Outline - Harvard Universityqpt.physics.harvard.edu/talks/goa.pdf · Outline 1. Quantum...

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

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

Ground state is a product of pairs

of entangled spins.

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 S-matrix

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

Quantum critical point with non-local

entanglement in spin wavefunction

∫d2xdτ

(c2 (∇x�ϕ)

2+ (∂τ �ϕ)

2+ s�ϕ2

(�ϕ2

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

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

Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.

H =∑〈ij〉

Jij�Si · �Sj

LGW theory for quantum criticality

S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

∫d2xdτ

(c2 (∇x�ϕ)

2+ (∂τ �ϕ)

2+ s�ϕ2

(�ϕ2

∫d2xdτ

(c2 (∇x�ϕ)

2+ (∂τ �ϕ)

2+ s�ϕ2

(�ϕ2

sNeel state

〈�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)

∫d2xdτ

(c2 (∇x�ϕ)

2+ (∂τ �ϕ)

2+ s�ϕ2

(�ϕ2

sNeel state

〈�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

(|↑↓〉 − |↓↑〉)

=1√2

(|↑↓〉 − |↓↑〉)

=1√2

(|↑↓〉 − |↓↑〉)

=1√2

(|↑↓〉 − |↓↑〉)

=1√2

(|↑↓〉 − |↓↑〉)

=1√2

(|↑↓〉 − |↓↑〉)

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

spin S = 1/2.

transformation

zα → eiθzα

spin S = 1/2.

transformation

zα → eiθzα

Naive expectation: Low energy spinon theory for “quantum dis-

ordering” a Neel state is

∫d2xdτ

[c2 |(∇x − iAx)zα|2 + |(∂τ − iAτ )zα|2 + s |zα|2

+ u(|zα|2

4e2(εμνλ∂νAλ)2

〈zα〉 �= 0

Neel state

〈zα〉 = 0

Spin liquid state with stable S = 1/2 zα

spinons, and a gapless U(1) photon Aμ

representing the topological order.

∫d2xdτ

+ u(|zα|2

Spin liquid state with stable S = 1/2 zα

spinons, and a gapless U(1) photon Aμ

representing the topological order.

〈zα〉 �= 0

Neel state

〈zα〉 = 0

∫d2xdτ

+ u(|zα|2

However, monopoles in the Aμ field will

proliferate because of the gap to zα exci-

tations, and lead to confinement of zα.

Outline

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).

Φ → e2iθΦ

Sz,Φ =

Φ → e2iθΦ

Sz,Φ =

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

+ |(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.

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

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

Characteristics of Z2 spin liquid

• 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).

Fisher, Phys. Rev. B 62, 7850 (2000))

Phys. Rev. Lett. 86, 1881 (2001).

Fisher, Phys. Rev. B 62, 7850 (2000))

Phys. Rev. Lett. 86, 1881 (2001).

Fisher, Phys. Rev. B 62, 7850 (2000))

Phys. Rev. Lett. 86, 1881 (2001).

Fisher, Phys. Rev. B 62, 7850 (2000))

Phys. Rev. Lett. 86, 1881 (2001).

Outline

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)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

∑a,μ

�ϕa · �ϕa+μ

Partition function on cubic lattice in spacetime

LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g

Z =∏a

∫d�ϕaδ

(�ϕ2

a − 1)exp

∑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

Z =∏a

∫d�ϕaδ

(�ϕ2

a − 1)exp

∑a,μ

�ϕa · �ϕa+μ + iSBerry

Partition function on cubic lattice

S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

Z =∏a

∫d�ϕaδ

(�ϕ2

a − 1)exp

∑a,μ

Partition function on cubic lattice

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

∑a,μ

Z =∏a

∫dzaαdAaμδ

(∑α

|zaα|2 − 1

× exp

∑a,μ

z∗aαeiAaµza+μ,α + i∑

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

∑�

cos (ΔμAaν − ΔνAaμ) + i∑

ηaAaτ

Ψvbs(i) =∑〈ij〉

�Si · �Sjei arctan(rj−ri)

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

U(1) spin liquid unstable to VBS order

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

+|(∂μ − 2iAμ)Φ|2 + s2|Φ|2 + u|Φ|4 + λΦ∗εαβzα∂xzβ + c.c.

]+ monopoles + SBerry

Monopole

fugacity

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

Neel state

〈zα〉 �= 0 , 〈Φ〉 = 0

Spiral state

〈zα〉 �= 0 , 〈Φ〉 �= 0

〈zα〉 = 0 , 〈Φ〉 = 0

〈zα〉 = 0 , 〈Φ〉 �= 0

Sz,Φ =

∫d2xdτ

[|(∂μ − iAμ)zα|2 + s1 |zα|2 + u

(|zα|2)2

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

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

Phase diagram of doped antiferromagnets

Hole density x

Neel VBS

A = (2π)2x/8

Phase diagram of lightly doped antiferromagnet

Neel VBS

A = (2π)2x/4

Holon metal

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

∑a,μ

z∗aαeiAaµza+μ,α + i∑

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

• 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.

∫dτ

∑s=±

∫♦

4π2f†

s (�k)

(∂τ − isAτ + ε(�k − s �A)

)fs(�k)

Neel VBS

A = (2π)2x/8

Holon metal

Neel VBS

A = (2π)2x/4

Neel VBS

A = (2π)2x/8

Holon metal

Neel VBS

A = (2π)2x/4

A new non-Fermi liquid phase:

The holon metal

An algebraic charge liquid.

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

Holon pairing leading to d-wave superconductivity

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

(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

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+ . . .

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

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:

∫d2rdτ

[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +

2|ψ|4

∫d2rdτ

[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +

2|ψ|4

∫d2rdτ

[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +

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

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

μSuperconductor

Scanning tunnelling microscopy

Ca1.90Na0.10CuO2Cl2

STM studies of the underdoped superconductor

Ca1.90Na0.10CuO2Cl2

Y. Kohsaka et al. Science 315, 1380 (2007)

Topograph

Intense Tunneling-Asymmetry (TA)

variation are highly similar

dI/dV Spectra

Ca1.90Na0.10CuO2Cl2

Topograph

Tunneling Asymmetry (TA)-map at E=150meV

Ca1.90Na0.10CuO2Cl2

Indistinguishable bond-centered TA contrast

with disperse 4a0-wide nanodomains

Ca1.88Na0.12CuO2Cl2, 4 KR map (150 mV)

TA Contrast is at oxygen site (Cu-O-Cu bond-centered)

Ca1.88Na0.12CuO2Cl2, 4 KR map (150 mV)

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

μSuperconductor

Insulator x =1/8

μSuperconductor

Insulator x =1/8

Outline

∫d2rdτ

[|∂τψ|2 + c2|�∇ψ|2 + s|ψ|2 +

2|ψ|4

0InsulatorSuperfluid

Quantumcritical

Wave oscillations of the condensate (classical Gross-

Pitaevski equation)

Quantumcritical

Dilute Boltzmann gas of particle and holes

Quantumcritical

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

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)

−iω

k2χ(k, ω)

−iω

k2χ(k, ω)

K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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

Imχ(k, ω)/k2

ImDχc

Dk2 − iω

Collisionless to hydrodynamic crossover of SYM3

Universal constants of SYM3

σ(ω) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

hK , �ω � kBT

hΘ1Θ2 , �ω kBT

χc =kBT

(hv)2Θ1

D =hv2

kBTΘ2

√2N3/2

Θ1 =8π2

√2N3/2

Θ2 =3

C. Herzog, JHEP 0212, 026 (2002)

Electromagnetic self-duality

Outline

Ca1.90Na0.10CuO2Cl2

Dope the antiferomagnets with charge carriers of density x by applying a chemical potential μ

μSuperconductor

Nernst measurements

Nernst experiment

μSuperconductor

Insulator x =1/8

μSuperconductor

Insulator x =1/8

Nernst measurements

For experimental applications, we must move away from the ideal CFT

• 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

Constitutive relations which follow from Lorentz transformation to moving frame

Single dissipative term allowed by requirement of positive entropy production. There is only one

independent transport co-efficient

• A magnetic field B

• An impurity scattering rate 1/ imp (its T dependence

follows from scaling arguments)

CFTCFT

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

)αxy = ΦsB (kBT )2

(2πτimp

)2ρ2 + ΦσΦε+P (kBT )3 �/2πτimp

Φ2ε+P (kBT )6 + B

2ρ2(2πτimp/�)2

B = Bφ0/(�v)2 ; ρ = ρ/(�v)2.

LSCO - Theory

LSCO - Experiments

N. P. Ong et al.

Only input parameters Output

LSCO - Theory

�v = 47 meV A

τimp ≈ 10−12 sωc = 6.2GHz · B

Only input parameters Output

LSCO - Theory

�v = 47 meV A

τimp ≈ 10−12 sωc = 6.2GHz · B

Similar to velocity estimates by A.V. Balatsky and Z-X. Shen, Science 284, 1137 (1999).

To the solvable supersymmetric, Yang-Mills theory CFT, we add

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

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