Why condensed matter at all?

Outline Motivation Quantum Hall Eect The Model Conclusion

AdS/QHE

A holographic approach to describing quantum Hall plateaux transitions

Allan Bayntun

National Taiwan University Taipei, Taiwan

December 31, 2010

Outline Motivation Quantum Hall Eect The Model Conclusion

Motivation

Strongly Correlated Electron Systems Quantum Hall Eect

QHE Properties Symmetries of QHE The Model

Quantum Hall-ography Conclusion

Outline Motivation Quantum Hall Eect The Model Conclusion

Why condensed matter at all?

• Many recent outstanding problems in condensed matter are related to strongly correlated electron systems

• Any theoretical tool capable of calculating quantities in strongly interacting systems (of real electrons) would be of great value to these problems

• Moreover, many experiments (and theories) in condensed matter are interested in critical phenomena of these systems

• These are very close to CFTs, and so the AdS/CFT dictionary could be the right tool

Outline Motivation Quantum Hall Eect The Model Conclusion

Why condensed matter at all?

• Many recent outstanding problems in condensed matter are related to strongly correlated electron systems

• Any theoretical tool capable of calculating quantities in strongly interacting systems (of real electrons) would be of great value to these problems

• Moreover, many experiments (and theories) in condensed matter are interested in critical phenomena of these systems

• These are very close to CFTs, and so the AdS/CFT dictionary could be the right tool

Outline Motivation Quantum Hall Eect The Model Conclusion

Why condensed matter at all?

• Many recent outstanding problems in condensed matter are related to strongly correlated electron systems

• Any theoretical tool capable of calculating quantities in strongly interacting systems (of real electrons) would be of great value to these problems

• Moreover, many experiments (and theories) in condensed matter are interested in critical phenomena of these systems

• These are very close to CFTs, and so the AdS/CFT dictionary could be the right tool

Outline Motivation Quantum Hall Eect The Model Conclusion

Why condensed matter at all?

• Many recent outstanding problems in condensed matter are related to strongly correlated electron systems

• Any theoretical tool capable of calculating quantities in strongly interacting systems (of real electrons) would be of great value to these problems

• Moreover, many experiments (and theories) in condensed matter are interested in critical phenomena of these systems

• These are very close to CFTs, and so the AdS/CFT dictionary could be the right tool

Outline Motivation Quantum Hall Eect The Model Conclusion

Hopes and Dreams of AdS/CMT

Graphene

Quantum Critical Theories

High-Tc Superconductivity

Quantum Hall Eect

Outline Motivation Quantum Hall Eect The Model Conclusion

Hopes and Dreams of AdS/CMT

Graphene

Quantum Critical Theories

High-Tc Superconductivity

Quantum Hall Eect

Outline Motivation Quantum Hall Eect The Model Conclusion

Hopes and Dreams of AdS/CMT

Graphene

Quantum Critical Theories

High-Tc Superconductivity

Quantum Hall Eect

Outline Motivation Quantum Hall Eect The Model Conclusion

Hopes and Dreams of AdS/CMT

Graphene

Quantum Critical Theories

High-Tc Superconductivity

Quantum Hall Eect

Outline Motivation Quantum Hall Eect The Model Conclusion

Hopes and Dreams of AdS/CMT

Graphene

Quantum Critical Theories

High-Tc Superconductivity

Quantum Hall Eect

Outline Motivation Quantum Hall Eect The Model Conclusion

Why the (Fractional) Quantum Hall Eect (FQHE)?

• Any strongly correlated system would be remarkable if completely described by AdS space.

• There are many possible gravity duals (even more from a string phenomenological setting)

• Best guide is to use symmetry principles to constrain our possible gravity duals.

• FQHE has a large amount of (fairly unique) symmetries and dualities

Outline Motivation Quantum Hall Eect The Model Conclusion

Why the (Fractional) Quantum Hall Eect (FQHE)?

• Any strongly correlated system would be remarkable if completely described by AdS space.

• There are many possible gravity duals (even more from a string phenomenological setting)

• Best guide is to use symmetry principles to constrain our possible gravity duals.

• FQHE has a large amount of (fairly unique) symmetries and dualities

Outline Motivation Quantum Hall Eect The Model Conclusion

Why the (Fractional) Quantum Hall Eect (FQHE)?

• Any strongly correlated system would be remarkable if completely described by AdS space.

• There are many possible gravity duals (even more from a string phenomenological setting)

• Best guide is to use symmetry principles to constrain our possible gravity duals.

• FQHE has a large amount of (fairly unique) symmetries and dualities

Outline Motivation Quantum Hall Eect The Model Conclusion

Why the (Fractional) Quantum Hall Eect (FQHE)?

• Any strongly correlated system would be remarkable if completely described by AdS space.

• There are many possible gravity duals (even more from a string phenomenological setting)

• Best guide is to use symmetry principles to constrain our possible gravity duals.

• FQHE has a large amount of (fairly unique) symmetries and dualities

Outline Motivation Quantum Hall Eect The Model Conclusion

The Classical Hall Eect

Lorentz force on charge carriers in a current.

Dene resistivity matrix as Ei = ρijJ^j

As T → 0, we expect the ohmic resistivity ρxx→ 0,

classically.

What about Hall resistivity?

0 = Fx = qEx− qv_yBz= qEx−^V_nJyBz

Since ρxyJ_y = E_x we see ρxy = ^B_ρ^z. The hall resistivity is linear in Bz.

Outline Motivation Quantum Hall Eect The Model Conclusion

The Classical Hall Eect

Lorentz force on charge carriers in a current.

Dene resistivity matrix as Ei = ρijJ^j

As T → 0, we expect the ohmic resistivity ρxx→ 0,

classically.

What about Hall resistivity?

0 = Fx = qEx− qv_yBz= qEx−^V_nJyBz

Since ρxyJ_y = E_x we see ρxy = ^B_ρ^z. The hall resistivity is linear in Bz.

Outline Motivation Quantum Hall Eect The Model Conclusion

The (Fractional) Quantum Hall Eect

Instead of a straight line, we see successive plateaux as we increase the magnetic eld.

These quantized conductivities (resistivities) are given by σ_xy = ρ⁻¹_xy = ρ

B ≡ ν^e_h²,

where ν is quantized to be a fraction in general with odd denominator.

Outline Motivation Quantum Hall Eect The Model Conclusion

The (Fractional) Quantum Hall Eect

Instead of a straight line, we see successive plateaux as we increase the magnetic eld.

These quantized conductivities (resistivities) are given by σ_xy = ρ⁻¹_xy = ρ

B ≡ ν^e_h²,

where ν is quantized to be a fraction in general with odd denominator.

Outline Motivation Quantum Hall Eect The Model Conclusion

The (Fractional) Quantum Hall Eect

Instead of a straight line, we see successive plateaux as we increase the magnetic eld.

These quantized conductivities (resistivities) are given by σ_xy = ρ⁻¹_xy = ρ

B ≡ ν^e_h²,

where ν is quantized to be a fraction in general with odd denominator.

Outline Motivation Quantum Hall Eect The Model Conclusion

The (Fractional) Quantum Hall Eect

Instead of a straight line, we see successive plateaux as we increase the magnetic eld.

These quantized conductivities (resistivities) are given by σ_xy = ρ⁻¹_xy = ρ

B ≡ ν^e_h²,

where ν is quantized to be a fraction in general with odd denominator.

Outline Motivation Quantum Hall Eect The Model Conclusion

Properties of FQHE

Semi-Circle Law

Semi-circles are swept out in the conductivity plane departing from plateau. Also, states ν = p/q and ν⁰ = r/sare only connected if and only if |ps − rq| = 1.

Outline Motivation Quantum Hall Eect The Model Conclusion

Properties of FQHE

Duality

Across the critical magnetic eld, the ohmic resistance enjoys the relationship

ρ_xx(ν_c− ∆ν) ∝ 1 ρxx(νc+ ∆ν)

Outline Motivation Quantum Hall Eect The Model Conclusion

Duality and the Modular Group - SL(2, Z)

• Dene a complex conductivity σ ≡ σxy+ iσ_xx

• These properties imply that σ has the relation σ → aσ + b

cσ + d

• This describes the duality relating the various plateaux for integers a, b, c, and d for |ad − bc| = 1

• The observed odd-denominator quantum hall states restricts c to be even

• This breaks SL(2, Z) to Γ0(2)

Outline Motivation Quantum Hall Eect The Model Conclusion

Duality and the Modular Group - SL(2, Z)

• Dene a complex conductivity σ ≡ σxy+ iσ_xx

• These properties imply that σ has the relation σ → aσ + b

cσ + d

• This describes the duality relating the various plateaux for integers a, b, c, and d for |ad − bc| = 1

• The observed odd-denominator quantum hall states restricts c to be even

• This breaks SL(2, Z) to Γ0(2)

Outline Motivation Quantum Hall Eect The Model Conclusion

Duality and the Modular Group - SL(2, Z)

• Dene a complex conductivity σ ≡ σxy+ iσ_xx

• These properties imply that σ has the relation σ → aσ + b

cσ + d

• This describes the duality relating the various plateaux for integers a, b, c, and d for |ad − bc| = 1

• The observed odd-denominator quantum hall states restricts c to be even

• This breaks SL(2, Z) to Γ0(2)

Outline Motivation Quantum Hall Eect The Model Conclusion

Duality and the Modular Group - SL(2, Z)

• Dene a complex conductivity σ ≡ σxy+ iσ_xx

• These properties imply that σ has the relation σ → aσ + b

cσ + d

• This describes the duality relating the various plateaux for integers a, b, c, and d for |ad − bc| = 1

• The observed odd-denominator quantum hall states restricts c to be even

• This breaks SL(2, Z) to Γ0(2)

Outline Motivation Quantum Hall Eect The Model Conclusion

Duality and the Modular Group - SL(2, Z)

• Dene a complex conductivity σ ≡ σxy+ iσ_xx

• These properties imply that σ has the relation σ → aσ + b

cσ + d

• This describes the duality relating the various plateaux for integers a, b, c, and d for |ad − bc| = 1

• The observed odd-denominator quantum hall states restricts c to be even

• This breaks SL(2, Z) to Γ0(2)

Outline Motivation Quantum Hall Eect The Model Conclusion

Duality and the Modular Group - SL(2, Z)

To further motivate this group, we can see RG ow lines consistent with commuting with the group Γ0(2)

From this, we see both the semicircles associated with moving away from a plateaux, as well as ow towards various

odd-denominator states as T → 0

Outline Motivation Quantum Hall Eect The Model Conclusion

Deep IR - Fractionally Charged Anoyons

The nal ingredient we would like is knowledge of the system in the IR. It is generally understood that the mechanism for fractional plateaux is from the charge-carries having fractional statistics.

This is accomplished by deforming an action by

S[ψ, A] → S[ψ, A + a] − e² 2ϑ

Z

d³x^µνλa_µ∂_νa_λ

ie we couple the system to a statistics eld with a Chern-Simons term

Outline Motivation Quantum Hall Eect The Model Conclusion

Deep IR - Fractionally Charged Anoyons

The nal ingredient we would like is knowledge of the system in the IR. It is generally understood that the mechanism for fractional plateaux is from the charge-carries having fractional statistics.

This is accomplished by deforming an action by

S[ψ, A] → S[ψ, A + a] − e² 2ϑ

Z

d³x^µνλa_µ∂_νa_λ

ie we couple the system to a statistics eld with a Chern-Simons term

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Review: Our CMT Ingredient List

We want a duality group relating the plateaux that the RG ow commutes with

• Look for bulk actions which are SL(2, R) invariant (which we can later break)

We need fractional statistics and therefore a Chern-Simons term on the boundary theory

• A bulk axion, χ, is simply the Chern-Simons coeecient on the boundary in the AdS/CFT dictionary

Finally, we require a nite Ohmic conductivity at nite

temperature in the boundary theory to see the semicircles in the conductivity plane

• Require a `probe' system that can dissipate energy into a background, avoiding a Drude peak in conductivity

Outline Motivation Quantum Hall Eect The Model Conclusion

Quantum Hall-ography - The Model

Background bulk action:

S_grav= − Z

d⁴x√

−g

 1 2κ²

 R − 6

L² + 1 2



∂_µφ∂^µφ + e^2φ∂_µχ∂^µχ

+ S_Lifshitz

With S_Lifshitzsuch that the background near-horizon metric is ds² = L²



−h(v)dt²

v^2z + dv²

v²h(v) +dx²+ dy² v²



with a temperature T = ^|h_4πv^0(vz−1^h)|

h and h(vh) = 0.

If we dene

τ = χ + ie^−φ,

this action has the familiar SL(2, R) symmetry found in string theory,

τ → τ⁰ = aτ + b

cτ + d |ad − bc| = 1

Outline Motivation Quantum Hall Eect The Model Conclusion

Quantum Hall-ography - The Model

Background bulk action:

S_grav= − Z

d⁴x√

−g

 1 2κ²

 R − 6

L² + 1 2



∂_µφ∂^µφ + e^2φ∂_µχ∂^µχ

+ S_Lifshitz

With S_Lifshitzsuch that the background near-horizon metric is ds² = L²



−h(v)dt²

v^2z + dv²

v²h(v) +dx²+ dy² v²



with a temperature T = ^|h_4πv^0(vz−1^h)|

h and h(vh) = 0. If we dene

τ = χ + ie^−φ,

this action has the familiar SL(2, R) symmetry found in string theory,

τ → τ⁰ = aτ + b

cτ + d |ad − bc| = 1

Outline Motivation Quantum Hall Eect The Model Conclusion

Quantum Hall-ography - The Model

Background bulk action:

S_grav= − Z

d⁴x√

−g

 1 2κ²

 R − 6

L² + 1 2



∂_µφ∂^µφ + e^2φ∂_µχ∂^µχ

+ S_Lifshitz

With S_Lifshitzsuch that the background near-horizon metric is ds² = L²



−h(v)dt²

v^2z + dv²

v²h(v) +dx²+ dy² v²



with a temperature T = ^|h_4πv^0(vz−1^h)|

h and h(vh) = 0. If we dene

τ = χ + ie^−φ,

this action has the familiar SL(2, R) symmetry found in string theory,

τ → τ⁰ = aτ + b

cτ + d |ad − bc| = 1

Outline Motivation Quantum Hall Eect The Model Conclusion

Quantum Hall-ography - The Model

Probe-brane action:

S_gauge= − T Z

d⁴x

q

−det gµν+ `²e^−φ/2Fµν
−√

−g



−1 4

Z d⁴x√

−gχF_µνF˜^µν

We use the standard AdS/CFT dictionary to calculate the current (and therefore conductivity)

J^a= δS_gauge δAa|_v=0,

which gives us an ohmic conductivity of σ^xx ∝ T^−2/z at low temperatures.

Outline Motivation Quantum Hall Eect The Model Conclusion

Conductivity plane

So have we reproduced the quantum hall eect?

Outline Motivation Quantum Hall Eect The Model Conclusion

Conductivity plane

So have we reproduced the quantum hall eect?

From this, the only plateaux-like behaviour is for σ = τ. But for Imτ → 0 this corresponds to strong coupling, where we expect SL(2, R)to be broken.

Outline Motivation Quantum Hall Eect The Model Conclusion

Conductivity plane

So have we reproduced the quantum hall eect?

However, we can map areas of weak coupling to regions of strong coupling by the surviving SL(2, R) symmetry (or one of the sub- groups). Here we can see plateaux-like behaviour.

Outline Motivation Quantum Hall Eect The Model Conclusion

A Bonus Gift!

From dimensional arguments, the conductivity will be of the functional form

σ = σ

 ρ T^2/z, B

T^2/z

 ,

which we can rewrite (for small changes in lling fraction) as

σ ' σ



∆ν, ∆B T^2/z



Experimentally, the exponent of T measured experimentally (∼ 0.42) closely corresponds to a dynamical critical exponent of z = 5, which is the exponent corresponding to our `probe' system being the background!

Outline Motivation Quantum Hall Eect The Model Conclusion

A Bonus Gift!

From dimensional arguments, the conductivity will be of the functional form

σ = σ

 ρ T^2/z, B

T^2/z

 ,

which we can rewrite (for small changes in lling fraction) as

σ ' σ



∆ν, ∆B T^2/z



Experimentally, the exponent of T measured experimentally (∼ 0.42) closely corresponds to a dynamical critical exponent of z = 5, which is the exponent corresponding to our `probe' system being the background!

Outline Motivation Quantum Hall Eect The Model Conclusion

A Bonus Gift!

From dimensional arguments, the conductivity will be of the functional form

σ = σ

 ρ T^2/z, B

T^2/z

 ,

which we can rewrite (for small changes in lling fraction) as

σ ' σ



∆ν, ∆B T^2/z



Experimentally, the exponent of T measured experimentally (∼ 0.42) closely corresponds to a dynamical critical exponent of z = 5, which is the exponent corresponding to our `probe' system being the background!

