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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
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
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
Lattice bosons
Cooper pairs in crystal
4He films on substrates
Josephson junction arrays
Atomic BEC in 3D optical lattice[Greiner et al., Nature (2002)]
BEC in cold bosonic atomsUltra-cold trapped 87Rb atoms form BECfirst observed 1995
Standing waves from laser superimpose an optical lattice (2002)
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
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
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
Detecting the supersolid
Supersolid has doubled unit cell: additional coherence peaksclear and unambiguous signal!
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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?
First simulation results: yes!
Batrouni, Scalettar, Zimany, Kampf, PRL (1995)evidence for supersolid at 3% dopingfinite superfluid density and solid structure factor
Later simulation results: maybe?Batrouni and Scalettar, PRL (2000)
evidence for first order transition in canonical energiesbut careful finite size scaling is still needed
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
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
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)
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!!
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
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!
Stability of triangular lattice supersolids
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
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
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
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
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
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
Supersolid for 1/3 < ρ < 2/3
particle doping ρ=1/3 solid
hole doping ρ=2/3 solid
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
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?
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