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Holographic defect entropies and information flows

Johanna Erdmenger
Max–Planck–Institut für Physik, München1Outline

Kondo models from holography– ModelJ.E., Hoyos, O’Bannon, Wu 1310.3271, JHEP 1312 (2013) 086– Entanglement entropy

J.E., Flory, Newrzella 1410.7811, JHEP 1501 (2015) 058
J.E., Flory, Hoyos, Newrzella, O’Bannon, Wu 1511.03666, Fortsch.Phys. 64 (2016)– Two-point functions J.E., Hoyos, O’Bannon, Papadimitriou, Probst, Wu in progress– Quantum quenches

J.E., Flory, Newrzella, Wu in progress2

Kondo models from gauge/gravity duality

Kondo models from gauge/gravity duality

Kondo effect:
Screening of a magnetic impurity by conduction electrons at low temperatures

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:

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. Simple model for a RG flow with dynamical scale generation (as in QCD)
2. Example for holographic g-theorem
3. Relation to Sachdev-Ye-Kitaev model
4. New applications of gauge/gravity duality to condensed matter physics:
Magnetic impurity coupled to strongly correlated electron system
Entanglement entropy, quantum quench
3

Kondo effect4

Kondo modelKondo model

Original Kondo model (Kondo 1964):
Magnetic impurity interacting with free electron gas

Kondo model

Original Kondo model (Kondo 1964):
Magnetic impurity interacting with free electron gas

Hamiltonian:
H=vF †~ · J~ ,ψ i∂xψ + λK vF δ(x)S2π1J~ = ψ † T~ ψ2

Decisive in development of renormalization groupIR fixed point, CFT approach Affleck, Ludwig ’90’s

5

Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU (N )Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU (N )In this case, interaction term simplifies introducing slave fermions:
S a = χ†T aχ
Totally antisymmetric representation: Young tableau with Q boxes
Constraint: χ†χ = Q

Kondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU (N )In this case, interaction term simplifies introducing slave fermions:
S a = χ†T aχ
Totally antisymmetric representation: Young tableau with Q boxes
Constraint: χ†χ = Q
Interaction: J aS a = (ψ †T aψ)(χ†T aχ) = OO†, where O = ψ †χKondo models from gauge/gravity duality
Gauge/gravity requires large N : Spin group SU (N )In this case, interaction term simplifies introducing slave fermions:
S a = χ†T aχ
Totally antisymmetric representation: Young tableau with Q boxes
Constraint: χ†χ = QInteraction: J aS a = (ψ †T aψ)(χ†T aχ) = OO†, where O = ψ †χScreened phase has condensate hOi
Parcollet, Georges, Kotliar, Sengupta cond-mat/9711192, PRB 58 (1998) 3794
Senthil, Sachdev, Vojta cond-mat/0209144, PRL 90 (2003) 216403

6

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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 liquidResults:
RG flow from perturbation by ‘double-trace’ operator
Dynamical scale generation, screening
Holographic superconductor: Condensate forms in AdS2Power-law scaling of conductivity in IR with real exponent
Holographic entanglement entropy from including backreaction
Quantum quench: Late-time behaviour dominated by quasinormal modes7

Kondo models from gauge/gravity duality
J.E., Hoyos, O’Bannon, Wu 1310.3271, JHEP 1312 (2013) 086

Kondo models from gauge/gravity duality
J.E., Hoyos, O’Bannon, Wu 1310.3271, JHEP 1312 (2013) 086

Top-down brane realization
N D3
N7 D7
N5 D50X
X
X

1X
X

2X

3X

45

67

8

9X
X

X
X

X
X

X
X

X
X

X

3-7 strings: Chiral fermions ψ in 1+1 dimensions
3-5 strings: Slave fermions χ in 0+1 dimensions
5-7 strings: Scalar (tachyon)
8Near-horizon limit and field-operator map

D3: AdS5 × S 5
D7: AdS3 × S 5 → Chern-Simons Aµ dual to J µ = ψ †σ µψ†YMadualtoχ
χ=qt
4
D5: AdS2 × S →Scalar dual to ψ †χOperator
Electron current JCharge Q = χ†χOperator O = ψ †χ⇔⇔⇔Gravity field
Chern-Simons gauge field A in AdS32d gauge field a in AdS22d complex scalar Φ9

Bottom-up gravity dual for Kondo model
Action:S = SEinstein−Hilbert + SCS + SAdS2 ,Z2NTr A ∧ dA + A ∧ A ∧ A ,SCS = −4π
3AdS3Z√1†
SAdS2 = −N d3x δ(x) −g Trf mnfmn + g mn (DmΦ) DnΦ + V (Φ†Φ)
4V (Φ) = M 2Φ†Φ

Metric:ds2 = gµν dxµdxν =2h(z) = 1 − z 2/zH,

1z2



2



dz− h(z) dt2 + dx2 ,h(z)T = 1/(2πzH )10‘Double-trace’ deformation by OO†

Boundary expansion
Φ = z 1/2(α ln z + β)
α = κβκ dual to double-trace deformation

Witten hep-th/0112258

‘Double-trace’ deformation by OO†

Boundary expansion
Φ = z 1/2(α ln z + β)
α = κβκ dual to double-trace deformation

Witten hep-th/0112258

Φ invariant under renormalization ⇒ Running coupling
κ0

κT =
Λ
1 + κ0 ln 2πT‘Double-trace’ deformation by OO†

Boundary expansion
Φ = z 1/2(α ln z + β)
α = κβκ dual to double-trace deformation

Witten hep-th/0112258

Φ invariant under renormalization ⇒ Running coupling
κ0

κT =
Λ
1 + κ0 ln 2πT
Dynamical scale generation
11

Kondo models from gauge/gravity duality
Scale generation

Divergence of Kondo coupling determines Kondo temperature TKTransition temperature to phase with condensed scalar: Tc
Tc < TK
12

Kondo models from gauge/gravity duality

RG flow

UV Strongly interacting
electrons

Deformation by
Kondo operator

Strongly interacting
electronsIR
Non-trivial condensate

13

Kondo models from gauge/gravity duality

Normalized condensate hOi ≡ κβ as function of the temperature

(a)

(b)

Mean field transition
hOi approaches constant for T → 014

Entanglement entropy for magnetic impurity(see poster by Mario Flory)

Including the backreaction using a thin brane and Israel junction conditions
Israel junction conditions Kµν − γµν K = − κ2 Tµν ⇔ Energy conditions
boundaryboundary

hypersurface
bulkbulk

identify points
J.E., Flory, Newrzella 1410.7811
Cf. previous work on holographic BCFT
Takayanagi; Fujita, Takayanagi, Tonni 2011; Nozaki, Takayanagi, Ugajin 201215

Entanglement entropy for magnetic impurity
AdS

B

AImpurity entropy:
Simp = Scondensed phase − Snormal phase

Subtraction also guarantees UV regularity

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Entanglement entropy for magnetic impurity J.E., Flory, Newrzella 1410.7811

Depending on the brane tension λ, the total space is enhanced or reduced

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Entanglement entropy for magnetic impurity J.E., Flory, Newrzella 1410.7811

0.0T/TCz/zH

0.2

01

0.4
0.6
0.8
1.0

- 0.06- 0.05- 0.04- 0.03x+ (z)/zH- 0.02- 0.01

0.00

The larger the condensate, the shorter the geodesic

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Impurity entropy from gauge/gravity dualityℒ imp=6 S imp/c
0.065
0.060
0.055
0.050
0.045
0.040

01

2

34ℓ/zH

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Entanglement entropy for magnetic impurity: Comparison to field theory
Field-theory result

Sorensen, Chang, Laflorencie, Affleck 2007
(Eriksson, Johannesson 2011)

π 2 ξK T
Simp(`) =coth(2π`T )6Entanglement entropy for magnetic impurity: Comparison to field theory
Field-theory result

Sorensen, Chang, Laflorencie, Affleck 2007
(Eriksson, Johannesson 2011)

π 2 ξK T
Simp(`) =coth(2π`T )6In our gravity approach:

J.E., Flory, Hoyos, Newrzella, O’Bannon, Wu 1511.03666

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Entanglement entropy for magnetic impurity: Comparison to field theory
On gravity side:
Impurity entropy from difference of entanglement entropies for constant tension
branesSimp(`) = SBH (` + D) − SBH (`)
c1
SBH (`) = lnsinh(2π`T )3πTEntanglement entropy for magnetic impurity: Comparison to field theory
On gravity side:
Impurity entropy from difference of entanglement entropies for constant tension
branesSimp(`) = SBH (` + D) − SBH (`)
c1
SBH (`) = lnsinh(2π`T )3πTFor D  `:2πDTSimp(`) ∼ D · ∂`SBH (`) =coth(2π`T )3Agrees with field theory result subject to identification D ∼ ξK21Kondo model: Example for holographic g-theorem
Formation of Kondo cloud corresponds todecrease in impurity degrees of freedomcSimp(` → ∞) = − x̃+(zH )3x̃+(z): Defect embedding scalarg-theorem:T·∂Simp(` → ∞)≥ 0∂TDue to null energy condition

