Yet another talk about the Holographic Model of the Quantum Hall - - PowerPoint PPT Presentation

Yet another talk about the Holographic Model of the Quantum Hall Effect

Matt Lippert (Crete)

With Niko Jokela (Technion & Haifa) & Matti Järvinen (Crete)

References

D3-D7’

• O. Bergman, G. Lifschytz, N. Jokela, MSL

Quantum Hall effect in a holographic model JHEP 1010 (2010) 063; arXiv:1003.4965 [hep-th]

• G. Lifschytz, N. Jokela, MSL

Magneto-roton excitation in a holographic quantum Hall fluid JHEP 1102 (2011) 104; arXiv: 1012.1230 [hep-th]

• O. Bergman, G. Lifschytz, N. Jokela, MSL

Striped Instability of a holographic Fermi-like liquid JHEP 10 (2011) 034; arXiv: 1106.3883 [hep-th]

D2-D8’

• N. Jokela, M. Järvinen, MSL

A holographic quantum hall model at integer filling JHEP 1105 (2011) 101; arXiv: 1101.3329 [hep-th]

• N. Jokela, M. Järvinen, MSL

Fluctuations of a holographic integer quantum Hall fluid arXiv: 1107.3836 [hep-th]

Outline

• 1. Introduction
• Motivation
• QHE review
• 2. D2-D8’ Model Construction
• 3. Embeddings and Conductivity
• 4. Phase Diagram
• 5. Fluctuations
• Normal Modes - Rotons
• Quasi-Normal Modes - Instability
• 6. Summary and Open Questions

Holographic fermions

Many phenomena involve strongly-coupled fermions:

• Chiral Symmetry Breaking
• Quantum Critical Points
• Fractional quantum Hall effect (FQHE)
• …

But interesting = difficult

Top-Down Holographic Approach:

• study concrete string models
• known field theory duals
• gives new effective theories
• look for universal features

Quantum Hall Effect (QHE)

e-

B

Experimental Setup:

e- in 2+1 d high magnetic field B low temperature T

Conductivity

Longitudinal: Hall: Filling fraction

Filling Fraction

ν ∈ Z Integer QHE ν ∈ Z Fractional QHE

QH states for particular values of ν

Open questions:

• microscopic description
• allowed ν’s
• Transitions between

# electrons # flux quanta ~

Dp-Dq Models

Two other examples:

• D3-D7’ Model

2+1-dim fermions, 3+1-dim gauge FQHE, ν = irrational, set by internal flux

• D2-D8’ Model

2+1-dim gauge + fermions IQHE, ν = 1 ∀ internal flux ≠ 0 Focus of this talk

Brane intersections with #ND=6

• fundamental fermions at intersection
• Dq probe in Dp background
• SUSY stability?

Example: Sakai Sugimoto model: D4-D8-D8

Rey Kraus et al Myers et al Hong & Yee …

D2-D8′ system

N D2 0 1 2 r ψ S2 S3

2+1 Mink. S6

D8 D8-brane embedding: wraps S2 × S3 ⊂ S6 solve for ψ(r) Stabilization lowest mode for ψ tachyonic (slipping mode) wrap magnetic flux on internal S2

Embeddings

Black Hole (BH)

r cos ψ r sin ψ

rT

Minkowski (MN)

r cos ψ r sin ψ

r0 D8 enters horizon D8 ends where S3 shrinks m m

Add charges and magnetic field

Charge density Magnetic field Chern-Simons

where C5 flux and B induce charge

Where’s the charge?

Charge density D radial displacement flux d(r)

r cos ψ r sin ψ

Total charge density: D = d∞ Split between two types: 1. Induced charge: Bc(rmin) 2. Ordinary charge: D - Bc(rmin) d

Induced Ordinary

Black Hole Embeddings

Metallic state

• gapless charged excitations
• conductivity (via Karch-O’Bannon)
• σxx ≠ 0
• σxy ≠ 0 even for B = 0 AHE

r cos ψ r sin ψ

spikey soln Solutions become spikey as

filling fraction per fermion

QH state

• no sources at tip, all charge induced
• ν = 1 independent of internal flux
• gap for charged excitations mg ~ r0
• conductivity
• σxx = 0
• σxy = ν/2π

Minkowski Embeddings

r cos ψ r sin ψ

r0

Phase Diagram

T B

nonspikey BH spikey BH spikey BH Critical Point MN 1st order

• Fixed D
• Phase Transition

MN/spikey BH nonspikey BH 1st order, end in critical point

2π D/N

MN to spikey BH transition

MN spikey BH is

• 1st order for B increasing
• At least 2nd order for B decreasing

MN solution indep. of µ CE ill-defined Use GCE, where d(µ) d µ ≥ 2nd order 1st order

Fluctuation Analysis

in four easy steps: 1. Choose better coordinates ρ(R) 2. Perturb fields, choose wavelike ansatz 3. Expand Coupled 2nd order ODEs

• Very long and gross
• Normalizability δρ = δaµ = 0 in UV
• Solve by shooting from IR
• Use determinant method

4. Solutions

• QH fluid (MN) normal modes
• Metal (BH) quasi-normal modes

Normal mdes of QH fluid

Neutral Spectrum

• gapped
• k=0, scalars (red) and

vectors (blue) decouple

• level crossings

ω0 T

Unstable branch

BH MN 2 MN solutions Above phase transition, two MN solutions 1. metastable 2. unstable lowest mode tachyonic Phase transition

T

ω0

2

T B

Dispersion

Magneto-Roton Dispersion

• ω* < ω0 at k* > 0
• lower mode near level crossings
• quasiparticle-quasihole dipole
• seen in experiments

e.g. for ν = 1/3 Hirjibehedin et al.

cond-mat/0407145

(k*, ω*) k ω Massive dispersion

• generic
• speed of sound cs indep. of mode #

k ω

Quasinormal Modes of BH

Longest-lived Mode

• Diffusive, hyrodynamical mode at small ω
• Zero sound (modified for T > 0)

Re ω Im ω k Zero-sound Hydro. ω

Collisionless Diffusion

NOP Instability

Maxwell-Axion theory in (3+1) dim Perturb around background F03 =E Linearized EOM + Bianchi give Plane wave ansatz dispersion relation Tachyonic for

NOP for D2-D8

For small enough T, Instability for kmin < k < kmax

Im ω

k Unstable True ground state Charge/spin density wave?

Summary

Top-down models of QHE Features:

• Quantized ν
• Mass gap
• Conductivities
• Fluctuations

Bugs:

• Only one QH state per model
• Limited choice of ν

Open Questions

QHE features

• multiple filling fractions & transitions
• impurities, plateaux
• boundaries, edge states
• connect to bottom-up models

e.g. Lee et al. arXiv/1008.1917

Modulated instability

• dependence on B
• striped ground state