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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
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.
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