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Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Ground state has long-range Néel order
Square lattice antiferromagnet
H =∑〈ij〉
Jij�Si · �Sj
Order parameter is a single vector field �ϕ = ηi�Si
ηi = ±1 on two sublattices
〈�ϕ〉 �= 0 in Neel state.
Antiferromagnetic (Neel) order in the insulator
No entanglement of spins
Weaken some bonds to induce spin
entanglement in a new quantum phase
Oxygen
SrCu2O3
Copper
Ground state is a product of pairs
of entangled spins.
Phase diagram as a function of the ratio of
exchange interactions, λ
λ
Ground state is a product of pairs
of entangled spins.
Excitation: S=1 triplon
Excitation: S=1 triplon
Excitation: S=1 triplon
Excitation: S=1 triplon
Excitation: S=1 triplon
G. Xu, C. Broholm, Yeong-Ah Soh, G. Aeppli, J. F. DiTusa, Y. Chen, M. Kenzelmann, C. D. Frost, T. Ito, K. Oka, and H. Takagi, Science 317, 1049 (2007).
Neutron scattering
Y2BaNiO5
Collision of triplons
Collision of triplons
Collision of triplons
-1
Collision S-matrix
Collision of triplons
Collision of triplons
G. Xu, C. Broholm, Yeong-Ah Soh, G. Aeppli, J. F. DiTusa, Y. Chen, M. Kenzelmann, C. D. Frost, T. Ito, K. Oka, and H. Takagi, Science 317, 1049 (2007).
Parameter free
prediction by K. Damle
and S. Sachdev, Phys.
Rev. B 57, 8307 (1998)
from multiple collisions
with universal,
quantum S-matrices
Neutron scattering linewidth
Y2BaNiO5
Phase diagram as a function of the ratio of
exchange interactions, λ
λλc
Quantum critical point with non-local
entanglement in spin wavefunction
Phase diagram as a function of the ratio of
exchange interactions, λ
λλc
Sϕ =
∫d2xdτ
[1
2
(c2 (∇x�ϕ)
2+ (∂τ �ϕ)
2+ s�ϕ2
)+ u
(�ϕ2
)2]
Landau-Ginzburg-Wilson Theory
Observation of longitudinal mode in
TlCuCl3Christian Ruegg, Bruce Normand, Masashige
Matsumoto, Albert Furrer, Desmond McMorrow,
Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,
Hannu Mutka, and Martin Boehm
Ground state has long-range Néel order
Square lattice antiferromagnet
H =∑〈ij〉
Jij�Si · �Sj
Order parameter is a single vector field �ϕ = ηi�Si
ηi = ±1 on two sublattices
〈�ϕ〉 �= 0 in Neel state.
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
H =∑〈ij〉
Jij�Si · �Sj
Square lattice antiferromagnet
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
H =∑〈ij〉
Jij�Si · �Sj
Square lattice antiferromagnet
LGW theory for quantum criticality
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
Sϕ =
∫d2xdτ
[1
2
(c2 (∇x�ϕ)
2+ (∂τ �ϕ)
2+ s�ϕ2
)+ u
(�ϕ2
)2]
LGW theory for quantum criticality
Sϕ =
∫d2xdτ
[1
2
(c2 (∇x�ϕ)
2+ (∂τ �ϕ)
2+ s�ϕ2
)+ u
(�ϕ2
)2]
sNeel state
sc
〈�ϕ〉 �= 0〈�ϕ〉 = 0
State with no broken symmetries. Fluc-
tuations of �ϕ about �ϕ = 0 realize a sta-ble S = 1 quasiparticle with energy εk =√
s + c2k2
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
LGW theory for quantum criticality
Sϕ =
∫d2xdτ
[1
2
(c2 (∇x�ϕ)
2+ (∂τ �ϕ)
2+ s�ϕ2
)+ u
(�ϕ2
)2]
sNeel state
sc
〈�ϕ〉 �= 0〈�ϕ〉 = 0
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
However, S = 1/2 antiferromagnets on
the square lattice have no such state.
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has
fractionalized S=1/2 excitations
=1√2
(|↑↓〉 − |↓↑〉)
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has
fractionalized S=1/2 excitations
=1√2
(|↑↓〉 − |↓↑〉)
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has
fractionalized S=1/2 excitations
=1√2
(|↑↓〉 − |↓↑〉)
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has
fractionalized S=1/2 excitations
=1√2
(|↑↓〉 − |↓↑〉)
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has
fractionalized S=1/2 excitations
=1√2
(|↑↓〉 − |↓↑〉)
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has
fractionalized S=1/2 excitations
=1√2
(|↑↓〉 − |↓↑〉)
There is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
Decompose the Neel order parameter into spinors
�ϕ = z∗α�σαβzβ
where �σ are Pauli matrices, and zα are complex spinors which carry
spin S = 1/2.
Key question: Can the zα become the needed S = 1/2 exci-
tations of a fractionalized phase ?
Effective theory for spinons must be invariant under the U(1) gauge
transformation
zα → eiθzα
Possible theory for fractionalization and topological order
Decompose the Neel order parameter into spinors
�ϕ = z∗α�σαβzβ
where �σ are Pauli matrices, and zα are complex spinors which carry
spin S = 1/2.
Key question: Can the zα become the needed S = 1/2 exci-
tations of a fractionalized phase ?
Effective theory for spinons must be invariant under the U(1) gauge
transformation
zα → eiθzα
Possible theory for fractionalization and topological order
Possible theory for fractionalization and topological order
Decompose the Neel order parameter into spinors
�ϕ = z∗α�σαβzβ
where �σ are Pauli matrices, and zα are complex spinors which carry
spin S = 1/2.
Key question: Can the zα become the needed S = 1/2 exci-
tations of a fractionalized phase ?
Effective theory for spinons must be invariant under the U(1) gauge
transformation
zα → eiθzα
Possible theory for fractionalization and topological order
Naive expectation: Low energy spinon theory for “quantum dis-
ordering” a Neel state is
Sz =
∫d2xdτ
[c2 |(∇x − iAx)zα|2 + |(∂τ − iAτ )zα|2 + s |zα|2
+ u(|zα|2
)2+
1
4e2(εμνλ∂νAλ)2
]
s
〈zα〉 �= 0
Neel state
Possible theory for fractionalization and topological order
〈zα〉 = 0
Spin liquid state with stable S = 1/2 zα
spinons, and a gapless U(1) photon Aμ
representing the topological order.
sc
Naive expectation: Low energy spinon theory for “quantum dis-
ordering” a Neel state is
Sz =
∫d2xdτ
[c2 |(∇x − iAx)zα|2 + |(∂τ − iAτ )zα|2 + s |zα|2
+ u(|zα|2
)2+
1
4e2(εμνλ∂νAλ)2
]
Spin liquid state with stable S = 1/2 zα
spinons, and a gapless U(1) photon Aμ
representing the topological order.
s
〈zα〉 �= 0
Neel state
Possible theory for fractionalization and topological order
〈zα〉 = 0
sc
Naive expectation: Low energy spinon theory for “quantum dis-
ordering” a Neel state is
Sz =
∫d2xdτ
[c2 |(∇x − iAx)zα|2 + |(∂τ − iAτ )zα|2 + s |zα|2
+ u(|zα|2
)2+
1
4e2(εμνλ∂νAλ)2
]
However, monopoles in the Aμ field will
proliferate because of the gap to zα exci-
tations, and lead to confinement of zα.
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Z2 gauge theory for fractionalization and topological order
• Discrete gauge theories do have deconfined phases in 2+1 dimensions.
• Find a collective excitation Φ with the gauge transformation
Φ → e2iθΦ
• Higgs state with 〈Φ〉 �= 0 is described by the fractionalized phase of a
Z2 gauge theory in the which the spinons zα carry Z2 gauge charges (E.
Fradkin and S. Shenker, Phys. Rev. D 19, 3682 (1979)).
• What is Φ in the antiferromagnet ? Its physical interpretation becomes
clear from its allowed coupling to the spinons:
Sz,Φ =
∫d2rdτ [λΦ∗εαβzα∂xzβ + c.c.]
From this coupling it follows that the states with 〈Φ〉 �= 0 have coplanarspin correlations.
Z2 gauge theory for fractionalization and topological order
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)A. Chubukov, T. Senthil, and S. Sachdev Phys. Rev. Lett. 72, 2089 (1994).
• Discrete gauge theories do have deconfined phases in 2+1 dimensions.
• Find a collective excitation Φ with the gauge transformation
Φ → e2iθΦ
• Higgs state with 〈Φ〉 �= 0 is described by the fractionalized phase of a
Z2 gauge theory in the which the spinons zα carry Z2 gauge charges (E.
Fradkin and S. Shenker, Phys. Rev. D 19, 3682 (1979)).
• What is Φ in the antiferromagnet ? Its physical interpretation becomes
clear from its allowed coupling to the spinons:
Sz,Φ =
∫d2rdτ [λΦ∗εαβzα∂xzβ + c.c.]
From this coupling it follows that the states with 〈Φ〉 �= 0 have coplanarspin correlations.
Z2 gauge theory for fractionalization and topological order
• Discrete gauge theories do have deconfined phases in 2+1 dimensions.
• Find a collective excitation Φ with the gauge transformation
Φ → e2iθΦ
• Higgs state with 〈Φ〉 �= 0 is described by the fractionalized phase of a
Z2 gauge theory in the which the spinons zα carry Z2 gauge charges (E.
Fradkin and S. Shenker, Phys. Rev. D 19, 3682 (1979)).
• What is Φ in the antiferromagnet ? Its physical interpretation becomes
clear from its allowed coupling to the spinons:
Sz,Φ =
∫d2rdτ [λΦ∗εαβzα∂xzβ + c.c.]
From this coupling it follows that the states with 〈Φ〉 �= 0 have coplanarspin correlations.
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)A. Chubukov, T. Senthil, and S. Sachdev Phys. Rev. Lett. 72, 2089 (1994).
Experimental realization: CsCuCl3
Neel state
Phase diagram of gauge theory of spinons
U(1) spin liquid unstable to confinement
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]
s1
s2
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
Neel state
Phase diagram of gauge theory of spinons
U(1) spin liquid unstable to confinement
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]
s1
s2
Neel state
Phase diagram of gauge theory of spinons
U(1) spin liquid unstable to confinement
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
s1
s2
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]
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)
• Two classes of gapped excitations:
– Bosonic spinons zα which carry Z2 gauge charge
– Z2 vortex associated with 2πn winding in phase of Φ. This vortex
appears as a π flux-tubes to spinons
• Ground state degeneracy is sensitive to topology: a Φ-vortex can be
inserted without energy cost in each “hole”: 4-fold degeneracy on a torus.
• Formulation in lattice model as an ‘odd’ Z2 gauge theory (R. Jalabert
and S. Sachdev, Phys. Rev. B 44, 686 (1991); T. Senthil and M. P. A.
Fisher, Phys. Rev. B 62, 7850 (2000))
• Structure identical to that found later in exactly solvable model: the Z2
toric code (A. Kitaev, quant-ph/9707021).
• Same states (without spinons) and Z2 gauge theories found to describe
liquid phases of quantum dimer models (R. Moessner and S. L. Sondhi,
Phys. Rev. Lett. 86, 1881 (2001).
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)
• Two classes of gapped excitations:
– Bosonic spinons zα which carry Z2 gauge charge
– Z2 vortex associated with 2πn winding in phase of Φ. This vortex
appears as a π flux-tubes to spinons
• Ground state degeneracy is sensitive to topology: a Φ-vortex can be
inserted without energy cost in each “hole”: 4-fold degeneracy on a torus.
• Formulation in lattice model as an ‘odd’ Z2 gauge theory (R. Jalabert
and S. Sachdev, Phys. Rev. B 44, 686 (1991); T. Senthil and M. P. A.
Fisher, Phys. Rev. B 62, 7850 (2000))
• Structure identical to that found later in exactly solvable model: the Z2
toric code (A. Kitaev, quant-ph/9707021).
• Same states (without spinons) and Z2 gauge theories found to describe
liquid phases of quantum dimer models (R. Moessner and S. L. Sondhi,
Phys. Rev. Lett. 86, 1881 (2001).
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)
• Two classes of gapped excitations:
– Bosonic spinons zα which carry Z2 gauge charge
– Z2 vortex associated with 2πn winding in phase of Φ. This vortex
appears as a π flux-tubes to spinons
• Ground state degeneracy is sensitive to topology: a Φ-vortex can be
inserted without energy cost in each “hole”: 4-fold degeneracy on a torus.
• Formulation in lattice model as an ‘odd’ Z2 gauge theory (R. Jalabert
and S. Sachdev, Phys. Rev. B 44, 686 (1991); T. Senthil and M. P. A.
Fisher, Phys. Rev. B 62, 7850 (2000))
• Structure identical to that found later in exactly solvable model: the Z2
toric code (A. Kitaev, quant-ph/9707021).
• Same states (without spinons) and Z2 gauge theories found to describe
liquid phases of quantum dimer models (R. Moessner and S. L. Sondhi,
Phys. Rev. Lett. 86, 1881 (2001).
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)
• Two classes of gapped excitations:
– Bosonic spinons zα which carry Z2 gauge charge
– Z2 vortex associated with 2πn winding in phase of Φ. This vortex
appears as a π flux-tubes to spinons
• Ground state degeneracy is sensitive to topology: a Φ-vortex can be
inserted without energy cost in each “hole”: 4-fold degeneracy on a torus.
• Formulation in lattice model as an ‘odd’ Z2 gauge theory (R. Jalabert
and S. Sachdev, Phys. Rev. B 44, 686 (1991); T. Senthil and M. P. A.
Fisher, Phys. Rev. B 62, 7850 (2000))
• Structure identical to that found later in exactly solvable model: the Z2
toric code (A. Kitaev, quant-ph/9707021).
• Same states (without spinons) and Z2 gauge theories found to describe
liquid phases of quantum dimer models (R. Moessner and S. L. Sondhi,
Phys. Rev. Lett. 86, 1881 (2001).
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)
• Two classes of gapped excitations:
– Bosonic spinons zα which carry Z2 gauge charge
– Z2 vortex associated with 2πn winding in phase of Φ. This vortex
appears as a π flux-tubes to spinons
• Ground state degeneracy is sensitive to topology: a Φ-vortex can be
inserted without energy cost in each “hole”: 4-fold degeneracy on a torus.
• Formulation in lattice model as an ‘odd’ Z2 gauge theory (R. Jalabert
and S. Sachdev, Phys. Rev. B 44, 686 (1991); T. Senthil and M. P. A.
Fisher, Phys. Rev. B 62, 7850 (2000))
• Structure identical to that found later in exactly solvable model: the Z2
toric code (A. Kitaev, quant-ph/9707021).
• Same states (without spinons) and Z2 gauge theories found to describe
liquid phases of quantum dimer models (R. Moessner and S. L. Sondhi,
Phys. Rev. Lett. 86, 1881 (2001).
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Neel state
Phase diagram of gauge theory of spinons
U(1) spin liquid unstable to confinement
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
s1
s2
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]
Neel state
Phase diagram of gauge theory of spinons
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
U(1) spin liquid unstable to confinement
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
s1
s2
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]
Partition function on cubic lattice in spacetime
LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g
Quantum theory for destruction of Neel order
Z =∏a
∫d�ϕaδ
(�ϕ2
a − 1)exp
(1
g
∑a,μ
�ϕa · �ϕa+μ
)
eiA/2
Partition function on cubic lattice in spacetime
LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g
Quantum theory for destruction of Neel order
Z =∏a
∫d�ϕaδ
(�ϕ2
a − 1)exp
(1
g
∑a,μ
�ϕa · �ϕa+μ
)
Coherent state path integral on cubic lattice in spacetime
Modulus of weights in partition function: those of a classical ferromagnet at a “temperature” g
Quantum theory for destruction of Neel order
Z =∏a
∫d�ϕaδ
(�ϕ2
a − 1)exp
(1
g
∑a,μ
�ϕa · �ϕa+μ + iSBerry
)
Partition function on cubic lattice
Quantum theory for destruction of Neel order
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Z =∏a
∫d�ϕaδ
(�ϕ2
a − 1)exp
(1
g
∑a,μ
�ϕa · �ϕa+μ + iSBerry
)
Partition function on cubic lattice
Quantum theory for destruction of Neel order
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Partition function expressed as a gauge theory of spinor degrees of freedom
Z =∏a
∫d�ϕaδ
(�ϕ2
a − 1)exp
(1
g
∑a,μ
�ϕa · �ϕa+μ + iSBerry
)
Z =∏a
∫dzaαdAaμδ
(∑α
|zaα|2 − 1
)
× exp
(1
g
∑a,μ
z∗aαeiAaµza+μ,α + i∑
a
ηaAaτ
)
Large g effective action for the Aaμ after integrating z μ
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).
Z =∏a,μ
∫dAaμ exp
(1
2e2
∑�
cos (ΔμAaν − ΔνAaμ) + i∑
a
ηaAaτ
)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Ψvbs(i) =∑〈ij〉
�Si · �Sjei arctan(rj−ri)
Neel state
Phase diagram of gauge theory of spinons
U(1) spin liquid unstable to confinement
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
s1
s2
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]
Neel state
Phase diagram of gauge theory of spinons
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
s1
s2
U(1) spin liquid unstable to VBS order
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+|(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]+ monopoles + SBerry
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
Monopole
fugacity
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
Neel state
Phase diagram of gauge theory of spinons
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
s1
s2
U(1) spin liquid unstable to VBS order
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+|(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]+ monopoles + SBerry
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)
Neel state
Phase diagram of gauge theory of spinons
〈zα〉 �= 0 , 〈Φ〉 = 0
Spiral state
〈zα〉 �= 0 , 〈Φ〉 �= 0
〈zα〉 = 0 , 〈Φ〉 = 0
〈zα〉 = 0 , 〈Φ〉 �= 0
Z2 spin liquid with bosonic spinons zα
s1
s2
U(1) spin liquid unstable to VBS order
Sz,Φ =
∫d2xdτ
[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u
(|zα|2)2
+1
4e2(εμνλ∂νAλ)2
+|(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.
]+ monopoles + SBerry
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
Probability distribution
of VBS order vbs 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).
Re[Ψvbs]
Im[Ψvbs]
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
Outline
1. Quantum “disordering” magnetic order Collinear order and confinement
2. Z2 spin liquids Noncollinear order and fractionalization
3. U(1) spin liquids Valence bond solid (VBS) order
4. Doped spin liquids Superconductors with topological order
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, arXiv:0706.2187, Nature Physics, in press.
La2CuO4
or
Phase diagram of doped antiferromagnets
Hole density x
s1g
Neel VBS
A = (2π)2x/8
Phase diagram of lightly doped antiferromagnet
Neel VBS
A = (2π)2x/4
s
sc
P
0
Neel
VBS
Holon metal
x
g
gc
Pictorial explanation of factor of 2:
• In the Neel phase, sublattice index is identical to spin index.
So for each valley and momentum, degeneracy of the hole
state is 2.
• In the VBS state, the sublattice index and the spin index are
distinct. So for each valley and momentum, degeneracy of
the hole state is 4.
• Begin with the representation of the quantum antiferromagnet
as the lattice CP1 model:
Sz = −1
g
∑a,μ
z∗aαeiAaµza+μ,α + i∑
a
ηaAaτ
• Write the electron operator at site r, cα(r) in terms of fermionicholon operators f±
cα(r) =
{f†+(r)zrα for r on sublattice A
εαβf†−(r)z∗rβ for r on sublattice B
Note that the holons fs have charge s under the U(1) gauge field
Aμ.
• Include the hopping between opposite sublattices (Shraiman-
Siggia term):
St = −t∑〈rr′〉
c†α(r)cα(r′) + h.c.
= −t∑〈rr′〉
(f†+(r)zrα)†εαβf†
−(r′)z∗rβ
• Complete theory for doped antiferromagnet:
S = Sz + Sf + St
• Choose the dispersion, ε(�k) of the f± in momentum space so that
its minima are at (±π/2,±π/2). To avoid double-counting, thesedispersions must be restricted to be within the diamond Brillouinzone.
Sf =
∫dτ
∑s=±
∫♦
d2k
4π2f†
s (�k)
(∂τ − isAτ + ε(�k − s �A)
)fs(�k)
Neel VBS
A = (2π)2x/8
Phase diagram of lightly doped antiferromagnet
s
sc
P
0
Neel
VBS
Holon metal
Neel VBS
A = (2π)2x/4
x
g
gc
Neel VBS
A = (2π)2x/8
Phase diagram of lightly doped antiferromagnet
s
sc
P
0
Neel
VBS
Holon metal
Neel VBS
A = (2π)2x/4
x
g
gc
A new non-Fermi liquid phase:
The holon metal
An algebraic charge liquid.
K1
K4
K3
K2
Area of each Fermi pocket,
A = (2π)2x/4.
The Fermi pocket will show sharp
magnetoresistance oscillations, but
it is invisible to photoemission.
• Ignore compactness in Aμ and Berry phase term.
• Neutral spinons zα are gapped.
• Charge e fermions fs form Fermi surfaces and carry charges s = ±1
under the U(1) gauge field Aμ.
• Quasi-long range order in a variety of VBS and pairing correlations.
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)
Quantum oscillations and
the Fermi surface in an
underdoped high Tc su-
perconductor (ortho-II or-
dered YBa2Cu3O6.5).
Holon pairing leading to d-wave superconductivity
First consider holon pairing in the Neel state, where holon=hole.
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
St
Holon pairing leading to d-wave superconductivity
K1
K4
K3
K2
Holon pairing leading to d-wave superconductivity
K1
K4
K3
K2
++
-
-
Gap
nodes
Holon pairing leading to d-wave superconductivity
K1
K4
K3
K2
++
-
-
Gap
nodes
We assume the same pairing holds across a transition involving loss
of long-range Neel order. The resulting phase is another algebraic
charge liquid - the holon superconductor. This superconductor
has gapped spinons with no electrical charge, and spinless, nodal
Bogoliubov-Dirac quasiparticles. The superconductivity does notgap the U(1) gauge field Aμ, because the Cooper pairs are gauge
neutral.
Low energy theory of holon superconductor
4 two-component Dirac quasiparticles coupled to a U(1) gauge field
Sholon superconductor =
∫dτd2r
[1
2e20
(εμνλ∂νAλ)2
+
4∑i=1
ψ†i (Dτ − ivF (∂x − iAx)τx − ivF (∂y − iAy)τy) ψi
]
Low energy theory of holon superconductor
External vector potential �A couples as
HA = �j · �A
where
jx = vF
(ψ†
3ψ3 − ψ†1ψ1
), jx = vF
(ψ†
4ψ4 − ψ†2ψ2
)are conserve charges of Sholon superconductor.
Fundamental property: The superfluid density, ρs, has the fol-
lowing x and T dependence:
ρs(x, T ) = cx −RkBT
where c is a non-universal constant and R is a universal constant
obtained in a 1/N expansion (N = 4 is the number of Dirac
fermions):
R = 0.4412 +0.307
N+ . . .
x
T
AFMott
Insulator AFMetal
AF+dSC
Algebraic charge liquid
Superconducting algebraic charge liquid
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.
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Ultracold 87Rb atoms - bosons
The insulator:
Excitations of the insulator:
Excitations of the insulator:
S =
∫d2rdτ
[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +
u
2|ψ|4
]
S =
∫d2rdτ
[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +
u
2|ψ|4
]
S =
∫d2rdτ
[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +
u
2|ψ|4
]
Superfluid-insulator transition at fractional filling f
f =1/2 Superfluid
Insulator
Possible continuous superfluid-insulator transition is described by a more complex CFT
L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. Sengupta, Physical Review B 71, 144508 (2005)
Superfluid-insulator transition at fractional filling f
f =1/16
Insulator
Possible continuous superfluid-insulator
transition is described by a more
complex CFT
(a) (b)
(c) (d)y
x
L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. Sengupta, Physical Review
B 71, 144508 (2005)
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Dope the antiferomagnets with charge carriers of density x by applying a chemical potential μ
T
μSuperconductor
T
μSuperconductor
Scanning tunnelling microscopy
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
STM studies of the underdoped superconductor
12 nm
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Y. Kohsaka et al. Science 315, 1380 (2007)
Topograph
Intense Tunneling-Asymmetry (TA)
variation are highly similar
dI/dV Spectra
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Y. Kohsaka et al. Science 315, 1380 (2007)
12 nm
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Y. Kohsaka et al. Science 315, 1380 (2007)
Topograph
12 nm
Tunneling Asymmetry (TA)-map at E=150meV
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Y. Kohsaka et al. Science 315, 1380 (2007)
12 nm
Tunneling Asymmetry (TA)-map at E=150meV
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Y. Kohsaka et al. Science 315, 1380 (2007)
12 nm
Tunneling Asymmetry (TA)-map at E=150meV
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Y. Kohsaka et al. Science 315, 1380 (2007)
Indistinguishable bond-centered TA contrast
with disperse 4a0-wide nanodomains
Ca1.88Na0.12CuO2Cl2, 4 KR map (150 mV)
4a0
12 nm
TA Contrast is at oxygen site (Cu-O-Cu bond-centered)
Y. Kohsaka et al. Science 315, 1380 (2007)
Ca1.88Na0.12CuO2Cl2, 4 KR map (150 mV)
4a0
12 nm
TA Contrast is at oxygen site (Cu-O-Cu bond-centered)
S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).
Evidence for VBS order - a valence bond supersolid
T
μSuperconductor
T
g
μSuperconductor
Insulator x =1/8
T
g
μSuperconductor
Insulator x =1/8
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
S =
∫d2rdτ
[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +
u
2|ψ|4
]
g
T
gc
0InsulatorSuperfluid
Quantumcritical
TKT
g
T
gc
0InsulatorSuperfluid
Quantumcritical
TKT
Wave oscillations of the condensate (classical Gross-
Pitaevski equation)
g
T
gc
0InsulatorSuperfluid
Quantumcritical
TKT
Dilute Boltzmann gas of particle and holes
g
T
gc
0InsulatorSuperfluid
Quantumcritical
TKT
CFT at T>0
D. B. Haviland, Y. Liu, and A. M. Goldman,
Phys. Rev. Lett. 62, 2180 (1989)
Resistivity of Bi films
M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)
Conductivity σ
σSuperconductor(T → 0) = ∞σInsulator(T → 0) = 0
σQuantum critical point(T → 0) ≈ 4e2
h
Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
Density correlations in CFTs at T >0
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)
Imχ(k, ω)/k2 ImK√
k2 − ω2
Collisionless to hydrodynamic crossover of SYM3
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)
Imχ(k, ω)/k2
ImDχc
Dk2 − iω
Collisionless to hydrodynamic crossover of SYM3
Universal constants of SYM3
σ(ω) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
4e2
hK , �ω � kBT
4e2
hΘ1Θ2 , �ω kBT
χc =kBT
(hv)2Θ1
D =hv2
kBTΘ2
K =
√2N3/2
3
Θ1 =8π2
√2N3/2
9
Θ2 =3
8π2
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)
C. Herzog, JHEP 0212, 026 (2002)
Electromagnetic self-duality
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
1. Superfluid-insulator transitionInteger and fractional filling
2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
3. SYM3 with N = 8 supersymmetry
4. Nernst effect in the cuprate superconductorsQuantum criticality and dyonic black holes
Outline
Ca1.90Na0.10CuO2Cl2
Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy
Dope the antiferomagnets with charge carriers of density x by applying a chemical potential μ
T
μSuperconductor
T
μSuperconductor
Nernst measurements
Nernst experiment
H
ey
Hm
T
g
μSuperconductor
Insulator x =1/8
T
g
μSuperconductor
Insulator x =1/8
Nernst measurements
For experimental applications, we must move away from the ideal CFT
e.g.
• A chemical potential μ
• A magnetic field BCFT
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
Conservation laws/equations of motion
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
Constitutive relations which follow from Lorentz transformation to moving frame
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
Single dissipative term allowed by requirement of positive entropy production. There is only one
independent transport co-efficient
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
For experimental applications, we must move away from the ideal CFT
e.g.
CFT
• A chemical potential μ
• A magnetic field B
CFT
For experimental applications, we must move away from the ideal CFT
e.g.
• A chemical potential μ
• A magnetic field B
• An impurity scattering rate 1/ imp (its T dependence
follows from scaling arguments)
CFTCFT
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
Transverse thermoelectric co-efficient(h
2ekB
)αxy = ΦsB (kBT )2
(2πτimp
�
)2ρ2 + ΦσΦε+P (kBT )3 �/2πτimp
Φ2ε+P (kBT )6 + B
2ρ2(2πτimp/�)2
,
where
B = Bφ0/(�v)2 ; ρ = ρ/(�v)2.
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
LSCO - Theory
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
LSCO - Experiments
N. P. Ong et al.
Only input parameters Output
LSCO - Theory
�v = 47 meV A
τimp ≈ 10−12 sωc = 6.2GHz · B
1T
(35K
T
)3
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
Only input parameters Output
LSCO - Theory
�v = 47 meV A
τimp ≈ 10−12 sωc = 6.2GHz · B
1T
(35K
T
)3
Similar to velocity estimates by A.V. Balatsky and Z-X. Shen, Science 284, 1137 (1999).
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)
To the solvable supersymmetric, Yang-Mills theory CFT, we add
• A chemical potential μ
• A magnetic field B
After the AdS/CFT mapping, we obtain the Einstein-Maxwell theory of a black hole with
• An electric charge
• A magnetic charge
The exact results are found to be in precise accord with
all hydrodynamic results presented earlier
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)