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A Dimensionally Deconstructed Holographic Superconductor Dylan Albrecht Crete Center for Theoretical Physics September 10, 2013 Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 1 / 21 AdS/CFT

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A Dimensionally Deconstructed Holographic Superconductor

Dylan Albrecht

Crete Center for Theoretical Physics

September 10, 2013

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 1 / 21

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SLIDE 2

AdS/CFT

Jumping right in..

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 2 / 21

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SLIDE 3

AdS/CFT

Jumping right in.. AdS/CFT is a specific duality: Anti-de Sitter Bulk, weakly-coupled gravitational theory in (d + 1)-dimensional spacetime ↔ Conformal Field Theory Boundary, strongly-coupled gauge theory in d-dimensional spacetime

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 2 / 21

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SLIDE 4

AdS/CFT

Jumping right in.. AdS/CFT is a specific duality: Anti-de Sitter Bulk, weakly-coupled gravitational theory in (d + 1)-dimensional spacetime ↔ Conformal Field Theory Boundary, strongly-coupled gauge theory in d-dimensional spacetime Provides a framework: Symmetries and quantum numbers match on both sides.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 2 / 21

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SLIDE 5

AdS/CFT

AdS metric (g): ds2 = 1 z2

  • ηµνdxµdxν − dz2

Fields in AdS, Φ(x, z), break up into two pieces: Normalizable ↔ O(x) Non-normalizable ↔ φ0(x) (E.g. A0µ(x) sources Jµ(x))

AdS Space

From Hartnoll arXiv:1106.4324

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 3 / 21

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SLIDE 6

AdS/CFT

Recipe for model building: AdS Fields Gauge fields in bulk Black hole ↔ ↔ ↔ ↔ CFT Operators Global Symmetry Finite temperature

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 4 / 21

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SLIDE 7

Holographic Superconductors?

Gubser (2008): AdS4 black hole can develop a condensate, spontaneously breaking a U(1) gauge symmetry.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 5 / 21

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SLIDE 8

Holographic Superconductors?

Gubser (2008): AdS4 black hole can develop a condensate, spontaneously breaking a U(1) gauge symmetry. → Black holes superconduct.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 5 / 21

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SLIDE 9

Holographic Superconductors?

Gubser (2008): AdS4 black hole can develop a condensate, spontaneously breaking a U(1) gauge symmetry. → Black holes superconduct. → Holographic interpretation?

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 5 / 21

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Constructing Holographic Superconductor

Ingredients [Hartnoll, Herzog, and Horowitz]: Consider U(1) gauge field F. Add charged scalar Ψ(x, z), dual to O, Cooper pair operator. Background for gauge field → Chemical potential. Black hole background → System at temperature.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 6 / 21

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SLIDE 11

Constructing Holographic Superconductor

Ingredients [Hartnoll, Herzog, and Horowitz]: Consider U(1) gauge field F. Add charged scalar Ψ(x, z), dual to O, Cooper pair operator. Background for gauge field → Chemical potential. Black hole background → System at temperature. We have an AdS4-Schwarzschild background (g): ds2 = 1 z2

  • f(z)dt2 − d

x2 − dz2 f(z)

  • ,

ǫ ≤ z ≤ zH where f(z) = 1 − (z/zH)3.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 6 / 21

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Holographic Superconductor

The action for a holographic superconductor: S =

  • d4x √g
  • |DΨ|2 − m2|Ψ|2 − 1

4FMNFMN

  • Strategy:

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 7 / 21

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SLIDE 13

Holographic Superconductor

The action for a holographic superconductor: S =

  • d4x √g
  • |DΨ|2 − m2|Ψ|2 − 1

4FMNFMN

  • Strategy:

Vary the temperature. Find nonvanishing solution for Ψ. → O = 0.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 7 / 21

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A Holographic Superconductor

Some features: O ∝ (1 − T/Tc)1/2 2∆ ≡

  • O ≈ 8.4Tc

Condensate O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 8 / 21

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SLIDE 15

A Holographic Superconductor

Some features: O ∝ (1 − T/Tc)1/2 2∆ ≡

  • O ≈ 8.4Tc

The normal phase has Re[σ(ω)] = 1. Delta function δ(ω)

Conductivity

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 9 / 21

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Deconstructing Superconductivity

What do I mean by dimensional deconstruction?

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 10 / 21

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Deconstructing Superconductivity

What do I mean by dimensional deconstruction? → Basically, turning the extra dimension into a lattice. → Many scalar fields, but now 3D model.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 10 / 21

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Deconstructing Superconductivity

What do I mean by dimensional deconstruction? → Basically, turning the extra dimension into a lattice. → Many scalar fields, but now 3D model.

One scalar at each site

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 10 / 21

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SLIDE 19

Deconstructing Superconductivity

Not so simple – need a “comparator”.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 11 / 21

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Deconstructing Superconductivity

Not so simple – need a “comparator”. ⇒ U(1) at each site, link field Σ “between” them.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 11 / 21

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Deconstructing Superconductivity

Not so simple – need a “comparator”. ⇒ U(1) at each site, link field Σ “between” them.

Moose diagram

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 11 / 21

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Deconstructing Superconductivity

The Lagrangian for the moose diagram: L =

N−1

  • j=2
  • −1

4(Fµν)j(Fµν)j + Zj|DµΨj|2

  • +

N−1

  • j=1
  • |DµΣj|2 − ZjVj
  • .

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 12 / 21

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SLIDE 23

Deconstructing Superconductivity

The Lagrangian for the moose diagram: L =

N−1

  • j=2
  • −1

4(Fµν)j(Fµν)j + Zj|DµΨj|2

  • +

N−1

  • j=1
  • |DµΣj|2 − ZjVj
  • .

Leaving out the details... Couplings change from site to site. Link fields get a vev and are “integrated out”.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 12 / 21

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SLIDE 24

Deconstructing Superconductivity

The Lagrangian for the moose diagram: L =

N−1

  • j=2
  • −1

4(Fµν)j(Fµν)j + Zj|DµΨj|2

  • +

N−1

  • j=1
  • |DµΣj|2 − ZjVj
  • .

Leaving out the details... Couplings change from site to site. Link fields get a vev and are “integrated out”.

  • S =
  • d4x √g
  • −1

4FMNFMN + |DΨ|2 − m2|Ψ|2

  • Similar strategy to continuum case: Solve set of equations for

nonvanishing Ψj.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 12 / 21

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Deconstructing Superconductivity

Boundary conditions: Chosen to best match continuum result. Nondynamical first site and last site.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 13 / 21

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Deconstructing Superconductivity

Boundary conditions: Chosen to best match continuum result. Nondynamical first site and last site. Continuum ingoing wave BC presents a challenge. → Ingoing wave BC closer to UV.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 13 / 21

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Deconstructing Superconductivity

What do we find?

Continuum O Deconstructed O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 14 / 21

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Deconstructing Superconductivity

What do we find? (N = 1000 and N = 100).

Continuum O Deconstructed O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 15 / 21

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Deconstructed O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 16 / 21

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Deconstructing Superconductivity

What do we find? (N = 1000 and N = 100).

Continuum O Deconstructed O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 17 / 21

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Deconstructing Superconductivity

What do we find? (N = 10 and N = 5).

Continuum O Deconstructed O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 18 / 21

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Deconstructed O

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 19 / 21

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To Conclude

Deconstruction provides a framework for building lower-dimensional models that can mimic higher-dimensional physcs. Some calculations are difficult to match in a natural way. Interpreting ingoing wave BCs. Excitons governed by hidden local symmetries?

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 20 / 21

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The End.

Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 21 / 21