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Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid State Research 1. Pomeranchuk instability (symmetry-breaking Fermi surface deformations) 2. Soft Fermi surface and Non-Fermi liquid behavior 3. Experimental signatures – cuprates Collaborators: C. J. Halboth, A. Neumayr (Aachen); V. Oganesyan (Princeton) D. Rohe, S. Andergassen, L. Dell’Anna, H. Yamase (Stuttgart) PRL 85, 5162 (2000); PRL 91, 066402 (2003); cond-mat/0502238

Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

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Page 1: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Fermi surface symmetry breaking

and Fermi surface fluctuations

W. Metzner, MPI for Solid State Research

1. Pomeranchuk instability

(symmetry-breaking Fermi surface deformations)

2. Soft Fermi surface and Non-Fermi liquid behavior

3. Experimental signatures – cuprates

Collaborators:

C. J. Halboth, A. Neumayr (Aachen); V. Oganesyan (Princeton)

D. Rohe, S. Andergassen, L. Dell’Anna, H. Yamase (Stuttgart)

PRL 85, 5162 (2000); PRL 91, 066402 (2003); cond-mat/0502238

Page 2: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

1. Pomeranchuk instability

RG flow of forward scattering

interactions (singlet part)

in 2D Hubbard model

Halboth + wm ’00

-60

-50

-40

-30

-20

-10

0

10

20

30

0.01 0.1 1Γ

(i1,

i 2,i 3

)/t

Λ/t

RG-Flow: µ=-0.2t (n=0.959), t’=-0.05, U=t;

s(1,1,1)s(1,2,1)s(1,3,1)s(1,4,1)s(1,5,1)s(1,6,1)s(1,7,1)s(1,8,1)

Forward scattering interaction fkFk′F

with attractive d-wave component;

small Fermi velocity vkF near saddle points of εk

⇒ d-wave Fermi surface deformations easy (low energy cost)

Page 3: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Symmetry-breaking Fermi surface deformation (”Pomeranchuk instability”)

Quasi-particle interactions:

k

kx

0

π

−π

y

0

attraction

−π π

repulsion

Reshaping of Fermi surface

k

kx

0

π

y

−π0−π π

Tetragonal symmetry broken !

Realization of ”nematic” electron liquid (→ Kivelson et al. ’98)

See also: Yamase, Kohno ’00 (tJ-model)

Page 4: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Phenomenological 2D lattice model:

H = Hkin + 12V

k,k′,q fkk′(q)nk(q)nk′(−q)

where nk(q) =∑

σ c†k−q/2,σ ck+q/2,σ

and only small momentum transfers q contribute (forward scattering)

Interaction with uniform repulsion and d-wave attraction:

fkk′(q) = u(q) + g(q) dk dk′

with dk = cos kx − cos ky and u(q) ≥ 0, g(q) < 0

(qualitatively as from RG)

yields Pomeranchuk instability

Similar model without u-term for isotropic (not lattice) system

by Oganesyan, Kivelson, Fradkin ’01

Page 5: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Mean-field phase diagram (u = 0):

Khavkine et al. ’04

Yamase et al. ’05

First order transition

at low temperature

Tricritical points

0

0.05

0.1

0.15

0.2

0.25

0.7 0.75 0.8 0.85 0.9 0.95 1

Tc2nd

TcPSTtri

Tc2nd""

t'/t = -1/6t"/t = 0g/t = 1.0u/t = 0

n

T/t

0

0.05

0.1

0.15

0.2

0.25

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3

Tc2nd

Tc1stTtri

Tc2nd""

t'/t = -1/6t"/t = 0g/t = 1.0u/t = 0

T/t

µ/t

(a)

(f)

ky

xk

µ= -0.4tT= 0.01t

(c)

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

ky

xk

µ= -0.9tT= 0.01t

(b)

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

(d)

0

0.05

0.1

0.15

0.2

0.25

0.3

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3

T=0.15tT=0.01t

µ/t

η/t

0.70.80.9

1

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3

n

T=0.15t

µ/t

(e)

n

T=0.01t

0.70.80.9

1

g/t = 1.0g/t = 0

g/t = 1.0g/t = 0

Page 6: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Linear response of nd = V −1∑

k dk〈nk〉 to perturbation Hd = −µd∑

k dknk :

d-wave compressibility κd =dnddµd

=κ0d

1 + gκ0d

Stoner factor S = (1 + gκ0d)−1

strongly enhanced even

at first order transition

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1S

-1

T/Ttri

g/t = 1.0, u/t = 0

g/t = 0.5, u/t = 10g/t = 0.5, u/t = 0

t'/t = -1/6, t"/t = 0

T

µ

⇒ Soft Fermi surface near transition

Page 7: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Phase diagrams for t′/t = −1/6, t′′/t = 1/5, g/t = 0.5 and u/t = 0, 1, 2:T/

t

0

0.01

0.02

0.03

0.04

0.05

0.06

-1.48 -1.44 -1.40 -1.36µ /t

Tc2nd

Tc1stTtri

Tc2nd""

t'/t = -1/6t"/t = 1/5g/t = 0.5u/t = 0

(a)

-1 -0.95 -0.9 -0.85 -0.8 -0.75T/

tµ /t

Tc2nd

Tc1stTtri

Tc2nd""

t'/t = -1/6t"/t = 1/5g/t = 0.5u/t = 1.0

(b)

-0.5 -0.4 -0.3 -0.2

T/t

µ /t

Tc2nd

Tc1stTtri

Tc2nd""

t'/t = -1/6t"/t = 1/5g/t = 0.5u/t = 2.0

(c)

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

T/t

n

Tc2nd

TcPSTtri

Tc2nd""

t'/t = -1/6t"/t = 1/5g/t = 0.5u/t = 0

(d)

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

T/t

n

Tc2nd

TcPSTtri

Tc2nd""

t'/t = -1/6t"/t = 1/5g/t = 0.5u/t = 1.0

(e)

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

T/t

n

Tc2nd

TcPSTtri

Tc2nd""

t'/t = -1/6t"/t = 1/5g/t = 0.5u/t = 2.0

(f)

0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

Quantum critical point for u/t ≥ 2

Page 8: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

2. Soft Fermi surface and non-FL behavior

Soft Fermi surface (near Pomeranchuk instability)

⇒ large response to anisotropic (d-wave) perturbations,

large Fermi surface fluctuations

Critical fluctuations near continuous Pomeranchuk transition,

quantum critical at T = 0, if transition remains continuous at low T

Non-Fermi liquid behavior in critical regime

wm, Rohe, Andergassen ’03

Page 9: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Origin of non-FL behavior:

Electrons see fluctuating Fermi surface

⇒ enhanced and anisotropic decay rates

0

k

kx

y

π

π

Fluctuations collective and overdamped;

not to be confused with:

• usual thermal smearing

• zero sound (propagating Fermi surface oscillation)

Page 10: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Dynamical effective interaction:

Γ = +f

+ff

. . .

Γkk′(q, ω) =u(q)

1− u(q) Π(q, ω)+

g(q)1− g(q) Πd(q, ω)

dk dk′

d-wave polarization function Πd0(q, ω) = −

∫ f(ξp+q/2)−f(ξp−q/2)

ω−(ξp+q/2−ξp−q/2) d2p

Critical point for Fermi surface symmetry breaking:

limq→0

g(q) Πd(q, 0) = 1

(while g(q) Πd(q, 0) < 1 for q 6= 0 )

Page 11: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Singular part near Pomeranchuk instability for small q and small ω/|q|

Γkk′(q, ω) ∼ g(0) dk dk′

(ξ0/ξ)2 + ξ02 |q|2 − i ωu|q|

Parameters:

Velocity u > 0 (related to ImΠd)

microscopic length scale ξ0

correlation length ξ, related to Stoner factor by S = (ξ/ξ0)2

No generic ”non-analytic” corrections in charge channel (Chubukov + Maslov ’03)

Temperature dependence of ξ determined by dangerously irrelevant

interaction of critical fluctuations (Millis ’93);

in quantum critical regime:

ξ(T ) ∝ 1√T× logarithmic corrections

Page 12: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Electron self-energy:

Leading order (Fock approximation)

Γ

Σ =

At quantum critical point (T = 0, ξ =∞):

ImΣ(kF , ω) =g d2

kF

4√

3πvkF

u1/3

ξ4/30

|ω|2/3 for ω → 0

• large anisotropic imaginary part

• maximal near van Hove points,

minimal near diagonal in Brillouin zone

⇒ no quasi-particles away from Brillouin zone diagonal

Page 13: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Above quantum critical point: Dell’Anna + wm ’05

ImΣ(kF , ω) for

T ≥ 0,

ξ(T ) ∝ T−1/2

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Im Σ

(kF,ω

)

ω

T = 0.03

T = 0.01

T = 0.002

T = 0

ImΣ(kF , 0) →g d2

kF

4vkF ξ20

T ξ(T ) ∝ T 1/2 × log. corr. for T → 0

Page 14: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

For k outside Fermi surface:

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ImΣ

(k,ω

) / (g

dk F

2 )

ω

T = 0.000T = 0.003T = 0.012T = 0.027T = 0.048

Page 15: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Selfconsistency (G instead of G0 in Fock term) yields no qualitative changes.

At least at T = 0 results for Σ also stable against vertex corrections

(cf. fermions coupled to gauge field, in particular Altshuler et al. ’94)

Page 16: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

3. Experimental signatures – cuprates

• Response to lattice distortions:

Strong reaction of electronic properties to slight

LTT lattice distortions observed in

La2−xBaxCuO4 (Axe et al. ’89)

Nd-doped LSCO (Buchner et al. ’94)

Increasing evidence for strong ab-anisotropy

of electronic properties of CuO2-planes in

YBCO (Lu et al. ’01, Hinkov et al. ’04)

• Linewidth in photoemission:

Large anisotropic decay rates for single-particle

excitations observed in optimally doped cuprates

Page 17: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

• Anisotropy in transport:

c-axis vs. ab-plane anisotropy in transport

follows naturally from anisotropic decay rate

with minima on the Brillouin zone diagonals

↔ ”cold spot” scenario (Ioffe + Millis ’98)

• Relation to (dynamical) stripes ?

Stripes break orientation and translation symmetry

• Raman scattering:

Signatures in B1g-channel ?

Spin-dependent d-wave Pomeranchuk instability proposed recently

for Sr3Ru2O7 (Grigera et al. ’04)

Page 18: Fermi surface symmetry breaking and Fermi surface ...corpes05/Presentations/Metzner-WS.pdf · Fermi surface symmetry breaking and Fermi surface fluctuations W. Metzner, MPI for Solid

Conclusions:

• Interactions can induce symmetry-breaking Fermi surface deformations:

Pomeranchuk instability

• Near Pomeranchuk instability soft Fermi surface

reacting strongly to anisotropic perturbations.

• Fluctuations of soft Fermi surface lead to

large anisotropic quasi-particle decay rates.