Ginsburg-Landau Theory of Solids and Supersolids Jinwu Ye Penn State University

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Ginsburg-Landau Theory of Solids and
SupersolidsJinwu Ye Penn State University

• Outline of the talk
• Introduction the experiment, broken
  symmetries..
• Ginsburg-Landau theory of a supersolid
• SF to NS and SF to SS transition
• NS to SS transition at T0
• 5. Elementary Excitations in SS
• 6. Vortex loop in SS
• 7. Debye-Waller factor, Density-Density
  Correlation
• Specific heat and entropy in SS
• Supersolids in other systems
• 10. Conclusions

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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,
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1.Introduction
Cosmology Astrophysics
General Relativity
Gravity
Symmetry and symmetry breaking at
different energy (or length ) scales
String Theory ???
Elementary particle physics
Quantum Mechanics Quantum Field Theory
Atomic physics Many body physics Condensed matter
physics Statistical mechanics
2d conformal field theory
Emergent quantum Phenomena !
Is supersolid a new emergent phenomenon ?
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P.W. Anderson More is different !
Emergent Phenomena !
Macroscopic quantum phenomena emerged in
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 ?
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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
Reciprocal lattice vectors
Density operator
Breaking translational symmetry
Lattice phonons
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Essentially all substances take solids except
What is a superfluid ?
A superfluid can flow without viscosity
Complex order parameter
Breaking Global U(1) symmetry
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 ?
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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
• J. Leggett, 1970, Non-Classical Rotational
  Inertial (NCRI) of supersolid He4
• 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
• Over the last 35 years, a number of experiments
  have been designed to search for the supersolid
  state without success.

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Torsional oscillator is ideal for the detection
of superfluidity
?f
f0
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Kim and Chan
By Torsional Oscillator experiments
Science 305, 1941 (2004)
Soild He4 at 51 bars
NCRI appears below 0.25K Strong
vmax dependence (above 14µm/s)
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Very recently, there are three experimental
groups one in US, two in Japan Confirmed ( ?
) PSUs 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

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PSUs experiments have rekindled great
theoretical interests in the possible supersolid
phase of He4

• Numerical ( Path Integral quantum Monte-Carlo )
  approach
• 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

Superfluid flowing in grain boundary ?

Supersolid ?
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Uncertainty Relation
Cannot measure position and momentum precisely
simultaneously !
Cannot measure phase and density precisely
simultaneously !
Superfluid phase order in the global phase
Solid
density order in local density
How to reconcile the two extremes into a
supersolid ?
Has to have total boson number
fluctuations !
The solid is not perfect has either vacancies or
interstitials whose BEC may lead to a supersolid
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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
Invariant under the U(1) symmetry
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(a) In liquid
(b) In superfluid
U(1) symmetry is broken
U(1) symmetry is respected
Both sides have the translational symmetry
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3d XY model to describe the ( cusp )
transition
Greywall and Ahlers, 1973 Ahler, 1971
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(2) GL theory of Liquid to solid transition
Density operator
Order parameter
Invariant under translational symmetry
is any vector
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In liquid
In solid
Translational symmetry is broken down to the
lattice symmetry
is any lattice vector
NL to NS transition Density OP at reciprocal
lattice vector cubic term, first order
NL to SF transition Complex OP at even order
terms 2nd order transition
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(3) GL theory of a supersolid
Density operator
Complex order parameter
Vacancy or interstitial
Invariant under
Under
Repulsive
Two competing orders
Interstitial case
Vacancy case
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In supersolid
In the NL, has a gap, can
be integrated out.
In the NL,
has a gap, can be integrated out
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3. The SF to NS or SS transition
In the SF state,
BEC breaks U(1)
symmetry
As , the roton minimum gets deeper
and deeper, as detected in neutron
scattering experiments
Feymann relation
The first maximum peak in the
roton minimum in
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Two possibilities
(1)
No SS
SF to NS transition
Commensurate Solid,
(2)
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
Lets focus on case (1) first
I will explicitly construct a Quantum
Ginzburg-Landau (QGL) action to describe SF to NS
transition
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3a. The SF to NS transition
Inside the SF
Where
Phase representation
Dispersion Relation
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Neglecting vortex excitations
Density representation
Feymann Relation
Structure function
Including vortex excitations QGL to describe SF
to NS transition
SF
NS
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(No Transcript)
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3b. The SF to SS transition
(2)
There is a SS !
SF to SS transition
Vacancies or Interstitials !
Feymann relation
The first maximum peak in the
roton minimum in
The lattice and Superfluid
density wave (SDW) formations happen
simultaneously
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Global phase diagram of He4 in case (2)
The phase diagram will be confirmed from the NS
side
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Add quantum fluctuations of lattice to
the Classical GL action
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4. NS to SS transition at
Lattice phonons
Density operator
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The effective action
strain tensor
elastic constants, 5 for hcp , 2 for
isotropic
For hcp
For isotropic solid
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Under RG transformation
Both and upper critical dimension
is
All couplings are irrelevant at
The NS to SS transition is the same universality
class as Mott to SF transition in a rigid lattice
with exact exponents
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Neglecting quantum fluctuations
Both and upper critical dimension
is
expansion in
M. A. De Moura, T. C. Lubensky, Y. Imry, and A.
Aharony,1976 D. Bergman and B. Halperin, 1976.
model ,
is irrelevant
NS to SS transition at finite is the
universality class

