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

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 Acknowledgement I thank References: Jinwu Ye, Phys. Rev. Lett. 97, , 2006 Jinwu Ye, cond-mat/ Jinwu Ye, cond-mat/ ,

1.Introduction Quantum Mechanics Quantum Field Theory Elementary particle physics Atomic physics Many body physics Condensed matter physics Statistical mechanics General Relativity Gravity String Theory ??? 2d conformal field theory Emergent quantum Phenomena ! Cosmology Astrophysics Symmetry and symmetry breaking at different energy (or length ) scales Is supersolid a new emergent phenomenon ?

P.W. Anderson: More is different ! Emergent Phenomena ! Macroscopic quantum phenomena emerged in ~ interacting atoms, electrons or spins 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 ? …………??? States of matter break different symmetries at low temperature 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 What is a solid ? At low temperatures, any matter has to have some orders which break some kinds of symmetries. A solid can not flow Breaking translational symmetry: Density operator: Lattice phonons Reciprocal lattice vectors

A supersolid is a new state which has both crystalline order and superfluid order. What is a superfluid ? Complex order parameter: Breaking Global U(1) symmetry: Superfluid phonons A superfluid can flow without viscosity What is a supersolid ? Essentially all substances take solids except Can a supersolid exist in nature, especially in He4 ?

The supersolid state was theoretically speculated in 1970: Andreev and I. Lifshitz, 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 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. Large quantum fluctuations in He4 make it possible

Torsional oscillator is ideal for the detection of superfluidity Be-Cu Torsion Rod Torsion Bob Containing helium Drive Detection f0f0 ff

NCRI appears below 0.25K Strong |v| max dependence (above 14µm/s) Kim and Chan Science 305, 1941 (2004) Soild He4 at 51 bars By Torsional Oscillator experiments

Possible phase diagram of Helium 4 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/ M. Kubota et al K. Shirahama et al 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: D. M. Ceperley, B. Bernu, Phys. Rev. Lett. 93, (2004); N. Prokof'ev, B. Svistunov, Phys. Rev. Lett. 94, (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}, (2006). (2) Phenomenological approach: Xi Dai, Michael Ma, Fu-Chun Zhang, Phys. Rev. B 72, (2005) P. W. Anderson, W. F. Brinkman, David A. Huse, Science 18 Nov (2006) A. T. Dorsey, P. M. Goldbart, J. Toner, Phys. Rev. Lett. 96, Superfluid flowing in grain boundary ? Supersolid ?

Superfluid: phase order in the global phase Uncertainty Relation: Cannot measure position and momentum precisely simultaneously ! Solid:density order in local density How to reconcile the two extremes into a supersolid ? The solid is not perfect: has either vacancies or interstitials whose BEC may lead to a supersolid Cannot measure phase and density precisely simultaneously ! Has to have total boson number fluctuations !

2. Ginsburg-Landau (GL) Theory of a supersolid (1) GL theory of Liquid to superfluid transition: Complex order parameter: Invariant under the U(1) symmetry: Construct GL theory: Expanding free energy in terms of order parameters near critical point consistent with the symmetries of the system

(a) In liquid:(b) In superfluid: U(1) symmetry is broken Both sides have the translational symmetry: U(1) symmetry is respected

3d XY model to describe the ( cusp ) transition Greywall and Ahlers, 1973 Ahler, 1971

(2) GL theory of Liquid to solid transition: Order parameter: Density operator: Invariant under translational symmetry: is any vector

In liquid: In solid: Translational symmetry is broken down to the lattice symmetry: NL to NS transition Density OP at reciprocal lattice vector cubic term, first order is any lattice vector NL to SF transition Complex OP at even order terms 2 nd order transition

Density operator : Complex order parameter : Repulsive: (3) GL theory of a supersolid: Invariant under: Two competing orders Interstitial caseVacancy case Vacancy or interstitial Under

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

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: (1) No SS There is a SS ! Which scenario will happen depends on the sign and strength of the coupling, will be analyzed further in a few minutes (2) Let’s focus on case (1) first: SF to NS transition SF to SS transition I will explicitly construct a Quantum Ginzburg-Landau (QGL) action to describe SF to NS transition Commensurate Solid, In-Commensurate Solid with either vacancies or interstitials

3a. The SF to NS transition Inside the SF: Where: Phase representation: Dispersion Relation:

Neglecting vortex excitations: Density representation: Feymann Relation: Structure function: NS: Including vortex excitations: QGL to describe SF to NS transition: SF:

3b. The SF to SS transition Feymann relation : The first maximum peak in the roton minimum in (2)There is a SS ! SF to SS transition The lattice and Superfluid density wave (SDW) formations happen simultaneously Vacancies or Interstitials !

Global phase diagram of He4 in case (2) The phase diagram will be confirmed from the NS side

Add quantum fluctuations of lattice to the Classical GL action:

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: 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: Both and upper critical dimension is

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, model, NS to SS transition at finite is the universality class is irrelevant A. T. Dorsey, P. M. Goldbart, J. Toner, Phys. Rev. Lett. 96,

(2) Lattice phonons (1) Superfluid phonons in the sector: Topological vortex loop excitations in 5. Excitations in a Supersolid 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:

Poles: Transverse modes in SS are the same as those in NS

(b) For hcp crystal: along Direction:

alongplane

along general direction One can not even define longitudinal and transverse matrix diagonization in Qualitative physics stay the same. The two longitudinal modes should be detected by in-elastic neutron scattering if SS indeed exists ! Smoking gun experiment to prove or dis-prove the existence of supersolid in Helium 4:

6. Debye-Waller factor in X-ray scattering Density operator at : The Debye-Waller factor: Taking the ratio: where

will approach from above as It can be shown The longitudinal vibration amplitude is reduced in SS compared to the NS with the same !

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. where

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

M. A. Adams, et.al, Phys. Rev. Lett. 98, (2007). Atom kinetic energy S.O. Diallo, et.al, cond-mat/ Mometum distribution function E. Blackburn, et.al, cond-mat/ Debye-Waller factor Recent Inelastic neutron scattering experiments:

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

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 Supersolid on Lattices can be realized on optical lattices ! Jinwu, Ye, cond-mat/ , bipartite lattices; cond-mat/ , frustrated lattices The physics of lattice SS is different than that of He4 SS Possible excitonic supersolid in electronic systems ? Electron + holeExciton is a boson Excitonic supersolid in EHBS ? There is also a roton minimum in electron-hole bilayer system Jinwu, Ye, PRL, Supersolids in other systems Cooper pair supersolid in high temperature SC ? Jinwu Ye, preprint to be submitted

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 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: