The quantum mechanics of two dimensional superfluids

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The quantum mechanics of two dimensional
superfluids
Physical Review B 71, 144508 and 144509
(2005), cond-mat/0502002
Leon Balents (UCSB) Lorenz Bartosch (Yale)
Anton Burkov (UCSB) Subir Sachdev (Yale)
Krishnendu Sengupta (Toronto)
Talk online Sachdev
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Outline

1. Bose-Einstein condensation and superfluidity
2. The superfluid-Mott insulator quantum phase
   transition
3. The cuprate superconductors Superfluids
   proximate to finite doping Mott insulators with
   VBS order ?
4. Vortices in the superfluid
5. Vortices in superfluids near the
   superfluid-insulator quantum phase
   transition The quantum order of the
   superconducting state evidence for vortex
   flavors

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I. Bose-Einstein condensation and superfluidity
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Superfluidity/superconductivity occur in

• liquid 4He
• metals Hg, Al, Pb, Nb, Nb3Sn..
• liquid 3He
• neutron stars
• cuprates La2-xSrxCuO4, YBa2Cu3O6y.
• M3C60
• ultracold trapped atoms
• MgB2

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The Bose-Einstein condensate A macroscopic
number of bosons occupy the lowest energy quantum
state
Such a condensate also forms in systems of
fermions, where the bosons are Cooper pairs of
fermions
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Velocity distribution function of ultracold 87Rb
atoms
M. H. Anderson, J. R. Ensher, M. R. Matthews, C.
E. Wieman and E. A. Cornell, Science 269, 198
(1995)
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• II. The superfluid-Mott insulator quantum phase
  transition

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Apply a periodic potential (standing laser beams)
to trapped ultracold bosons (87Rb)
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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).
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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).
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Superfluid-insulator quantum phase transition at
T0
V010Er
V03Er
V00Er
V07Er
V013Er
V014Er
V016Er
V020Er
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Superfluid-insulator quantum phase transition at
T0
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Bosons at filling fraction f 1
Weak interactions superfluidity
Strong interactions Mott insulator which
preserves all lattice symmetries
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Bosons at filling fraction f 1
Weak interactions superfluidity
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Bosons at filling fraction f 1
Weak interactions superfluidity
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Bosons at filling fraction f 1
Weak interactions superfluidity
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Bosons at filling fraction f 1
Weak interactions superfluidity
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Bosons at filling fraction f 1
Strong interactions insulator
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Bosons at filling fraction f 1/2
Weak interactions superfluidity
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)
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Bosons at filling fraction f 1/2
Weak interactions superfluidity
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)
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Bosons at filling fraction f 1/2
Weak interactions superfluidity
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)
23
Bosons at filling fraction f 1/2
Weak interactions superfluidity
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)
24
Bosons at filling fraction f 1/2
Weak interactions superfluidity
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)
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Bosons at filling fraction f 1/2
Strong interactions insulator
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)
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Bosons at filling fraction f 1/2
Strong interactions insulator
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)
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Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
28
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
29
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
30
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
31
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
32
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
33
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
34
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
35
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
36
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
37
Insulating phases of bosons at filling fraction f
1/2
Valence bond solid (VBS) order
Valence bond solid (VBS) order
Charge density wave (CDW) order
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)
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Superfluid-insulator transition of bosons at
generic filling fraction f
The transition is characterized by multiple
distinct order parameters (boson condensate,
VBS/CDW order) Traditional (Landau-Ginzburg-Wilso
n) view Such a transition is first order, and
there are no precursor fluctuations of the order
of the insulator in the superfluid.
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Superfluid-insulator transition of bosons at
generic filling fraction f
The transition is characterized by multiple
distinct order parameters (boson condensate,
VBS/CDW order) Traditional (Landau-Ginzburg-Wilso
n) view Such a transition is first order, and
there are no precursor fluctuations of the order
of the insulator in the superfluid. Recent
theories Quantum interference effects can
render such transitions second order, and the
superfluid does contain VBS/CDW fluctuations.
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989).
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev
and M.P.A. Fisher, Science 303, 1490 (2004).
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III. The cuprate superconductors
Superfluids proximate to finite doping Mott
insulators with VBS order ?
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La2CuO4
La
O
Cu
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La2CuO4
Mott insulator square lattice antiferromagnet
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La2-dSrdCuO4
Superfluid condensate of paired holes
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Many experiments on the cuprate superconductors
show

• Tendency to produce modulations in spin singlet
  observables at wavevectors (2p/a)(1/4,0) and
  (2p/a)(0,1/4).
• Proximity to a Mott insulator at hole density d
  1/8 with long-range charge modulations at
  wavevectors (2p/a)(1/4,0) and (2p/a)(0,1/4).

