Solid State Communications 150 (2010) 2040–2044

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Solid State Communications

journal homepage: www.elsevier.com/locate/ssc

Supersolidity for hard-core-bosons coupled to optical phononsSanjoy Datta, Sudhakar Yarlagadda ∗

CAMCS and TCMP Div., Saha Institute of Nuclear Physics, 1/AF Salt Lake, Kolkata-64, India

a r t i c l e i n f o

Article history:Received 14 July 2010Accepted 9 August 2010by Y.E. LozovikAvailable online 15 August 2010

Keywords:A. SupersolidsD. PhononsD. Phase transitions

a b s t r a c t

The coexistence of diagonal long-range order (DLRO) and off-diagonal long-range order (ODLRO),manifested in some systems, is still a theoretical enigma.Here,wepresent a novelmicroscopicmechanismfor supersolidity or the homogeneous coexistence of charge-density-wave state (an example of DLRO)and superfluidity/superconductivity (a realization of ODLRO). We derive an effective d-dimensionalHamiltonian for a system of hard-core-bosons coupled to optical phonons in a lattice. At non-half-fillings,a superfluid/superconductor to a supersolid transition occurs at intermediate boson–phonon couplings,while at strong-couplings the system phase separates. We demonstrate explicitly that the presence ofnext-nearest-neighbor hopping and nearest-neighbor repulsion leads to supersolidity.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The competition or cooperation among diverse electronicphases is a subject of immense interest. Coexistence of charge-density-wave (CDW) and superconductivity is realized in a varietyof systems such as the perovskite-type bismuthates (i.e., BaBiO3doped with K or Pb) [1], quasi-two-dimensional-layered dichalco-genides (e.g., 2H–TaSe2, 2H–TaS2, and 2H–NbSe2) [2], etc. Anotherinteresting example of DLRO-ODLRO [i.e., CDWand superfluid (SF)]homogeneous concurrence is helium-4. What is special about theabove systems is that they defy the usual expectation that thecompeting CDWand superfluid/superconducting orders occurmu-tually exclusively. Here, DLRO breaks a continuous translationalinvariance into a discrete translational symmetry, whereas ODLRObreaks a global U(1) phase rotational invariance [3].

Although it has been conjectured long ago that solid helium-4 may become a supersolid (SS) [4,5], only recently Chan andKim claimed to have observed signature of a SS [6]. However,conventional SS ideas have been challenged and an alternatepicture of a superglass state in solid He-4 has been proposed[7]. Nevertheless, recent shear modulus measurements seem toreaffirm the earlier supersolidity picture [8]. It is imperative tonote that phonons do exist in solid helium-4 and their role in thepossiblemanifestation of supersolidity needs to be understood [9].

In the bismuthate systems, the observed valence skipping ofthe bismuth ion is explained by invoking non-linear screeningwhich is said to produce a large attractive interaction resulting inthe formation of local pairs or hard-core-bosons (HCB) [10,11]. Inthe superconducting regime (that occurs upon doping), whether

∗ Corresponding author. Tel.: +91 33 2337 0379; fax: +91 33 2337 4637.E-mail address: y.sudhakar@saha.ac.in (S. Yarlagadda).

0038-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2010.08.009

the Cooper pairs are local or not is an unsettled issue. However,phonons (corresponding to the cooperative breathing mode of theoxygen octahedra surrounding the Bismuth ion) are expected toplay a role in the formation of the two long-range orders.

In dichalcogenides such as NbSe2, where homogeneous coex-istence of the two long-range orders has been unambiguously es-tablished [12], phonons are claimed to be relevant for the SS phase[13].

Theoretically, to understand supersolidity, bosonic modelswere studied in different kinds of lattice structures [14–17].In contrast to the above treatments, we employ a microscopicapproach to study the quantum phase transitions exhibited bya minimal Bose–Holstein (BH) lattice model where HCB arecoupled to optical phonons. Without addressing the suitability ofa model for any particular observable system, it is well worthunderstanding how phonons may be relevant to supersolidity.

Starting with our BH model, we derive an effective d-dimensional Hamiltonian for HCB by using a transparent non-perturbative technique. The region of validity of our effectiveHamiltonian is governed by the small parameter ratio of theadiabaticity t/ω0 and the boson–phonon (b–p) coupling g . Themost interesting feature of this effective Hamiltonian is thatit contains an additional next-nearest-neighbor (NNN) hoppingcompared to the Heisenberg xxz-model involving only nearest-neighbor (NN) hopping and NN repulsion [18,19]. We employeda modified Lanczos algorithm [20] (on lattices of sizes 4 × 4,√18 ×

√18,

√20 ×

√20, and 4 × 6) and found that (except for

the extreme anti-adiabatic limit) the BH model shows supersolidityat intermediate b–p coupling strengths whereas the xxz-modelproduces only a phase-separated (PS) state. In Fig. 1, we presentthe calculations for a 4 × 4 lattice. Our main results are depictedin Fig. 1(a), where supersolidity manifested by local pairs (as HCB)implies homogeneous coexistence of CDW and superconductivity.

S. Datta, S. Yarlagadda / Solid State Communications 150 (2010) 2040–2044 2041

a b

Fig. 1. Quantum phase diagram at various particle numbersNp , t/ω0 = 1.0 and (a) NNN hopping J2 > 0 in our effective BH Hamiltonian (Eq. (5)) and (b) J2 = 0 (xxz-model).Supersolidity occurs only in (a). The continuous lines in (a) and (b) are guides for the eye.

2. Effective Bose–Holstein Hamiltonian

We start with a system of spinless HCB coupled with opticalphonons on a square lattice. This system is described by a BHHamiltonian [21]

H = −t−j,δ

bĎj bj+δ + ω0

−j

aĎj aj + gω0

−j

nj(aj + aĎj ), (1)

where δ corresponds to nearest-neighbors, ω0 is the opticalphonon frequency, bj (aj) is the destruction operator for HCB(phonons), and nj ≡ bĎj bj. Then, we perform the Lang–Firsov(LF) transformation [22,23] on this Hamiltonian and this producesdisplaced simple harmonic oscillators and dresses the hoppingparticles with phonons. Under the LF transformation given byeSHe−S

= H0 +H ′ with S = −g∑

i ni(ai − aĎi ), bj and aj transformlike fermions and phonons in the Holsteinmodel. This is due to theunique commutation properties of HCB given by

[bi, bj] = [bi, bĎj ] = 0, for i = j,

{bi, bĎi } = 1. (2)

Next, we take the unperturbed Hamiltonian H0 to be [23]

H0 = −J1−j,δ

bĎj bj+δ + ω0

−j

aĎj aj − g2ω0

−j

nj, (3)

and the perturbation H ′ to be

H ′= −J1

−j,δ

bĎj bj+δ{SjĎ+S

j− − 1}, (4)

where Sj± = exp[±g(aj − aj+δ)], J1 = t exp(−g2), and g2ω0 is

the polaronic binding energy. We then follow the same steps asin Ref. [23] (including assuming t exp(−g2) ≪ ω0) to get thefollowing effective Hamiltonian in d-dimensions for our BHmodel

He = −g2ω0

−j

nj − J1−j,δ

bĎj bj+δ

− J2−

j,δ,δ′=δ

bĎj+δ′bj+δ − 0.5Jz−j,δ

nj(1 − nj+δ), (5)

where Jz ≡ (J21/ω0)[4f1(g) + 2f2(g)] and J2 ≡ (J21/ω0)f1(g)with f1(g) ≡

∑∞

n=1 g2n/(n!n) and f2(g) ≡

∑∞

n=1∑

∞

m=1 g2(n+m)/

[n!m!(n + m)]. The NNN hopping term −J2∑

j,δ,δ′=δ bĎj+δ′bj+δ

corresponds to hopping twice along the bonds that connect theNN sites in the lattice. Thus, NNN hopping (for a lattice constantof unit length) corresponds to a distance of

√2 along the diagonal

and 2 along the x- or y-axes. The effective Hamiltonian for fermions

(as shown in Ref. [23]) contains an extra correlated hopping termJ2

∑j,δ,δ′=δ 2njc

Ďj+δ′cj+δ (with cj being the destruction operator for

fermions). This is because the commutation relations for fermionsare different from those of HCB given in Eq. (2). Herewewould alsolike to point out that, as shown in Ref. [24], the small parameter forour perturbation theory is t/(gω0) for g > 1.

Our effective Hamiltonian (of Eq. (5)), upon making theconnection bĎ = S+, b = S−, and bĎb = Sz + 0.5, yieldsan anisotropic NNN Heisenberg model. When boson–phononcoupling is zero (g = 0), we get J2 = Jz = 0, and the Hamiltonianis an XY -model. For J2 = 0 and Jz = 0 (which would be a goodapproximation for g ≫ 1), we get the xxz-model or the anisotropicHeisenberg model. For the xxz-model, numerical results give aphase separated state and not a supersolid state as shown in thephase diagram of Fig. 1(b). The phase diagram of the xxz-modelis well understood and our calculations are in accord with theestablished picture. Upon including J2 (i.e., NNN hopping), onegets supersolidity as shown numerically in the phase diagram ofFig. 1(a) (and with details given in Section 4).

3. Long-range orders

Diagonal long-range order (DLRO) can be characterized by thestructure factor defined in terms of the particle density operatorsas follows:

S(q) =1N

−i,j

eiq·(Ri−Rj)(⟨ninj⟩ − ⟨ni⟩⟨nj⟩). (6)

Off-diagonal long-range order (ODLRO) [3], in an interactingBose–Einstein condensate (BEC), is characterized by the orderparameter ⟨b0⟩ =

√⟨n0⟩eiθ where n0 is the occupation number

for the k = 0 momentum state or the BEC. It is useful to define thegeneral one-particle density matrix

ρ(i, j) = ⟨bĎi bj⟩ =1N

−k,q

ei(k·Ri−q·Rj)⟨bĎkbq⟩, (7)

where ⟨⟩ denotes ensemble average. Eq. (7) gives the BECfraction as

nb =⟨n0⟩

Np=

−i,j

ρ(i, j)NNp

. (8)

In general, to find nb, one constructs the generalized one-particledensity matrix ρ and then diagonalizes it to find out the largesteigenvalue.

To characterize a SF, an important quantity is the SF fractionns which is calculated as follows. Spatial variation in the phase

2042 S. Datta, S. Yarlagadda / Solid State Communications 150 (2010) 2040–2044

a b

-14

-12

-10

-8

-6

-4

-2

0

3 4 5 6 7 8

Fre

e E

nerg

y/J 1

Np

t/ω0=0.1J2 > 0

g = 2.1

2.2

2.3

2.4

2.5

3.0

-30

-25

-20

-15

-10

-5

0

3 4 5 6 7 8

Fre

e E

nerg

y/J 2

Np

J1 = 0 g ≥ 1.3

g = 1.2

Fig. 2. Free energy, at different fillings in a 4 × 4 lattice, for different values of g when (a) J1 > 0, J2 = 0, and t/ω0 = 0.1; and (b) J1 = 0, J2 = 0, and t/ω0 = 1.0.

of the SF order parameter will increase the free energy of thesystem. As long as the phase variation is small, the entire freeenergy change can be attributed to the change in kinetic energyonly. We use a linear phase variation θ(x) = θ0x/L with θ0 beinga small angle and L the linear dimension in the x-direction. Thisis done by imposing twisted boundary conditions (TBC) on theNp-particle wave function. The phase twist in the wavefunctioncan be mapped onto the Hamiltonian by a unitary transformation(see Ref. [25]). The eigenenergies of the new twisted Hamiltonianwith periodic boundary conditions are the same as those of theoriginal Hamiltonian with twisted boundary conditions. The phasevariation (taken to be the same for BEC and SF order parameters) isintroduced in our twisted Hamiltonian by modifying the hoppingterms with bj → bj exp[−iθ0x ·Rj/L]. At T = 0 K, we can write thechange in energy to be

E[θ0] − E[0] =12mNpns

hm ∇θ(x)2 . (9)

Then, the SF fraction is given by [26,25,15]

ns =

N

Npteff

E[θ0] − E[0]

θ20

, (10)

where teff = h2 /2m. For our Hamiltonian in Eq. (5), we findteff = J1 + 8J2.

4. Results and discussion

We employ the mean field analysis (MFA) of Robaszkiewiczet al. [18] to study the phase transitions dictated by the effectiveHamiltonian of Eq. (5). We obtain the following expression for theSF-PS (SF-CDW) phase boundary at non-half-filling (half-filling):

Jz2J1

−3J2J1

=1 + (2n − 1)2

1 − (2n − 1)2. (11)

Eq. (11) leads to the samephase diagramas that (for the xxz-model)in Ref. [18] but with Jeffz = Jz/2−3J2 as the y-coordinate instead ofJeffz = Jz/2. Thus, within mean field, NNN hopping does not changethe qualitative features of the phase diagram; it only increases thecritical value of Jz/J1 atwhich the transition from the SF state to thePS or CDW state occurs. However, as demonstrated below by thenumerics at non-half-fillings, MFA fails to capture the supersolidphase.

We studied the stability of the phases by examining the natureof the free energy versusNp curves at different b–p couplings g (seeFigs. 2 and 3) and by using an analysis equivalent to the Maxwellconstruction. For the system at a given Np, g , and t/ω0, if the freeenergy point P lies above the straight line joining the nearest twostable points Q and R (lying on either side of P) on the same free

-18-16-14-12-10

-8-6-4-2 0

3 4 5 6 7 8F

ree

Ene

rgy/

J1

Np

t/ω0=1.0

g = 3.0 = 2.8 = 2.6 = 2.4 = 2.3 = 2.2 = 2.1 = 2.0 = 1.9

J2 > 0

Fig. 3. Plot of free energy, for different number of particles in a 4 × 4 cluster, atvarious values of g when J1 = 0, J2 = 0, and t/ω0 = 1.0.

energy curve, then the system at P breaks up into two phasescorresponding to points Q and R.

For the range of parameters that we considered (i.e., 0.1 ≤

t/ω0 ≤ 1 and g > 1), the small parameter t/(gω0) < 1. Thebehavior of the system for t/ω0 = 0.1 is the same for both J2 = 0and 0 = J2/J1 [=(J1/ω0)f1(g)] because J2/J1 is negligible in thelatter case. Furthermore, for J2 = 0, the behavior of the systemas a function of g is qualitatively the same for all values of t/ω0 asthere is only one dimensionless parameter Jz/J1 involved in Eq. (5).

We will now analyze together, in one plot, the quanti-ties nb, ns, and the normalized structure factor S∗(π, π) =

S(π, π)/Smax(π, π)where Smax(π, π) corresponds to all the parti-cles occupying only one sub-lattice. At half-filling, we first observethat the system is either a pure CDW or a pure SF. For a half-filledsystem (i.e., Np = 8) at J2 = 0 and t/ω0 = 0.1 (t/ω0 = 1.0), wecan see from Fig. 4(a) (Fig. 4(b)) that the system undergoes a sharp(first order) transition to an insulating CDW state at gc ≈ 2.15(gc ≈ 1.55). At g = gc , while there is a sharp rise in S∗(π, π),there is also a concomitant sharp drop in both the condensationfraction nb and the SF fraction ns. Furthermore, while ns actuallygoes to zero, nb remains finite (as follows from Eq. (8)) at a value1/N = 1/16 which is an artifact of the finiteness of the system.Larger values of t/ω0 for a half-filled system leads to lower valuesof gc . This is in accordance with the MFA phase boundary equation(11) and the fact that (Jz/J1) × (ω0/t) [(J2/J1) × (ω0/t)] is mono-tonically increasing (decreasing) function of g for g > 1.

Away from half-filling, the system shows markedly differentbehaviors compared to the half-filled situation. For Np ≤ 4 inthe 4 × 4 lattice considered here, without actually presenting thedetails of the calculations, we first note that there is no evidence ofa phase transition (for both J2 = 0 and J2 = 0).

In Fig. 4(c) drawn forNp = 5 and t/ω0 = 0.1, although S∗(π, π)displays a CDW transition at a critical value gc = 2.45, ns doesnot go to zero even at large values of g considered. Furthermore,we see clearly from Fig. 2(a) that, above this critical value of g ,

S. Datta, S. Yarlagadda / Solid State Communications 150 (2010) 2040–2044 2043

g g

g g

a b

c d

Fig. 4. Comparative plots of S∗(π, π), nb , and ns , in a 4 × 4 system, when J1 = 0, J2 = 0, and (a) t/ω0 = 0.1, Np = 8; (b) t/ω0 = 1.0, Np = 8; (c) t/ω0 = 0.1, Np = 5; and(d) t/ω0 = 1.0, Np = 5.

a b

g g

Fig. 5. Comparison of S∗(π, π), nb , and ns (in a 4 × 4 lattice) when t/ω0 = 1.0, J2 = 0 but J1 = 0 and for (a) Np = 8; and (b) Np = 5.

the curvature of the free energy curves suggests that the systemat Np = 5 is an inhomogeneous mixture of the CDW state and SFstate. Thus away fromhalf-filling, at small values of the adiabaticityt/ω0, our HCB-systemundergoes a transition from a SF state to a PSstate at a critical b–p coupling strength (similar to the xxz-modelin Fig. 1(b)).

However for t/ω0 not too small, when NNN hopping is present,the system shows a strikingly new behavior for a certain region ofthe g-parameter space. Let us consider the system at Np = 5,t/ω0 = 1.0, J1 = 0, and J2 = 0. Fig. 4(d) shows that, aboveg ≈ 1.85, the system enters a CDW state (as can be seen from thestructure factor); however, it continues to have a SF character asreflected by the finite value of ns. Furthermore, Fig. 3 reveals thatthe curve is concave, i.e., the system is PS, only above g = 2.0.This simultaneous presence of CDW and SF states, without anyinhomogeneity (for 1.85 < g < 2.1), implies that the system isa supersolid. Similarly, for six and seven particles as well, we findthat the system undergoes transition from a SF state to a SS stateand then to a PS state. This behavior is displayed in Fig. 1(a).

Finally, we shall present the interesting case of J1 = 0 and J2 =

0 as a means of understanding the SS phase in the phase diagramof Fig. 1(a). The physical scenario, when J1 can be negligibly small

compared to J2, has been addressed in Ref. [24] for a cooperativeelectron–phonon interaction in an one-dimensional system.WhenJ1 = 0, for large values of nearest-neighbor repulsion, it isquite natural that all the particles will occupy a single sub-lattice.However, the dramatic jump (at a critical value of g), from an equaloccupation of both sub-lattices to a single sub-lattice occupation,is quite unexpected (see Fig. 5). For a half-filled system, above acritical point, all the particles get localized which results in aninsulating state. This can be seen from Fig. 5(a). One can see that (atg ≈ 1.23) the structure factor dramatically jumps to its maximumvalue, while ns drops to zero and nb takes the limiting value of1/N = 1/16 for reasons discussed earlier. This shows that aboveg ≈ 1.23, the system is in an insulating state with one sub-latticebeing completely full. However, away from half-filling, the systemconducts perfectly while occupying a single sub-lattice because ofthe presence of holes in the sub-lattice. For instance, from Fig. 5(b)drawn for Np = 5, we see that the structure factor jumps to itsmaximum value at g ≈ 1.26, whereas ns drops to a finite valuewhich remains constant above g = 1.26. We see from Fig. 2(b),based on the curvature of the free energy curves, that the five-particle system does not phase separate both above and below theCDW transition. In fact, this single-phase stability is true for any

2044 S. Datta, S. Yarlagadda / Solid State Communications 150 (2010) 2040–2044

filling. This means that the system, at any non-half filling and atJ1 = 0, exhibits supersolidity above a critical value of g!

We will now justify that the calculated phase diagram inFig. 1(a) is not a finite size artifact. For a 4 × 6 system as well,calculations were performed for selective number of particles andthe results are in agreement with the phase diagram for a 4 × 4system. For instance, for Np = 7 in the 4 × 6 system, SS occurs atg ≈ 1.85 and PS occurs at g ≈ 2.1. For the Lanczos techniqueemployed by us, 4 × 6 is the biggest computable system size(especially when trying to understand PS states which do notrespect the symmetries of the Hamiltonian). At larger adiabaticityt/ω0 values, we also find that the structure factor, superfluidfraction ns, and BEC fraction nb undergo a more sharp change (asg is varied) for our effective Hamiltonian than for the xxz-modelwhose well-known phase diagram is reproduced by the 4 × 4system (see Fig. 1(b)).

5. Conclusions

We demonstrated that our BH model displays supersolidity in2D due to NNN hopping and NN repulsion. We believe that ourapproach does provide insight into the role played by phononsin the supersolidity phenomenon. Our effective Hamiltonian ofEq. (5) should be realizable in 2D molecular conductors. Inmolecular conductors, strong-coupling between electrons andintermolecular-phonons also produces HCB; such HCB couple tointra-molecular phonons [27]. In three dimensions, supersolidityis more achievable (i) in general, due to an increase in the ratioof NNN and NN coordination numbers, and (ii) in particular forsystems with cooperative breathing mode where the ratio of NNNand NN hoppings is enhanced.

Acknowledgements

S. Datta thanks Arnab Das for very useful discussions onimplementation of Lanczos algorithm. S. Yarlagadda thanksK. Sengupta, S. Sinha, A.V. Balatsky, P.W. Anderson, G. Baskaran,R.J. Cava, I. Mazin, M. Randeria, and C.M. Varma for valuable

discussions. This research was supported in part by the NationalScience Foundation under Grant No. PHY05-51164 at KITP.

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