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Relation to Sachdev-Ye-Kitaev model
Sachdev-Ye-Kitaev model: Gaussian random couplings Jαβ,γδSachdev+Ye 1993, Kitaev 2015

NXX †1††H =χαχαJαβ,γδ χαχβ χγ χδ − µ3/2(2N ) α,β,γ,δ=1αMay be obtained from two-dimensional model as follows:(Bray, Moore J. Phys. C 1980; Georges, Parcollet, Sachdev PRB 63 92001)
Reduction to single site by averaging over disorder

HS = −X~i · S~jJij S(ij)J2Seff = −2NZ β000~ ) · S(τ~ 0) ,dτ dτ Q(τ − τ )S(τ1 ~~ 0)iQ(τ − τ ) = 2 hS(τ )S(τN0Use Abrikosov fermions χ as before, S a = χ†T aχ, and take large N limit(see also Maldacena, Stanford arXiv:1604.07818)
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Relation to Sachdev-Ye-Kitaev model

Similarly in Parcollet, Georges, Kotliar, Sengupta cond-mat/9711192:
Reduction of large N -Kondo model to single-site modelby integrating out conduction electrons

Relation to Sachdev-Ye-Kitaev model

Similarly in Parcollet, Georges, Kotliar, Sengupta cond-mat/9711192:
Reduction of large N -Kondo model to single-site modelby integrating out conduction electrons⇒ Spectral asymmetry of Green’s functions

Relation to Sachdev-Ye-Kitaev model

Similarly in Parcollet, Georges, Kotliar, Sengupta cond-mat/9711192:
Reduction of large N -Kondo model to single-site modelby integrating out conduction electrons⇒ Spectral asymmetry of Green’s functions

Sachdev 1506.05111, Phys. Rev. X 5, 041025 (2015):
Spectral asymmetry also observed in SYK model
related to entropy of AdS2 black holeωs =qT ∂S~ ∂Q

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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 hOO †i∂Sωs = qT~ ∂Q

see also Sachdev 1506.05111, AdS2 black hole (fermions)25Two-point function in holographic Kondo model

J.E., Hoyos, O’Bannon, Papadimitriou, Probst, Wu in progress
Reduction to single-site automatic since equation of motion for 3d Chern-Simons field decouples
from EOM’s for 2d fields
Im <O† O>
ω-105-5

10TK-1

T/TK = 3.3978, zH = 0.8-2-3

T/TK = 2.7183, zH = 1.0
T/TK = 1.8122, zH = 1.5
T/TK = 1.3591, zH = 2.0-4-526

Time dependence

Allow for time dependence of the Kondo coupling and study response of the condensate

Time dependence

Allow for time dependence of the Kondo coupling and study response of the condensate
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)Time dependence

Allow for time dependence of the Kondo coupling and study response of the condensate
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 asymptotically small or large
Timescales governed by quasinormal modes

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28

1086Quantum quenches inholographic Kondo model4

To and from condensed phase
2050

100

150

200

Timescales determined byquasinormal modes

0.14
0.12
0.10
0.08

J.E., Flory, Newrzella, Strydom, Wu

0.06
0.04
0.02
0.00

50

100

150

200

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Timescales in quantum quench

500

1000

1500

2000

2500

3000

0.002

0.004

0.006

0.008

Unstable
QNM

10.0

Separatetime scales9.5

9.0

8.5
500

1000

1500

2000

2500

3000

30

Timescales in quantum quench

10203040506051015202530

2.0

Quasinormal
ringing

1.5

1.0Quench timescale

0.5

0.0

10203040506031

Timescales in quantum quench

Equilibration determined by quasinormal modes, which depend on T
Cf. Bayat, Bose, Johannesson, Sodano Phys. Rev. B 92, 155141 (2015) :Quench in two-impurity Kondo model in spin-chain approach:Late-time behaviour dominated by single-frequency oscillations,independent of energy released

32

Conclusions and outlook

Kondo model:
Magnetic impurity coupled to strongly coupled system
Entanglement entropy– In agreement with g-theorem– Reproduces large N field theory result for large `– Geometrical realization of Kondo correlation length
Two-point functions– Spectral asymmetry– Relation to SYK model
Quantum quenches– Dominated by quasinormal modes
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