Phase structure of a defect field theory with a domain wall. V. Filev IMI, Bulgarian Academy of Sciences Vienna July 2018 Filev (IMI) Defect field theory with a domain wall. MMNGST 18 1 / 40
Outline AdS/CFT correspondence 1 Adding flavours D3/D7 Karch & Katz Meson melting phase transition Critical behaviour D0/D4 system 2 Lower dimensional correspondence Berkooz-Douglas matrix model Comparison Defect field theory with a domain wall 3 Introducing probe D5-branes Introducing the domain wall Critical point Filev (IMI) Defect field theory with a domain wall. MMNGST 18 2 / 40
AdS/CFT correspondence Type IIB String Theory on Nc D3 AdS 5 × S 5 N = 4 SU ( N c ) SUSY YM Gubser-Klebanov-Polyakov-Witten formula: d d x φ 0 ( x ) �O ( x ) � � CFT = Z string [ φ 0 ( x )] � � e Filev (IMI) Defect field theory with a domain wall. MMNGST 18 3 / 40
AdS/CFT correspondence Type IIB String Theory on Nc D3 AdS 5 × S 5 N = 4 SU ( N c ) SUSY YM Gubser-Klebanov-Polyakov-Witten formula: d d x φ 0 ( x ) �O ( x ) � � CFT = Z string [ φ 0 ( x )] � � e Filev (IMI) Defect field theory with a domain wall. MMNGST 18 3 / 40
Adding flavours D3/D7 Karch & Katz Generalizing the correspondence Nc D3 m Nf D7 0 1 2 3 4 5 6 7 8 9 · · · · · · D3 - - - - D7 - - - - - - - - · · Adding N f massive N = 2 Hypermultiplets: 0 � d 2 θ ˜ m q = m / 2 πα ′ m q Q Q → SYM with Filev (IMI) Defect field theory with a domain wall. MMNGST 18 4 / 40
Adding flavours D3/D7 Karch & Katz Generalizing the correspondence Nc D3 m Nf D7 0 1 2 3 4 5 6 7 8 9 · · · · · · D3 - - - - D7 - - - - - - - - · · Adding N f massive N = 2 Hypermultiplets: 0 � d 2 θ ˜ m q = m / 2 πα ′ m q Q Q → SYM with Filev (IMI) Defect field theory with a domain wall. MMNGST 18 4 / 40
String spectrum pure N =4 SYM 3-3 strings adjoint of SU ( N c ) 3-7 strings Q i fundamental chiral field Q i anti-fundamental chiral field ˜ 7-3 strings gauge field on the D7 brane 7-7 strings frozen by infinite volume Filev (IMI) Defect field theory with a domain wall. MMNGST 18 5 / 40
Probe approximation N f ≪ N c The probe is described by a Dirac-Born-Infeld action � d 7 ξ e − Φ � || G ab − 2 πα ′ F ab || S ∝ The profile of the D-brane encodes the fundamental condensate of theory. The semi-classical fluctuations correspond to meson-like excitations. The D-brane gauge field can describe: external electromagnetic field, chemical potential, electric current etc. Numerous applications: thermal and quantum phase transitions, chiral symmetry breaking, magnetic catalysis etc. Filev (IMI) Defect field theory with a domain wall. MMNGST 18 6 / 40
Probe approximation N f ≪ N c The probe is described by a Dirac-Born-Infeld action � d 7 ξ e − Φ � || G ab − 2 πα ′ F ab || S ∝ The profile of the D-brane encodes the fundamental condensate of theory. The semi-classical fluctuations correspond to meson-like excitations. The D-brane gauge field can describe: external electromagnetic field, chemical potential, electric current etc. Numerous applications: thermal and quantum phase transitions, chiral symmetry breaking, magnetic catalysis etc. Filev (IMI) Defect field theory with a domain wall. MMNGST 18 6 / 40
Probe approximation N f ≪ N c The probe is described by a Dirac-Born-Infeld action � d 7 ξ e − Φ � || G ab − 2 πα ′ F ab || S ∝ The profile of the D-brane encodes the fundamental condensate of theory. The semi-classical fluctuations correspond to meson-like excitations. The D-brane gauge field can describe: external electromagnetic field, chemical potential, electric current etc. Numerous applications: thermal and quantum phase transitions, chiral symmetry breaking, magnetic catalysis etc. Filev (IMI) Defect field theory with a domain wall. MMNGST 18 6 / 40
Probe approximation N f ≪ N c The probe is described by a Dirac-Born-Infeld action � d 7 ξ e − Φ � || G ab − 2 πα ′ F ab || S ∝ The profile of the D-brane encodes the fundamental condensate of theory. The semi-classical fluctuations correspond to meson-like excitations. The D-brane gauge field can describe: external electromagnetic field, chemical potential, electric current etc. Numerous applications: thermal and quantum phase transitions, chiral symmetry breaking, magnetic catalysis etc. Filev (IMI) Defect field theory with a domain wall. MMNGST 18 6 / 40