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The cuprate superconductor Ca2-xNaxCuO2Cl2
T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M.
Azuma, M. Takano, H. Takagi, and J. C.
Davis, Nature 430, 1001 (2004).
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Many experiments on the cuprate superconductors
show

• Tendency to produce modulations in spin singlet
  observables at wavevectors (2p/a)(1/4,0) and
  (2p/a)(0,1/4).
• Proximity to a Mott insulator at hole density d
  1/8 with long-range charge modulations at
  wavevectors (2p/a)(1/4,0) and (2p/a)(0,1/4).

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Many experiments on the cuprate superconductors
show

• Tendency to produce modulations in spin singlet
  observables at wavevectors (2p/a)(1/4,0) and
  (2p/a)(0,1/4).
• Proximity to a Mott insulator at hole density d
  1/8 with long-range charge modulations at
  wavevectors (2p/a)(1/4,0) and (2p/a)(0,1/4).

Superfluids proximate to finite doping Mott
insulators with VBS order ?
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Experiments on the cuprate superconductors also
show strong vortex fluctuations above Tc
Measurements of Nernst effect are well explained
by a model of a liquid of vortices and
anti-vortices
N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S.
Uchida, Annalen der Physik 13, 9 (2004). Y. Wang,
S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S.
Uchida, and N. P. Ong, Science 299, 86 (2003).
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• Main claims
• There are precursor fluctuations of VBS order in
  the superfluid.
• There fluctuations are intimately tied to the
  quantum theory of vortices in the superfluid

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IV. Vortices in the superfluid
Magnus forces, duality, and point vortices as
dual electric charges
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Excitations of the superfluid Vortices
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Observation of quantized vortices in rotating 4He
E.J. Yarmchuk, M.J.V. Gordon, and R.E. Packard,
Observation of Stationary
Vortex Arrays in Rotating Superfluid Helium,
Phys. Rev. Lett. 43, 214 (1979).
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Observation of quantized vortices in rotating
ultracold Na
J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W.
Ketterle, Observation of Vortex Lattices in
Bose-Einstein Condensates, Science 292, 476
(2001).
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Quantized fluxoids in YBa2Cu3O6y
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).
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Excitations of the superfluid Vortices
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 ?
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In ordinary fluids, vortices experience the
Magnus Force
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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)
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V. Vortices in superfluids near the
superfluid-insulator quantum phase transition
The quantum order of the superconducting state
evidence for vortex flavors
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A3
A1A2A3A4 2p f where f is the boson filling
fraction.
A2
A4
A1
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Bosons at filling fraction f 1

• At f1, the magnetic flux per unit cell is 2p,
  and the vortex does not pick up any phase from
  the boson density.
• The effective dual magnetic field acting on
  the vortex is zero, and the corresponding
  component of the Magnus force vanishes.

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Bosons at rational filling fraction fp/q
Quantum mechanics of the vortex particle in a
periodic potential with f flux quanta per unit
cell
Space group symmetries of Hofstadter Hamiltonian
The low energy vortex states must form a
representation of this algebra
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Vortices in a superfluid near a Mott insulator at
filling fp/q
Hofstadter spectrum of the quantum vortex
particle with field operator j
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Vortices in a superfluid near a Mott insulator at
filling fp/q
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Mott insulators obtained by condensing vortices
Spatial structure of insulators for q2 (f1/2)
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Field theory with projective symmetry
Spatial structure of insulators for q4 (f1/4 or
3/4)
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Vortices in a superfluid near a Mott insulator at
filling fp/q
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Vortices in a superfluid near a Mott insulator at
filling fp/q
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Vortex-induced LDOS of Bi2Sr2CaCu2O8d integrated
from 1meV to 12meV at 4K
Vortices have halos with LDOS modulations at a
period 4 lattice spacings
b
Prediction of VBS order near vortices K. Park
and S. Sachdev, Phys. Rev. B 64, 184510 (2001).
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).
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Measuring the inertial mass of a vortex
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Measuring the inertial mass of a vortex
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• Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple
  (usually q) flavors
• The lattice space group acts in a projective
  representation on the vortex flavor space.
• These flavor quantum numbers provide a
  distinction between superfluids they constitute
  a quantum order
• Any pinned vortex must chose an orientation in
  flavor space. This necessarily leads to
  modulations in the local density of states over
  the spatial region where the vortex executes its
  quantum zero point motion.

The Mott insulator has average Cooper pair
density, f p/q per site, while the density of
the superfluid is close (but need not be
identical) to this value