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

1. 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

2. 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,

3. 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 ?

4. 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 ?

5. 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

6. 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 ?

7. 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.

8. Torsional oscillator is ideal for the detection of superfluidity Be-Cu Torsion Rod Torsion Bob Containing helium Drive Detection f f0

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

10. 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

11. 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 ?

12. 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

13. 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:

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

15. 3d XY model to describe the ( cusp ) transition Greywall and Ahlers, 1973 Ahler, 1971

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

17. 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

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

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

20. 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

21. Two possibilities: (1) SF to NS transition No SS 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 Let’s focus on case (1) first: I will explicitly construct a Quantum Ginzburg-Landau (QGL) action to describe SF to NS transition

22. 3a. The SF to NS transition Inside the SF: Where: Phase representation: Dispersion Relation:

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

25. 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

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

27. Add quantum fluctuations of lattice to the Classical GL action:

28. 4. NS to SS transition at Lattice phonons: Density operator:

29. The effective action: strain tensor elastic constants, 5 for hcp , 2 for isotropic For hcp: For isotropic solid:

30. 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:

31. 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

32. 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 !

33. (a) For isotropic solid:

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

35. (b) For hcp crystal: Direction: along

36. along plane

37. 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 !

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

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

40. 7. Vortex loops in supersolid Vortex loop representation: 6 component anti-symmetric tensor gauge field vortex loop current Gauge invariance: Coulomb gauge:

41. Longitudinal couples to transverse Vortex loop density: Vortex loop current: Integrating out , instantaneous density-density int.

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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: