A Dimensionally Deconstructed Holographic Superconductor Dylan - PowerPoint PPT Presentation

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

AdS/CFT Jumping right in.. AdS/CFT is a specific duality: Anti-de Sitter Conformal Field Theory Bulk, weakly-coupled Boundary, ↔ gravitational theory in strongly-coupled gauge ( d + 1)-dimensional theory in d -dimensional spacetime spacetime Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 2 / 21

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

AdS/CFT AdS metric ( g ): AdS Space ds 2 = 1 � η µν dx µ dx ν − dz 2 � z 2 Fields in AdS, Φ( x , z ) , break up into two pieces: Normalizable ↔ O ( x ) Non-normalizable ↔ φ 0 ( x ) (E.g. A 0 µ ( x ) sources J µ ( x ) ) From Hartnoll arXiv:1106.4324 Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 3 / 21

AdS/CFT Recipe for model building: ↔ CFT AdS ↔ Operators Fields ↔ Global Symmetry Gauge fields in bulk ↔ Finite temperature Black hole Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 4 / 21

Holographic Superconductors? Gubser (2008): AdS 4 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

Holographic Superconductors? Gubser (2008): AdS 4 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

Holographic Superconductors? Gubser (2008): AdS 4 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

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

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 AdS 4 -Schwarzschild background ( g ): x 2 − dz 2 � � ds 2 = 1 f ( z ) dt 2 − d � , ǫ ≤ z ≤ z H z 2 f ( z ) where f ( z ) = 1 − ( z / z H ) 3 . Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 6 / 21

Holographic Superconductor The action for a holographic superconductor: d 4 x √ g � � � | D Ψ | 2 − m 2 | Ψ | 2 − 1 4 F MN F MN S = Strategy: Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 7 / 21

Holographic Superconductor The action for a holographic superconductor: d 4 x √ g � � � | D Ψ | 2 − m 2 | Ψ | 2 − 1 4 F MN F MN S = Strategy: Vary the temperature. Find nonvanishing solution for Ψ . → �O� � = 0. Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 7 / 21

A Holographic Superconductor Condensate � O � Some features: �O� ∝ ( 1 − T / T c ) 1 / 2 � 2 ∆ ≡ �O� ≈ 8 . 4 T c Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 8 / 21

A Holographic Superconductor Conductivity Some features: �O� ∝ ( 1 − T / T c ) 1 / 2 � 2 ∆ ≡ �O� ≈ 8 . 4 T c The normal phase has Re [ σ ( ω )] = 1. Delta function δ ( ω ) Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 9 / 21

Deconstructing Superconductivity What do I mean by dimensional deconstruction? Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 10 / 21

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

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

Deconstructing Superconductivity Not so simple – need a “comparator”. Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 11 / 21

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

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

Deconstructing Superconductivity The Lagrangian for the moose diagram: N − 1 N − 1 � − 1 � � | D µ Σ j | 2 − Z j V j � � 4 ( F µν ) j ( F µν ) j + Z j | D µ Ψ j | 2 � L = + . j = 2 j = 1 Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 12 / 21

Deconstructing Superconductivity The Lagrangian for the moose diagram: N − 1 N − 1 � − 1 � � | D µ Σ j | 2 − Z j V j � � 4 ( F µν ) j ( F µν ) j + Z j | D µ Ψ j | 2 � L = + . j = 2 j = 1 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

Deconstructing Superconductivity The Lagrangian for the moose diagram: N − 1 N − 1 � − 1 � � | D µ Σ j | 2 − Z j V j � � 4 ( F µν ) j ( F µν ) j + Z j | D µ Ψ j | 2 � L = + . j = 2 j = 1 Leaving out the details... Couplings change from site to site. Link fields get a vev and are “integrated out”. � d 4 x √ g � �� � − 1 4 F MN F MN + | D Ψ | 2 − m 2 | Ψ | 2 S = Similar strategy to continuum case: Solve set of equations for nonvanishing Ψ j . Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 12 / 21

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

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

Deconstructing Superconductivity What do we find? Continuum �O� Deconstructed �O� Dylan Albrecht (CCTP) A Dimensionally Deconstructed Holographic Superconductor September 10, 2013 14 / 21

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

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

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

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

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