A. T. Dorsey, P. M. Goldbart, J. Toner, Phys.
Rev. Lett. 96, 055301
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5. Excitations in a Supersolid
(1) Superfluid phonons in the sector

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 !
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(a) For isotropic solid
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Poles
Transverse modes in SS are the same as those in NS
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(b) For hcp crystal
Direction
along
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along
plane
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along general direction
One can not even define longitudinal and
transverse
matrix diagonization in
Qualitative physics stay the same.
Smoking gun experiment to prove or dis-prove the
existence of supersolid in Helium 4
The two longitudinal modes
should be detected by in-elastic neutron
scattering if SS indeed exists !
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6. Debye-Waller factor in X-ray scattering
Density operator at
The Debye-Waller factor
where
Taking the ratio
where
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It can be shown
The longitudinal vibration amplitude is reduced
in SS compared to the NS with the same
!
will approach from above as
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7. Vortex loops in supersolid
Vortex loop representation
6 component anti-symmetric tensor gauge field
vortex loop current
Gauge invariance
Coulomb gauge
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Longitudinal couples to transverse
Vortex loop density
Vortex loop current
Integrating out , instantaneous
density-density int.
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Vortex loop density-density interaction is the
same as that in SF with the same parameters
!
is solely determined by
Critical behavior of vortex loop close to
in SF is studied in
G.W. Williams, 1999
Integrating out , retarded
current-current int.
where
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The difference between SS and SF
Equal time at
Vortex current-current int. in SS is stronger
than that in SF with the same parameters
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Recent Inelastic neutron scattering experiments
M. A. Adams, et.al, Phys. Rev. Lett. 98, 085301
(2007). Atom kinetic energy S.O. Diallo,
et.al, cond-mat/0702347 Mometum distribution
function E. Blackburn, et.al,
cond-mat/0702537 Debye-Waller factor
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8. Specific heat and entropy in SS
Specific heat in NS
No reliable calculations on yet
Specific heat in SF
Specific heat in SS
is the same as that in the NS
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Entropy in SS
Excess entropy due to vacancy condensation
due to the lower branch
Molar Volume
is the gas constant
3 orders of magnitude smaller than
non-interacting Bose gas
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9. Supersolids in other systems
Supersolid on Lattices can be realized on optical
lattices !
Cooper pair supersolid in high temperature SC ?
Jinwu, Ye, cond-mat/0503113, bipartite lattices
cond-mat/0612009, frustrated lattices
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
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10. Conclusions

• 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
• 3. Construct explicitly the QGL of SF to NS
  transition
• SF to SS is a simultaneous formation of SDW and
  normal lattice
• Vacancy induced supersolid is certainly possible
• 6. NS to SS is a 3d XY with much narrower
  critical regime than NL to SF
• Elementary Excitations in SS, vortex loops in SS
• 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