisticated microscopic calcu- switching”betweenthenormalandsupersolidstates,the

1 lationshave,however,providedconstraintstoanytheory transition at zero temperature is generically first order.
v for supersolid in 4He. The NMR experiments [8, 9] on
We start with the lattice as defined by the periodicity
3
solid 3He rule out a non-negligible zero point vacancy
7 of crystalline helium. On each lattice site, we consider
concentration. The energy of a vacancy in solid 4He was
3 two single-particle localized states. The lower energy
1 estimatedtobeabout10K bythex-rayscatteringexper- state (a-state), with energy −ǫ , has its maximum on
a
0 iment[10]andtobe15K inatheoreticalcalculation[11]. the lattice site. The other state (b-state), with a higher
5 The energy of a pure interstitial state has recently been energy −ǫ +∆, is less localized and has maxima dis-
0 calculated to be about 48±5K [12]. Assuming that the a
tributed with hcp symmetry away from the lattice site.
/
t observedsupersolidityisagenuinebulkphenomenon,the Becauseofthestronglyrepulsivecores,eachofthisstate
a
m quandary is thus how defects with such high activation canholdatmostone4Heatom,i.e. foreachstate,helium
energies can condense at low temperature.
behavesashardcorebosons. Ontheotherhand,because
-
d In this Letter, we propose a possible solution to this of the spatial separation between the a and b states, an
n quandary. Inadditiontovacanciesandinterstitials,there atom in the a-state repels one in the b-state with a large
o is a third type of defects, with relatively low excitation but weaker strength U. An atom on one site can tun-
c
energy,which correspondsto a bound state of a vacancy nel to a neighboring site from a to a-state, b to b -state,
:
v andaninterstitial,henceforthcalledan”exciton”. While and a to b state with hopping amplitudes ta, tb, and tab
Xi such excitons do not carry mass and will not contribute respectively, taken all to be real and non-negative. In
tosupersolidphenomenalikethoseobservedbyKimand the DF state, we have one helium atom occupying the
r
a Chan[1,2],theycanfacilitatesupersoliditybytwomech- a-state on each lattice site. Due to the hard core condi-
anisms. First, vacancies can condense above a back- tion, the 4He atoms in this state are immobile, and this
groundofexcitonseasierthanabovethedefectfree(DF) state is a normal solid, which is consistent with the as-
normal state, so that the condensation energy can more sumption that defects are necessary for supersolid. To
than compensate for the exciton excitation energy. Sec- study defects, it is convenient to consider the DF state
ond, in a background formed of a coherent mixture of asthedefectvacuumstate,fromwhichvacanciesandin-
the DF state and the exciton, the effective kinetic en- terstitials can be created. The term vacuum will always
ergy of vacancies and interstitials can be enhanced due refer to the DF state henceforth in this article. Let a†
i
to constructive interference between hopping processes be the creation operator for the vacancy by removing a
involving the DF state and those involving the exciton. helium atom fromthis vacuum state at the site i, and b†
i
2
the creation operator for the interstitial by creating an
atomtothe b-stateonthe site. Note thatthe interstitial
defined this waycanbe viewedas aquantum generaliza-
tionoftheclassicalinterstitial. Thehardcoreconditions
specifiedaboveimpliestheseoperatorsarealsohardcore
boson operators. Our model Hamiltonian for defects in
solid helium is then,
H = ǫ n +ǫ n −Un n
X a j,a b j,b j,a j,b
j
− (t a†a +t b†b +t a†b†+h.c.) (1)
X a i j b i j ab i j
<ij>
In the above equation, n = a†a , n = b†b , and ǫ
j,a j j j,b j j a
(previously defined) and ǫ are the energy cost to create
b
a helium vacancy and an interstitial respectively. The
repulsion U between He atoms translates into an attrac-
tion of strength U between a vacancy and an interstitial
on the same site. The energy of a localized exciton is
∆ =ǫ +ǫ −U. For simplicity, in what follows we will
a b
consider t =0.
ab
To illuminate the effects of excitons, let’s first discuss
the non-interacting case U = 0. In this limit, the va-
cancies and interstitials are decoupled. For ǫ → ∞,
b
our model is reduced to the vacancy model proposed by
Andreeve and Lifshitz [6]. Let z(= 12) be the number
of the nearest neighbors in a hcp lattice, the onset of
Bosecondensationofvacancies(interstitials)isgivenex-
actly by ǫ −zt = 0−, which coincides with hav-
a(b) a(b)
ing zeroactivationenergyfor a singledefect, a condition
that is not supportedby experiments[10] andtheoretical
estimates[11,12]for4He. Thisisthequandarywestated
in the introduction.
We now consider the case U > 0. In this case, a va-
FIG. 1: (a) Mean field theory phase diagram of Hamiltonian
cancy and aninterstitial tend to bind together to form a (1)inthesymmetriccasewithǫa =ǫb =1andU =1.9. V-SF
local exciton. If the exciton energy ∆ is small (large U), (I-SF):vacancy(interstitial)superfluidphase. (b)Sameasin
thepresenceoftheexcitonsinthegroundstateenhances (a) in the asymmetric case ǫa = 1, ǫb = 4 and U = 4.8. See
the kinetic energy of the vacancy (or interstitial), which Fig. 2forthesnapshotsofthesephasesinrealspace. (c)The
may lead to the condensation of these defects. Since we variational energy of the symmetric model with ǫa = ǫb = 1
are primarily interested in the large U case, it is impor- and U = 1.9 as the function of the SF order parameter for
tanttotreattheon-siteattractiveinteractionaccurately. t = ta = tb = 0.056 (normal solid phase), t = 0.06 (at the
phase boundary) and t = 0.064 (VI-SF phase). (d) Same as
In order to do so, we use the single-site mean field ap-
in (c) with ta =tb=0.06 for various values of U.
proximation (MFA) by decoupling the kinetic terms as
a†a ≈ a¯(a +a†)−a¯2,
i j j i
b†b ≈ ¯b(b +b†)−¯b2 (2) theMFA,wefoundfivedifferentphases,characterizedby
i j j i the order parameters a¯ and¯b together with the vacuum,
where the spatiallyuniformBose condensationorderpa- vacancy, interstitial, and exciton defect densities n ,n ,
0 V
rameters a¯ =< a > and ¯b =< b > are determined n , and n . They are (1) The normal DF solid phase.
i i I ex
self-consistently. The single site mean field Hamiltonian (2)The vacancysuperfluidphase (V-SF(A)), where only
for the hard core bosons are then solved exactly. This thevacanciescondense(a¯6=0)abovetheDFbackground
MFA gives the correct exact conditions for onset of su- (n = n = 0). This phase is the same as the vacancy
I ex
perfluidity at U =0. stateofAndreeveandLifshitz. (3)Thecorrespondingin-
The ground state phase diagrams obtained within the terstitial superfluid phase (I-SF). (4) An alternative va-
MFA in the parameter space t and t are shown in Fig. cancy superfluid phase (V-SF(B)), where the vacancies
a b
1(a)for ǫ =ǫ =1, U =1.9 andin Fig. 1(b)for ǫ =1, condense above a background of excitons. (5) The VI-
a b a
ǫ = 4, and U = 4.8. They represent a more symmet- SFphase,wherewehavebothvacanciesandinterstitials
b
ricandastronglyasymmetriccasesrespectively. Within condensing above a backgroundof a coherentmixture of
3
at low T in 4He. In the hatched regime, the DF state
is metastable but not the global minimum energy state.
In Fig. 1(c) and 1(d), we illustrate this by plotting the
variational energy E(a¯) based on the MFA [13] for the
symmetric model with t = t = t, so that a¯ = ¯b, with
a b
increasing t (Fig.1(c)) and U (Fig.1(d)). The existence
of two minima are clearly seen. With increasing t or U,
the Bose condensed state becomes lower in energy than
the normal state in the hatched areas. The transition is
first order due to ”vacuum switching”: the normal state
at a¯ = 0 is the DF state with no excitons (n = 0),
ex
whilethebackgroundthatvacanciesandintersitialscon-
dense above to form the VI-SF is the one with a finite
density of excitons (n 6=0). In what follows we discuss
ex
in more details the physics of the asymmetric and sym-
metric cases, focusing on the parameter regimes of the
hatched areas.
Due to the high He atom density and strong repul-
sive cores, solid 4He should fit the very asymmetric case
with ǫ /ǫ >> 1 in our model (Fig. 1(b)). Because
b a
the b-state is less localized compared to the a-state and
because of the exponentialdependence of the overlapin-
tegral on state size, we expect t >> t . For small t ,
b a a
the phase in the hatched region is the V-SF(B) phase.
To understand how this phase comes about, we consider
first t = 0. Here, the activation energies of a vacancy
a
and an exciton areǫ and ∆ respectively independent of
a
FIG. 2: Schematic illustration of snapshots of five phases t , so the DF state is locally stable. Since t =0, vacan-
b a
shown in fig.1(a) and (b) in terms of the occupation of he- cies cannot hop, and there is no possibility of vacancy
lium atoms at the lattice sites. A lattice site is represented
condensation above the DF state. However, if we take
by a numbered block. A solid dot represents a helium atom.
a background of excitons on every site, then vacancies
At each site, emptiness represents a vacancy, occupation at
can now hop with amplitude t , and the exciton state
the lower (higher) level represents a vacuum (exciton), and b
can be unstable with respect to a vacancy condensation
occupation at the both levels represents an interstitial.
that gives ¯b 6= 0. The onset of this V-SF(B) instability
is zt = ǫ −∆. For t greater than this value, E(¯b) has
b a b
a double minima behavior similar to that shown in Fig
the vacuumandthe exciton. Asnapshotofeachofthese 1(c) for the symmetric case, with E(¯b = 0) initially the
phasesintermsoftheoccupationofheliumatomsinreal
lowerenergy. However,ast increases(butstilllessthan
b
space is illustrated in Fig. 2. the value necessary for spontaneous generation of inter-
Toappreciatethephasediagramsandthenewphysics stitials), the vacancy condensation energy can become
arisingfromtheinteractionU,weshowthephasebound- largeenoughto overcomethe requiredvacancyandexci-
aries of the non-interacting model (U = 0) by dashed ton activation energies given by n ǫ +(1−n )∆, and
V a V
lines in Fig. 1(a) and (b). As remarked earlier, these a first order transition from the normal DF state to the
boundaries coincide with the vanishing of the activation V-SF(B)stateoccurs. Notethatthetransitionisaccom-
energy of an isolated vacancy or interstitial. Since U panied by a ”vacuum switching”, in that the vacancies
plays no role if only vacancies or only intersititials are arecondensingnotabovetheDFstate,butabovetheex-
present, these are also the phase boundaries for normal citon state. The V-SF(B) has ¯b 6= 0, and n +n = 1.
V ex
solidtoV-SF(A)orI-SFtransitionevenforU >0. Thus, As t increases,it becomes advantageousto mix in some
a
for such SF states, their corresponding normal (i.e. un- vacuum component to allow vacancy hopping through
condensed) states are not the DF state but contain a t , so that both a¯ and¯b are non-zero. Depending on the
a
finite density of defects, which is not the case for super- value ǫ , some interstitial condensation will also occur.
b
solid 4He. The most striking of our results is that the The V-SF(B) phase then makes a second order transi-
interplay between vacancies (and/or intersitials) and ex- tion into the VI-SF, characterized by both a¯ and ¯b 6= 0,
citon defects can lead to SF even in parameter regimes andn n n all6=0.Wewilldiscussthisphasefurther
V, I, ex
wherethenormalstateisstableagainstthegenerationof in the symmetric case, where it will feature more promi-
uncondensed defects (hatched areas in the figures). We nently. In this strongly asymmetric case, whether we
propose this as the reconciliation between activated de- take the SF state to be the V-SF(B) or the VI-SF state,
fect behavior at high temperature (T) and supersolidity we have n >> n . Thus, a prediction of our theory is
V I
4
that while the normalstate is a commensuratesolid, the MFT, there are four supersolid phases in terms of the
supersolid will have incommensurate density, which can order parameters a¯ and ¯b. However, there is no differ-
be confirmed by neutron scattering experiments in the ence between them in the type of off diagonal ordering
supersolid phase. Furthermore, because of the presence of the underlying 4He atoms. Thus, they are not really
of excitons in the supersolid (i.e. the He atom resides in distinct phases in the sense of different symmetry states,
the less localized b− state rather than the a− state, or but rather differ in the physics behind the condensation
some linear combination of the two), the local He den- as discussed above. Indeed, if fluctuations are included
sity inaunit cellwillchangewiththe transitioninto the or if we consider non-zero t , then the sharp distinction
ab
supersolid. This can be confirmed from the form factor between these ”phases” becomes crossover.
of neutron scattering or with a local probe. Following Leggett’s paper [4], we can derive the SF
Next we look at the symmetric case. Although this density detected by the NCRI experiment, which is
probably does not describe solid 4He, it may be applica-
ble to a system of trapped bosons in an optical lattice, ρ =ρ m m⋆+ρ m m⋆, (3)
S a 0/ a b 0/ b
wheretheperiodicpotentialisimposedexternallyrather
than internally generated. In such a system, the ratio where m⋆ and m⋆ are the effective mass of the vacancy
a b
ǫ /ǫ can be tuned by tuning the optical potential. The
b a and interstitial bands respectively determined by the
interesting phase here is the VI-SF phase in the hatched
hopping integral t and t and the lattice structure, and
a b
region. Again, the key question is what causes the addi- m is the bare mass of 4He. For the purpose of illus-
0
tional non-trivial SF solution since neither vacancy nor tration, we will assume that m⋆ = m at the value of
b 0
the interstitial alone can condense at T = 0. Unlike the
t = 1/3. Because the transition at T = 0 is first order,
b
asymmetric case, where n in this phase basically plays
I the SFdensityhasanabruptjump atthe criticality. For
no role, and indeed →0 as ǫ →∞, here, n ,n ,n ,n
b 0 V I ex reasonablevalues ofparametersǫ =1,ǫ =4, t =0.07
a b a
areallnon-negligibleinthis SFphase. Within the MFA, andt =0.2,wefindthejumptobe∼9%,whichisabout
b
the eigenstates are direct product of single-site states.
anorderlargerthantheSFdensitymeasuredinKimand
In this ground state, the state on each site is a coher-
Chan’s experiments[2]. The discrepancy is partly due to
ent mixture of the DF state, the vacancy, the intersti-
the simplicity of the model. For instance, the critical SF
tial, and the exciton. This coherence allows constructive density ρ is found to be ∼ 3% by including an on-site
S
interference between the various hopping processes,thus interband hopping t0 = 0.12. Furthermore, the quan-
ab
enhancingtheeffectofkineticenergyoverthatwhenonly
tumphasefluctuationsmentionedabovewilldecreaseρ
S
vacanciesorinterstitials arepresentor whenthey hopin
atcriticalitybythedualeffectsofdirectlydecreasingthe
a background of only DF states or only exciton states.
orderparametersandbyreductionofthecriticalvalueU
c
Thiscoherenteffectis bestunderstoodbyexaminingthe
duetostabilizingoftheSFphaserelativetotheDFstate.
highly symmetric case of ǫ = ǫ = ǫ, t = t = t and
b a b a We also note that the experiments of Kim and Chan are
U = 2ǫ (so that ∆ = 0+). By symmetry, we also have
a performedongranularratherthansingle crystal,sothat
a¯ = ¯b. Because of the symmetry between the vacancy
theobservedρ maybegovernedbyJosephsoneffectand
S
andtheinterstitial,andbetweenthevacuumandtheex-
therefore can be considerably smaller than the intrinsic
citon states, the MF energy EMF(a¯) can be analytically value of homogenous supersolid 4He.
found to be ǫ − q ǫ 2+(2zta¯)2 + 2zta¯2, from which We have seen that the T=0 transition should be first
2 (cid:0)2(cid:1)
the condition for the SF state is found to be ǫ−2zt<0. order. A careful examination of the finite T transition
Compared with the SF condition for vacancy only con- should include also the effect of phase fluctuations. The
densation above the DF background, we see that the ef- finite T transition as well as the collective excitations in
fective hopping integralincreases fromt to 2t due to the the various SF phases of our model will be discussed in
coherentmixture of all 4 states. For this artificialhighly a future publication.
symmetriccase,thenormaltoVI-SFtransitionissecond Inconclusion,wehaveproposedasolutiontohow4He
order due to ∆ being infinitesimal. As ∆ increases, the can become a supersolid at low T when both vacancies
transition becomes first order. This is what is shown in and interstitials have relatively high activation energies
Fig 1c as the transition is crossed by increasing t. Alter- inthenormalsolid. Inourtheory,thepresenceoflowen-
natively, the transition can be crossed by increasing U, ergyboundvacancy-interstitialdefectsfacilitatethecon-
as shown in Fig 1(d). densation of vacancies. In this theory, the normal state
Although our results so far are obtained using the is a commensurate solid while the supersolid is incom-
MFA, we believe the central conclusion, that supersolid mensurate. Furthermore,the local helium density in the
can occur even when defects like vacancies and intersti- unit cell has different profiles in the two phases. These
tialshaverelativelyhighactivationenergiesiscorrect. In predictions can be tested experimentally.
the lattice model of (1), the excitations in the supersolid This work is partially supported by the RGC of Hong
phase aregaplessand the excitations in the normalsolid Kong. We wish to thank M. H. W. Chan for bringing
are gapful. Therefore, the quantum fluctuations are ex- our attention to this interesting topic as well as many
pected to further stabilize the supersolid phase. In our stimulating discussions.
5
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[6] A.F. Andreev and I.M. Lifshitz, Soviet Phys. JETP 29, [13] In these variational calculations shown in Figs.1c, and
1107 (1969). 1d, for given mean fields a¯ =¯b, we first solve the mean
[7] N.Prokofev and B. Svistunov,cond-matt/0409472. fieldHamiltonianof(1)toobtainthelowestenergystate
[8] R.A.Guyer,Phys. Rev.Lett. 26 174, (1971). |Ψ(a¯)i. Wethen calculate E(a¯)=hΨ(a¯)|H |Ψ(a¯)i.
[9] M.W. Meisel, Physica B 178 121, (1992).