magnetic impurities and universality in ads/cmt johanna ... filerg flow from perturbation by...
TRANSCRIPT
![Page 1: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/1.jpg)
.
.
Magnetic impurities and universality in AdS/CMT
Johanna Erdmenger
Max–Planck–Institut fur Physik, Munchen
1
![Page 2: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/2.jpg)
Outline
1. Kondo models from holography
Model J.E., Hoyos, O’Bannon, Wu 1310.3271Quantum quenches J.E., Flory, Newrzella, Wu in progressEntanglement entropy J.E., Flory, Newrzella 1410.7811
J.E., Flory, Hoyos, Newrzella, O’Bannon, Wu 1511.03666Two-point functions J.E., Hoyos, O’Bannon, Papadimitriou, Probst, Wu in progress
2. S-Wave Superconductivity in Anisotropic Holographic InsulatorsJ.E., Herwerth, Klug, Meyer, Schalm 1501.07615
Scalar condenses in helical Bianchi VII backgroundHomes’ Law(See also talk by Keun-Young Kim on Thursday; Kim, Kim, Park 1604.06205)
2
![Page 3: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/3.jpg)
Kondo models from gauge/gravity duality
![Page 4: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/4.jpg)
Kondo models from gauge/gravity duality
Kondo effect:
Screening of a magnetic impurity by conduction electrons at low temperatures
![Page 5: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/5.jpg)
Kondo models from gauge/gravity duality
Kondo effect:
Screening of a magnetic impurity by conduction electrons at low temperatures
Motivation for study within gauge/gravity duality:
![Page 6: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/6.jpg)
Kondo models from gauge/gravity duality
Kondo effect:
Screening of a magnetic impurity by conduction electrons at low temperatures
Motivation for study within gauge/gravity duality:
1. Kondo model: Simple model for a RG flow with dynamical scale generation
![Page 7: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/7.jpg)
Kondo models from gauge/gravity duality
Kondo effect:
Screening of a magnetic impurity by conduction electrons at low temperatures
Motivation for study within gauge/gravity duality:
1. Kondo model: Simple model for a RG flow with dynamical scale generation
2. New applications of gauge/gravity duality to condensed matter physics:
Magnetic impurity coupled to strongly correlated electron systemEntanglement entropyQuantum quenchRelation to Sachdev-Ye-Kitaev model
3
![Page 8: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/8.jpg)
Kondo effect
4
![Page 9: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/9.jpg)
Kondo model
![Page 10: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/10.jpg)
Kondo model
Original Kondo model (Kondo 1964):Magnetic impurity interacting with free electron gas
![Page 11: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/11.jpg)
Kondo model
Original Kondo model (Kondo 1964):Magnetic impurity interacting with free electron gas
Hamiltonian:
H =vF2πψ†i∂xψ + λKvFδ(x)~S · ~J , ~J = ψ†
1
2~Tψ
Decisive in development of renormalization groupIR fixed point, CFT approach Affleck, Ludwig ’90’s
5
![Page 12: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/12.jpg)
Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU(N)
![Page 13: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/13.jpg)
Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU(N)
In this case, interaction term simplifies introducing slave fermions:
Sa = χ†T aχ
Totally antisymmetric representation: Young tableau with Q boxes
Constraint: χ†χ = Q
![Page 14: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/14.jpg)
Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU(N)
In this case, interaction term simplifies introducing slave fermions:
Sa = χ†T aχ
Totally antisymmetric representation: Young tableau with Q boxes
Constraint: χ†χ = Q
Interaction: JaSa = (ψ†T aψ)(χ†T aχ) = OO†, where O = ψ†χ
![Page 15: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/15.jpg)
Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU(N)
In this case, interaction term simplifies introducing slave fermions:
Sa = χ†T aχ
Totally antisymmetric representation: Young tableau with Q boxes
Constraint: χ†χ = Q
Interaction: JaSa = (ψ†T aψ)(χ†T aχ) = OO†, where O = ψ†χ
Screened phase has condensate 〈O〉
Parcollet, Georges, Kotliar, Sengupta cond-mat/9711192Senthil, Sachdev, Vojta cond-mat/0209144
6
![Page 16: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/16.jpg)
Kondo models from gauge/gravity duality
J.E., Hoyos, O’Bannon, Wu 1310.3271, JHEP 1312 (2013) 086
Coupling of a magnetic impurity to a strongly interacting non-Fermi liquid
![Page 17: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/17.jpg)
Kondo models from gauge/gravity duality
J.E., Hoyos, O’Bannon, Wu 1310.3271, JHEP 1312 (2013) 086
Coupling of a magnetic impurity to a strongly interacting non-Fermi liquid
Results:
RG flow from perturbation by ‘double-trace’ operator
Dynamical scale generation
Holographic superconductor: Condensate forms in AdS2
Power-law scaling of conductivity in IR with real exponent
Screening, phase shift
7
![Page 18: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/18.jpg)
Field-operator map
From top-down construction involving D3, D5 and D7 branes:
Operator Gravity fieldElectron current J ⇔ Chern-Simons gauge field A in AdS3
Charge Q = χ†χ ⇔ 2d gauge field a in AdS2
Operator O = ψ†χ ⇔ 2d complex scalar Φ
8
![Page 19: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/19.jpg)
Bottom-up gravity dual for Kondo model
Action:S = SEinstein−Hilbert + SCS + SAdS2,
SCS = −N4π
∫AdS3
Tr
(A ∧ dA+
2
3A ∧A ∧A
),
SAdS2 = −N∫d3x δ(x)
√−g[
1
4Trfmnfmn + gmn (DmΦ)
†DnΦ + V (Φ†Φ)
]V (Φ) = M2Φ†Φ
Metric:
ds2 = gµνdxµdxν =
1
z2
(dz2
h(z)− h(z) dt2 + dx2
),
h(z) = 1− z2/z2H , T = 1/(2πzH)
9
![Page 20: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/20.jpg)
‘Double-trace’ deformation by OO†
Boundary expansion
Φ = z1/2(α ln z + β)
α = κβ
κ dual to double-trace deformation Witten hep-th/0112258
![Page 21: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/21.jpg)
‘Double-trace’ deformation by OO†
Boundary expansion
Φ = z1/2(α ln z + β)
α = κβ
κ dual to double-trace deformation Witten hep-th/0112258
Φ invariant under renormalization⇒ Running coupling
κT =κ0
1 + κ0 ln(
Λ2πT
)
![Page 22: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/22.jpg)
‘Double-trace’ deformation by OO†
Boundary expansion
Φ = z1/2(α ln z + β)
α = κβ
κ dual to double-trace deformation Witten hep-th/0112258
Φ invariant under renormalization⇒ Running coupling
κT =κ0
1 + κ0 ln(
Λ2πT
)Dynamical scale generation
10
![Page 23: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/23.jpg)
Kondo models from gauge/gravity duality
Scale generation
Divergence of Kondo coupling determines Kondo temperature TK
Transition temperature to phase with condensed scalar: Tc
Tc < TK
11
![Page 24: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/24.jpg)
Kondo models from gauge/gravity duality
RG flow
UV
IR
Strongly interacting
electrons
Deformation by
Kondo operator
Non-trivial condensate
Strongly interacting
electrons
12
![Page 25: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/25.jpg)
Kondo models from gauge/gravity duality
Normalized condensate 〈O〉 ≡ κβ as function of the temperature
(a) (b)
Mean field transition
〈O〉 approaches constant for T → 0
13
![Page 26: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/26.jpg)
Time dependence
Allow for time dependence of the Kondo coupling and study response of thecondensate
![Page 27: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/27.jpg)
Time dependence
Allow for time dependence of the Kondo coupling and study response of thecondensate
Examples for time dependence of the Kondo coupling:
Gaussian pulse in IR
Quench from condensed to normal phase (IR to UV)
Quench from normal to condensed phase (UV to IR)
![Page 28: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/28.jpg)
Time dependence
Allow for time dependence of the Kondo coupling and study response of thecondensate
Examples for time dependence of the Kondo coupling:
Gaussian pulse in IR
Quench from condensed to normal phase (IR to UV)
Quench from normal to condensed phase (UV to IR)
Observations:
Different timescales depending on whether the condensate is asymptoticallysmall or large
Timescales governed by quasinormal modes
14
![Page 29: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/29.jpg)
15
![Page 30: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/30.jpg)
50 100 150 2000
2
4
6
8
10
50 100 150 2000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Quantum quenches inholographic Kondo modelTo and from condensed phase
Timescales determined byquasinormal modes
J.E., Flory, Newrzella, Strydom, Wu
cf. Quench in s-wave holographic superconductor,Bhaseen, Gauntlett, Simons, Sonner, Wiseman PRL 2012
16
![Page 31: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/31.jpg)
Timescales in quantum quench
500 1000 1500 2000 2500 3000
0.008
0.006
0.004
0.002
500 1000 1500 2000 2500 3000
8.5
9.0
9.5
10.0
Separate time scales
Unstable QNM
17
![Page 32: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/32.jpg)
Timescales in quantum quench
10 20 30 40 50 60
30
25
20
15
10
5
10 20 30 40 50 600.0
0.5
1.0
1.5
2.0
Quench timescale
Quasinormal ringing
18
![Page 33: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/33.jpg)
Quasinormal modes
-0.2 -0.1 0.0 0.1 0.2
-0.2
-0.1
0.0
0.1
0.2
ωR
2 π T
ωⅈ
2πT
19
![Page 34: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/34.jpg)
Kondo model: Two-point functions
Parcollet, Georges, Kotliar, Sengupta cond-mat/9711192: Large N Kondo model
Spectral asymmetry ωs: Particle-hole symmetry broken
−ImGR for bosonic 〈OO†〉
ωs = qT~∂S∂Q see also Sachdev 1506.05111, AdS2 black hole (fermions)
20
![Page 35: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/35.jpg)
Two-point function in holographic Kondo model
J.E., Hoyos, O’Bannon, Papadimitriou, Probst, Wu in progress
-10 -5 5 10
ω
TK
-5
-4
-3
-2
-1
Im <O†O>
T/TK = 3.3978, zH = 0.8
T/TK = 2.7183, zH = 1.0
T/TK = 1.8122, zH = 1.5
T/TK = 1.3591, zH = 2.0
21
![Page 36: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/36.jpg)
Entanglement entropy for magnetic impurity
Including the backreaction using a thin brane and Israel junction conditions
Israel junction conditions Kµν − γµνK = −κ2 Tµν ⇔ Energy conditions
identify points
boundary boundary
hypersurface
bulkbulk
J.E., Flory, Newrzella 1410.7811
22
![Page 37: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/37.jpg)
Entanglement entropy for magnetic impurity
AB
AdS
Impurity entropy:
Simp = Scondensed phase − Snormal phase
Subtraction also guarantees UV regularity
23
![Page 38: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/38.jpg)
Entanglement entropy for magnetic impurity J.E., Flory, Newrzella 1410.7811
Depending on the brane tension λ, the total space is enhanced or reduced
24
![Page 39: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/39.jpg)
Entanglement entropy for magnetic impurity J.E., Flory, Newrzella 1410.7811
- 0.06 - 0.05 - 0.04 - 0.03 - 0.02 - 0.01 0.001.0
0.8
0.6
0.4
0.2
0.0
x+(z)/zH
1
/TC
0
T
z/z H
The larger the condensate, the shorter the geodesic
25
![Page 40: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/40.jpg)
Impurity entropy from gauge/gravity duality
0 1 2 3 40.040
0.045
0.050
0.055
0.060
0.065
ℓ/zH
ℒ imp=6 S imp/c
26
![Page 41: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/41.jpg)
Entanglement entropy for magnetic impurity: Comparison to field theory
Field theory result Sorensen, Chang, Laflorencie, Affleck 2007(Eriksson, Johannesson 2011)
∆Simp(`) =π2ξKT
6coth(2π`T ) + C0
![Page 42: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/42.jpg)
Entanglement entropy for magnetic impurity: Comparison to field theory
Field theory result Sorensen, Chang, Laflorencie, Affleck 2007(Eriksson, Johannesson 2011)
∆Simp(`) =π2ξKT
6coth(2π`T ) + C0
In our gravity approach: J.E., Flory, Hoyos, Newrzella, O’Bannon, Wu 1511.03666
27
![Page 43: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/43.jpg)
Entanglement entropy for magnetic impurity: Comparison to field theory
On gravity side:
Impurity entropy from difference of entanglement entropies for constant tensionbranes
∆Simp(`) = c0 + SBH(`+D)− SBH(`)
SBH(`) =c
3ln
(1
πεTsinh(2π`T )
)
![Page 44: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/44.jpg)
Entanglement entropy for magnetic impurity: Comparison to field theory
On gravity side:
Impurity entropy from difference of entanglement entropies for constant tensionbranes
∆Simp(`) = c0 + SBH(`+D)− SBH(`)
SBH(`) =c
3ln
(1
πεTsinh(2π`T )
)
For D � `:
∆Simp(`) ∼ c0 +D · ∂`SBH(`) = c0 +2πDT
3coth(2π`T )
Agrees with field theory result subject to identification D ∼ ξK
28
![Page 45: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/45.jpg)
Universal properties of superconductors
Universality: IR fixed point determines physical propertiesMacroscopic properties do not depend on microscopic degrees of freedom
![Page 46: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/46.jpg)
Universal properties of superconductors
Universality: IR fixed point determines physical propertiesMacroscopic properties do not depend on microscopic degrees of freedom
Example: Universal result from gauge/gravity duality:
Shear viscosity over entropy density:η
s=
1
4π
~kB
![Page 47: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/47.jpg)
Universal properties of superconductors
Universality: IR fixed point determines physical propertiesMacroscopic properties do not depend on microscopic degrees of freedom
Example: Universal result from gauge/gravity duality:
Shear viscosity over entropy density:η
s=
1
4π
~kB
Planckian dissipator: relaxation time τ = ~kBT
Damle, Sachdev 1997
29
![Page 48: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/48.jpg)
Universal properties of superconductors
Is there a similiar universal result for applications of the duality within condensedmatter physics?
30
![Page 49: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/49.jpg)
Universal properties of superconductors
Candidate: Homes’ relationρs(T = 0) = C σDC(Tc)Tc
C. Homes et al, Nature 2004
31
![Page 50: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/50.jpg)
Universal properties of superconductors
Homes’ relation ρs(T = 0) = C σDC Tc
general form may be deduced from Planckian dissipation Zaanen 2004
![Page 51: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/51.jpg)
Universal properties of superconductors
Homes’ relation ρs(T = 0) = C σDC Tc
general form may be deduced from Planckian dissipation Zaanen 2004
J.E., Herwerth, Klug, Meyer, Schalm arXiv:1501.07615:
Investigation of C in a family of gauge/gravity duality models
In particular region of parameter space:
C ≈ 6.2
High-Tc in (ab)-plane and BCS superconductors in ‘dirty limit’: C = 8.1,High-Tc superconductors in c-plane: C = 4.4
32
![Page 52: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/52.jpg)
Universal properties of superconductors
Holography: J.E., Kerner Muller 2012
Conditions for identifying ρs:
![Page 53: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/53.jpg)
Universal properties of superconductors
Holography: J.E., Kerner Muller 2012
Conditions for identifying ρs:
Translation symmetry broken ⇒ Drude peak
![Page 54: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/54.jpg)
Universal properties of superconductors
Holography: J.E., Kerner Muller 2012
Conditions for identifying ρs:
Translation symmetry broken ⇒ Drude peak
Applicability of sum rules:
All normal state degrees of freedom condense at T = 0
![Page 55: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/55.jpg)
Universal properties of superconductors
Holography: J.E., Kerner Muller 2012
Conditions for identifying ρs:
Translation symmetry broken ⇒ Drude peak
Applicability of sum rules:
All normal state degrees of freedom condense at T = 0
Weak momentum relaxation is not enough
Horowitz, Santos 2013
![Page 56: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/56.jpg)
Universal properties of superconductors
Holography: J.E., Kerner Muller 2012
Conditions for identifying ρs:
Translation symmetry broken ⇒ Drude peak
Applicability of sum rules:
All normal state degrees of freedom condense at T = 0
Weak momentum relaxation is not enough
Horowitz, Santos 2013
Use background with helical symmetry
33
![Page 57: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/57.jpg)
Universal properties of superconductors
Background: Helical Bianchi VII symmetryDonos, Gauntlett 2011; Donos, Hartnoll 2012
Model with broken translation symmetry:
ω2-one form fieldin yz-plane
x
34
![Page 58: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/58.jpg)
Gauge/gravity duality with helical symmetry
Background: (Hartnoll, Donos)
35
![Page 59: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/59.jpg)
S-wave superconductivity in helical symmetry background
![Page 60: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/60.jpg)
S-wave superconductivity in helical symmetry background
Add charged scalar:
![Page 61: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/61.jpg)
S-wave superconductivity in helical symmetry background
Add charged scalar:
All charged degrees of freedom condense at T = 0
36
![Page 62: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/62.jpg)
Universal properties of superconductors
Phase diagram Tc as function of helix frequency
37
![Page 63: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/63.jpg)
σDCTc
ρs
Homes’ relation for q = 6 & κ = 0
p/µ
ρs
σD
CTc
Homes’ constant @ κ = 0
1.8
2.4
4.2
3.8
4.24.2
4.4
J.E., Herwerth, Klug, Meyer, Schalm 1501.07615
38
![Page 64: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/64.jpg)
Homes’ constant and comparison with phase diagram in normal phase
1 2 3 4
1
2
3
4
5
6
J.E., Meyer, Schalm, Shock in progress
Homes’ relation holds in region of phase diagram which is insulating(incoherent metal) in normal phase
39
![Page 65: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/65.jpg)
Sum rules
BCS Holography for helical background
![Page 66: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/66.jpg)
Sum rules
BCS Holography for helical background
(related work by Kim, Kim, Park)
40
![Page 67: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/67.jpg)
Conclusions and outlook
Kondo model:
Magnetic impurity coupled to strongly coupled system
Quantum quench
Entanglement entropy
Two-point and spectral functions
S-wave superconductor in Bianchi VII background:
Homes’ Relation
41
![Page 68: Magnetic impurities and universality in AdS/CMT Johanna ... fileRG flow from perturbation by ‘double-trace’ operator Dynamical scale generation Holographic superconductor: Condensate](https://reader031.vdocument.in/reader031/viewer/2022011811/5e1769720e46a70b7702b34c/html5/thumbnails/68.jpg)
Advertising our book
42