Instituut-Lorentz
Faces of umklapp in holographyor why do we like nasty periodic solutions so much
Alexander Krikun(Instituut Lorentz, Leiden)
Gauge/Duality Gravity 2018Wurzburg
References
arXiv:1512.02465 T.Andrade, A.K. JHEP 1605 (2016) 039
arXiv:1701.04625 T.Andrade, A.K. JHEP 1703 (2017) 168
arXiv:1710.05791 T.Andrade, A.K., Nature Physics (2018)K.Schalm and J.Zaanen
arXiv:1710.05801 A.K.
arXiv:1809.xxxxx F.Balm, A.K.,arXiv:1810.xxxxx A. Romero-Bermudez,
K.Schalm and J.Zaanen
A. Krikun: Faces of umklapp in holography
In collaboration with
Jan Zaanen,
Koenraad Schalm
Tomas Andrade (Barcelona)
Floris Balm,
Aurelio Romero-Bermudez
A. Krikun: Faces of umklapp in holography
Outline
1. Brilluin zone and Umklapp
2. Lock in: discommensuraitions
3. Lock in: 2D patterns
4. Fermionic response: Umklapp 2.0
A. Krikun: Faces of umklapp in holography
Periodic potential and Brilluin zone
Position space
D2ψ(x) + V (x)ψ(x) = 0
Momentum space
p2ψ(p)+V (q)ψ(p−q) = 0
V(q)
The spectrum is organized in orbits
Ψ(k) ∼ {ψ(k), ψ(k + q), ψ(k + 2q), . . . }, k ∈ [0, 2π}
Bloch wave function
Ψ(x) = e ikx ψ(x), ψ(x) ≡ ψ(x + 2π/q)
A. Krikun: Faces of umklapp in holography
Umklapp scattering of particles
-2 -1 0 1 2 3 4
0.5
1.0
1.5
2.0
A. Krikun: Faces of umklapp in holography
Umklapp scattering of particles
-2 -1 0 1 2 3 4
0.5
1.0
1.5
2.0
A. Krikun: Faces of umklapp in holography
Umklapp scattering of particles
-2 -1 0 1 2 3 4
0.5
1.0
1.5
2.0
A. Krikun: Faces of umklapp in holography
Commensurate lock inas an umklapp effect
A. Krikun: Faces of umklapp in holography
Spontaneous breaking of translations
S =
∫d4x√−g(R − 1
2(∂ψ)2 − τ(ψ)
4F 2 − V (ψ)
)−1
2
∫ϑ(ψ)F∧F
Donos, Gaunltett; 1106.2004, JHEP08(2011)140
A. Krikun: Faces of umklapp in holography
Spontaneous TSB in the Brilluin zone
µ(x) = µ0(1 + A cos(qx)
)Horowitz, Santos, Tong; 1209.1098 JHEP11(2012)102
A. Krikun: Faces of umklapp in holography
Commensurate enhancement
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Andrade, Krikun; 1701.04625 JHEP03(2017)168
A. Krikun: Faces of umklapp in holography
Lock in plateaux
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△△
0.4 0.6 0.8 1.0 1.2 1.4 1.6
1
2
1
3
2
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■
◆
▲
A. Krikun: Faces of umklapp in holography
Nonlinear lock in
A. Krikun: Faces of umklapp in holography
Nonlinear Lock in
0% 20% 40% 60% 80% 100%
0.05
0.1
0.15
0.21.2 1.4 1.6 1.8 2.
Doping
Tem
pera
ture(ρ)
p0/k
1/1
Mott
4
3
3aCDW
3/2
2aCDW
5
3
2/1
Strange metal
Andrade, Krikun, Schalm, Zaanen;1710.05791, Nat.Phys.(2018)
A. Krikun: Faces of umklapp in holography
Nonlinear Lock in
T ∼ 0.1Tc ≈ 0.01µ0
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Lattice
Amplitude
● 0.
■ 0.7
0% 20% 40% 60% 80% 100%
4
3
3/2
1
2
5
3
1. 1.2 1.4 1.6 1.8 2.
Doping
p/k
p0/k
A. Krikun: Faces of umklapp in holography
Discommensuration
Two scales for one Brilluin zone.
A. Krikun: Faces of umklapp in holography
DiscommensurationDiscommensurations account for the phase shift by π/2
A A A AB B B B
-2
-1
0
1
2
A A A AB B B B
-2
-1
0
1
2
A. Krikun: Faces of umklapp in holography
Discommensuration
And they form discommensuration lattices
-8 a -4 a 0 4 a 8 a
4a periodic stripes
Mesaros, A. et al. PNAS 113,
12661–12666 (2016).
Relevant to experiments!
A. Krikun: Faces of umklapp in holography
2D lock in
A. Krikun: Faces of umklapp in holography
Stripes and Checkerboards
Withers, 1407.1085, JHEP09(2014)102
A. Krikun: Faces of umklapp in holography
Varma loop currents
Balm, Krikun, R.-Bermudez, Schalm and Zaanen, in progress
A. Krikun: Faces of umklapp in holography
Umklapp ofHolographic Fermions
A. Krikun: Faces of umklapp in holography
Fermi pockets[Γf eµf (x)
(∂µ +
1
4ωabµ(x)ηacσ
cb − ieAµ(x)
)−m
]Ψ = 0
No overlap
Fermi pockets form
A. Krikun: Faces of umklapp in holography
Holographic fermions: weak lattice
Ling et al. 1304.2128, JHEP07(2013)045
A. Krikun: Faces of umklapp in holography
Holographic fermions: strong lattice
Balm, Krikun, R.-Bermudez, Schalm and Zaanen, in progress
A. Krikun: Faces of umklapp in holography
Nodal-antinodal dychotomy
Zhou, Yoshida, Shen, PRL.92.187001
A. Krikun: Faces of umklapp in holography
Conclusion
I Periodic lattices display interesting phenomenologywhich is relevant to experimental observations
I Umklapp effects can only be seen in spatially dependentbackgrounds (homogeneous lattices don’t have it)
I Holography brings new twist to the well known umklappphysics
A. Krikun: Faces of umklapp in holography