New Vistas on Quantum Matter Opened by Dipolar Fermions

922 views

Published on

Invited talk at the annual meeting of the UK Condensed Matter - Cold Atoms interface research network, St. Andrews, September 2010

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
922
On SlideShare
0
From Embeds
0
Number of Embeds
61
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • Fermi gas: Particles with well-defined momentum Indistinguishable particles (quantum) Homogeneous and isotropic fluid Mott/Wigner: Localised particles Distinguishable particles (classical) Crystal with broken translational and rotational symmetries
  • Very subtle form of symmetry breaking. SHOW CATO SANDFORD’S SIMULATION HERE!
  • mu* = -1.9 t and mu* = -1.5 t
  • Type notes here
  • Draw “bare” dispersion relation.
  • 2+1/2 order??
  • Indeed the OKF hamiltonian leads to 1st-order (you need to include cubic terms in the dispersion relation to make it 2nd-order).
  • New Vistas on Quantum Matter Opened by Dipolar Fermions

    1. 1. ISIS Facility, STFC School of Rutherford Appleton Laboratory Physical Sciences New Vistas on Quantum Matter Opened by Dipolar Fermions Jorge Quintanilla University of Kent & Rutherford Appleton Laboratory Collaborators: Sam T. Carr (Karlsruhe) Joseph J. Betouras (Loughborough) Andy J. Schofield (Birmingham) Masud Haque (MPI Dresden) Chris Hooley (St. Andrews) Ben J. Powell (Queensland) Funding: STFC, SEPNet 2010 Annual Meeting of the UK Cold-atom/Condensed Matter Physics Network, St. Andrews, 9th-10th Sept. 2010
    2. 2. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations SEPNet
    3. 3. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations SEPNet
    4. 4. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations SEPNet
    5. 5. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations SEPNet
    6. 6. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations ISIS
    7. 7. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations STRONG CORRELATIONS
    8. 8. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hubbard Model
    9. 9. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hubbard Model
    10. 10. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hubbard Model Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 276, No. 1365 (Nov. 26, 1963), pp. 238-257
    11. 11. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hubbard Model Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 276, No. 1365 (Nov. 26, 1963), pp. 238-257 “A theory of correlations [...] will be mainly concerned with understanding [...] the balance between band-like and atomic-like behaviour.”
    12. 12. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Strongly correlated quantum matter 2 p V r  r' 2 p  2m  2m  V r  r' many many pairs of pairs of particles particles particles particles kinetic energy interaction energy  
    13. 13. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Strongly correlated quantum matter 2 p V r  r' 2 p  2m  2m  V r  r' many many pairs of pairs of particles particles particles particles kinetic energy interaction energy λ ~ rs ~ Å > pz   Fermi surface py px
    14. 14. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Strongly correlated quantum matter 2 p V r  r' 2 p  2m  2m  V r  r' many many pairs of pairs of particles particles particles particles kinetic energy interaction energy λ ~ rs ~ Å > pz   Wigner Fermi crystal surface Mott py px insulator
    15. 15. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Strongly correlated quantum matter 2 p V r  r' 2 p  2m  2m  V r  r' many many pairs of pairs of particles particles particles particles kinetic energy interaction energy λ ~ rs ~ Å > Fermi liquid theory: pz   Wigner Fermi crystal surface •Effective mass m* •Fermi momentum pF Mott py px •Landau parameters insulator F0, F1, F2, …
    16. 16. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Strongly correlated quantum matter 2 p V r  r' 2 p  2m  2m  V r  r' many many pairs of pairs of particles particles particles particles kinetic energy interaction energy λ ~ rs ~ Å > pz  STRONGLY  Wigner Fermi crystal surface CORRELATED ELECTRON SYSTEMS Mott py px insulator
    17. 17. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Cuprates La2CuO4 Parameter: Cu O Cu O O 1 x  Number of electrons per Cu O Cu CuO2 plaquette ( E.g. La2-xSrxCuO4 ) 25-30% holes / CuO2  1 electron / CuO2 pz •High-temperature Antiferromagnetic Fermi superconductivity, Mott insulator liquid •stripes, [Tranquada et al., Nature (1995)] py px •Non-Fermi liquid, [Hussey et al.] •pseudo-gap,…
    18. 18. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Two-dimensional quantum wells Parameter: B   n electrons / n magnetic field lines = hcn el /eB ν >> 1 ν << 1 Two-dimensional Fermi liquid  •quantum Hall effect, Wigner • fractional quantum Hall crystal effect, py px •anisotropic state. [ M.B. Santos et al., [ M.P. Lilly et al., PRL (1999) ] Phys.Rev.Lett. 68, 1188 [ V. Senz et al., PRL (2000); (1992) ] Y. Y. Proskuryakov et al., PRL (2001) ]
    19. 19. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Exact solution of the Hubbard model Consider the Hubbard model in D = 1: Phase diagram known exactly... U/t Mott insulator Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 Luttinger liquid (1968); 21, 192 (1968). f 0 1 2 But an exact solution is not available for 1<D<∞.
