The quantum phase transition between a superfluid and an...
Transcript of The quantum phase transition between a superfluid and an...
![Page 1: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/1.jpg)
The quantum phase transition between a superfluid and an insulator:
applications to trapped ultracold atoms and the cuprate superconductors.
![Page 2: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/2.jpg)
The quantum phase transition between a superfluid and an insulator:
applications to trapped ultracold atoms and the cupratesuperconductors.
Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard)
Predrag Nikolic (Harvard) Subir Sachdev (Harvard)
Krishnendu Sengupta (HRI, India)
Talk online at http://sachdev.physics.harvard.edu
![Page 3: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/3.jpg)
OutlineOutlineI. Bose-Einstein condensation and superfluidity.
II. The superfluid-insulator quantum phase transition.
III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition.
V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.
VI. Experimental tests in the cuprates.
![Page 4: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/4.jpg)
I. Bose-Einstein condensation and superfluidity
![Page 5: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/5.jpg)
Superfluidity/superconductivity occur in:
• liquid 4He• metals Hg, Al, Pb, Nb,
Nb3Sn…..• liquid 3He• neutron stars• cuprates La2-xSrxCuO4,
YBa2Cu3O6+y….• M3C60• ultracold trapped atoms• MgB2
![Page 6: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/6.jpg)
The Bose-Einstein condensate:
A macroscopic number of bosons occupy the lowest energy quantum state
yk
xk ( )( )2 2
Pair wavefunction in cupra
s:
te
x yk kΨ = − ↑↓ − ↓↑
0S =
Such a condensate also forms in systems of fermions, where the bosons are Cooper pairs of fermions:
![Page 7: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/7.jpg)
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)
Velocity distribution function of ultracold 87Rb atoms
![Page 8: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/8.jpg)
( ) The wavefunction of the condensate
Superfluid velocity
(for non-Galilean invaria
Superflow:
i
s
e
m
θ
θ
Ψ → Ψ
=
r
v ∇nt superfluids,
the co-efficient of is modified)θ∇
![Page 9: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/9.jpg)
Excitations of the superfluid: Vortices
![Page 10: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/10.jpg)
E.J. Yarmchuk, M.J.V. Gordon, and R.E. Packard,
Observation of Stationary VortexArrays in Rotating Superfluid Helium, Phys. Rev. Lett. 43, 214 (1979).
Observation of quantized vortices in rotating 4He
![Page 11: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/11.jpg)
J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, 476 (2001).
Observation of quantized vortices in rotating ultracold Na
![Page 12: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/12.jpg)
Quantized fluxoids in YBa2Cu3O6+y
J. C. Wynn, D. A. Bonn, B.W. Gardner, Yu-Ju Lin, Ruixing Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).
![Page 13: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/13.jpg)
OutlineOutlineI. Bose-Einstein condensation and superfluidity.
II. The superfluid-insulator quantum phase transition.
III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition.
V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.