Outline Motivation Quantum Hall Eect The Model Conclusion

SL(2, R) - From Bulk Fields to Boundary Conductivities

Dene

G^µν = −2 δS δF_µν. SL(2, R)acts as

Gµν →aG_µν− b ˜Fµν

F_µν →c ˜G_µν+ dF_µν τ →aτ + b

cτ + d

AdS/CFT says

√−gG^va|_v=0= J^a. Transforming the conductivity equation

J^a= σ^abE_b, by SL(2, R) induces

σ → aσ + b cσ + d.

Any SL(2, R) invariant theory (with the symmetry group acting on the Maxwell eld) has an SL(2, R) invariant boundary conductivity!

Outline Motivation Quantum Hall Eect The Model Conclusion

SL(2, R) - From Bulk Fields to Boundary Conductivities

Dene

G^µν = −2 δS δF_µν. SL(2, R)acts as

Gµν →aG_µν− b ˜Fµν

F_µν →c ˜G_µν+ dF_µν τ →aτ + b

cτ + d

AdS/CFT says

√−gG^va|_v=0= J^a. Transforming the conductivity equation

J^a = σ^abE_b, by SL(2, R) induces

σ → aσ + b cσ + d.

Any SL(2, R) invariant theory (with the symmetry group acting on the Maxwell eld) has an SL(2, R) invariant boundary conductivity!

Outline Motivation Quantum Hall Eect The Model Conclusion

SL(2, R) - From Bulk Fields to Boundary Conductivities

Dene

G^µν = −2 δS δF_µν. SL(2, R)acts as

Gµν →aG_µν− b ˜Fµν

F_µν →c ˜G_µν+ dF_µν τ →aτ + b

cτ + d

AdS/CFT says

√−gG^va|_v=0= J^a. Transforming the conductivity equation

J^a = σ^abE_b, by SL(2, R) induces

σ → aσ + b cσ + d.

Any SL(2, R) invariant theory (with the symmetry group acting on the Maxwell eld) has an SL(2, R) invariant boundary conductivity!

Outline Motivation Quantum Hall Eect The Model Conclusion

Summary

• AdS/CFT is an emerging tool to compute strongly correlated electron systems in condensed matter.

• One system in condensed matter that has resisted

theoretical tools is the fractional quantum hall eect. Every reason to try to apply hall-ography.

• We have found a plausible model in AdS/CFT that captures some essential features of the quantum hall eect as well as a possible experimental prediction.

• Even if this is the wrong model, there's recourse in the very general arguments with SL(2, R) as a symmetry that should be present in any QHE model.

Outline Motivation Quantum Hall Eect The Model Conclusion

Summary

• AdS/CFT is an emerging tool to compute strongly correlated electron systems in condensed matter.

• One system in condensed matter that has resisted

theoretical tools is the fractional quantum hall eect. Every reason to try to apply hall-ography.

• We have found a plausible model in AdS/CFT that captures some essential features of the quantum hall eect as well as a possible experimental prediction.

• Even if this is the wrong model, there's recourse in the very general arguments with SL(2, R) as a symmetry that should be present in any QHE model.

Outline Motivation Quantum Hall Eect The Model Conclusion

Summary

• AdS/CFT is an emerging tool to compute strongly correlated electron systems in condensed matter.

• One system in condensed matter that has resisted

theoretical tools is the fractional quantum hall eect. Every reason to try to apply hall-ography.

• We have found a plausible model in AdS/CFT that captures some essential features of the quantum hall eect as well as a possible experimental prediction.

• Even if this is the wrong model, there's recourse in the very general arguments with SL(2, R) as a symmetry that should be present in any QHE model.

Outline Motivation Quantum Hall Eect The Model Conclusion

Summary

• AdS/CFT is an emerging tool to compute strongly correlated electron systems in condensed matter.

• One system in condensed matter that has resisted

theoretical tools is the fractional quantum hall eect. Every reason to try to apply hall-ography.

• We have found a plausible model in AdS/CFT that captures some essential features of the quantum hall eect as well as a possible experimental prediction.

• Even if this is the wrong model, there's recourse in the very general arguments with SL(2, R) as a symmetry that should be present in any QHE model.

Outline Motivation Quantum Hall Eect The Model Conclusion

Acknowledgements

Cli Burgess Brian Dolan Sung-Sik Lee In addition to several people in the condensed

matter-holography community that have contributed great reviews on the subject (Sean Hartnoll, Joe Polchinski, John McGreevy, Gary Horowitz, etc.)