Transcript (3) Supersolid

Quantum phase transitions from Solids to Supersolids
in bi-partite lattices
Jinwu Ye
Penn State University
Outline of the talk:
1. Introduction to Boson Hubbard Model and supersolids on lattices
2. Boson -Vortex duality in boson Hubbard Model
3. Extended Hubbard Model at honeycomb and square lattice near
half filling
4. Charge Density Wave supersolid
5. Valence Bond Supersolid
6. Implications to QMC
7. Conclusions
References:
1. The mobility of dual vortices in square, honeycomb, triangular, Kagome and
Dice lattices.
Longhua Jiang and Jinwu Ye, J. Phys, Condensed Matter. 18 (2006) 6907-6922
2. Quantum phase transitions from solids to supersolids in bipartite lattices
Jinwu Ye, cond-mat/0503113. Revised version, submitted to Phys. Rev. Lett.
3. Supersolids and solids to Supersolids transitions in frustrated lattices ,
Jinwu Ye, cond-mat/0612009,
4. The Extended Bose Hubbard Model on the Two Dimensional Honeycomb
Lattice
Jing Yu Gan, Yu Chuan Wen, Jinwu Ye, Tao Li, Shi-Jie Yang, Yue Yu,
Phys. Rev. B 75, 214509 (2007)
1.
Introduction to extended boson Hubbard model
and supersolid on a lattice
Extended Boson Hubbard Model (EBHM):
t : nearest neighbor hopping
U , V1 ,V2
......
: onsite, nn and nnn interactions
Ring exchange interaction:
Ultra-cold alkali gases:
Superfluid to Mott insulator transition in optical lattice,
Greiner et al, 2002
(1) Superfluid:
Bose-Einstein condensation (BEC)
Off-diagonal long range order:
 bi  0
(2) Charge density wave:
Diagonal order in boson density:

i i
ni  b b
(3) Supersolid:
Both off-diagonal and diagonal long range order
Supersolids on lattices are very different than possible
supersolids in Helium 4
Kim and Chan, 2005
Rather complete quantum phenomenology was worked out in
Jinwu Ye, Phys. Rev. Lett. 97, 125302 (2006).
Jinwu Ye, cond-mat/0705.0770, submitted to Nature Physics
Jinwu Ye, cond-mat/0603269, submitted to Phys. Rev. B.
See my seminar at KITP-C on Aug. 8, 2007
p
2. Boson-Vortex duality in EBHM at filling factor f  q
At Integer fillings f  n , Superconductor to Mott insulator
transition ( SIT ) was studied in:
•
Onsite interaction only:
Fisher, Weichman, Grinstein and Fisher, 1989
•
Long-range Coulomb interaction:
Fisher and Grinstein, 1988;
Jinwu Ye, 1998
Boson-Vortex duality at f  1
Number of particles= number of holes

Particles ~

Holes ~

relativistic field theory for
S   d 2 rd[|   |2  |  r |2  s |  |2 u |  |4 ]
Insulator: <

> =0;
Superfluid <

>

0


Excitations of the superfluid:
(1) Gapless Phonons: represented by a dual U(1) gauge field A
(2) Gapful Vortices: vortex operator 
Number of vortices = number of antivortices
 relativistic field theory for 
1
S vor   d 2 rd[| (   iA ) |2  sd |  |2 ud |  |4  (  A ) 2 ]
4
Superfluid: <

> =0;
Insulator: <

>
0
Boson-Vortex duality and SIT at f  1
SIT at cold alkali gas: Greiner et al, 2002
Boson quasi-particle excitations ~

in the insulator :
S   d 2 rd[|   |2  |  r |2  s |  |2 u |  |4 ]
is dual to
Vortex excitations ~  + phonon excitations
A in the superfluid:
1
S vor   d 2 rd[| (   iA ) |2  sd |  |2 ud |  |4  (  A ) 2 ]
4
Insulator:
<
> =0
<  >0

Superfluid:
<
>  0
<  > = 0

Dasgupta and Halperin, 1981

The boson acquires a 2
phase when it encircles a
vortex
A vortex in the vortex field is
the original boson, the vortex
acquires a 2 phase when
it encircles a boson
The average strength of the dual magnetic field on the vortex field
= density of boson= f flux quanta per plaquette
Balents, Bartosch, Burkov, Sachdev, Sengupta, 2005
Bosons at filling factor f=p/q hopping on a direct lattice
is dual to
Vortices hopping on the dual lattice subject to a fluctuating dual " magnetic
field“ whose average strength is f=p/q flux quanta per dual plaquette.
This is similar to
Hofstadter problem of electrons moving in a lattice in the presence of f=p/q
flux quanta per plaquette.
Hofstadter , 1976
The spectrum can be classified by Magnetic Space Group ( MSG)
MSG dictates that there are at least q -fold degenerate minima which
forms a q dimensional representation of the MSG.
The effective theory in terms of these q vortices should be invariant
under this MSG.
Balents, Bartosch, Burkov, Sachdev, Sengupta, 2005
Taking q  16, 32,........ and apply to study
CDW state in high temperature superconductors
Two holes in the DVM developed by
Balents et.al 2005
(1) It can NOT be applied to the CDW side:
In the CDW side, the boson density takes, for example,
a checkboard ( or stripe ) Order.
The average strength of the dual magnetic field on the vortex field
= density of boson
The average strength of the dual magnetic field per dual plaquette
should also be taken the same pattern !
Not as uniform as f  p / q
(2) As shown in:
Longhua Jiang and Jinwu Ye, 2006
The bandwidth of the dual vortex band
w  Ae cq
The DVM completely breaks down when
q  16, 32,........
1. I take a completely different route:
Taking q  2 , so the DVM works, then moving slightly away
from q  2 , the return is interesting and important:
Supersolids could exist only at in-commensurate fillings !
2. We will also extend the DVM also to the CDW side
3. EBHM on honeycomb lattice near half filling
Bosons hopping on honeycomb lattice at filling factor f=p/q
is dual to
Vortices hopping on triangular lattice subject to a fluctuating magnetic
field whose average strength is f=p/q per triangular plaquette
The Hamiltonian of the vortices is:
+ interaction terms
tv is the vortex hopping amplitude
The space group of the triangular lattice:
(1) Point group: C6v ~ D6 , 12 elements with 2 generators R / 3 , I1
(2) Two translations
T1 ,T2
The Magnetic Space Group (MSG ) are generated by:
Translations and Rotations:
Reflections:
T1 T2  Td , T1 T2   2 T2 T1 ;   2 f
K is the complex congugate
They all commute with the Hamiltonian H
The rotations around the two direct
lattice points A and B:
Both also commute with H
In the following, we focus on q=2 where the energy band is:
There are two minima at
are labeled by  
where the two eigenmodes
  transform under the MSG as:
The quadratic terms of the effective action is the
scalar electrodynamics:
The most general quartic term invariant under all
the transformations is:
Moving slightly away from half filling f=1/2 corresponds to
1