VI. Experimental tests in the cuprates.
![Page 14: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/14.jpg)
II. The superfluid-insulator quantum phase transition
![Page 15: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/15.jpg)
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)
Velocity distribution function of ultracold 87Rb atoms
![Page 16: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/16.jpg)
Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb)
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
![Page 17: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/17.jpg)
Momentum distribution function of bosons
Bragg reflections of condensate at reciprocal lattice vectors
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
![Page 18: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/18.jpg)
Superfluid-insulator quantum phase transition at T=0
V0=0Er V0=7Er V0=10Er
V0=13Er V0=14Er V0=16Er V0=20Er
V0=3Er
![Page 19: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/19.jpg)
Bosons at filling fraction f = 1
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Weak interactions: superfluidity
Strong interactions: Mott insulator which preserves all lattice
symmetries
![Page 20: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/20.jpg)
Bosons at filling fraction f = 1
Weak interactions: superfluidity
0Ψ ≠
![Page 21: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/21.jpg)
Bosons at filling fraction f = 1
Weak interactions: superfluidity
0Ψ ≠
![Page 22: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/22.jpg)
Bosons at filling fraction f = 1
Weak interactions: superfluidity
0Ψ ≠
![Page 23: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/23.jpg)
Bosons at filling fraction f = 1
Weak interactions: superfluidity
0Ψ ≠
![Page 24: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/24.jpg)
Strong interactions: insulator
Bosons at filling fraction f = 1
0Ψ =
![Page 25: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/25.jpg)
Bosons at filling fraction f = 1/2
Weak interactions: superfluidity
0Ψ ≠
![Page 26: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/26.jpg)
Weak interactions: superfluidity
Bosons at filling fraction f = 1/2
0Ψ ≠
![Page 27: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/27.jpg)
Weak interactions: superfluidity
Bosons at filling fraction f = 1/2
0Ψ ≠
![Page 28: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/28.jpg)
Weak interactions: superfluidity
Bosons at filling fraction f = 1/2
0Ψ ≠
![Page 29: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/29.jpg)
Weak interactions: superfluidity
Bosons at filling fraction f = 1/2
0Ψ ≠
![Page 30: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/30.jpg)
Strong interactions: insulator
Bosons at filling fraction f = 1/2
0Ψ =
![Page 31: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/31.jpg)
Strong interactions: insulator
Bosons at filling fraction f = 1/2
0Ψ =
![Page 32: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/32.jpg)
Strong interactions: insulator
Bosons at filling fraction f = 1/2
0Ψ =
Insulator has “density wave” order
![Page 33: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/33.jpg)
Superfluid
Insulator
Interactions between bosons
?
Charge density Charge density wave (CDW) orderwave (CDW) order
Bosons on the square lattice at filling fraction f=1/2
![Page 34: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/34.jpg)
Superfluid
Insulator
Interactions between bosons
?
Charge density Charge density wave (CDW) orderwave (CDW) order
Bosons on the square lattice at filling fraction f=1/2
![Page 35: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/35.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
12( + )=
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 36: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/36.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
12( + )=
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 37: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/37.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
12( + )=
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 38: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/38.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
12( + )=
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 39: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/39.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 40: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/40.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 41: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/41.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 42: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/42.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 43: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/43.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) orderSuperfluid
Insulator
Interactions between bosons
?
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
![Page 44: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/44.jpg)
OutlineOutlineI. Bose-Einstein condensation and superfluidity.
II. The superfluid-insulator quantum phase transition.
III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition.
V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.
VI. Experimental tests in the cuprates.
![Page 45: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/45.jpg)
III. The cuprate superconductors and their proximity to a superfluid-insulator transition
![Page 46: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/46.jpg)
Cu
O
LaLa2CuO4
![Page 47: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/47.jpg)
Mott insulator: square lattice antiferromagnet
jiij
ij SSJH ⋅= ∑><
La2CuO4
![Page 48: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/48.jpg)
Superfluid: condensate of paired holes
La2-δSrδCuO4
0S =
![Page 49: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/49.jpg)
T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).
The cuprate superconductor Ca2-xNaxCuO2Cl2
![Page 50: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/50.jpg)
The cuprate superconductor Ca2-xNaxCuO2Cl2
Evidence that holes can form an insulating state with period ≈ 4 modulation in the density
T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).
![Page 51: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/51.jpg)
Nature 431, 1078 (2004); cond-mat/0604101
Resonant X-ray scattering evidence that the modulated state has one hole pair per unit cell.
Sr24-xCaxCu24O41
![Page 52: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/52.jpg)
Nature 431, 1078 (2004); cond-mat/0604101
Sr24-xCaxCu24O41
Similar to the superfluid-insulator transition of bosons at fractional filling
![Page 53: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/53.jpg)
OutlineOutlineI. Bose-Einstein condensation and superfluidity.
II. The superfluid-insulator quantum phase transition.
III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition.
V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.
VI. Experimental tests in the cuprates.
![Page 54: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/54.jpg)
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition
![Page 55: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/55.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
Superfluid
Interactions between bosons
Insulator
?
12( + )=
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
0Ψ ≠
![Page 56: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/56.jpg)
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
12( + )=
( ) .Can define a common CDW/VBS order using a generalized "density" ieρ ρ= ∑ Q rQ
Q
r
Insulating phases of bosons at filling fraction f = 1/2
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
All insulators have 0 and 0 for certain ρΨ = ≠Q Q
![Page 57: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/57.jpg)
Landau-Ginzburg-Wilson approach to multiple order parameters:
[ ][ ]
charge int
2 41 1
2 4
charge 2 2
22int
sc
sc
F F F F
F r u
F r u
F v
ρ
ρ ρ ρ
ρ
⎡ ⎤= Ψ + +⎣ ⎦
Ψ = Ψ + Ψ +
⎡ ⎤ = + +⎣ ⎦
= Ψ +
Q
Q Q Q
Q
Distinct symmetries of order parameters permit couplings only between their energy densities
![Page 58: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/58.jpg)
Predictions of LGW theoryFirst order transitionΨ
1 2r r−Superconductor
ρQ
Charge-ordered insulator
![Page 59: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/59.jpg)
Predictions of LGW theory
(Supersolid)
Coexistence
1 2r r−
First order transitionΨ
1 2r r−Superconductor
Superconductor Charge-ordered insulator
Ψ ρQ
ρQ
Charge-ordered insulator
![Page 60: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/60.jpg)
Predictions of LGW theory
(Supersolid)
Coexistence
1 2r r−
First order transitionΨ
1 2r r−Superconductor
Superconductor Charge-ordered insulator
Ψ ρQ
ρQ
Charge-ordered insulator
1 2r r−Superconductor
Ψ ρQ( topologically ordered)
" "
0, 0sc
Disordered
ρ
≠
Ψ = =Q
Charge-ordered insulator
![Page 61: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/61.jpg)
Predictions of LGW theory
(Supersolid)
Coexistence
1 2r r−
First order transitionΨ
1 2r r−Superconductor
Superconductor Charge-ordered insulator
Ψ ρQ
ρQ
Charge-ordered insulator
1 2r r−Superconductor
Ψ ρQ( topologically ordered)
" "
0, 0sc
Disordered
ρ
≠
Ψ = =Q
Charge-ordered insulator
![Page 62: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/62.jpg)
OutlineOutlineI. Bose-Einstein condensation and superfluidity.
II. The superfluid-insulator quantum phase transition.
III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition.
V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.
VI. Experimental tests in the cuprates.
![Page 63: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/63.jpg)
V. Beyond the LGW paradigm: continuous transitions with multiple order parameters
![Page 64: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/64.jpg)
Central question:In two dimensions, we can view the vortices as
point particle excitations of the superfluid. What is the quantum mechanics of these “particles” ?
Excitations of the superfluid: Vortices and anti-vortices
![Page 65: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/65.jpg)
In ordinary fluids, vortices experience the Magnus Force
FM
( ) ( ) ( )mass density of air velocity of ball circulationMF = i i
![Page 66: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/66.jpg)
![Page 67: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/67.jpg)
Dual picture:The vortex is a quantum particle with dual “electric”
charge n, moving in a dual “magnetic” field of strength = h×(number density of Bose particles)
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989)
![Page 68: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/68.jpg)
![Page 69: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/69.jpg)
![Page 70: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/70.jpg)
Bosons on the square lattice at filling fraction f=p/q
![Page 71: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/71.jpg)
Bosons on the square lattice at filling fraction f=p/q
![Page 72: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/72.jpg)
Bosons on the square lattice at filling fraction f=p/q
![Page 73: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/73.jpg)
A Landau-forbidden continuous transitions
1 2r r−Superfluid Charge-ordered insulator
Ψ ρQ
Vortices in the superfluid have associated quantum numbers which determine the local “charge order”, and their proliferation in the superfluid can lead to a continuous transition to a charge-ordered insulator
![Page 74: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/74.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
Superfluid
Interactions between bosons
Insulator
?
12( + )=
Bosons on the square lattice at filling fraction f=1/2
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
0Ψ ≠
![Page 75: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/75.jpg)
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
Superfluid Insulator
12( + )=
Bosons on the square lattice at filling fraction f=1/2
0Ψ ≠
“Aharanov-Bohm” or “Berry” phases lead to surprising kinematic duality relations between seemingly distinct orders. These phase factors allow for continuous quantum phase transitions in situations where such transitions are forbidden by Landau-Ginzburg-Wilson theory.
