Yejin HuhHarvard

Ettore VicariPisa

Electronic quasiparticles and competing orders in the

cuprate superconductorsAndrea Pelissetto

Rome

Subir SachdevHarvard

Gapless nodal quasiparticles in d-wave superconductors

Brillouin zone

EDC MDCx=0.145

J. Chang, M. Shi, S. Pailhes, M. Maansson, T. Claesson, O. Tjernberg, A. Bendounan, L. Patthey, N. Momono, M. Oda, M. Ido, C. Mudry, and J. Mesot, arXiv:0708.2782

Photoemission spectra of La2-xSrxCuO4

x=0.145

Photoemission spectra of La2-xSrxCuO4

J. Chang, M. Shi, S. Pailhes, M. Maansson, T. Claesson, O. Tjernberg, A. Bendounan, L. Patthey, N. Momono, M. Oda, M. Ido, C. Mudry, and J. Mesot, arXiv:0708.2782

J. W. Alldredge, Jinho Lee, K. McElroy, M. Wang, K. Fujita, Y. Kohsaka, C. Taylor, H. Eisaki, S. Uchida, P. J. Hirschfeld, and J. C. Davis

Nature Physics 4, 319 (2008)

Scanning tunneling microscopy of BSCCO

Good fit with!!"#$%&%"%#%

Needed: Quantum critical point with nodal

quasiparticles part of the critical theory

T

rrc

0

QUANTUM CRITICAL

dx2-y2

superconductorsuperconducting state X

TX

Tc

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Neutron Scattering-LSCO

Vignolle et al., Nature Phys. 07Christensen et al., PRL 04Hayden et al., Nature 04Tranquada et al., Nature 04

Brillouin zone

K

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

T=0

SC to SC+SDW quantum critical point

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, arXiv:0712.2181

SC to SC+SDW quantum critical point

Si(r, τ) = Re[eiK1·r Φ1i(r, τ) + eiK2·r Φ2i(r, τ)

].

K1 =(

2π

a

) (12− ϑ,

12

), K2 =

(2π

a

) (12,12− ϑ

),

Brillouin zone

K1

K2SDW order parameters

Tx Ty R I TΦ1i −e−iϑΦ1i −Φ1i Φ2i Φ∗1i −Φ1i

Φ2i −Φ2i −e−iϑΦ2i Φ∗1i Φ∗2i −Φ2i

SDW field theory

LΦ = |∂τΦ1|2 + v21 |∂xΦ1|2 + v2

2 |∂yΦ1|2

+|∂τΦ2|2 + v22 |∂xΦ2|2 + v2

1 |∂yΦ2|2 + r(|Φ1|2 + |Φ2|2)

+u1

2(|Φ1|4 + |Φ2|4) +

u2

2(|Φ2

1|2 + |Φ22|2)

+w1|Φ1|2|Φ2|2 + w2|Φ1 · Φ2|2 + w3|Φ∗1 · Φ2|2

Most general theory invariant under spin rotation, square lattice space group, and time-reversal symmetries

SDW field theory

LΦ = |∂τΦ1|2 + v21 |∂xΦ1|2 + v2

2 |∂yΦ1|2

+|∂τΦ2|2 + v22 |∂xΦ2|2 + v2

1 |∂yΦ2|2 + r(|Φ1|2 + |Φ2|2)

+u1

2(|Φ1|4 + |Φ2|4) +

u2

2(|Φ2

1|2 + |Φ22|2)

+w1|Φ1|2|Φ2|2 + w2|Φ1 · Φ2|2 + w3|Φ∗1 · Φ2|2

x-translationsy-translations lattice

rotations

spinrotations

Symmetries:U(1)⊗U(1)⊗Z4⊗O(3)

Stable fixed point in a 6-loop RG analysis:w∗

1 = u∗1 − u∗2, w∗2 = w∗

3 = u∗2, v∗1 = v∗2

M. De Prato, A. Pelissetto, and E. Vicari Phys. Rev. B 74, 144507 (2006).

O(4)⊗O(3) invariant theory for ϕai, with a = 1 . . . 4 an O(4) index,and i = 1 . . . 3 an O(3) index, and

Φ1i = ϕ1i + iϕ2i, Φ2i = ϕ3i + iϕ4i.

SDW field theory

Symmetries:U(1)⊗U(1)⊗Z4⊗O(3)

x-translations

y-translations latticerotations

spinrotations

SDW field theory

x-translations

y-translations latticerotations

spinrotations

Symmetries:O(4)⊗O(3)

SDW field theory

Properties of O(4)⊗O(3) fixed point:

The following 9 order parameters have divergent fluctuations atthe spin-density wave ordering transition with the same exponentγ:

• The real ‘nematic’ order parameter, φ ≡∑

i

(|Φ1i|2 − |Φ2i|2

)

which measures breaking of Z4 symmetry

• Charge density waves at 2K1 and 2K2:∑

i Φ21i and

∑i Φ2

2i

• Charge density waves at K1±K2:∑

i Φ1iΦ2i and∑

i Φ∗1iΦ2i

At the quantum critical point, the susceptibilities of these ordersall diverge as χ ∼ T−γ , with

γ ={

0.90(36) MZM, 6 loops0.80(54) d = 3 MS, 5 loops

A. Pelissetto, S. Sachdev and E. Vicari, arXiv:0802.0199

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

LΨ = Ψ†1

(∂τ − i

vF√2(∂x + ∂y)τz − i

v∆√2(−∂x + ∂y)τx

)Ψ1

+Ψ†2

(∂τ − i

vF√2(−∂x + ∂y)τz − i

v∆√2(∂x + ∂y)τx

)Ψ2.

Brillouin zone

KΨh Ψh

Ψh

Ψh

Ψ1

Ψ1Ψ2

Ψ2

0 π−π

0

π

−π

Φα

Φα

P

K

Wavevector mismatch suggests SDW order and nodal quasiparticles are decoupled

Coupling of quasiparticles to SDW order

Coupling of quasiparticles to SDW order

KΨh Ψh

Ψh

Ψh

Ψ1

Ψ1Ψ2

Ψ2

0 π−π

0

π

−π

Φα

Φα

P

K

No “Yukawa” coupling ΦΨ†Ψ

Coupling of quasiparticles to SDW order

KΨh Ψh

Ψh

Ψh

Ψ1

Ψ1Ψ2

Ψ2

0 π−π

0

π

−π

Φα

Φα

P

K

Possible higher order coupling ~ ΦΨ†ΨΦΨ†Ψ2

L1 = λ1

(|Φ1|2 + |Φ2|2

) (Ψ†

1τzΨ1 + Ψ†

2τzΨ2

)

L2 = λ2

(|Φ1|2 − |Φ2|2

) (Ψ†

1τxΨ1 + Ψ†

2τxΨ2

)

L3 = εijk

[

(Φ∗

1jΦ1k + Φ∗2jΦ2k

) (−λ3Ψ†

2τxσiΨ2 + λ′

3Ψ†1τ

zσiΨ1

)

+(Φ∗

1jΦ1k − Φ∗2jΦ2k

) (λ3Ψ†

1τxσiΨ1 − λ′

3Ψ†2τ

zσiΨ2

)].

A. Pelissetto, S. Sachdev and E. Vicari, arXiv:0802.0199

Coupling of quasiparticles to SDW order

Higher - order couplings allowed by symmetry:

Energy-energy coupling

L1 = λ1

(|Φ1|2 + |Φ2|2

) (Ψ†

1τzΨ1 + Ψ†

2τzΨ2

)

L2 = λ2

(|Φ1|2 − |Φ2|2

) (Ψ†

1τxΨ1 + Ψ†

2τxΨ2

)

L3 = εijk

[

(Φ∗

1jΦ1k + Φ∗2jΦ2k

) (−λ3Ψ†

2τxσiΨ2 + λ′

3Ψ†1τ

zσiΨ1

)

+(Φ∗

1jΦ1k − Φ∗2jΦ2k

) (λ3Ψ†

1τxσiΨ1 − λ′

3Ψ†2τ

zσiΨ2

)].

A. Pelissetto, S. Sachdev and E. Vicari, arXiv:0802.0199

Coupling of quasiparticles to SDW order

Higher - order couplings allowed by symmetry:

Nematic coupling

L1 = λ1

(|Φ1|2 + |Φ2|2

) (Ψ†

1τzΨ1 + Ψ†

2τzΨ2

)

L2 = λ2

(|Φ1|2 − |Φ2|2

) (Ψ†

1τxΨ1 + Ψ†

2τxΨ2

)

L3 = εijk

[

(Φ∗

1jΦ1k + Φ∗2jΦ2k

) (−λ3Ψ†

2τxσiΨ2 + λ′

3Ψ†1τ

zσiΨ1

)

+(Φ∗

1jΦ1k − Φ∗2jΦ2k

) (λ3Ψ†

1τxσiΨ1 − λ′

3Ψ†2τ

zσiΨ2

)].

A. Pelissetto, S. Sachdev and E. Vicari, arXiv:0802.0199

Coupling of quasiparticles to SDW order

Higher - order couplings allowed by symmetry:

Spiral spin order coupling

dim[λ1] =1ν− 2 = −0.9(2)

dim[λ2] =(γ − 1)

2=

{−0.05(18) MZM, 6 loops−0.10(27) d = 3 MS, 5 loops

dim[λ3, λ′3] =

{−0.84(8) MZM, 6 loops−0.76(8) d = 3 MS, 5 loops

A. Pelissetto, S. Sachdev and E. Vicari, arXiv:0802.0199

Coupling of quasiparticles to SDW order

Scaling dimensions of these couplings:

A. Pelissetto, S. Sachdev and E. Vicari, arXiv:0802.0199

Coupling of quasiparticles to SDW order

Coupling of nematic order is nearly marginal:

Quantum-critical features appear in fermion spectrum via coupling to nematic fluctuations of spin density

wave order.

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Nematic order in YBCOV. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)

Nematic order in YBCO

V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)

Ca1.90Na0.10CuO2Cl2

Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy

STM studies of the underdoped superconductor

Intense Tunneling-Asymmetry (TA) variation are highly similar

dI/dV SpectraCa1.90Na0.10CuO2Cl2

Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy

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)

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

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

S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998).M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999).

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

4a0

12 nmBroken lattice translation and rotation symmetries:

valence bond supersolid or supernematic

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Outline

1. SDW order in LSCO Emergent O(4) symmetry

2. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to

(composite) nematic order

3. Nematic order in YBCO and BSCCO Broken lattice symmetry but no spin order

4. Theory of the onset of nematic order in a d-wave superconductor

Infinite anisotropy fixed point

Tn

TcT

1/λ1/λc

QuantumCritical

φ/v∆

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000)E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arXiv:0705.4099

r

dx2−y2 superconductordx2−y2 superconductor+ nematic order

rc

r

dx2−y2 superconductor

rc

dx2−y2 ± ssuperconductor

Order parameter - s pairing amplitude ∼ φAlso φ ∼ |Φ1|2 − |Φ2|2

SΨ =∫

d2k

(2π)2T

∑

ωn

Ψ†1a (−iωn + vF kxτz + v∆kyτx) Ψ1a

+∫

d2k

(2π)2T

∑

ωn

Ψ†2a (−iωn + vF kyτz + v∆kxτx) Ψ2a.

S0φ =

∫d2xdτ

[12(∂τφ)2 +

c2

2(∇φ)2 +

r

2φ2 +

u0

24φ4

];

SΨφ =∫

d2xdτ[λ0φ

(Ψ†

1aτxΨ1a + Ψ†2aτxΨ2a

) ],

M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Ising theory for nematic ordering

SΨ =∫

d2k

(2π)2T

∑

ωn

Ψ†1a (−iωn + vF kxτz + v∆kyτx) Ψ1a

+∫

d2k

(2π)2T

∑

ωn

Ψ†2a (−iωn + vF kyτz + v∆kxτx) Ψ2a.

S0φ =

∫d2xdτ

[12(∂τφ)2 +

c2

2(∇φ)2 +

r

2φ2 +

u0

24φ4

];

SΨφ =∫

d2xdτ[λ0φ

(Ψ†

1aτxΨ1a + Ψ†2aτxΨ2a

) ],

Free nodal quasiparticles

Field theory for dSC to dSC+nematic transition

SΨ =∫

d2k

(2π)2T

∑

ωn

Ψ†1a (−iωn + vF kxτz + v∆kyτx) Ψ1a

+∫

d2k

(2π)2T

∑

ωn

Ψ†2a (−iωn + vF kyτz + v∆kxτx) Ψ2a.

S0φ =

∫d2xdτ

[12(∂τφ)2 +

c2

2(∇φ)2 +

r

2φ2 +

u0

24φ4

];

SΨφ =∫

d2xdτ[λ0φ

(Ψ†

1aτxΨ1a + Ψ†2aτxΨ2a

) ],

M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Yukawa coupling is now permitted and is strongly relevant

RG analysis close to 3 dimensions yields runaway flow to strong coupling

Field theory for dSC to dSC+nematic transition

Integrating out the fermions yields an effective action for the nematic order parameter

Expansion in number of fermion spin components Nf

Sφ =Nf

v∆vFΓ

[λ0φ(x, τ);

v∆

vF

]

+Nf

2

∫d2xdτ

(rφ2(x, τ)

)

+ irrelevant terms

where Γ is a non-local and non-analytic functional of φ.

The theory has only 2 couplings constants: r and v∆/vF .

Integrating out the fermions yields an effective action for the nematic order parameter

Expansion in number of fermion spin components Nf

Sφ =Nf

2

∫

k,ω|φ(k, ω)|2

[r

+λ2

0

8vF v∆

(ω2 + v2

F k2x√

ω2 + v2F k2

x + v2∆k2

y

+ (x↔ y)

)]

+higher order terms which cannot be neglected

E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arXiv:0705.4099

The 1/Nf expansion has only one coupling constantat criticality: v∆/vF .

The RG has the structure:

dynamic critical exponent : z = 1 +1

NfF1(v∆/vF )

fermion anomalous dimension : ηf =1

NfF2(v∆/vF )

RG flow equation :d(v∆/vF )

d"=

1Nf

F3(v∆/vF )

where we have computed the functions F1,2,3(v∆/vF ).

Renormalization group analysisCouplings are local in the fermion action, so perform RG on fermion self energy

The RG flow is to v∆/vF → 0 with

d(v∆/vF )d!

= − 4π2Nf

(v∆/vF )2 ln(

0.46987(v∆/vF )

)

This implies that at the critical point, as T → 0 we have theasymptotic result

v∆

vF=

π2Nf

41

ln(

ΛT

)ln

[0.1904

Nfln

(ΛT

)] ,

and a more precise result is obtained by numerically integratingthe RG equation.

Renormalization group analysis

Renormalization group analysis

In the limit v∆/vF → 0, the effective action becomes

Sφ =Nf

v∆vFΓ

[λ0φ(x, τ);

v∆

vF

]

+Nf

2

∫d2xdτ

(rφ2(x, τ)

)

+ irrelevant terms

Renormalization group analysis

In the limit v∆/vF → 0, the effective action becomes

Sφ ∼ N

v∆

[terms that have at most a divergence

which is a power of ln(vF /v∆)]

⇒ The theory is controlled by the expansion parameterv∆/Nf , and so results are asymptotically exact even forNf = 2.

E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arXiv:0705.4099

Fermion spectral functionsφ fluctuations broaden the fermion spectral functions

except in a wedge near the nodal points

Conclusions

1. Theories for damping of nodal quasiparticles in cuprates

2. SDW theory for LSCO yields (nearly) quantum critical spectral functions with arbitrary values of

3. Nematic theory for YBCO/BSCCO has a fixed point with = 0 which is approached logarithmically. The theory is expressed as an expansion in

4. Theories yield “Fermi arc” spectra.

The RG flow is to v∆/vF → 0 with

d(v∆/vF )d!

= − 4π2Nf

(v∆/vF )2 ln(

0.46987(v∆/vF )

)

This implies that at the critical point, as T → 0 we have theasymptotic result

v∆

vF=

π2Nf

41

ln(

ΛT

)ln

[0.1904

Nfln

(ΛT

)] ,

and a more precise result is obtained by numerically integratingthe RG equation.

The RG flow is to v∆/vF → 0 with

d(v∆/vF )d!

= − 4π2Nf

(v∆/vF )2 ln(

0.46987(v∆/vF )

)

This implies that at the critical point, as T → 0 we have theasymptotic result

v∆

vF=

π2Nf

41

ln(

ΛT

)ln

[0.1904

Nfln

(ΛT

)] ,

and a more precise result is obtained by numerically integratingthe RG equation.

The RG flow is to v∆/vF → 0 with

d(v∆/vF )d!

= − 4π2Nf

(v∆/vF )2 ln(

0.46987(v∆/vF )

)

This implies that at the critical point, as T → 0 we have theasymptotic result

v∆

vF=

π2Nf

41

ln(

ΛT

)ln

[0.1904

Nfln

(ΛT

)] ,

and a more precise result is obtained by numerically integratingthe RG equation.