Probe approximation N f ≪ N c The probe is described by a Dirac-Born-Infeld action � d 7 ξ e − Φ � || G ab − 2 πα ′ F ab || S ∝ The profile of the D-brane encodes the fundamental condensate of theory. The semi-classical fluctuations correspond to meson-like excitations. The D-brane gauge field can describe: external electromagnetic field, chemical potential, electric current etc. Numerous applications: thermal and quantum phase transitions, chiral symmetry breaking, magnetic catalysis etc. Filev (IMI) Defect field theory with a domain wall. MMNGST 18 6 / 40
Meson melting phase transition Consider the AdS-black hole background: − u 4 − u 4 x 2 + u 2 R 2 R 2 u 2 dt 2 + u 2 du 2 + u 2 d Ω 2 ds 2 0 R 2 d � = 5 . u 4 − u 4 0 3 + sin 2 θ d φ 2 5 = d θ 2 + cos 2 θ d Ω 2 d Ω 2 Filev (IMI) Defect field theory with a domain wall. MMNGST 18 7 / 40
Meson melting phase transition Consider the AdS-black hole background: − u 4 − u 4 x 2 + u 2 R 2 R 2 u 2 dt 2 + u 2 du 2 + u 2 d Ω 2 ds 2 0 R 2 d � = 5 . u 4 − u 4 0 3 + sin 2 θ d φ 2 5 = d θ 2 + cos 2 θ d Ω 2 d Ω 2 Filev (IMI) Defect field theory with a domain wall. MMNGST 18 7 / 40
Critical behaviour Dp/Dq Consider a general Dp/Dq system ( D. Mateos, R. Myers, R. Thomson 2007 ) and the parametrisation: 8 − p = d θ 2 + sin 2 θ d Ω 2 n + cos 2 θ d Ω 7 − p − n d Ω 2 Next we zoom in at the near horizon geometry: � L � p − 3 � 7 − p θ = y � u 0 4 u = u 0 + π T z 2 , � 4 ˜ � , x = x L u 0 L To obtain the metric: ds 2 = − ( 2 π T ) 2 z 2 dt 2 + dz 2 + dy 2 + y 2 d Ω 2 x 2 + . . . n + d � ˜ Filev (IMI) Defect field theory with a domain wall. MMNGST 18 8 / 40
Critical behaviour Dp/Dq Consider a general Dp/Dq system ( D. Mateos, R. Myers, R. Thomson 2007 ) and the parametrisation: 8 − p = d θ 2 + sin 2 θ d Ω 2 n + cos 2 θ d Ω 7 − p − n d Ω 2 Next we zoom in at the near horizon geometry: � L � p − 3 � 7 − p θ = y � u 0 4 u = u 0 + π T z 2 , � 4 ˜ � , x = x L u 0 L To obtain the metric: ds 2 = − ( 2 π T ) 2 z 2 dt 2 + dz 2 + dy 2 + y 2 d Ω 2 x 2 + . . . n + d � ˜ Filev (IMI) Defect field theory with a domain wall. MMNGST 18 8 / 40
Critical behaviour Dp/Dq Consider a general Dp/Dq system ( D. Mateos, R. Myers, R. Thomson 2007 ) and the parametrisation: 8 − p = d θ 2 + sin 2 θ d Ω 2 n + cos 2 θ d Ω 7 − p − n d Ω 2 Next we zoom in at the near horizon geometry: � L � p − 3 � 7 − p θ = y � u 0 4 u = u 0 + π T z 2 , � 4 ˜ � , x = x L u 0 L To obtain the metric: ds 2 = − ( 2 π T ) 2 z 2 dt 2 + dz 2 + dy 2 + y 2 d Ω 2 x 2 + . . . n + d � ˜ Filev (IMI) Defect field theory with a domain wall. MMNGST 18 8 / 40
Critical behaviour Dp/Dq cont. The resulting EOM for Minkowski embeddings is: y 2 ) = 0 z y ¨ y + ( y ˙ y − nz )( 1 + ˙ It has scaling symmetry: if y ( z ) is a solution so is y ( µ z ) /µ . And a critical solution y = √ n z , linearising we obtain: √ n z + T − 1 y = 2 [ a sin ( α log Tz ) + b cos ( α Tz )] , n ( Tz ) � 4 ( n + 1 ) − n 2 / 2. Under the scaling symmetry the with α = constants a , b transform as: � cos ( α log µ ) � a � 1 � � a � sin ( α log µ ) → n b − sin ( α log µ ) cos ( α log µ ) b 2 + 1 µ This is a double spiral signalling a discrete self-similar structure, which forces the phase transition into a first order one. This behaviour is universal (depends only on n ). Filev (IMI) Defect field theory with a domain wall. MMNGST 18 9 / 40
Critical behaviour Dp/Dq cont. The resulting EOM for Minkowski embeddings is: y 2 ) = 0 z y ¨ y + ( y ˙ y − nz )( 1 + ˙ It has scaling symmetry: if y ( z ) is a solution so is y ( µ z ) /µ . And a critical solution y = √ n z , linearising we obtain: √ n z + T − 1 y = 2 [ a sin ( α log Tz ) + b cos ( α Tz )] , n ( Tz ) � 4 ( n + 1 ) − n 2 / 2. Under the scaling symmetry the with α = constants a , b transform as: � cos ( α log µ ) � a � 1 � � a � sin ( α log µ ) → n b − sin ( α log µ ) cos ( α log µ ) b 2 + 1 µ This is a double spiral signalling a discrete self-similar structure, which forces the phase transition into a first order one. This behaviour is universal (depends only on n ). Filev (IMI) Defect field theory with a domain wall. MMNGST 18 9 / 40
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