Entanglement entropy in a holographic model .2cm of the ... · Geometric equations of a similar...

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Entanglement entropy in a holographic model of the Kondo effect Mario Flory Max-Planck-Institut f¨ ur Physik Gauge/Gravity Duality 2015 The Galileo Galilei Institute, Florence Mario Flory Entanglement entropy & Kondo 1 / 16

Transcript of Entanglement entropy in a holographic model .2cm of the ... · Geometric equations of a similar...

Page 1: Entanglement entropy in a holographic model .2cm of the ... · Geometric equations of a similar form as Einstein equations, extrinsic curvature tensors (K+ ij) instead of intrinsic

Entanglement entropy in a holographic model

of the Kondo effect

Mario Flory

Max-Planck-Institut fur Physik

Gauge/Gravity Duality 2015

The Galileo Galilei Institute, Florence

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Overview

Part I: The holographic Kondo model

I The Kondo effect

I Bottom up bulk model

Part II: Including backreaction

I Israel junction conditions

I General results for AdS3/BCFT2

I Including Chern-Simons fields

Part III: Entanglement entropy for Kondo model

I Numerical results

I Qualitative discussion

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Part I: The holographic Kondo model

Field theory side:

I Spin-spin interaction of electrons with a localised magnetic impurity.

I Can be mapped to 1 + 1 dimensional conformal system [Affleck et. al.

1991].

I At low temperature, electrons form a bound state around impurity, the

Kondo cloud .

Holographic gravity side: [Erdmenger et. al.: 1310.3271]

I Dual gravity model has 2 + 1 (bulk-) dimensions.

I Localised spin impurity is represented by co-dimension one hypersurface

(”brane”) extending from boundary into the bulk.

I Finite T is implemented by BTZ black hole background.

I Kondo cloud is described by condensation of scalar field Φ.

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The holographic Kondo model

S = SCS [A]−∫

d3xδ(x)√−g(14 f mnfmn + γmn(DmΦ)†DnΦ + V (Φ†Φ)

)Mario Flory Entanglement entropy & Kondo 4 / 16

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The holographic Kondo model

How can we obtain information about the Kondo cloud from our model?

Kondo cloud is formed by anti-aligned spins

⇒ expect imprint on entanglement entropy SEE , e.g. entanglement of state

|Ψ〉 = 1N (|↑↑↑ ↓↓ ...〉 − |↓↓↓ ↑↑ ...〉) does not vanish.

SEE is determined by spacelike geodesics [Ryu, Takayanagi, 2006]

⇒ to calculate it, we need backreaction on the geometry.

What is the backreaction of an infinitely thin hypersurface carrying

energy-momentum? Israel junction conditions!

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Part II: Including backreaction

In electromagnetism: To describe field around an infinitely thin charged surface

Σ, integrate Maxwells equations in a box around Σ:

⇒ ~E|| continuous, ~E⊥ discontinuous on Σ

In gravity: To describe backreaction of an infinitely thin massive surface,

integrate Einsteins equations in a box

⇒ Israel junction conditions [Israel, 1966]:

(K+ij − γijK

+)− (K−ij − γijK−) = −κSij

Sij : energy momentum tensor on the brane, γij : induced metric,

K±: extrinsic curvatures depending on embedding.

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Israel junction conditions

With mirror symmetry (K+ = −K−): K+ij − γijK+ = −κ2 Sij (∗)

⇒ Embedding (location of the brane) will not be x ≡ 0 anymore, but a dynamical

function x(z) with (∗) its own equations of motion.

identify points

boundary boundary

hypersurface

bulkbulk

With (∗) we arrive at a general setting for the study of AdS/boundary CFT

correspondence proposed by Takayanagi et. al.: [Takayanagi 2011, Fujita et. al.

2011, Nozaki et. al. 2012].

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Israel junction conditions

K+ij − γijK+ = −κ2 Sij

curvature = energy momentum

Geometric equations of a similar form as Einstein equations, extrinsic curvature

tensors (K+ij ) instead of intrinsic ones (Rµν).

General questions:

Impact of energy conditions on possible geometries?

Find exact solutions for simple toy models of Sij?

Investigate Kondo model?

Answers in [Erdmenger, M.F., Newrzella: 1410.7811].

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Possible geometries

Utilising the barrier theorem [Engelhardt, Wall: 1312.3699], we can constrain the

possible geometries allowed by different energy conditions.

Whether or not a brane Q bends back to the boundary or goes deep into the bulk

depends on whether Sij satisfies or violates WEC and SEC.

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Junction conditions for Chern-Simons field

Our Kondo model contains both the metric field and a Chern-Simons field in

the bulk. Assume CS field to be U(1) in simplest case.

Similarly to the metric, we get junctions conditions for the CS field along the

hypersurface Q (located at η ≡ 0) if it carries a current in its worldvolume.

Split up field: A ∼ θ(η)A+ + θ(−η)A− + δ(η)A0.

Q

A

A0

+

A-

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Junction conditions for Chern-Simons field

Q

A

A0

+

A-

With Dm ≡ (A+||m + A−||m )/2 (projected mean value),

Cm ≡ A+||m − A−||m (projected discontinuity),

and A0µ = A0nµ (component localised on Q is normal)

we find: εim(Ci + ∂iA

0)

= 2πJm [γ,Φ, a,D]

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Part III: Entanglement entropy for Kondo model

Sbrane [am,Φ] = −∫

dVbrane

(14 f mnfmn + γmn(DmΦ)†DnΦ + V (Φ†Φ)

)Due to Yang-Mills field am, SEC is violated everywhere in the bulk.

Hence brane starts at boundary and falls into black hole, does not turn

around and bend back to boundary.

Preliminary numerical results:

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Numerical resultsPreliminary results on entanglement entropy: Difference of SEE (`) relative to

solution with Φ ≡ 0.

0.5 1.0 1.5 2.0 2.5

0.30

0.25

0.20

0.15

0.10

0.05

0.00

[Erdmenger, M.F., Hoyos, Newrzella, O’Bannon, Wu: work in progress]

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DiscussionSome of the features of these results follow directly from the energy conditions

and geometric considerations.

Entanglement entropy for given ` decreases as Kondo cloud forms, because

Φ satisfies NEC, brane bends to the right.

As `→∞, curves go to a constant.

The fall-off towards this constant value is for large ` exponential, due to

simple geometric arguments:

∆SEE (`)∼−−→ c0 + c1(T )T

(1 + 2e−4π`T + ...

)Qualitative agreement with results of field theory calculations [Affleck et. al.

2007, 2009; Eriksson, Johannesson 2011]:

∆SEE (`) = c0 +π2ξKT

6vcoth

(2π`T

v

)→ c0 +

π2ξKT

6v

(1 + 2e−

4π`Tv + ...

)v : Fermi velocity, ξK : Kondo scale

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Summary and Outlook

We studied a holographic model of the Kondo effect.

Gravity dual involves thin brane carrying energy-momentum.

Backreaction of the brane is described by Israel junction conditions.

We obtained general results constraining possible geometries of the brane by

energy conditions [Erdmenger et. al. 1410.7811].

These results may also be applicable to holographic duals of BCFTs

[Takayanagi, 2011] or the Hall effect [Melnikov et. al, 2012] involving thin

branes.

Specific Kondo model will be solved numerically, results on entanglement

entropy can be compared to field theory literature.

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Thank you for your attention

Munich 2013 Florence 2015

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Back up slides...

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Entanglement entropy

Entanglement entropy SEE (A) defines the entropy of a subsystem A with respect

to the total system A ∪ B.

SEE (A) = −TrA[ρA log(ρA)]

with reduced density matrix ρA ≡ TrB [ρA∪B ].

[see e.g. Nielsen, Chuang: Quantum Computation and Quantum Information]

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Holographic entanglement entropyIn AdS/CFT correspondence: bulk spacetime M, boundary ∂M.

SEE (A) = Area(EA)4GN

where EA is a spacelike extremal surface in the bulk.

→ Generalisation of Bekenstein-Hawking entropy formula

[Ryu, Takayanagi, 2006]

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A simple form

Our brane has 1 + 1 dimensions, hence there are only two distinct null directions.

Define a basis of symmetric (0,2)-tensors:

Sij ≡S

2γij + SLli lj + SR ri rj = trace part + traceless parts

Static case: no energy flux from left to right, hence SL = SR ≡ SL/R .

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A simple form

Sij ≡S

2γij + SLli lj + SR ri rj = trace part + traceless parts

Doing this decomposition on both sides, the tensorial equation

K+ij − γijK+ = −κ2 Sij

becomes the set of scalar equations

K = κ2 S , KL = κ

2 SL, KR = κ2 SR

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Energy conditions

Null energy condition (NEC)

Sijmimj ≥ 0 ∀ mimi = 0 ⇒ SL,SR ≥ 0

Weak energy condition (WEC)

Sijmimj ≥ 0 ∀ mimi < 0⇒ SL,SR ≥ 0, S ≤ 2

√SLSR

Strong energy condition (SEC)

(Sij − Sγij)mimj ≥ 0 ∀ mimi < 0 ⇒ SL,SR ≥ 0, S ≥ −2√

SLSR

This SEC will be of much phenomenological importance.

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Energy conditions and conservation of energy momentum

In the static case (SL = SR ≡ SL/R), SEC reads:

SL,SR ≥ 0, S + 2SL/R ≥ 0.

Energy-momentum conservation ∇iSij = 0 implies for embeddings in Poincare

background:

∂z(S + 2SL/R

)=

4

zSL/R .

By NEC, the right hand side is positive, hence S + 2SL/R can only grow with z .

When NEC holds and the SEC is satisfied near the boundary z = 0, it is satisfied

everywhere in the bulk.

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Possible geometries

Barrier Theorem (Engelhardt, Wall arXiv:1312.3699)

Let Q be a hypersurface splitting the spacetime N in two parts N± with

boundaries M± such that K+ij v iv j ≤ 0 for any vector field v i on Q. Then any

spacelike extremal surface Υ which is anchored in M+ remains in N+.

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Possible geometriesWith the junction conditions, we can express the assumption made in the barrier

theorem in terms of energy conditions:

WEC and SEC satisfied on Q ⇒ K+ij v iv j ≤ 0 ∀v i

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Possible geometries

Whether or not a brane Q bends back to the boundary or goes deep into the bulk

depends on whether Sij satisfies or violates WEC and SEC.

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Exact analytical solutions

We first studied simple models for Sij and obtained some exact analytical

solutions to the junction conditions for:

Perfect fluids:

Sij = (ρ+ p)uiuj + pγij with p = a · ρ, a ∈ R.

As the special case thereof with a = 1: The free massless scalar φ with

Sij = ∂iφ∂jφ−1

2γij(∂φ)2.

The U(1) Yang-Mills field ai in the absence of sources:

Sij = −1

4f mnfmnγij + γmnfmi fnj = −1

2γijC

2.

All of these were studied in AdS and BTZ backgrounds.

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Exact analytical solutionsFor the free massless scalar φ with Sij = ∂iφ∂jφ− 1

2γij(∂φ)2, we obtain

x(z) =cz3

32F1

(1

2,

3

4;

7

4; c2z4

).

with 2F1(a, b; c; d) the hypergeometric function. WEC and SEC are satisfied,

hence the brane bends back to the boundary.

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