Outline

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Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
3
Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
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Square lattice antiferromagnet
Ground state has long-range Néel order
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Square lattice antiferromagnet
Destroy Neel order by perturbations which
preserve full square lattice symmetry e.g.
second-neighbor or ring exchange.
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Square lattice antiferromagnet
Destroy Neel order by perturbations which
preserve full square lattice symmetry e.g.
second-neighbor or ring exchange.
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LGW theory for quantum criticality
S. Chakravarty, B.I. Halperin, and D.R. Nelson,
Phys. Rev. B 39, 2344 (1989)
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LGW theory for quantum criticality
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B
49, 11919 (1994)
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LGW theory for quantum criticality
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B
49, 11919 (1994)
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
Liquid of valence bonds has fractionalized
S1/2 excitations
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
Liquid of valence bonds has fractionalized
S1/2 excitations
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
Liquid of valence bonds has fractionalized
S1/2 excitations
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
Liquid of valence bonds has fractionalized
S1/2 excitations
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
Liquid of valence bonds has fractionalized
S1/2 excitations
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There is no state with a gapped, stable S1
quasiparticle and no broken symmetries
Liquid of valence bonds has fractionalized
S1/2 excitations
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Possible theory for fractionalization and
topological order
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Possible theory for fractionalization and
topological order
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Possible theory for fractionalization and
topological order
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Possible theory for fractionalization and
topological order
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Possible theory for fractionalization and
topological order
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Possible theory for fractionalization and
topological order
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Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
24
Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
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Z2 gauge theory for fractionalization and
topological order
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Z2 gauge theory for fractionalization and
topological order
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev, Phys. Rev. B 45, 12377 (1992)
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Z2 gauge theory for fractionalization and
topological order
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev, Phys. Rev. B 45, 12377 (1992)
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Experimental realization CsCuCl3
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Phase diagram of gauge theory of spinons
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Phase diagram of gauge theory of spinons
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Phase diagram of gauge theory of spinons
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Characteristics of Z2 spin liquid
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Characteristics of Z2 spin liquid
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Characteristics of Z2 spin liquid
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Characteristics of Z2 spin liquid
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Characteristics of Z2 spin liquid
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
38
Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
39
Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
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Phase diagram of gauge theory of spinons
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
41
Phase diagram of gauge theory of spinons
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991) S. Sachdev and N. Read, Int. J. Mod. Phys.
B 5, 219 (1991)
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Quantum theory for destruction of Neel order
Partition function on cubic lattice in spacetime
LGW theory weights in partition function are
those of a classical ferromagnet at a
temperature g
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Missing ingredient Spin Berry Phases
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Quantum theory for destruction of Neel order
Partition function on cubic lattice in spacetime
LGW theory weights in partition function are
those of a classical ferromagnet at a
temperature g
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Quantum theory for destruction of Neel order
Coherent state path integral on cubic lattice in
spacetime
Modulus of weights in partition function those
of a classical ferromagnet at a temperature g
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Quantum theory for destruction of Neel order
Partition function on cubic lattice
S. Sachdev and K. Park, Annals of Physics, 298,
58 (2002)
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Quantum theory for destruction of Neel order
Partition function on cubic lattice
Partition function expressed as a gauge theory of
spinor degrees of freedom
S. Sachdev and K. Park, Annals of Physics, 298,
58 (2002)
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Large g effective action for the Aaµ after
integrating zaµ
This theory can be reliably analyzed by a duality
mapping. The gauge theory is in a confining
phase, and there is VBS order in the ground
state. (Proliferation of monopoles in the
presence of Berry phases).
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989). S. Sachdev and R. Jalabert, Mod. Phys.
Lett. B 4, 1043 (1990). K. Park and S. Sachdev,
Phys. Rev. B 65, 220405 (2002).
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or
?
g
0
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Monopole fugacity
Arovas-Auerbach state
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Phase diagram of S1/2 square lattice
antiferromagnet
or
g
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev
and M.P.A. Fisher, Science 303, 1490 (2004).
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Phase diagram of gauge theory of spinons
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Phase diagram of gauge theory of spinons
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Phase diagram of gauge theory of spinons
Critical U(1) spin liquid
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Large scale Quantum Monte Carlo studies
A.W. Sandvik, S. Daul, R. R. P. Singh, and D. J.
Scalapino, Phys. Rev. Lett. 89, 247201
(2002) A.W. Sandvik and R.G. Melko, Phys. Rev. E
72, 026702 (2005).
A.W. Sandvik, Phys. Rev. Lett. 98, 2272020
(2007). R.G. Melko and R.K. Kaul, arXiv0707.2961
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Easy-plane model
Spin stiffness
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Easy-plane model
Valence bond solid (VBS) order in expectation
values of plaquette and exchange terms
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SU(2) invariant model
Strong evidence for a continuous deconfined
quantum critical point
T. Senthil, A. Vishwanath, L. Balents,
S. Sachdev and M.P.A. Fisher, Science 303, 1490
(2004).
A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007).
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SU(2) invariant model
R.G. Melko and R.K. Kaul, arXiv0707.2961
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SU(2) invariant model
Probability distribution of VBS order ? at
quantum critical point
Emergent circular symmetry is evidence for U(1)
photon and topological order
A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007).
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Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
72
Outline
1. Quantum disordering magnetic order
Collinear order and confinement 2. Z2 spin
liquids Noncollinear order and
fractionalization 3. Gapless U(1) spin liquids
Deconfined criticality 4. Doped spin
liquids Superconductors with topological
order
73
Hole dynamics in an antiferromagnet across a
deconfined quantum critical point, R. K. Kaul,
A. Kolezhuk, M. Levin, S. Sachdev, and T.
Senthil, Physical Review B 75 , 235122 (2007)
Algebraic charge liquids and the underdoped
cuprates, R. K. Kaul, Y. B. Kim, S. Sachdev, and
T. Senthil, arXiv0706.2187
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Phase diagram of gauge theory of spinons
Critical U(1) spin liquid
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Phase diagram of gauge theory of spinons and
holons
Add a finite concentration of charge carriers
Critical U(1) spin liquid
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Phase diagram of doped antiferromagnets
or
La2CuO4
Hole density x
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Phase diagram of lightly doped antiferromagnet
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Phase diagram of lightly doped antiferromagnet
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Phase diagram of lightly doped antiferromagnet
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A new non-Fermi liquid phase The holon metal An
algebraic charge liquid.
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N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J.
Levallois, J.-B. Bonnemaison, R. Liang,
D. A. Bonn, W. N. Hardy, and L. Taillefer,
Nature 447, 565 (2007)
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• Holon pairing leading to d-wave
  superconductivity

• First consider holon pairing in the Neel state,
  where holonhole.
• This was studied in V. V. Flambaum, M. Yu.
  Kuchiev, and O. P. Sushkov, Physica C 227, 267
  (1994) V. I. Belincher et al., Phys. Rev. B 51,
  6076 (1995). They found p-wave pairing of holons,
  induced by spin-wave exchange from the sublattice
  mixing term . This corresponds to d-wave
  pairing of physical electrons

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• Holon pairing leading to d-wave
  superconductivity

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• Holon pairing leading to d-wave
  superconductivity

-
Gap nodes


-
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• Holon pairing leading to d-wave
  superconductivity

-
Gap nodes


-
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Low energy theory of holon superconductor
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Low energy theory of holon superconductor
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Holon-hole metal
Holon-hole superconductor
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Conclusions
1. Theory for Z2 and U(1) spin liquids in quantum
antiferromagnets, and evidence for their
realization in model spin systems. 2. Algebraic
charge liquids appear naturally upon adding
fermionic carriers to spin liquids with bosonic
spinons. These are conducting states with
topological order. 3. The holon
metal/superconductor, obtained by doping a
Neel-ordered insulator, matches several observed
characteristics of the underdoped cuprates.