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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 1 Γ

    (i 1,

    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 �k

    ⇒ 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

    −π 0−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

    TcPS Ttri

    Tc2nd""

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

    Tc1st Ttri

    Tc2nd""

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

    T/ t

    µ/t

    (a)

    (f)

    ky

    xk

    µ= -0.4t T= 0.01t

    (c)

    -3

    -2

    -1

    0

    1

    2

    3

    -3 -2 -1 0 1 2 3

    ky

    xk

    µ= -0.9t T= 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=0.15t T=0.01t

    µ/t

    η /t

    0.7 0.8 0.9

    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

    0.7 0.8 0.9

    1

    g/t = 1.0 g/t = 0

    g/t = 1.0 g/t = 0

  • Linear response of nd = V −1 ∑

    k dk〈nk〉 to perturbation Hd = −µd ∑

    k dknk :

    d-wave compressibility κd = dnd dµd

    = 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 1 S

    -1

    T/Ttri

    g/t = 1.0, u/t = 0

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

    Tc1st Ttri

    Tc2nd""

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

    (a)

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

    t µ /t

    Tc2nd

    Tc1st Ttri

    Tc2nd""

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

    (b)

    -0.5 -0.4 -0.3 -0.2

    T/ t

    µ /t

    Tc2nd

    Tc1st Ttri

    Tc2nd""

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

    (c)

    0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

    T/ t

    n

    Tc2nd

    TcPS Ttri

    Tc2nd""

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

    (d)

    0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

    T/ t

    n

    Tc2nd

    TcPS Ttri

    Tc2nd""

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

    (e)

    0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

    T/ t

    n

    Tc2nd

    TcPS Ttri

    Tc2nd""

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

    lim q→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 d2kF

    4 √

    3πvkF

    u1/3

    ξ 4/3 0

    |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

    (k F,ω

    )

    ω

    T = 0.03

    T = 0.01

    T = 0.002

    T = 0

    ImΣ(kF , 0) → g d2kF

    4vkF ξ 2 0

    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

    Im Σ

    (k ,ω

    ) / (g

    d k F

    2 )

    ω

    T = 0.000 T = 0.003 T = 0.012 T = 0.027 T = 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 disto