Russian Academy of Sciences
Landau Institute for Theoretical Physics
Mesoscopic fluctuations of the single-particle Green’s function
at Anderson transitions with Coulomb interaction
Igor Burmistrov
"Workshop: Anderson Localization in Topological Insulators", PSC, Daejeon, South Korea
1/20
in collaboration with
Igor Gornyi (Karlsruhe Inst. of Technology)Alexander Mirlin (Karlsruhe Inst. of Technology)Eugene Repin (Delft Univ. of Technology / Landau Inst.)
details can be found in
Phys. Rev. Lett. 111, 066601 (2013)Phys. Rev. B 89, 035430 (2014)Phys. Rev. B 91, 085427 (2015)Phys. Rev. B 93, 205432 (2016)arXiv:1609.0XXXX
2/20
Introduction: non-interacting multifractality in mesoscopic fluctuations of LDOS at Anderson transition
[Wegner (1980,1987); Kravtsov, Lerner (1985); Pruisken(1985); Castellani, Peliti (1986)]
local density of states (LDOS) in the cube of size L
ρ(E, r ) =∑
α
∣∣ψα (r )∣∣2δ(E − εα )
where ψα (r ) and εα w. f. and energy for a given disorder realization
scaling of the moments of LDOS⟨
[ρ(E, r )]q⟩
dis∼ L−∆q , q = 0, 1, 2, . . .
multifractality: the exponent ∆q 6 0 is nonlinear function of q
spatial correlations of LDOS
〈ρ(E, r )ρ(E, r + R)〉dis ∼ (R /L)∆2 , R ≪ L
examples for Anderson transitions in d = 2:∆2 = −0.34 (class AII, spin-orbit coupling)∆2 = −0.52 (class A, integer qHe)∆2 = −1/4 (class C, spin qHe)
[see for a review, Evers&Mirlin (2008)]3/20
Introduction: experiments on non-interacting multifractality
ultrasound speckle in the system of randomly packed Al beads[Faez et al., 2009]
localization of light in an array of dielectric nano-needles[Mascheck et al., 2012]
4/20
Introduction: non-interacting multifractality of the single-particle Green’s function
scaling of other combinations of wave functions, e.g.,
⟨∣∣∣ψα (r1)ψβ (r2)− ψβ (r1)ψα (r2)∣∣∣2⟩
dis∼ L−µ2 µ2 > 0.
determines the scaling of dephasing rate due to short-rangeinteraction at the Anderson transition in class A, τ−1
φ ∼ T 1+2µ2 /d
[Lee, Wang (1996); Wang, Fisher, Girvin, Chalker (2000)]
[Burmistrov, Bera, Evers, Gornyi, Mirlin (2011)]
exact relations between anomalous dimensions of very differentcorrelation functions of single-particle Green’s functions at Andersontransitions
[Gruzberg, Ludwig, Mirlin, Zirnbauer (2013); Gruzberg, Mirlin, Zirnbauer (2013)]
5/20
Introduction: LDOS suppression at the Fermi energy due to Coulomb interaction
[Altshuler, Aronov, Lee (1980), Finkelstein (1983), Castellani, DiCastro, Lee, Ma (1984)]
[Nazarov (1989), Levitov, Shytov (1997), Kamenev, Andreev (1999)]
zero bias anomaly in d = 2 (L = ∞)
⟨ρ(E, r )
⟩dis ∼ exp
(−
14πg ln
(|E|τ
)ln |E|
D2κ
4τ
)
where g is a bare conductance in units e2/h, D diffusion coefficient,
τ elastic mean free time, κ = e2ρ0/ε inverse static screening length.
zero-bias anomaly in d = 2 + ε at Anderson transitions
⟨ρ(E, r )
⟩dis ∼ |E|β , β = O(1)
in the absence of interaction average LDOS is non-critical (β = 0) for Wigner-Dyson classes
[see for a review, Finkelstein (1990), Kirkpatrick&Belitz (1994)]6/20
Introduction: differential conductance maps from scanning tunneling microscopy experiments
2DEG in InSb at B = 12 T[Morgenstern et al. (2012)]
Ga1−xMnxAs with x = 1.5%[Richardella et al. (2010)]
NbN films nearsuperconducting transition
[Noat et al. (2013)]
7/20
Introduction: numerics with Coulomb interaction
• numerics for moments of Kohn-Sham w. f. within DFT[Harashima, Slevin (2012, 2013)]
• numerics for local DOS correlation function[Amini, Kravtsov, Muller (2013)]
〈ρHF (E, r )ρHF (E, r + R)〉〈ρHF (E, r )〉2
made from Hartree-Fock w.f.
8/20
Introduction: the question
how presence of interaction affects mesoscopic fluctuationsof the single-particle Green’s function?
9/20
Model: hamiltonian H = H0 + Hdis + Hint
free electrons in d dimensions
H0 =∫
dd r ψ(r )[−
∇2
2m
]ψ(r ),
scattering off white-noise random potential
Hdis =∫
dd r ψ(r )V (r )ψ(r ), 〈V (r )V (0)〉 = 12πρ0τ
δ(r )
effective electron-electron interaction:
Hint = 12
∫dd r1d
d r2 U(|r1 − r2|)ψσ (r1)ψσ (r1)ψσ ′ (r2)ψσ ′ (r2)
disorder-averaged moments of the LDOS
⟨[ρ(E, r )
]q⟩
dis=
⟨[−
1π
ImGR (E, r , r)]q⟩
dis, GR (r t ; r ′t ′) = −iθ(t − t ′)
⟨ψσ (r t), ψσ (r ′t ′)
⟩
standard assumptions (diffusive regime): chemical potential µ ≫ 1/τ ≫ T , |E |
10/20
Model: field-theory approach
[Efetov, Larkin, Khmelnitskii (1980); Finkelstein (1983); Baranov, Pruisken, Skoric (1999)]
nonlinear sigma-model action (class AI)
S[Q] = −g
32
∫dr tr(∇Q)2 + 4πTZω
∫dr tr ηQ −
πT
4∑
α,n,r ,j
∫dr Γj tr Iαn trjQ tr Iα−ntrjQ
where the matrix field Q (in Matsubara, replica, spin and Nambu spaces)satisfies
Q2(r ) = 1, trQ = 0, Q = C TQTQ, C = it12
the matrices are as follows
Λαβnm = sgn n δnmδαβ, ηαβnm = n δnmδ
αβ, (Iγk )αβnm = δn−m,kδαβδαγ , trj = τr⊗σj
matrix field Q ∈ Sp(8NrNm)/Sp(4NrNm) × Sp(4NrNm), Nr → 0 and Nm → ∞
Γj = Γs , Γt , Γt , Γt stand for the interaction amplitudes, Finkelstein parameter Zω describes
renormalization of frequency. Coulomb interaction corresponds to γ = Γs /Zω = −1
11/20
Results: pure scaling operators in the interacting theory - I
operator linear in Q
K1(E) = 14 ReP+
1 (E), P1(iεn) = spQααnn (r )
where εn = πT (2n + 1) is the fermionic Matsubara frequencies, iεn → E + i0+
operators bilinear in Q:
K2(E, E ′) = 164
∑
p1,p2=±
p1p2Pp1p22 (E, E ′)
P2(iεn, iεm) = spQα1α1nn (r ) spQα2α2
mm (r ) + µ2 sp[Qα1α2nm (r )Qα2α1
mn (r )], α1 6= α2
iεn → E + ip10+ and iεm → E ′ + ip20+
12/20
Results: pure scaling operators in the interacting theory - II
operators of higher order in Q:
Kq(E1, . . . , Eq) = 123q
∑
p1,...pq=±
q∏
j=1pj
Pp1,...,pqq (E1, . . . , Eq)
Pq(iεn1 , . . . , iεnq ) =∑
k1,...,kq
µk1,...,kq
⟨Ak1,...,kq
⟩,
Ak1,...,kq =kq∏
r=k1
sp[Q
αj1 αj2nj1 nj2
Qαj2 αj3nj2 nj3
. . . Qαjr αj1njr nj1
], αj 6= αk
k1 + k2 + · · · + kq = q, k1 > k2 > · · · > kq > 1
13/20
Results: mesoscopic fluctuations of the single-particle Green’s function
disorder-averaged LDOS
〈K1(E)〉S =⟨ρ(E, r )
⟩
dis/ρ0
disorder-averaged second moment of LDOS
〈K(−2)2 (E, E ′)〉S =
⟨ρ(E, r )ρ(E ′, r )
⟩
dis/ρ2
0
non-local correlations of the Green’s function (λF ≪ |r − r ′| ≪ l):
〈K(µ2)2 (E, E ′)〉S ∝
⟨ImGR (E, rr ) ImGR (E ′, r ′r ′) − µ2 ImGR (E, rr ′) ImGR (E ′, r ′r )
⟩
dis
non-local correlations of the LDOS (λF ≪ |r − r ′| ≪ l):
〈K(1)2 (E,E )〉S ∝
⟨3ρ(E, r )ρ(E, r ′) − ρ(E, r )ρ(E, r ′)
⟩
dis
14/20
Reminder: pure scaling operators in the noninteracting theory
[Hof&Wegner (1986), Wegner (1987)]
averaging over rotations in replica space; energies do not mixed, e.g.
K(µ2 )2 → Tr ΛQ Tr ΛQ + µ2 Tr(ΛQ)2
classification of operators by irreducible rep. of the symmetric groupSp(8Nr )
anomalous dimension of pure scaling operators
ζ (q)µ (t) = µ2,1,...,1t + ζ(3)c3t4 + O(t5), t = 2/(πg).
µ2,1,...,1 and c3 are computed from the group theory, e.g.
µ2,1,...,1 -2 1 -6 -1 3c3 3/2 -3/2 21/2 3 -12
15/20
Results: pure scaling operators in the interacting theory - III
two-loop computations in d = 2 + ε dimensions: K(µ)q = Z q/2 m
(µ)q
−d lnm(µ)
q
d ln L = ζ (µ)q (t, γj ) = µ2,1,...,1
[t + [c(γs) + 3c(γt )]t2/2
]+ a
(q)µ
εt2 + . . .
c(γ) = 2 + 2 + γ
γli2(−γ) + 1 + γ
2γ ln2(1 + γ)
a(µ)q = 0 ⇔ µ2
2,1,...,1 + µ2,1,...,1 = 3µ3,1,...,1 + 2µ2,2,...,1 + q(q − 1)
a(q)µ is independent of γ and, consequently, vanishes for the same sets of µ2,1,...,1, . . . as in
the noninteracting case. In particular, a(2)µ ∝ (µ2
2 + µ2 − 2) vanishes for µ2 = −2 and µ2 = 1.
one-loop computations for the average LDOS: K1 = Z 1/2
−d ln Zd ln L = ζ = −t [ln(1 + γs) + 3 ln(1 + γt )] + O(t2)
[Finkelstein (1983), Castellani, DiCastro, Lee, Ma (1984)]
N.B.: one-loop background field method is insensitive to interaction and fixes µ2,1,...,1, . . . to
their noninteracting values
N.B.: two-loop computation for q > 3 is not enough to fix coefficients µ2,1,...,1, µ3,1,...,1, . . . , µq16/20
Results: background field renormalization - I
backround field method (slow U and fast Q fields)
QαβnmQ
γηmn →
[U−1QU
]αβ
nm
[U−1QU
]γη
mn
in the absence of interactions correlation functions of Q field arefixed by the symmetry of the action:
〈Qαβ±±Q
γη±±〉0 = a0δ
αβδγη + b0δαηδβγ ,
〈Qαβ++Q
γη−−〉0 = c0δ
αβδγη,
〈Qαβ±∓Q
γη∓±〉0 = f0δ
αηδβγ
17/20
Results: background field renormalization - II
one-loop approximation for fast fields (for r = 0, 3 and j = 0, 1, 2, 3):⟨
[Qrj (p)]α1β1n1m1 [Qrj (−p)]β2α2
m2n2
⟩= 2
gδα1α2δβ1β2δn12,m12Dp(iΩε
nm)[δn1n2 −
32πTΓjg
δα1β1D (j)p (iΩε
nm)],
where n12 = n1 − n2 > 0, m12 = m1 − m2 > 0, Ωεnm = εn1 − εm1 ≡ εn2 − εm2 , and Qrj = sp[Qtrj ]/4.
