fermi surface symmetry breaking and fermi surface ...corpes05/presentations/metzner-ws.pdf · fermi...

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

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)

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)

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

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

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

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

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

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)

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 )

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

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

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

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

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)

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

• 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)

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.