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Ginsburg-Landau Theory of Solids and Supersolids Jinwu 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 T=0

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ginsburg landau theory of solids and supersolids jinwu ye penn state university
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 T=0

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



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


Jinwu Ye, Phys. Rev. Lett. 97, 125302, 2006

Jinwu Ye, cond-mat/07050770

Jinwu Ye, cond-mat/0603269,

1 introduction



General Relativity


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 ?


P.W. Anderson: More is different !

Emergent Phenomena !

Macroscopicquantum 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 thesupersolid 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

Reciprocal lattice vectors

Density operator:

Breaking translational symmetry:

Lattice phonons


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 ?


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.

Torsional oscillator is ideal for the detection of superfluidity


Torsion Rod

Torsion Bob








Kim and Chan

By Torsional Oscillator experiments

Science 305, 1941 (2004)

Soild He4 at 51 bars

NCRI appears below


Strong |v|maxdependence

(above 14µm/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

  • 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 ?


Uncertainty Relation:

Cannot measure position and momentum precisely simultaneously !

Cannot measure phase and density precisely simultaneously !

Superfluid: phase order in the global phase


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

2 ginsburg landau gl theory of 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:

Invariant under the U(1) symmetry:


(a) In liquid:

(b) In superfluid:

U(1) symmetry is respected

U(1) symmetry is broken

Both sides have the translational symmetry:


3d XY model to describe the ( cusp ) transition

Greywall and Ahlers, 1973

Ahler, 1971


(2) GL theory of Liquid to solid transition:

Density operator:

Order parameter:

Invariant under translational symmetry:

is any vector


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


even order terms

2nd order transition


(3) GL theory of a supersolid:

Density operator:

Complex order parameter:

Vacancy or interstitial

Invariant under:


Two competing orders


Vacancy case

Interstitial case


In supersolid:

In the NL, has a gap, can be integrated out.

In the NL, has a gap, can be integrated out

3 the sf to ns or ss transition
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


Two possibilities:


SF to NS transition


Commensurate Solid,


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

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
3a. The SF to NS transition

Inside the SF:


Phase representation:

Dispersion Relation:


Neglecting vortex excitations:

Density representation:

Feymann Relation:

Structure function:

Including vortex excitations: QGL to describe SF to NS transition:



3b the sf to ss transition
3b. The SF to SS transition


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


Global phase diagram of He4 in case (2)

The phase diagram will be confirmed from the NS side

4 ns to ss transition at
4. NS to SS transition at

Lattice phonons:

Density operator:


The effective action:

strain tensor

elastic constants, 5 for hcp , 2 for isotropic

For hcp:

For isotropic solid:


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:


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

5 excitations in a supersolid
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 !



Transverse modes in SS are

the same as those in NS


(b) For hcp crystal:







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 !

6 debye waller factor in x ray scattering
6. Debye-Waller factor in X-ray scattering

Density operator at :

The Debye-Waller factor:


Taking the ratio:



It can be shown

The longitudinal vibration amplitude is reduced in SS

compared to the NS with the same !

will approach from above as

7 vortex loops in supersolid
7. Vortex loops in supersolid

Vortex loop representation:

6 component anti-symmetric tensor gauge field

vortex loop current

Gauge invariance:

Coulomb gauge:


Longitudinal couples to transverse

Vortex loop density:

Vortex loop current:

Integrating out , instantaneous density-density int.


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.



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


Recent Inelastic neutron scattering experiments:

M. A. Adams,, Phys. Rev. Lett. 98, 085301 (2007).

Atom kinetic energy

S.O. Diallo,, cond-mat/0702347

Mometum distribution function

E. Blackburn,, cond-mat/0702537

Debye-Waller factor

8 specific heat and entropy in ss
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


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

9 supersolids in other systems
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 differentthan 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
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: