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Supersolid phases of (lattice) bosons Matthias Troyer (ETH Zürich) Hebert, Batrouni, Scalettar, Schmid, Troyer, Dorneich, PRB 65, 014513 (2002) Schmid, Todo, Troyer, Dorneich, PRL 88, 167208 (2002) Schmid & Troyer, PRL 93, 067003 (2004) Sengupta, Pryadko, Alet, Schmid, Troyer, PRL 94, 207202 (2005) Wessel & Troyer, PRL 95, (2005, in press) alps.comp-phys.org

Matthias Troyer- Supersolid phases of (lattice) bosons

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Page 1: Matthias Troyer- Supersolid phases of (lattice) bosons

Supersolid phases of (lattice) bosons

Matthias Troyer (ETH Zürich)

Hebert, Batrouni, Scalettar, Schmid, Troyer, Dorneich, PRB 65, 014513 (2002)Schmid, Todo, Troyer, Dorneich, PRL 88, 167208 (2002)

Schmid & Troyer, PRL 93, 067003 (2004)Sengupta, Pryadko, Alet, Schmid, Troyer, PRL 94, 207202 (2005)

Wessel & Troyer, PRL 95, (2005, in press)

alps.comp-phys.org

Page 2: Matthias Troyer- Supersolid phases of (lattice) bosons

Can a supersolid exist?A supersolid show simultaneously density wave order (broken translational symmetry) and superfluidity

Experiment on HeliumKim and Chan (2004): evidence for superfluidity in solid Helium?

Theory: can Helium be supersolid?Penrose and Onsager (1956): noAndreev and Lifshitz (1969), Chester (1970), Legget (1970): yesLegget (2004): maybeAnderson, Brinkman, Huse (2005): yes

Page 3: Matthias Troyer- Supersolid phases of (lattice) bosons

Numerical simulations of supersolids

continuum models for HeliumCeperley (2004): no superfluidity in perfect Helium crystalsProkof ’ev and Svistunov (2005): supersolidity requires defectsbut no final answer yet

supersolids on latticeseasier to investigate since the lattice is rigidefficient algorithms exist

relevance of lattice supersolidsphysical insight gained can be applied to continuum supersolidscan be realized in physical systems

Page 4: Matthias Troyer- Supersolid phases of (lattice) bosons

Lattice bosons

Cooper pairs in crystal

4He films on substrates

Josephson junction arrays

Atomic BEC in 3D optical lattice[Greiner et al., Nature (2002)]

Page 5: Matthias Troyer- Supersolid phases of (lattice) bosons

BEC in cold bosonic atomsUltra-cold trapped 87Rb atoms form BECfirst observed 1995

Standing waves from laser superimpose an optical lattice (2002)

Page 6: Matthias Troyer- Supersolid phases of (lattice) bosons

describes bosonic atoms in optical lattice well understood without the trap: Fisher et al, PRB 1989

Boson-Hubbard model

Phase diagram (V=0)

Ut /

1=n

2=n

0=n

U/µIncompressible Mott-insulator

Integer filling

superfluid

H = !t!

!i,j"

"

b†i bj + b†jbi

#

+ U!

i

ni(ni ! 1)/2 ! µ!

i

ni

Page 7: Matthias Troyer- Supersolid phases of (lattice) bosons

Quantum phase transition in trapped atoms

Experiment: Greiner et al, Nature (2002)Coherence vanishes as atoms enter Mott-insulating phaseMeasurements of momentum distribution

by taking real space image after expansion of the gas cloud

M. Greiner et al, Nature (2002)

increasing U/t

Page 8: Matthias Troyer- Supersolid phases of (lattice) bosons

Super solids with longer range repulsion

Extended Bose Hubbard model, e.g. strong dipolar interactions in Chromium condensates

shows simultaneously solid order and superfluidity

solid (crystal)caused by large V

doped solid:interstitials

supersolid:superfluid interstitials

H = !t!

!i,j"

(a†iaj + a†

jai) ! µ!

i

ni + U!

i

ni(ni ! 1)/2+V!

!i,j"

ninj

Page 9: Matthias Troyer- Supersolid phases of (lattice) bosons

Detecting the supersolid

Supersolid has doubled unit cell: additional coherence peaksclear and unambiguous signal!

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Page 10: Matthias Troyer- Supersolid phases of (lattice) bosons

Can a supersolid exist in lattice models?

Nearest neighbor repulsion modelBatrouni et al (1995): yes for hard-core bosonsvan Otterlo et al (1995): yes for hard and soft-core bosonsSørensen et al (1996): yes for soft-core bosonsBatrouni et al (2000): no for hard-core bosons

Supersolid in cold atomsGoral, Santos and Lewenstein (2002): yes with dipolar interactionsBüchler and Blatter (2003): yes in boson-fermion mixtures

What is the truth?

Page 11: Matthias Troyer- Supersolid phases of (lattice) bosons

First simulation results: yes!

Batrouni, Scalettar, Zimany, Kampf, PRL (1995)evidence for supersolid at 3% dopingfinite superfluid density and solid structure factor

Page 12: Matthias Troyer- Supersolid phases of (lattice) bosons

Later simulation results: maybe?Batrouni and Scalettar, PRL (2000)

evidence for first order transition in canonical energiesbut careful finite size scaling is still needed

Page 13: Matthias Troyer- Supersolid phases of (lattice) bosons

Efficient Cluster algorithms for quantum systems

Which system sizes can be studied?

Open source codes available at http://alps.comp-phys.org/

temperature Metropolis modern algorithms

1 16’000 spins 16’000’000 spins

0.1 200 spins 1’000’000 spins

0.1 32 bosons 10’000 bosons

0.005 ––– 50’000 spins

Page 14: Matthias Troyer- Supersolid phases of (lattice) bosons

Simulations with 100 x more particles clearly show phase separation at first order phase transition instead of supersolid

Hebert, Batrouni, Scalettar, Schmid, MT, Dorneich, PRB (2002) Schmid, Todo, MT, Dorneich, PRL (2002)

New simulations: no!

0.650.60.550.5Density ρ

solid

superfluid

Page 15: Matthias Troyer- Supersolid phases of (lattice) bosons

Supersolids versus phase separation

solid doped solid supersolid

!! = !"2t2

V

doped particles gain energy by forming a domain wall

!! = !"t < !"2t2

V

G. Schmid PhD thesis, ETH (2004)Sengupta, Pryadko, Alet, MT , Schmid, PRL (2005)

Page 16: Matthias Troyer- Supersolid phases of (lattice) bosons

Stabilizing the supersolid

It matters how we dope the solidU>>4V: particles go onto empty sublattice and phase separate4V>> U: particles go onto “occupied” sublattice and form supersolid

solid supersoliddopants on same sublattice!

V ! U/4 > t

H = !t!

!i,j"

(a†iaj + a†

jai) ! µ!

i

ni + U!

i

ni(ni ! 1)/2+V!

!i,j"

ninj

qualitatively different supersolid!!

Page 17: Matthias Troyer- Supersolid phases of (lattice) bosons

Phase diagram

Soft-core bosons with U/t = 20 P. Sengupta, L. Pryadko, F. Alet, MT, G. Schmid, PRL (2005)

supersolid possible at ρ>1/2 and small on-site repulsion U < 4V

no supersolid at density ρ<1/2 (superfluid domain walls)

0.00 0.25 0.50 0.75 1.00

!

2

4

6

8

10

V

SF PS SS PS

SF

PSM

IC

DW

II

CD

W I

U=20

t=1

Page 18: Matthias Troyer- Supersolid phases of (lattice) bosons

Two alternative routes to supersolids

doped solid striped supersolid

add next nearest neighbor hopping: H = H ! t!

!

""i,j##

(a†iaj + a

†jai)

striped solid

add next nearest neighbor repulsion: H = H + V!

!

""i,j##

ninj

E = !!4t!

Page 19: Matthias Troyer- Supersolid phases of (lattice) bosons

Stability of triangular lattice supersolids

Page 20: Matthias Troyer- Supersolid phases of (lattice) bosons

Triangular lattcie hard-core bosons

Classical limit: V >> t

all other densities: infinitely degenerate ground stateshow is the degeneracy lifted by quantum fluctuations?

ρ=0

ρ=1/3 ρ=2/3

ρ=1

Page 21: Matthias Troyer- Supersolid phases of (lattice) bosons

Mean-field calculations

Murthy, Arovas, Auerbach, PRB (1997)factor S!k" at the Neel vector !and properly proportional tothe lattice volume", and a nonzero value of the superfluiddensity #s . Again, next-nearest neighbor V! can stabilize astriped supersolid phase with anisotropic #s . One can alsoobtain Mott insulating phases with fractional filling in the

presence of next-nearest neighbor interactions.

In this paper, we will investigate the properties of the

model in Eq. !2" on frustrated two-dimensional lattices. Weare motivated by the fascinating interplay between frustra-

tion, quantum fluctuations, order, and disorder which has

been seen in quantum magnetism.

Frustration enhances the effects of quantum fluctuations.

Indeed, as early as 1973, Fazekas and Anderson23,24 raised

the possibility that for such systems, quantum fluctuations

might destroy long-ranged antiferromagnetic order even at

zero temperature. In many cases, frustration leads to an infi-

nite degeneracy at the classical !or mean field" level not as-sociated with any continuous symmetry of the Hamiltonian

itself. In these cases, it is left to quantum !or thermal" fluc-tuations to lift this degeneracy and select a unique ground

state,25,26 sometimes with long-ranged order. Our models ex-

hibit both a depletion !but not unambiguous destruction" oforder due to quantum fluctuations, as well as the phenom-

enon of ‘‘order by disorder.’’

In our work, we will choose the units of energy to be J ! ,

writing $%t/V!J!/2J ! , and h%H/J ! . We will be follow-

ing closely the analysis of the anisotropic triangular lattice

antiferromagnet by Kleine, Muller-Hartmann, Frahm, and

Fazekas !KMFF",27 who performed a mean-field !S!&limit" and spin-wave theory !order 1/S corrections to mean-field" analysis. Contemporaneously with KMFF, Chubukovand Golosov28 derived the spin-wave expansion for an iso-

tropic Heisenberg antiferromagnet in a magnetic field, while

Sheng and Henley29 obtained the spin-wave theory for the

anisotropic antiferromagnet in the absence of a field.

The mean-field phase diagram is shown in Fig. 1 !both thetriangular and kagome lattices have the same mean-field

phase diagram up to a rescaling of h". Notice that the super-solid phase appears in a broad region of $ and filling. The

reason the supersolid is so robust is that the lattice frustrates

a full condensation into a solid. Generically, frustrated lat-

tices might be good places to look for this phase.

Let us briefly concentrate on h!0 before describing theentire phase diagram. We will be assuming a three sublattice

structure throughout. The mean-field state is then described

by three polar and three azimuthal angles: ('A ,'B ,'C ,(A ,(B ,(C), and is invariant under uniform rotation of the

azimuths.

Due to the ferromagnetic coupling in the x-y spin direc-

tions the mean-field solution is always coplanar. Just as in

KMFF, there is a one-parameter family of degenerate mean-

field solutions in the zero-field case !originally found by Mi-yashita and Kawamura30". The A sublattice polar angle 'Amay be chosen as the free parameter; spin-wave theory

!SWT" is necessary to lift the degeneracy and uncover thetrue ground state. Figure 2 shows the ground-state energy in

SWT as a function of 'A for the triangular lattice at $!0.25.Using SWT we also compute the fluctuations of the spins,

and the consequent quantum-corrected magnetization and the

solid and ODLRO order parameters. Figure 3 illustrates

these quantities in mean field and to leading order in SWT

!where S has been set equal to 1/2" as a function of $ for thetriangular lattice. It is clear that the quantum corrected Sz is

very close to zero for all $, reflecting the fact that at h!0the lattice is half-filled. Two sublattices acquire large correc-

tions due to quantum fluctuations !even in the Ising limit$!0", while the third has only small quantum corrections.

This is very similar to the fully antiferromagnetic case stud-

ied by KMFF. Therefore, even at S!1/2 the solid order sur-vives. The off-diagonal order parameter Sx is reduced in

FIG. 1. Mean-field phase diagram for the triangular lattice. The

kagome lattice phase diagram differs only by a rescaling of h .

Heavy lines denote first-order transitions, light lines second-order

transitions, and dashed lines denote linear instabilities.

FIG. 2. Ground-state energy of the triangular lattice at $!0.25as a function of 'A . The minimum is quadratic.

55 3105SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED . . .

ρ=1/3

ρ=2/3

ρ=0

ρ=1super-fluid

supersolid

t / V

chem

ical

pot

entia

l

Page 22: Matthias Troyer- Supersolid phases of (lattice) bosons

Recent investigations

May 11, 2005: three preprints, to be published in PRLHeidarian & Damle, cond-mat/0505257

Melko, Paramekanti, Burkov, Vishwanath, Sheng, Balents, cond-mat/0505258

Wessel & MT, cond-mat/0505298

July 26, 2005: one follow-up paper, submitted to PRLBoninsegni & Prokof ’ev, cond-mat/0507620

Quantum Monte Carlo and analytical argumentsall find supersolidnot complete agreement on nature of supersolid

Page 23: Matthias Troyer- Supersolid phases of (lattice) bosons

Quantum Monte CarloPhase diagram similar to mean-field calculations

0 0.1 0.2 0.3 0.4 0.5t/V

-2

0

2

4

6

8

µ/V

full

empty

superfluid

solid ρ=2/3

solid ρ=1/3

supersolid

0 0.1 0.2 0.3 0.4 0.5t/V

0

0.2

0.4

0.6

0.8

1

ρ superfluid

solid ρ=2/3

solid ρ=1/3

supersolid

PSPS

PS

PS

canonical grand-canonical

Page 24: Matthias Troyer- Supersolid phases of (lattice) bosons

Domain wall instability

Doping ρ > 2/3 or ρ < 1/3superfluid domain walls have lower energy than uniform supersolidphase separation instead of supersolid

Energy ! "t2/V Energy ! "t

Page 25: Matthias Troyer- Supersolid phases of (lattice) bosons

Domain wall instability

Doping ρ > 2/3 or ρ < 1/3superfluid domain walls have lower energy than uniform supersolidphase separation instead of supersolid

3 4 5 6 7 8µ/V

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

ρ

t/V=0.1t/V=0.2t/V=0.3

0.1 0.15 0.2 0.25t/V

0.60.620.640.66

ρ

µ/V=4

jump in densityat 1st order phase transition

Page 26: Matthias Troyer- Supersolid phases of (lattice) bosons

Supersolid for 1/3 < ρ < 2/3

particle doping ρ=1/3 solid

hole doping ρ=2/3 solid

Page 27: Matthias Troyer- Supersolid phases of (lattice) bosons

Numerical results: supersolid!

Simulations show both density wave order and superfluidity

0 0.05 0.1 0.15 0.2

1/L

0

0.05

0.1

0.15

0.2

/V=3t/V=0.1

S / 2

S(Q) / N

Page 28: Matthias Troyer- Supersolid phases of (lattice) bosons

Summary & ConclusionsSimulations provide high-accuracy results for large bosonic systemsSupersolids in square and cubic lattices

unstable towards formation of superfluid domain walls

need weak on-site interaction to stabilize a supersolid

longer range interactions give striped supersolid (relevant for high-Tc cuprates?)

Supersolids on triangular latticetwo different supersolids stable in wide filling regime

can be realized in Chromium BEC on optical lattices

Lessons for Helium supersolidsSuperfluid domain walls are main instability for supersolids

do the Helium experiments just see superfluid grain boundaries?

Page 29: Matthias Troyer- Supersolid phases of (lattice) bosons

Thanks to my collaborators

ETH Zürich Guido Schmid

Université de NiceGeorge BatrouniFrederic Hebert

CEA Saclay Fabien Alet

University of TokyoSynge Todo

UMass AmherstNikolay Prokof ’evBoris Svistunov

UC RiversideLeonid Pryadko Pinaki Sengupta

Universität StuttgartStefan Wessel