D −1p (iωn) = p2 + h2 + 16Zω|ωn |/g, [D s/t
p (iωn)]−1 = p2 + h2 + 16(Zω + Γs/t )|ωn |/g.
one-loop approximation for fast fields (for r = 1, 2 and j = 0, 1, 2, 3):⟨
[Qrj (p)]α1β1n1m1 [Qrj (−p)]β2α2
m2n2
⟩= 2
gδα1α2δβ1β2δn1n2δm1m2 Dp(iΩε
nm).
one-loop renormalization(
A2[Q]A1,1[Q]
)→
[1 + 2(Z 1/2 − 1)
] (A2[Q0]A1,1[Q0]
)+ t ln LM2
(A2[Q0]A1,1[Q0]
),
where the mixing matrix
M2 =
(−1 12 0
)
with eigenvalues −2 and 1.18/20
Results: background field renormalization - III
n′n′′ < 0, m′m′′ < 0 and n′m′′ > 0 (for r = 0, 3 and j = 0, 1, 2, 3):⟨
[Qrj ]α ′
1α′′1
n′n′′ [Qrj ]α ′
2α′′2
m′m′′
⟩= Zδα
′1α
′′1 δα
′2α
′′2 δn′−n′′,m′′−m′B1
[δn′m′′ + TC1δ
α ′1α
′′1],
n′n′′ < 0, m′m′′ < 0 and n′m′′ > 0 (for r = 1, 2 and j = 0, 1, 2, 3):⟨
[Qrj ]α ′
1α′′1
n′n′′ [Qrj ]α ′
2α′′2
m′m′′
⟩= ZB1δ
α ′1α
′′1 δα
′2α
′′2 δn′m′′δn′′m′
n′n′′ < 0 and m′m′′ > 0〈Q
α ′1α
′′1
n′n′′ Qα ′
2α′′2
m′m′′ 〉f ∝ T .
n′n′′ > 0, m′m′′ > 0, and n′m′ > 0
〈[Qrj ]α ′
1α′′1
n′n′′ [Qrj ]α ′
2α′′2
m′m′′ 〉f = Zδr0δj0δn′n′′δm′m′′δα′1α
′′1 δα
′2α
′′2 + B2δn′m′′δm′n′′δα
′1α
′′2 δα
′2α
′′1
−B24
sp[trjCtTrjC ] × δn′m′δn′′m′′δα
′1α
′2δα
′′1 α
′′2 .
n′n′′ > 0, m′m′′ > 0, and n′m′ < 0 〈Qα ′
1,α′′1
n′n′′ Qα ′
2α′′2
m′m′′ 〉f = ZB3Λα ′
1α′′1
n′n′′ Λα ′
2α′′2
m′m′′ .(
A2[Q]A1,1[Q]
)→ Z (1 + B3)/2
(A2[Q0 ]A1,1[Q0 ]
)+ 2Z (B1 − B2)M2
(A2[Q0 ]A1,1[Q0]
).
19/20
Conclusions:
multifractality in single-particle Green’s functions does exist ininteracting disordered systems
multifractal exponents are different from ones at the non-interactingcritical point
20/20
