2. coupled dimer antiferromagnet 3. honeycomb lattice: semi...

1. Introduction to the Hubbard model Superexchange and antiferromagnetism

2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point

3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model

4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover

5. Hubbard model as a SU(2) gauge theorySpin liquids, valence bond solids: analogies with SQED and SYM

Outline

Monday, June 14, 2010

6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations

7. Instabilities near the SDW critical point d-wave superconductivity and other orders

8. Global phase diagram of the cuprates Competition for the Fermi surface

Outline

H = −�

tijc†iαcjα + U

�ni↑ −

��ni↓ −

�− µ

c†iαciα

tij → “hopping”. U → local repulsion, µ → chemical potential

Spin index α =↑, ↓

niα = c†iαciα

c†iαcjβ + cjβc

†iα = δijδαβ

ciαcjβ + cjβciα = 0

The Hubbard Model

Will study on the honeycomb and square lattices

H = −�

tijc†iαcjα + U

�ni↑ −

��ni↓ −

�− µ

c†iαciα

tij → “hopping”. U → local repulsion, µ → chemical potential

Spin index α =↑, ↓

niα = c†iαciα

c†iαcjβ + cjβc

†iα = δijδαβ

ciαcjβ + cjβciα = 0

The Hubbard Model

Will study on the honeycomb and square lattices

Fermi surface+antiferromagnetism

Hole states

occupied

Electron states

occupied

The electron spin polarization obeys�

�S(r, τ)�

= �ϕ(r, τ)eiK·r

where K is the ordering wavevector.

Fermi surfaces in electron- and hole-doped cupratesHole states

occupied

Electron states

occupied

ΓEffective Hamiltonian for quasiparticles:

H0 = −�

tijc†iαciα ≡

εkc†kαckα

with tij non-zero for first, second and third neighbor, leads to satisfactory agree-

ment with experiments. The area of the occupied electron states, Ae, from

Luttinger’s theory is

�2π

2(1− p) for hole-doping p

2π2(1 + x) for electron-doping x

The area of the occupied hole states, Ah, which form a closed Fermi surface and

so appear in quantum oscillation experiments is Ah = 4π2 −Ae.

Spin density wave theory

In the presence of spin density wave order, �ϕ at wavevector K =(π,π), we have an additional term which mixes electron states withmomentum separated by K

Hsdw = �ϕ ·�

k,α,β

c†k,α�σαβck+K,β

where �σ are the Pauli matrices. The electron dispersions obtainedby diagonalizing H0 +Hsdw for �ϕ ∝ (0, 0, 1) are

Ek± =εk + εk+K

��εk − εk+K

�+ ϕ2

This leads to the Fermi surfaces shown in the following slides forelectron and hole doping.

Increasing SDW order

ΓΓΓ

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Hole pockets

Electron pockets

Hole-doped cuprates

ΓΓΓ

Hole pockets

Electron pockets

Hole-doped cuprates

ΓΓΓ

Hole pockets

Electron pockets

Hole-doped cuprates

Hot spots

ΓΓΓ

Hole pockets

Electron pockets

Hole-doped cuprates

Fermi surface breaks up at hot spotsinto electron and hole “pockets”

Hole pockets

Hot spots

ΓΓΓ

Hole pockets

Electron pockets

Hole-doped cuprates

Fermi surface breaks up at hot spotsinto electron and hole “pockets”

Hot spots

arXiv:0912.3022

Fermi liquid behaviour in an underdoped high Tc superconductor

Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich

Evidence for small Fermi pockets

ΓΓΓ

Hole pockets

Electron pockets

Large Fermi surface breaks up intoelectron and hole pockets

Hole-doped cuprates

ΓΓΓ

Hole pockets

Electron pockets

Hole-doped cuprates

�ϕ fluctuations act on thelarge Fermi surface

Start from the “spin-fermion” model

Z =�DcαD�ϕ exp (−S)

S =�

dτ�

c†kα

�∂

∂τ− εk

�ckα

− λ

�dτ

c†iα�ϕi · �σαβciβeiK·ri

dτd2r

(∇r �ϕ)2 +�ζ2

(∂τ �ϕ)2 +s

2�ϕ2 +

4�ϕ4

� = 1

� = 2

� = 4

� = 3

Low energy fermionsψ�

1α, ψ�2α

� = 1, . . . , 4

Lf = ψ�†1α

�ζ∂τ − iv�

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

v�=11 = (vx, vy), v�=1

2 = (−vx, vy)Monday, June 14, 2010

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

ψ2 fermionsoccupied

ψ1 fermionsoccupied

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

“Hot spot”

“Cold” Fermi surfaces

Order parameter: Lϕ =1

2(∇r �ϕ)

�ζ2

(∂τ �ϕ)2

2�ϕ2

4�ϕ4

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

2(∇r �ϕ)

�ζ2

(∂τ �ϕ)2

2�ϕ2

4�ϕ4

“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†

1α�σαβψ�2β + ψ�†

2α�σαβψ�1β

Hertz theoryIntegrate out fermions and obtain non-local corrections to Lϕ

Lϕ =12

�ϕ2�q2 + γ|ω|

�/2 ; γ =

2πvxvy

Exponent z = 2 and mean-field criticality (upto logarithms)

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

2(∇r �ϕ)

�ζ2

(∂τ �ϕ)2

2�ϕ2

4�ϕ4

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

Integrate out fermions and obtain non-local corrections to Lϕ

Lϕ =12

�ϕ2�q2 + γ|ω|

�/2 ; γ =

2πvxvy

Exponent z = 2 and mean-field criticality (upto logarithms)

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

2(∇r �ϕ)

�ζ2

(∂τ �ϕ)2

2�ϕ2

4�ϕ4

OK in d = 3, but higher order terms contain an

infinite number of marginal couplings in d = 2

Hertz theory

Perform RG on both fermions and �ϕ,using a local field theory.

Lf = ψ�†1α

1 · ∇r

�ψ�

1α + ψ�†2α

2 · ∇r

�ψ�

2(∇r �ϕ)

�ζ2

(∂τ �ϕ)2

2�ϕ2

4�ϕ4

Outline

Superconductivity by SDW fluctuation

exchange

Physical Review B 34, 8190 (1986)

Spin density wave theory in hole-doped cuprates

ΓΓΓ Γ

Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates

David Pines, Douglas Scalapino

Fermions at the large Fermi surface exchangefluctuations of the SDW order parameter �ϕ.

Pairing by SDW fluctuation exchange

We now allow the SDW field �ϕ to be dynamical, coupling to elec-trons as

Hsdw = −�

k,q,α,β

�ϕq · c†k,α�σαβck+K+q,β .

Exchange of a �ϕ quantum leads to the effective interaction

Hee = −12

p,γ,δ

k,α,β

Vαβ,γδ(q)c†k,αck+q,βc†p,γcp−q,δ,

where the pairing interaction is

Vαβ,γδ(q) = �σαβ · �σγδχ0

ξ−2 + (q−K)2,

with χ0ξ2 the SDW susceptibility and ξ the SDW correlation length.

BCS Gap equation

In BCS theory, this interaction leads to the ‘gapequation’ for the pairing gap ∆k ∝ �ck↑c−k↓�.

∆k = −�

�3χ0

ξ−2 + (p− k−K)2

�∆p

ε2p + ∆2

Non-zero solutions of this equation require that∆k and ∆p have opposite signs when p− k ≈ K.

d-wave pairing of the large Fermi surface

�ck↑c−k↓� ∝ ∆k = ∆0(cos(kx)− cos(ky))

� = 1

� = 2

� = 4

� = 3

Low energy fermionsψ�

1α, ψ�2α

� = 1, . . . , 4

d-wave pairing in the theory of hotspots

Hot spots have

strong instability to

d-wave pairing near

SDW critical point.

This instability is

stronger than the

BCS instability of a

Fermi liquid.

Pairing order parameter: εαβ�ψ31αψ

11β − ψ3

2αψ12β

d-wave pairing in the theory of hotspots

ψ31 ψ3

2 ψ31 ψ3

2 ψ31

d-wave Cooper pairing instability in particle-particle channel

+�ϕ

Similar theory applies to the pnictides, and leads to s± pairing.

Q = ( !, 0)

Emergent Pseudospin symmetry

Continuum theory of hotspots in invariant under:

�ψ�↑

ψ�†↓

�→ U �

�ψ�↑

ψ�†↓

where U � are arbitrary SU(2) matrices which can bedifferent on different hotspots �.

ψ31 ψ3

2 ψ31 ψ3

2 ψ31

d-wave Cooper pairing instability in particle-particle channel

+�ϕ

ψ31 ψ3

2 ψ31 ψ3

2 ψ31

Bond density wave (with local Ising-nematic order) instability in particle-hole channel

+�ϕ

d-wave pairing has apartner instabilityin the particle-hole

channel

Density-wave order parameter:

�ψ3†1αψ

11α − ψ3†

2αψ12α

Single ordering wavevector Q:�c†k−Q/2,αck+Q/2,α

Φ(cos kx − cos ky)

No modulations on sites. Modulated bond-densitywave with local Ising-nematic ordering:�

c†k−Q/2,αck+Q/2,α

�= Φ(cos kx − cos ky)

“Bond density” measures amplitude for electrons to be

in spin-singlet valence bond:

VBS order

No modulations on sites. Modulated bond-densitywave with local Ising-nematic ordering:�

c†k−Q/2,αck+Q/2,α

�= Φ(cos kx − cos ky)

“Bond density” measures amplitude for electrons to be

in spin-singlet valence bond:

VBS order

M. J. Lawler, K. Fujita,

Jhinhwan Lee,

A. R. Schmidt,

Y. Kohsaka, Chung Koo

Kim, H. Eisaki,

S. Uchida, J. C. Davis,

J. P. Sethna, and

Eun-Ah Kim, preprint

STM measurements of Z(r), the energy asymmetry

in density of states in Bi2Sr2CaCu2O8+δ.

Jhinhwan Lee,

A. R. Schmidt,

Kim, H. Eisaki,

J. P. Sethna, and

Jhinhwan Lee,

A. R. Schmidt,

Kim, H. Eisaki,

J. P. Sethna, and

Strong anisotropy of electronic states between

x and y directions:Electronic

“Ising-nematic” order

Outline

Antiferro-magnetism

Fermi surface

d-wavesupercon-ductivity

Antiferro-magnetism

Fermi surface

FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Theory of quantum criticality in the cuprates

Underlying SDW ordering quantum critical pointin metal at x = xm

Increasing SDW orderIncreasing SDW order

Antiferro-magnetism

Fermi surface

Spin density wave

Fermi surface

Spin density wave

FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Underlying SDW ordering quantum critical pointin metal at x = xm

Increasing SDW orderIncreasing SDW order

LargeFermi

surface

StrangeMetal

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

Onset of d-wave superconductivity

hides the critical point x = xm

Fluctuating, paired Fermi

pockets

LargeFermi

surface

StrangeMetal

pairing fluctuationsE. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

pockets

LargeFermi

surface

StrangeMetal

pairing fluctuationsE. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

pockets

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gap

Thermallyfluctuating

pockets

E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

surface

StrangeMetal

Spin gap

Thermallyfluctuating

V. Galitski andS. Sachdev, Phys.Rev. B 79, 134512(2009).

E. G. Moon andS. Sachdev, Phys.Rev. B 80, 035117(2009)

pockets

Physics of competition: d-wave SC and SDW“eat up” same pieces of the large Fermi surface.

surface

StrangeMetal

Spin gapThermallyfluctuating

d-waveSC

pockets

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

surface

StrangeMetal

d-waveSC

pockets

M"Normal"

SDW(Small Fermi

pockets)

SC+SDW

surface

StrangeMetal

d-waveSC

pockets

Quantum oscillations

M"Normal"

SDW(Small Fermi

pockets)

SC+SDW

surface

StrangeMetal

d-waveSC

pockets

J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann,

H. M. Roennow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess,

M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda, M. Ido, and

J. Mesot, Phys. Rev. Lett. 102, 177006

(2009).

J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow,

D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono,

M. Oda, M. Ido, and J. Mesot, Physical Review B 78, 104525 (2008).

M"Normal"

SDW(Small Fermi

pockets)

SC+SDW

surface

StrangeMetal

d-waveSC

pockets

G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223

Similar phase diagram for CeRhIn5

Similar phase diagram for the pnictides

Ishida, Nakai, and HosonoarXiv:0906.2045v1

0 0.02 0.04 0.06 0.08 0.10 0.12

AFM Ort/

S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, arXiv:0911.3136.

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