    20. 20. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations SOFT QUANTUM MATTER AND THE POMERANCHUK INSTABILITY
    21. 21. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] classical temperature ideal normal liquid solid gas liquid crystals state quantum STRONGLY Fermi Fermi CORRELATED Wigner gas liquid ELECTRON crystal/Mott SYSTEMS insulator correlations
    22. 22. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid Wigner crystal
    23. 23. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid Wigner crystal
    24. 24. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid Wigner crystal
    25. 25. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid Wigner “stripes” crystal
    26. 26. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid nematic Fermi Wigner “stripes” liquid crystal
    27. 27. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid nematic Fermi Wigner “stripes” liquid crystal
    28. 28. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter [ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ] /sci/physics/theory/research/simulation/ ] [http://www2.warwick.ac.uk/fac Pictures: Mike Allen Fermi liquid nematic Fermi Wigner “stripes” liquid crystal
    29. 29. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations The Pomeranchuk instability [ Pomeranchuk (1958) ] l0 (s-wave) Stoner Magnetism l 1 (p-wave)  standing current …  l  2 (d-wave) l3 nematic (f-wave) 
    30. 30. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Instability condition [ Pomeranchuk (1958) ] n(k)  0 Arbitrary Fermi surface deformation: n(k)  0 Quasiparticle energy:    Landau parameters: f k,k'  hv F  Fl cosl k   k '  (2D) l 0 Pomeranchuk Instability condition: E  0  Fl   2 
    31. 31. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations MEAN FIELD THEORIES OF THE POMERANCHUK INSTABILITY
    32. 32. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? Many ordered states of electrons can be l q described in terms of pair formation... ... and condensation.
    33. 33. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    34. 34. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    35. 35. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    36. 36. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    37. 37. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    38. 38. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    39. 39. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    40. 40. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    41. 41. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition q≠0 l=0 FFLO spin- and charge- state density waves q=0 l≠0 unconventional pairing Pomeranchuk superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    42. 42. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the order parameter? l q particle-particle particle-hole q=0 l=0 s-wave Stoner ferromagnet, superconductor gas-liquid transition The order parameter is * q≠0 l=0 FFLO spin- and charge- ∫dθp cos(lθp) n(p) state density waves l = 1,2,3,... q=0 l≠0 unconventional pairing Pomeranchuk * Expression for the case of a 2D continuum superconductor instability q≠0 l≠0 FFLO + “d”-density waves unconventional pairing
    43. 43. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Microscopic description [ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ] [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] 1) Microscopic model: free fermions + isotropic interaction  p 2  H     V r  r' 2m  all pairs of many particles particles 2) Trial ground state:   c ˆ  k, 0  k 0   variationally: 3) Determine (k)  H   minimum 
    44. 44. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Instability condition [ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ] [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] 4 v F E  0  Vl kF ,kF   (2D) kF   
    45. 45. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Topological Fermi surface transitions [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] Same recipe: interactions with sharp length scale r0 > rs : ~ V(r) V(r) r r r0 r0 g/r0ε0 kFr0 kFr0
    46. 46. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations POMERANCHUK INSTABILITY AND DECONFINEMENT
    47. 47. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Pomeranchuk on a lattice [ JQ, C. Hooley, B.J. Powell, A.J. Schofield & M. Haque, Physica B, 403, 1279-1281 (2008). ] Theory can be generalised to crystal lattices: t2 u0 u2 t1 t3 … + u1 u2 … Interactions beyond on-site ⇒ band-structure renormalisation: t1 , t2 , t3 ,… → t1* , t2* , t3* ,…
    48. 48. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Pomeranchuk on a lattice [ JQ, C. Hooley, B.J. Powell, A.J. Schofield & M. Haque, Physica B, 403, 1279-1281 (2008). ] Example: square lattice with t1 ≠0 , u1 ≠0 (u0 won’t do!) Empty Half-filled band band Band V1   filling large V1  Order parameter: t * /V1 c 1c j j small V1  
    49. 49. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Confinement C.f. the “confinement hypothesis” 1D 2D t’ / t 0 ( t’ / t )crit [ David G. Clarke, S. P. Strong, and P. W. Anderson, Phys. t’ Rev. Lett. 72, 3218 (1994) ] t
    50. 50. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Confinement C.f. the “confinement hypothesis” 1D 2D t’ / t 0 ( t’ / t )crit [ David G. Clarke, S. P. Strong, and P. W. Anderson, Phys. t’ Rev. Lett. 72, 3218 (1994) ] t Latest evidence: functional RG ( t → 0 limit ) [Sascha Ledowski and Peter Kopietz (2007) ]
    51. 51. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Confinement C.f. the “confinement hypothesis” 1D 2D t’ / t 0 ( t’ / t )crit [ David G. Clarke, S. P. Strong, and P. W. Anderson, Phys. t’ Rev. Lett. 72, 3218 (1994) ] t Latest evidence: functional RG ( t → 0 limit ) [Sascha Ledowski and Peter Kopietz (2007) ]
    52. 52. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations 1D → 2D [ J. Quintanilla, S.T. Carr, J.J. Betouras, PRA 79, 031601(R) (2009) ] What about the opposite: can interactions restore 2D behaviour? 1D 2D V 0 Vcrit V t Model:
    53. 53. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations MAKING SOFT QUANTUM MATTER
    54. 54. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter with dipolar fermions Use dipolar fermions (e.g. 40K87Rb molecules* or 161/163Dy**). * K.-K. Ni et al., Science 322, 231 (2008). ** M. Lu, S. H. Youn, & B. L. Lev, PRL 104, 063001 (2010). Applied field polarises the fermions. Load onto quasi-1D optical lattice. Align chains at the “magic angle” to the applied field. J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    55. 55. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter with dipolar fermions By tuning the ratio of the lattice constants, a/b, can make in-plane interaction strongly anisotropic: J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    56. 56. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Soft quantum matter with dipolar fermions By tuning the ratio of the lattice constants, a/b, can make in-plane interaction strongly anisotropic: V(k) 2Vcos(ky) J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    57. 57. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Phase diagram, large a >> b J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    58. 58. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Phase diagram, large a >> b meta-nematic transition J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    59. 59. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Phase diagram, large a >> b J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    60. 60. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Phase diagram, large a >> b crystallisation J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    61. 61. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Phase diagram, a ~ b crystallisation   ,     stripes  0,    J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009) 
    62. 62. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations A controlled realisation of the soft quantum matter scenario J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    63. 63. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations A controlled realisation of the soft quantum matter scenario Weak- coupling J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)
    64. 64. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Nature of the meta-nematic transition S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    65. 65. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Nature of the meta-nematic transition S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    66. 66. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Nature of the meta-nematic transition Vol. 11, pp. 1130-1135 (1960) S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    67. 67. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Nature of the meta-nematic transition S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    68. 68. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Nature of the meta-nematic transition The “2+½-order” Lifshitz transition is the quantum critical endpoint of the 1st-order meta-nematic transition. S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    69. 69. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Nature of the meta-nematic transition The “2+½-order” Lifshitz transition is the quantum critical endpoint of the 1st-order meta-nematic transition. It is a non-analytic transition (in the sense of BCS theory): C.f. Y. Yamaji, T. Misawa & M. Imada, JPSJ 75, 094719 (2006) (t-t’ Hubbard model). S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    70. 70. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Finite temperature S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    71. 71. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Effect of the trapping potential S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    72. 72. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Effect of the trapping potential S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    73. 73. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Effect of the trapping potential S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)
    74. 74. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations CONCLUSIONS •Soft quantum matter can be realised, in a controlled way (both theoretically and experimentally), using ultra-cold dipolar fermions in a suitable optical lattice. •This should enable us to establish the extent to which soft quantum matter can be a useful framework for understanding strongly-correlated materials. •As always, we learn more than we expected as we go along.
    75. 75. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations THANKS!
    76. 76. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations ADDITIONAL SLIDES
    77. 77. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Aside: The Pomeranchuk state is uniform ! ky   c  ˆk 0  k 0 ˆ ˆ ckck  1 N(r)   c rc r  ˆ ˆ ˆ ˆ ckck  0   e ipq .r  c p c q  ˆ ˆ  kx   p,q  p,q  c p c p  ˆ ˆ    cp cp  N ˆ ˆ  p 
    78. 78. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hartree-Fock
    79. 79. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations RPA
    80. 80. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Vertex corrections = 0 because of the integration with respect to py. ≠ 0, but ky-independent, < X0 by a factor ~ (V/t||)2 and not divergent near kx=2kF .
    81. 81. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Direct quantum simulation Another talk... See • Campo, Capelle, ˆ Quintanilla & Hooley, H PRL 99, 240403 (2007) . • Hooley & Quintanilla, PRL 93, 080404 (2004). [ See also Quintanilla & optical Hooley, Physics World (June 2009).] lattices
    82. 82. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the mechanism? http://www.warwick.ac.uk/go/theory/researc [ Picture: Mike Allen, h/simulation/ ]
    83. 83. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations What is the mechanism? px  py ˆ2 ˆ2 2 px py  ˆ ˆ [ Oganseyan, Kivelson and Fradkin, PRB (2001); Recipe: Qr   r    r   Khavine, Chung, Oganesyan & Kee, PRB (2004); ˆ ˆ 2 px py py  px  ˆ2 ˆ2 Kee & Kim, PRB (2005). ]  ~ ( F2(q) ≈ F2 δq,0 )
    84. 84. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hidden order URu2Si2 Sr3Ru2O7 [ Palstra et al., [ S.A.Grigera et al., PRL 55, 2727 (1985) ] Science 306, 1154 (2004) ]
    85. 85. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hidden order URu2Si2 Sr3Ru2O7 [ Palstra et al., [ S.A.Grigera et al., PRL 55, 2727 (1985) ] Science 306, 1154 (2004) ]
    86. 86. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Hidden order URu2Si2 Sr3Ru2O7 [Varma & Zhu, PRL 96, 036405 (2006)] [ S.A.Grigera et al., [ Palstra et al., [ Science 306, et al., (2004) ; S.A.Grigera 1154 PRL 55, 2727 (1985) ] R. A. Borzi et1154 (2004) ] Science 306, al., Science 315, 214 (2007). ]
    87. 87. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Microscopic description [ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ] [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] Ground state energy:  2 k 2  nV E     k k  * N  2m k    2 Momentum-dependent chemical potential: 1   k    * V k  k'Nk' 2 k'
    88. 88. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Microscopic description [ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ] [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] Contact potential: V(r) |k-k’| r Potential with a length scale: V(r) ~ 1/r0 |k-k’| r0 r
    89. 89. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Instability condition [ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ] [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] Recover Landau theory... 1 ... with Landau parameters given by Fl  hv F l,0  Vl kF ,kF  ... in terms amplitudes on Fermi surface of partial wave (2D) decomposition of the interaction potential:   V k  k'  Vl k,k'coslk  k'  l 0
    90. 90. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Recipe for Pomeranchuk [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] Interactions with sharp features at a finite length scale r0 > rs : ~ V(r) g/r0ε0 r r0 (3D) kFr0
    91. 91. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Recipe for Pomeranchuk [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] Interactions with sharp features at a finite length scale r0 > rs : ~ V(r) r r0 (3D)
    92. 92. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Recipe for Pomeranchuk [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] A screening length is not sharp enough! (3D)
    93. 93. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Topological Fermi surface transitions [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] Our theory not only tells us when the dispersion relation becomes anisotropic - it gives the full form of that dispersion relation: 2 k 1  k     3    , 'V k  k'N k', ' V 2m L k', ' This allows us to identify other Fermi surface shape instabilities, apart from Pomeranchuk. 
    94. 94. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Topological Fermi surface transitions [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] g/r0ε0 = 0, 20, kFr0 97.66, 150
    95. 95. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Topological Fermi surface transitions [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] kFr0 g/r0ε0 = 0, 20, kFr0 97.66, 150
    96. 96. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Topological Fermi surface transitions [ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ] m*/m m*/m g/r0ε0
    97. 97. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Describing the ordered state [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] To describe ordered state need to describe interaction away from Fermi surface: 1 Vl k,kF   V  V ' k  kF   V '' k  kF   ... 2 2 Transition can be second  order... kF
    98. 98. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Into the ordered state [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] To describe ordered state need to describe interaction away from Fermi surface: 1 Vl k,kF   V  V ' k  kF   V '' k  kF   ... 2 2 ... as well as first order:  kF
    99. 99. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Not always an instability! [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ]
    100. 100. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations Not always an instability! [ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ] Q.Q interaction is here!
    101. 101. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations
    102. 102. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations
    103. 103. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations
    104. 104. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations
    105. 105. St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations

    ×