Download Supersolid

Ginsburg-Landau Theory of Solids and Supersolids
Jinwu Ye
Penn State University
Outline of the talk:
1. Introduction: the experiment, broken symmetries…..
2. Ginsburg-Landau theory of a supersolid
3. SF to NS and SF to SS transition
4. NS to SS transition at T=0
5. Elementary Excitations in SS
6. Vortex loop in SS
7. Debye-Waller factor, Density-Density Correlation
8. Specific heat and entropy in SS
9. Supersolids in other systems
10. Conclusions
Acknowledgement
I thank
P. W. Anderson, T. Clark, M. Cole, B. Halperin,
J. K. Jain , T. Leggett and G. D. Mahan especially
Moses Chan and Tom Lubensky for many helpful
discussions and critical comments
References:
Jinwu Ye, Phys. Rev. Lett. 97, 125302, 2006
Jinwu Ye, cond-mat/07050770
Jinwu Ye, cond-mat/0603269,
1.Introduction
19
~ 10 Gev
General Relativity
String Theory ???
Cosmology
Astrophysics
Gravity
Symmetry and symmetry breaking at different
energy (or length ) scales
Quantum Mechanics
Quantum Field Theory
2d conformal field theory
~ mK
Elementary particle physics~ 10 2 Gev
Atomic physics
Many body physics
Condensed matter physics
Statistical mechanics
Emergent quantum Phenomena !
Is supersolid a new emergent phenomenon ?
P.W. Anderson: More is different !
Emergent Phenomena !
Macroscopic quantum phenomena emerged in ~ 10 23
interacting atoms, electrons or spins
States of matter break different symmetries at low temperature
•Superconductivity
•Superfluid
•Quantum Hall effects
•Quantum Solids
•Supersolid ?
•……..???
• High temperature superconductors
• Mott insulators
• Quantum Anti-ferromagnets
• Spin density wave
• Charge density waves
• Valence bond solids
• Spin liquids ?
•…………???
Is the supersolid a new state of matter ?
What is a liquid ?
A liquid can flow with some viscosity
It breaks no symmetry, it exists only at high temperatures
At low temperatures, any matter has to have some orders which
break some kinds of symmetries.
What is a solid ?
A solid can not flow

G:
Reciprocal lattice vectors
 

Density operator: n( x )  n0   n  eiG x
G


G
Breaking translational symmetry:
H 2O, H 2 ,.......
Lattice phonons

u
4
Essentially all substances take solids except
3
He, He
What is a superfluid ?
A superfluid can flow without viscosity
Complex order parameter:
 | | e
Breaking Global U(1) symmetry:

i
Superfluid phonons

What is a supersolid ?
A supersolid is a new state which has both crystalline order
and superfluid order.
Can a supersolid exist in nature, especially in He4 ?
Large quantum fluctuations in He4 make it possible
The supersolid state was theoretically speculated in 1970:
Andreev and I. Lifshitz, 1969. Bose-Einstein Condensation (BEC) of
vacancies leads to supersolid, classical hydrodynamices of vacancies.
G. V. Chester, 1970, Wavefunction with both BEC and crystalline order, a
supersolid cannot exist without vacancies or interstitials
A. J. Leggett, 1970, Non-Classical Rotational Inertial (NCRI) of
supersolid He4
I  I cl (1   s /  ),  s /   10 4
quantum exchange process of He atoms can also lead to a supersolid even
in the absence of vacancies,
W. M. Saslow , 1976, improve the upper bound
s

 10
2
Over the last 35 years, a number of experiments have been designed to search
for the supersolid state without success.
Torsional oscillator is ideal for
the detection of superfluidity
Be-Cu
Torsion Rod
Torsion Bob
Containing
helium
f
Drive
Detection
 o  2
I
K
f0
I  I cl (1   s /  )
Kim and Chan
By Torsional Oscillator experiments
Science 305, 1941 (2004)
Soild He4 at 51 bars
1 ~ 2 % NCRI appears below
0.25K
Strong |v|max dependence
(above 14µm/s)
I
  2
K
I  I cl (1   s /  )
Very recently, there are three experimental groups one in US, two in Japan
Confirmed ( ? ) PSU’s experiments.
Torsional Oscillator experiments
• A.S. Rittner and J. D. Reppy, cond-mat/0604568
• M. Kubota et al
• K. Shirahama et al
Possible phase diagram
of Helium 4
Other experiments:
• Ultrasound attenuation
• Specific heat
• Mass flow
• Melting curve
• Neutron Scattering
PSU’s experiments have rekindled great theoretical interests in
the possible supersolid phase of He4
(1) Numerical ( Path Integral quantum Monte-Carlo ) approach:
Superfluid flowing
in grain boundary ?
•
D. M. Ceperley, B. Bernu, Phys. Rev. Lett. 93, 155303 (2004);
•
N. Prokof'ev, B. Svistunov, Phys. Rev. Lett. 94, 155302 (2005);
•
E. Burovski, E. Kozik, A. Kuklov, N. Prokof'ev, B. Svistunov, Phys. Rev. Lett., 94,165301(2005).
•
M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer,
•
Phys. Rev. Lett. {\bf 97}, 080401 (2006).
(2) Phenomenological approach:
•
Xi Dai, Michael Ma, Fu-Chun Zhang, Phys. Rev. B 72, 132504 (2005)
•
P. W. Anderson, W. F. Brinkman, David A. Huse, Science 18 Nov. 2005 (2006)
•
A. T. Dorsey, P. M. Goldbart, J. Toner, Phys. Rev. Lett. 96, 055301
•

Supersolid ?
Uncertainty Relation:
[ x, p ]  i, x  p  1
Cannot measure position and momentum precisely simultaneously !
[ Nb ,  ]  i, Nb    1
Cannot measure phase and density precisely simultaneously !
Superfluid: phase order in the global phase


Solid: density order in local density n(x )
How to reconcile the two extremes into a supersolid ?
Has to have total boson number
Nb
fluctuations !
The solid is not perfect: has either vacancies or interstitials
whose BEC may lead to a supersolid
2. Ginsburg-Landau (GL) Theory of a supersolid
Construct GL theory:
Expanding free energy in terms of order parameters near
critical point consistent with the symmetries of the system
(1) GL theory of Liquid to superfluid transition:
Complex order parameter:
 | | e
i
|  |3
t  T  Tc
Invariant under the U(1) symmetry:
  
(a) In liquid:
t  0,    0
U(1) symmetry is respected
(b) In superfluid:
t  0,    0
U(1) symmetry is broken

 
Both sides have the translational symmetry:  ( x )   ( x  a )
3d XY model to describe the

( cusp ) transition
Greywall and Ahlers, 1973
Ahler, 1971


Cv ~| t | ,  s ~| t |
  0.013  0,   0.672
(2) GL theory of Liquid to solid transition:
Density operator:
Order parameter:
 


'
iG  x
e
n( x )  n0  n( x )  
n
G


nG , G  0
rG  r  c(G  k ) , S (k ) ~ 1 / rG
2
G
2 2
r
Invariant under translational symmetry:
 

 
n( x )  n( x  a ), nG  nG eiG a


n( x )  n( x )

a
is any vector
In liquid:
r  0,  nG  0
In solid:
r  0,  nG  0
Translational symmetry is broken down to the lattice symmetry:
   
a  R, G  R  2

R
is any lattice vector
 

 
n( x )  n( x  R), nG  nG eiG  R  nG
NL to SF transition
Complex OP
at k  0
even order terms
2nd order transition
NL to NS transition

Density OP nG , G  0
at reciprocal lattice vector
cubic term,
first order

G
(3) GL theory of a supersolid:
 

iG  x
e
n( x )  
n
G

Density operator:
 (x) :
G
Complex order parameter:
Vacancy or interstitial
f  f LSF  f L NS  f int
f int  gn( x) |  ( x) |2 v(n( x)) 2 |  ( x) |2 ......
Invariant under:
Under

 

 
n( x )  n( x  a ),  ( x )   ( x  a ),
   ei
n( x)  n( x), g   g , v  v
Two competing orders
g  0 : Vacancy case
 Repulsive:   0
g  0 : Interstitial case
In supersolid:
In the NL, t
 nG  0,    0
 0,  has a gap, can be integrated out.
In the NL,  n( x)  n0 , n( x) has a gap, can be integrated out
?
3. The SF to NS or SS transition
In the SF state,
t  0, r  0
BEC:   1  a  0 breaks U(1) symmetry
kr ~
2
a
~
2

3.17 A
As P  , the roton minimum gets deeper and deeper,  r 
as detected in neutron scattering experiments
Feymann relation:
q2
 (q) ~
2mSn (q)
The first maximum peak in S n (q)  the roton minimum in
 (q )
Two possibilities:
(1)
   0
Commensurate Solid,
(2)
SF to NS transition
No SS
   0
In-Commensurate Solid with either vacancies or interstitials
There is a SS !
SF to SS transition
Which scenario will happen depends on the sign and strength of
the coupling
, will be analyzed further in a few minutes
g
Let’s focus on case (1) first:
I will explicitly construct a Quantum Ginzburg-Landau (QGL)
action to describe SF to NS transition
3a. The SF to NS transition
Inside the SF:
Where:
Phase representation:
Dispersion Relation:

Neglecting vortex excitations:
Density representation:
Feymann Relation:
Structure function:
Including vortex excitations: QGL to describe SF to NS transition:
SF:
NS:
3b. The SF to SS transition
(2)
   0
:
1
There is a SS !
SF to SS transition
Vacancies or Interstitials !
2
Feymann relation:
q2
 (q) ~
2mSn (q)
The first maximum peak in S n (q)  the roton minimum in
n

The
lattice and
Superfluid density wave (SDW)
formations happen simultaneously
 (q )
Global phase diagram of He4 in case (2)
? ?
q2
q 
2mSq
The phase diagram will be confirmed from the NS side
Add quantum fluctuations of lattice to the
Classical GL action:


n( x )  n( x , ),

1
 n ( n ) 2
2

 ( x )   ( x , ),




  ( x , ) ( x , )

n( x , )  ( x , ) ( x , )
4. NS to SS transition at
Lattice phonons:
Density operator:
 
u ( x , )
T 0
The effective action:
u 

1
( u     u )
2
strain tensor
elastic constants, 5 for hcp , 2 for isotropic
For hcp: a  az n n  a (  n n )
   z n n    (   n n )
For isotropic solid:
a  a  ,   
Under RG transformation:
 '   / b z , x '  x / b,
 '   / Z , u'  u / Z
z  2, Z  b  d / 2
g '  gb 2 d ,
 n'  b  2  n ,
a0'  b  d / 21a0 ,
a1'  b1 d / 2 a1
r b r
'
2
Both
g
and
a1
upper critical dimension is d u  2
All couplings are irrelevant at
d 3
The NS to SS transition is the same universality class as
Mott to SF transition in a rigid lattice with exact exponents:
z  2,  1 / 2,  0
Neglecting quantum fluctuations:
x '  x / b,
g '  gb 4d ,
 '   / Z , u'  u / Z
a1'  b 2d / 2 a1
Z  b ( 2 d ) / 2
r '  b2r
Both
g
  4d
and
a1
upper critical dimension is
du  4
expansion in
M. A. De Moura, T. C. Lubensky, Y. Imry, and A. Aharony;,1976; D.
Bergman and B. Halperin, 1976.
3d XY model
  0.012  0 ,
a1
is irrelevant
NS to SS transition at finite T is the 3d XY universality class
A. T. Dorsey, P. M. Goldbart, J. Toner, Phys. Rev. Lett. 96, 055301
5. Excitations in a Supersolid
(1) Superfluid phonons in the

sector:

u
Topological vortex loop excitations in
(2) Lattice phonons


Well inside the SS, the low energy effective action is :
In NS, diffuse mode of vacancies:
P. C. Martin, O. Parodi, and P. S. Pershan, 1972
In SS, the diffusion mode becomes the SF
 mode !
(a) For isotropic solid:
a  a  ,   
Poles:
2  v2 q 2
a ~ 0.1, v ~ 110% v p , v ~ 90% vs
Transverse modes in SS are
the same as those in NS
v  v p  vs  v , v2  v2  v 2p  vs2 , v v  v p vs
vlp2  (  2 ) / n , vlp ~ 450  500 m / s
vtp2   / n , vtp ~ 230  320 m / s
vs2   s /  , vs ~ 360 m / s
(b) For hcp crystal:

q
along

z
Direction:
q z  0, q x  q y  0
uz  ul , K33    2, az  a, vlp2  K33 / n
vt2  K 44 / 4  n , vt ~ 250 m / s

q
along
xy
plane
vlp ~ 540 m / s
vs2   s /  , vs ~ 360 m / s
q z  0, q x  0, q y  0
xy
K11    2, K12  , vlp2  K11 / n , vlp ~ 455 m / s
2
vtxy
 ( K11  K12 ) / 2n , vtxy ~ 225 m / s
vtz2  K 44 / 4  n , vtz ~ 255 m / s
vs2   s /  , vs ~ 360 m / s

q
along general direction
One can not even define longitudinal and transverse
4 4
matrix diagonization in
u x , u y , u z ,
Qualitative physics stay the same.
Smoking gun experiment to prove or dis-prove the existence
of supersolid in Helium 4:
2
2 2


v
The two longitudinal modes 
 q should be detected
by in-elastic neutron scattering if SS indeed exists !
6. Debye-Waller factor in X-ray scattering

Density operator at G :
The Debye-Waller factor:
1


 G 2 u 2 
I (G)  e 3
I 0 (G)
where
 u 2  ul2  2  ut2 
Taking the ratio:
where
It can be shown
(u )l  0
The longitudinal vibration amplitude is reduced in SS
compared to the NS with the same  n ,  ,  !

 (G )
will approach from
1
above as
T  T3dxy
7. Vortex loops in supersolid
Vortex loop representation:
K 0   , K   s
a   a
v
j
6 component anti-symmetric tensor gauge field
1

     
2
vortex loop current
Gauge invariance:
a  a        
Coulomb gauge:
 a  0, at  i  q a / q
Longitudinal
ul
couples to transverse
v
0
Vortex loop density:
j
Vortex loop current:
v
j
at
1

      
2
1

  [ 0 ,   ]
2
Integrating out a0 , instantaneous density-density int.
Vortex loop density-density interaction is the same as that in SF
with the same parameters  ,  s !
T3 dxy
is solely determined by
s
Critical behavior of vortex loop close to
T3 dxy in SF is studied in
G.W. Williams, 1999
Integrating out
where
at
, retarded current-current int.
The difference between SS and SF:
Equal time at T  0
 '
c
Dt ( x  x ,  0)    ' 2
(x  x )
c0
Vortex current-current int. in SS is stronger than that in SF
with the same parameters  ,  s
Recent Inelastic neutron scattering experiments:
M. A. Adams, et.al, Phys. Rev. Lett. 98, 085301 (2007).
Atom kinetic energy
T  70 mK  400mK
S.O. Diallo, et.al, cond-mat/0702347

Mometum distribution function
n( k )
n0  0, T  80mK  500mK
E. Blackburn, et.al, cond-mat/0702537
Debye-Waller factor
T  140 mK  1K
8. Specific heat and entropy in SS
Specific heat in NS:
No reliable calculations on Cvan yet
Specific heat in SF:
Specific heat in SS:
C
tp
is the same as that in the NS
Entropy in SS:
Excess entropy due to
vacancy condensation:
S  0
due to the lower branch
Molar Volume
S ~ 10 5 R
v0 ~ 20 cm3 / mole
R  N0kB
v  vlp
TSS ~ 100 mK
is the gas constant
3 orders of magnitude smaller than non-interacting Bose gas
9. Supersolids in other systems
Supersolid on Lattices can be realized on optical lattices !
Jinwu, Ye, cond-mat/0503113, bipartite lattices;
cond-mat/0612009, frustrated lattices
Cooper pair supersolid
in high temperature SC ?
The physics of lattice SS is different than that of He4 SS
Possible excitonic supersolid in electronic systems ?
Electron + hole
 Exciton is a boson
There is also a roton minimum in electron-hole bilayer system
Excitonic supersolid in EHBS ?
Jinwu Ye, preprint to be submitted
There is a magneto-roton in Bilayer quantum Hall systems,
for very interesting physics similar to He4 in BLQH, see:
Jinwu, Ye and Longhua Jiang, PRL, 2007
Jinwu, Ye, PRL, 2006
10. Conclusions
1.
A simple GL theory to study all the 4 phases from a unified picture
2
If a SS exists depends on the sign and strength of the coupling
g
3. Construct explicitly the QGL of SF to NS transition
4.
SF to SS is a simultaneous formation of SDW and normal lattice
5.
Vacancy induced supersolid is certainly possible
6. NS to SS is a 3d XY with much narrower critical regime than NL to SF
7.
Elementary Excitations in SS, vortex loops in SS
8.
Debye-Waller factors, densisty-density correlation, specific heat, entropy
9. X-ray scattering patterns from SS-v and SS-i
No matter if He4 has the SS phase, the SS has deep and wide
scientific interests. It could be realized in other systems: