Quantum%20Phase%20Transitions

1
Quantum Phase Transitions Subir Sachdev Talks
online at http//sachdev.physics.harvard.edu
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What is a phase transition ?
A change in the collective properties of a
macroscopic number of atoms
3
What is a quantum phase transition ?
Change in the nature of entanglement in a
macroscopic quantum system.
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Entanglement
Hydrogen atom
Hydrogen molecule
_

Superposition of two electron states leads to
non-local correlations between spins
5
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

6
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

7
Chinese Terracotta warriors (479-221 BC)
M. Jaime et al., Phys. Rev. Lett. 93, 087203
(2004)
8
Han Purple BaCuSi2O6
Each Cu2 has a single free electron spin
M. Jaime et al., Phys. Rev. Lett. 93, 087203
(2004)
9
Weak magnetic field
Han Purple BaCuSi2O6
M. Jaime et al., Phys. Rev. Lett. 93, 087203
(2004)
10
Strong magnetic field
Han Purple BaCuSi2O6
Magnetic field, H
M. Jaime et al., Phys. Rev. Lett. 93, 087203
(2004)
11
Han Purple BaCuSi2O6
SPM
XY-AFM
QPM
M. Jaime et al., Phys. Rev. Lett. 93, 087203
(2004)
12
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

13
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

14
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

15
Han Purple BaCuSi2O6
Each Cu2 has a single free electron spin.
Vary the ratio J/J
M. Jaime et al., Phys. Rev. Lett. 93, 087203
(2004)
16
Spin gap
Neel
J/J
Vary the ratio J/J
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Spin gap
Neel
J/J
Vary the ratio J/J
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Spin wave
J/J
Vary the ratio J/J
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Spin gap
Neel
J/J
Vary the ratio J/J
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Spin gap
Neel
Triplet magnon
J/J
Vary the ratio J/J
21
Spin gap
Neel
J/J
Vary the ratio J/J
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Spin gap
Neel
J/J
Vary the ratio J/J
S. Chakravarty, B.I. Halperin, and D.R. Nelson,
Phys. Rev. B 39, 2344 (1989)
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Quantum Criticality
J/J
Vary the ratio J/J
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411
(1992).
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Quantum Criticality
Thermal excitations interact via a universal S
matrix.
Spin gap
Neel
J/J
Vary the ratio J/J
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411
(1992).
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Decoherence time
Spin gap
Neel
J/J
Vary the ratio J/J
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411
(1992).
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Quantum critical transport
Spin diffusion constant
where Q is a universal number
Spin gap
Neel
J/J
Vary the ratio J/J
A.V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B
49, 11919 (1994).
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Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

28
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

29
Trap for ultracold 87Rb atoms
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M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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The Bose-Einstein condensate in a periodic
potential
Lowest energy state for many atoms
Large fluctuations in number of atoms in each
potential well superfluidity (atoms can flow
without dissipation)
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Breaking up the Bose-Einstein condensate
Lowest energy state for many atoms
By tuning repulsive interactions between the
atoms, states with multiple atoms in a potential
well can be suppressed. The lowest energy state
is then a Mott insulator it has negligible
number fluctuations, and atoms cannot flow
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Velocity distribution of 87Rb atoms
Superfliud
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Velocity distribution of 87Rb atoms
Insulator
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Non-zero temperature phase diagram
Insulator
Superfluid
Depth of periodic potential
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Non-zero temperature phase diagram
Dynamics of the classical Gross-Pitaevski equation
Insulator
Superfluid
Depth of periodic potential
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Non-zero temperature phase diagram
Dilute Boltzmann gas of particle and holes
Insulator
Superfluid
Depth of periodic potential
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Non-zero temperature phase diagram
No wave or quasiparticle description
Insulator
Superfluid
Depth of periodic potential
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Resistivity of Bi films
D. B. Haviland, Y. Liu, and A. M. Goldman, Phys.
Rev. Lett. 62, 2180 (1989)
M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)
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Non-zero temperature phase diagram
Insulator
Superfluid
Depth of periodic potential
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Non-zero temperature phase diagram
Collisionless-to hydrodynamic crossover of a
conformal field theory (CFT)
Insulator
Superfluid
Depth of periodic potential
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714
(1997).
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Non-zero temperature phase diagram
Needed Cold atom experiments in this regime
Collisionless-to hydrodynamic crossover of a
conformal field theory (CFT)
Insulator
Superfluid
Depth of periodic potential
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714
(1997).
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Hydrodynamics of a conformal field theory (CFT)
Maldacenas AdS/CFT correspondence relates the
hydrodynamics of CFTs to the quantum gravity
theory of the horizon of a black hole in Anti-de
Sitter space.
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Hydrodynamics of a conformal field theory (CFT)
Maldacenas AdS/CFT correspondence relates the
hydrodynamics of CFTs to the quantum gravity
theory of the horizon of a black hole in Anti-de
Sitter space.
Holographic representation of black hole physics
in a 21 dimensional CFT at a temperature equal
to the Hawking temperature of the black hole.
31 dimensional AdS space
Black hole
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Hydrodynamics of a conformal field theory (CFT)
Hydrodynamics of a CFT
Waves of gauge fields in a curved background
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Hydrodynamics of a conformal field theory (CFT)
The scattering cross-section of the thermal
excitations is universal and so transport
co-efficients are universally determined by kBT
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714
(1997).
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Hydrodynamics of a conformal field theory (CFT)
For the (unique) CFT with a SU(N) gauge field and
16 supercharges, we know the exact diffusion
constant associated with a global SO(8) symmetry
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son,
hep-th/0701036
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Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