f

f

adding a small dual magnetic field
2 in the action:
r > 0, superfluid state,
<  l > =0, l  
r < 0, insulating state,
<  l >  0 for at least one l
1.  1  0 : Ising limit
2.  1  0 : Easy-plane limit
It turns out that the action only works in the SF and VBS sides,
But breaks down in the CDW side
The SF is stable against the changing of chemical potential
In the following, we focus on the insulating side.
4. Charge density wave supersolid
In the Ising limit, at mean field:
Density Wave order parameter in the Ising limit:
A uniform saddle point does not work in the CDW side
Saddle point for the dual gauge field:
for sublattice A
for sublattice B
A and B could be checkboard or stripe order
There is only one vortex minimum in such a staggered
dual magnetic field with   1/ 2
Slightly away from 1/2 filling, the effective action
inside the CDW is:
It has a structure identical to q=1 component Ginzburg-Landau
model for a type-II superconductor in a magnetic field
a
A
is always massive, so does not appear in above Eqn.
Phase diagram of Type II superconductor
in external magnetic field:
D. Nelson, 1989
Fisher, Fisher and Huse, 1991
f  1 / 2  f
f n
IC-CDW
C-CDW
1
(1 1)[ ]
2
Fisher, Weichman,
Grinstein and Fisher, 1989
Ising limit
The CDW could also be stripe order
In the CDW state:
  b  0
b
A
is massive
In the CDW-SS state:   b  0
b
A
becomes gapless which stands for the superfluid mode
The transition from the CDW to the CDW-SS is in the
same universality class as that from the SF to Mott insulator,
Therefore have exact critical exponents:
z  2,  1 / 2,  0
The superfluid density in the CDW-SS scales as:
with a logarithmic correction
The CDW-SS has the same diagonal
density order as the CDW at f  1 / 2
The transition from the CDW-SS
to the IC-CDW is first order
The transition from the SF
to the CDW-SS is first order
The IC-CDW can be stabilized only
by very long-range interactions
Similar phase diagram also holds for square lattice
It is important to compare with microscopic calculations such
as quantum monte-carlo (QMC)
QMC simulations by:
F. Hebert, et.al, Phys. Rev. B, 65, 014513 (2001).
Hard core
U  ,V1  0
Soft core case:
U  ,V1  0,V2  0
P. Sengupta, et.al. 2005
( ,  ) SS is stable
The stripe solid to stripe-SS transition is 2nd order with
in the stripe supersolid, but scale in the same way
with different coefficients
5.Valence bond supersolid
In the easy-plane limit:
Valence Bond Solid (VBS) order parameter:
At mean field:
The uniform saddle point
SF and the VBS
K AB  cos(Q  x   ), Q 
holds in both
2
(1,1),        , x  dual lattice
3
Sixth order terms are needed
to determine the relative phase
Only C3   cos 3 can fix the relative phase
It turns out both signs of

  0,   

are equivalent
The other two VBS can be obtained by
A/ B
2 / 3
R
P.W. Anderson; Science 235, 1196(1987)
A ring exchange term :
is needed to stabilize the VBS order
C3   cos 3 is relevant, so the SF to VBS transition is 1st order
Slightly away from half-filling inside the VBS:

sector is massive, so can be integrated out
Assuming   0,   
u~  2 0
It is also identical to q=1 component GL
model for a type-II superconductor in a magnetic field
All the discussions on the Ising case follow
In the VBS:
  a     b  0,
A is massive
In the VB-SS:
  a   b  0
Gapless superfluid mode:
A
In square lattice:
The lowest order term coupling the two phases is
  0 (  0)
qs
VBS has Columnar dimer ( plaquette ) pattern
C is irrelevant, so the SF to the
VBS is a second order transition
through so called deconfined QCP
discovered by
O. Motrunich and A. Vishwanath, 2004
T. Senthil et. al, 2004
There are strong numerical evidence that there is no such
deconfined QCP, the SF to the VBS is weakly 1st order
Anatoly Kuklov, Nikolay Prokof'ev, Boris Svistunov, M. Troyer, con-mat/0602466
6. Implications on future QMC
(1) In square lattice: Stripe solid to stripe supersolid in the
hard core case, Checkboard solid to checkboard supersolid
in soft case to test the universality class by finite size scaling
z  2,  1 / 2,  0
K: tuning parameter:

Static structure factor:
Equal time

: Q is the ordering wave vector

K

b
Similar quantities can be defined for a bond
ij
i b j  h.c.
(2) Do the same things in the honeycome lattice
(3) With ring-exchange term, test the VBS to VB-SS transition
Soft core boson in honeycomb lattice
Jing Yu Gan, Yu Chuan Wen, Jinwu Ye,
Tao Li, Shi-Jie Yang, Yue Yu, Phys. Rev. B 75, 214509 (2007)
7. Conclusions
(1) Map out the global phase diagram in a unified theory:
SF, CDW( VBS), CDW-SS (VB-SS), IC-CDW (IC-VBS)
(2) Discover universality class of solids to supersolids transitions
(3) Propose a novel kind of supersolid: valence bond supersolid
(4) Several important implications on QMC
These results should have direct impacts on ongoing experiments
of cold atoms loaded on optical lattices. It should also shed some
lights on possible supersolid Helium 4.