![Page 76: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/76.jpg)
OutlineOutlineI. Bose-Einstein condensation and superfluidity.
II. The superfluid-insulator quantum phase transition.
III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.
IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition.
V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.
VI. Experimental tests in the cuprates.
![Page 77: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/77.jpg)
VI. Experimental tests in the cuprates
![Page 78: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/78.jpg)
STM around vortices induced by a magnetic field in the superconducting stateJ. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan,
H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).
-120 -80 -40 0 40 80 1200.0
0.5
1.0
1.5
2.0
2.5
3.0
Regular QPSR Vortex
Diff
eren
tial C
ondu
ctan
ce (n
S)
Sample Bias (mV)
Local density of states (LDOS)
1Å spatial resolution image of integrated
LDOS of Bi2Sr2CaCu2O8+δ
( 1meV to 12 meV) at B=5 Tesla.
S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).
![Page 79: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/79.jpg)
100Å
b7 pA
0 pA
Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K
J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).
Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings
Prediction of periodic LDOS modulations near vortices: K. Park and S. Sachdev, Phys. Rev. B 64, 184510 (2001).
![Page 80: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/80.jpg)
![Page 81: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/81.jpg)
![Page 82: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/82.jpg)
Influence of the quantum oscillating vortex on the LDOS
/ vω ω2
2 1F
v
mvα ω= =
![Page 83: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/83.jpg)
Influence of the quantum oscillating vortex on the LDOS
/ vω ω2
2 1F
v
mvα ω= =
No zero bias peak.
![Page 84: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/84.jpg)
Influence of the quantum oscillating vortex on the LDOS
/ vω ω2
2 1F
v
mvα ω= =
Resonant feature near the vortex oscillation frequency
![Page 85: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/85.jpg)
Influence of the quantum oscillating vortex on the LDOS
-120 -80 -40 0 40 80 1200.0
0.5
1.0
1.5
2.0
2.5
3.0
Regular QPSR Vortex
Diff
eren
tial C
ondu
ctan
ce (n
S)
Sample Bias (mV)
I. Maggio-Aprile et al. Phys. Rev. Lett. 75, 2754 (1995)S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).
/ vω ω2
2 1F
v
mvα ω= =
![Page 86: The quantum phase transition between a superfluid and an ...qpt.physics.harvard.edu/talks/ucsc.pdf · applications to trapped ultracold atoms and the cuprate superconductors. The](https://reader036.fdocuments.in/reader036/viewer/2022081603/5f04857e7e708231d40e621a/html5/thumbnails/86.jpg)
Conclusions• Quantum zero point motion of the vortex provides a natural
explanation for LDOS modulations observed in STM experiments.
• Size of modulation halo allows estimate of the inertial mass of a vortex
• Direct detection of vortex zero-point motion may be possible in inelastic neutron or light-scattering experiments
• The quantum zero-point motion of the vortices influences the spectrum of the electronic quasiparticles, in a manner consistent with LDOS spectrum
• “Aharanov-Bohm” or “Berry” phases lead to surprising kinematicduality relations between seemingly distinct orders. These phasefactors allow for continuous quantum phase transitions in situations where such transitions are forbidden by Landau-Ginzburg-Wilson theory.
Conclusions• Quantum zero point motion of the vortex provides a natural
explanation for LDOS modulations observed in STM experiments.
• Size of modulation halo allows estimate of the inertial mass of a vortex
• Direct detection of vortex zero-point motion may be possible in inelastic neutron or light-scattering experiments
• The quantum zero-point motion of the vortices influences the spectrum of the electronic quasiparticles, in a manner consistent with LDOS spectrum
• “Aharanov-Bohm” or “Berry” phases lead to surprising kinematicduality relations between seemingly distinct orders. These phasefactors allow for continuous quantum phase transitions in situations where such transitions are forbidden by Landau-Ginzburg-Wilson theory.