49
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

50
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

51
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

52
Valence bonds in benzene
Resonance in benzene leads to a symmetric
configuration of valence bonds (F. Kekulé, L.
Pauling)
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Valence bonds in benzene
Resonance in benzene leads to a symmetric
configuration of valence bonds (F. Kekulé, L.
Pauling)
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Valence bonds in benzene
Resonance in benzene leads to a symmetric
configuration of valence bonds (F. Kekulé, L.
Pauling)
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Temperature-doping phase diagram of the cuprate
superconductors
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Antiferromagnetic (Neel) order in the insulator
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Induce formation of valence bonds by e.g.
ring-exchange interactions
A. W. Sandvik, cond-mat/0611343
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As in H2 and benzene, each electron wants to pair
up with another electron and form a valence bond

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62

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Entangled liquid of valence bonds (Resonating
valence bonds RVB)

P. Fazekas and P.W. Anderson, Phil Mag 30, 23
(1974).
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Valence bond solid (VBS)

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

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Valence bond solid (VBS)

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

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Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

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Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

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Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

69
Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

70
Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

71
Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

72
Valence bond solid (VBS) More
possibilities for entanglement with nearby states

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). R. Moessner
and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).

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Excitations of the RVB liquid

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Excitations of the RVB liquid

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Excitations of the RVB liquid

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Excitations of the RVB liquid

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Excitations of the RVB liquid

Electron fractionalization
Excitations carry spin S1/2 but no charge
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Excitations of the VBS

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Excitations of the VBS

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Excitations of the VBS

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Excitations of the VBS

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Excitations of the VBS

Free spins are unable to move apart no
fractionalization, but confinement
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Phase diagram of square lattice antiferromagnet
A. W. Sandvik, cond-mat/0611343
84
Phase diagram of square lattice antiferromagnet
VBS order
Neel order
K/J
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev
and M.P.A. Fisher, Science 303, 1490 (2004).
85
Phase diagram of square lattice antiferromagnet
VBS order
Neel order
K/J
RVB physics appears at the quantum critical point
which has fractionalized excitations deconfined
criticality
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev
and M.P.A. Fisher, Science 303, 1490 (2004).
86
Phase diagram of square lattice antiferromagnet
VBS order
Neel order
K/J
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev
and M.P.A. Fisher, Science 303, 1490 (2004).
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Quantum criticality of fractionalized
excitations
K/J
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Phases of nuclear matter
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Observation of a valence bond solid (VBS)
XPd(dmit)22
M. Tamura et al., J. Phys. Soc. Jpn. 75, 093701
(2006)
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Observation of a valence bond solid (VBS)
RVB (?)
Pressure-temperature phase diagram of
ETMe3PPd(dmit)22
Y. Shimizu et al. cond-mat/0612545
91
Temperature-doping phase diagram of the cuprate
superconductors
92
Temperature-doping phase diagram of the cuprate
superconductors
VBS order
Neel order
K/J
Deconfined quantum critical point (DQCP)
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Temperature-doping phase diagram of the cuprate
superconductors
Supercon-ducting algebraic holon liquid
Neel order d-wave supercon-ductivity
Neel order
DQCP
d-wave supercon-ductivity
Hole concentration
R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil,
to appear
94
Temperature-doping phase diagram of the cuprate
superconductors
Quantum critical phases with enhanced VBS
correlations
Supercon-ducting algebraic holon liquid
Neel order d-wave supercon-ductivity
Neel order
DQCP
d-wave supercon-ductivity
Hole concentration
R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil,
to appear
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Temperature-doping phase diagram of the cuprate
superconductors
STM in zero field
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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C.
Lupien, T. Hanaguri, M. Azuma, M. Takano, H.
Eisaki, H. Takagi, S. Uchida, and J. C. Davis,
Science 315, 1380 (2007)
97
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C.
Lupien, T. Hanaguri, M. Azuma, M. Takano, H.
Eisaki, H. Takagi, S. Uchida, and J. C. Davis,
Science 315, 1380 (2007)
98
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C.
Lupien, T. Hanaguri, M. Azuma, M. Takano, H.
Eisaki, H. Takagi, S. Uchida, and J. C. Davis,
Science 315, 1380 (2007)
99
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C.
Lupien, T. Hanaguri, M. Azuma, M. Takano, H.
Eisaki, H. Takagi, S. Uchida, and J. C. Davis,
Science 315, 1380 (2007)
100
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C.
Lupien, T. Hanaguri, M. Azuma, M. Takano, H.
Eisaki, H. Takagi, S. Uchida, and J. C. Davis,
Science 315, 1380 (2007)
101
Glassy Valence Bond Solid (VBS)
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C.
Lupien, T. Hanaguri, M. Azuma, M. Takano, H.
Eisaki, H. Takagi, S. Uchida, and J. C. Davis,
Science 315, 1380 (2007)
102
Temperature-doping phase diagram of the cuprate
superconductors
Glassy Valence Bond Solid (VBS)
103
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

104
Outline
Quantum phase transitions

1. Spin ordering in Han purple
2. Entanglement at the critical point physical
   consequences at non-zero temperatures (a)
   Double-layer antiferromagnet (b)
   Superfluid-insulator transition (c)
   Hydrodynamics via mapping to quantum theory of
   black holes.
3. Entanglement of valence bonds
4. Conclusions

105
Conclusions

• Studies of new materials and trapped ultracold
  atoms are yielding new quantum phases, with novel
  forms of quantum entanglement.
• Some materials are of technological importance
  e.g. high temperature superconductors.
• Real-world studies on the entanglement of large
  numbers of qubits insights may be important for
  quantum cryptography and quantum computing.
• Tabletop laboratories for the entire universe
  quantum mechanics of black holes, quark-gluon
  plasma, neutrons stars, and big-bang physics.