Quantum phase transitions

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Quantum phase transitions of correlated
electrons and atoms
Subir Sachdev Harvard University
See also Quantum phase transitions of correlated
electrons in two dimensions, cond-mat/0109419.
Quantum Phase Transitions Cambridge University
Press
2
What is a quantum phase transition ?
Non-analyticity in ground state properties as a
function of some control parameter g
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Why study quantum phase transitions ?
gc
g

• Critical point is a novel state of matter
  without quasiparticle excitations

• Critical excitations control dynamics in the
  wide quantum-critical region at non-zero
  temperatures.

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• Outline
• Quantum Ising Chain
• Landau-Ginzburg-Wilson theory
  Mean field theory and
  the evolution of the excitation
  spectrum.
  
• Superfluid-insulator transition Boson Hubbard
  model at integer filling.
• Bosons at fractional filling Beyond the
  Landau-Ginzburg-Wilson paradigm.
• Quantum phase transitions and the Luttinger
  theorem Depleting the Bose-Einstein condensate
  of trapped ultracold atoms see talk by
  Stephen Powell

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I. Quantum Ising Chain
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I. Quantum Ising Chain
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Experimental realization
LiHoF4
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Weakly-coupled qubits
Ground state
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Weakly-coupled qubits
Quasiparticle pole
Three quasiparticle continuum
3D
Structure holds to all orders in 1/g
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Strongly-coupled qubits
Ground states
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Strongly-coupled qubits
Two domain-wall continuum
2D
Structure holds to all orders in g
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Entangled states at g of order unity
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Critical coupling
No quasiparticles --- dissipative critical
continuum
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S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411
(1992). S. Sachdev and A.P. Young, Phys. Rev.
Lett. 78, 2220 (1997).
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• Outline
• Quantum Ising Chain
• Landau-Ginzburg-Wilson theory
  Mean field theory and
  the evolution of the excitation
  spectrum.
  
• Superfluid-insulator transition Boson Hubbard
  model at integer filling.
• Bosons at fractional filling Beyond the
  Landau-Ginzburg-Wilson paradigm.
• Quantum phase transitions and the Luttinger
  theorem Depleting the Bose-Einstein condensate
  of trapped ultracold atoms see talk by
  Stephen Powell

18
II. Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum
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• Outline
• Quantum Ising Chain
• Landau-Ginzburg-Wilson theory
  Mean field theory and
  the evolution of the excitation
  spectrum.
  
• Superfluid-insulator transition Boson Hubbard
  model at integer filling.
• Bosons at fractional filling Beyond the
  Landau-Ginzburg-Wilson paradigm.
• Quantum phase transitions and the Luttinger
  theorem Depleting the Bose-Einstein condensate
  of trapped ultracold atoms see talk by
  Stephen Powell

22
III. Superfluid-insulator transition
Boson Hubbard model at integer filling
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Bose condensation 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)
24
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).
26
Superfluid-insulator quantum phase transition at
T0
V010Er
V03Er
V00Er
V07Er
V013Er
V014Er
V016Er
V020Er
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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).
28
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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The Superfluid-Insulator transition
Boson Hubbard model
M.PA. Fisher, P.B. Weichmann, G. Grinstein,
and D.S. Fisher Phys. Rev. B 40, 546 (1989).
34
What is the ground state for large U/t ?
Typically, the ground state remains a superfluid,
but with superfluid density density
of bosons
The superfluid density evolves smoothly from
large values at small U/t, to small values at
large U/t, and there is no quantum phase
transition at any intermediate value of U/t.
(In systems with Galilean invariance and at zero
temperature, superfluid densitydensity of
bosons always, independent of the strength of the
interactions)
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What is the ground state for large U/t ?
Incompressible, insulating ground states, with
zero superfluid density, appear at special
commensurate densities
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Excitations of the insulator infinitely
long-lived, finite energy quasiparticles and
quasiholes
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Excitations of the insulator infinitely
long-lived, finite energy quasiparticles and
quasiholes
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Excitations of the insulator infinitely
long-lived, finite energy quasiparticles and
quasiholes
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Insulating ground state
Continuum of two quasiparticles
one quasihole
Similar result for quasi-hole excitations
obtained by removing a boson
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Entangled states at of order unity
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B
49, 11919 (1994)
gc
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Crossovers at nonzero temperature
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411
(1992). K. Damle and S. Sachdev Phys. Rev. B 56,
8714 (1997).
44

• Outline
• Quantum Ising Chain
• Landau-Ginzburg-Wilson theory
  Mean field theory and
  the evolution of the excitation
  spectrum.
  
• Superfluid-insulator transition Boson Hubbard
  model at integer filling.
• Bosons at fractional filling Beyond the
  Landau-Ginzburg-Wilson paradigm.
• Quantum phase transitions and the Luttinger
  theorem Depleting the Bose-Einstein condensate
  of trapped ultracold atoms see talk by
  Stephen Powell

45
IV. Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson paradigm
L. Balents, L. Bartosch, A. Burkov, S. Sachdev,
K. Sengupta, Physical Review B 71, 144508 and
144509 (2005), cond-mat/0502002, and
cond-mat/0504692.
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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)
47
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)
48
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)
49
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)
50
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)
51
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)
52
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)
53
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)
54
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)
55
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)
56
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)
57
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)
58
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)
59
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)
60
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)
61
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)
62
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)
63
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)
64
Ginzburg-Landau-Wilson approach to multiple order
parameters
Distinct symmetries of order parameters permit
couplings only between their energy densities
S. Sachdev and E. Demler, Phys. Rev. B 69, 144504
(2004).
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Predictions of LGW theory
First order transition
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Predictions of LGW theory
First order transition
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Excitations of the superfluid Vortices
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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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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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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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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