Conformal defects in Supergravity Piotr Witkowski based on work - PowerPoint PPT Presentation

Conformal defects in Supergravity Piotr Witkowski based on work with R. Janik and J. Jankowski ArXiv:1503.08459 Faculty of Physics, Astronomy and Applied Computer Science Jagiellonian University Max-Planck-Institut f¨ ur Physik Munich, 30 VI 2015

1 Motivation and previous studies 2 First approach and ”emergent” Supergravity 3 Full Solutions 4 Conclusions and further directions 5 Bibliography

Motivation and previous studies Main Motivation: AdS/CMT AdS/CMT “purely AdS” solutions – translational invariance spoils computations of transport properties (like DC conductivity) a solution proposed by [G. T. Horowitz, J. E. Santos, D. Tong] – translational invariance broken by introduction of spatially modulated scalar field (”Holographic lattice”) extensively investigated (ex. [M. Blake, D. Tong, D. Vegh] , [A. Donos, J.P. Gauntlett] ) the lattice is mimicked by a spatially spread (“wide”) source ∼ cos( kx ) Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 3 / 18

Motivation and previous studies Our modifications: 1 idea: replace “wide” source with local, point- or line-like source ∼ δ ( x ) 2 use solutions with local sources to study point-like defect 3 try to obtain lattice constructed from such defects – source ∼ � δ ( x − nx l ) – holographic realisation of the Kronig-Penney model n of condensed matter physics Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 4 / 18

First approach and ”emergent” Supergravity Framework: AdS 4 /CFT 3 action 1 � � | g | ( R − 1 / 2 ∇ a φ ∇ a φ − V ( φ )) S = 16 π G n potential V ( φ ) = − 6 − φ 2 ⇔ cosmological constant & mass m 2 = − 2 φ ( x , y , z ) ⇔ operator O ( x , y ) of dimension ∆ = 2, deforming CFT near-boundary asymptotic of scalar φ 1 ( x , y ) z + φ 2 ( x , y ) z 2 + ... (in Poincar´ φ 1 ( x , y ) φ ( x , y , z ) = φ 1 ( x , y ) e coordinates) the operator deforms CFT by a shift of Lagrangian: φ 1 ( x , y ) L = L CFT 3 + φ 1 ( x , y ) φ 1 ( x , y ) O ( x , y ) Its expectation value reads �O� = φ 2 ( x , y ) focus on single defect concentrated along some line (eg. x=0) Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 5 / 18

First approach and ”emergent” Supergravity Goal: φ 1 ( x , y ) Implement the boundary condition of type φ 1 ( x , y ) φ 1 ( x , y ) = ηδ ( x ) (Dirac delta on line x = 0) in Einstein equations generated by given Action Previous works with discontinuous BCs in (super)gravity φ 1 φ 1 φ 1 = θ ( x ) and m φ = 0 analytical Janus solutions [D. Bak, M. Gutperle, S. Hirano] φ 1 = θ ( x ) and m φ = 0 at T > 0 numerical and analytical Janus black φ 1 φ 1 holes in d = 2 + 1 [D. Bak, M. Gutperle, R. A. Janik] φ 1 = δ ( x ) and m 2 φ 1 φ 1 φ = − 2 with SUSY, analytical and scale invariant [E. D’Hoker et al. ] Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 6 / 18

First approach and ”emergent” Supergravity Goal: φ 1 ( x , y ) Implement the boundary condition of type φ 1 ( x , y ) φ 1 ( x , y ) = ηδ ( x ) (Dirac delta on line x = 0) in Einstein equations generated by given Action Previous works with discontinuous BCs in (super)gravity φ 1 φ 1 φ 1 = θ ( x ) and m φ = 0 analytical Janus solutions [D. Bak, M. Gutperle, S. Hirano] φ 1 = θ ( x ) and m φ = 0 at T > 0 numerical and analytical Janus black φ 1 φ 1 holes in d = 2 + 1 [D. Bak, M. Gutperle, R. A. Janik] φ 1 = δ ( x ) and m 2 φ 1 φ 1 φ = − 2 with SUSY, analytical and scale invariant [E. D’Hoker et al. ] → various non-trivial p -forms Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 6 / 18

First approach and ”emergent” Supergravity Goal: φ 1 ( x , y ) Implement the boundary condition of type φ 1 ( x , y ) φ 1 ( x , y ) = ηδ ( x ) (Dirac delta on line x = 0) in Einstein equations generated by given Action Previous works with discontinuous BCs in (super)gravity φ 1 φ 1 φ 1 = θ ( x ) and m φ = 0 analytical Janus solutions [D. Bak, M. Gutperle, S. Hirano] φ 1 = θ ( x ) and m φ = 0 at T > 0 numerical and analytical Janus black φ 1 φ 1 holes in d = 2 + 1 [D. Bak, M. Gutperle, R. A. Janik] φ 1 = δ ( x ) and m 2 φ 1 φ 1 φ = − 2 with SUSY, analytical and scale invariant [E. D’Hoker et al. ] → various non-trivial p -forms → hard to generalise to black hole case ( T > 0) Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 6 / 18

First approach and ”emergent” Supergravity Goal: φ 1 ( x , y ) Implement the boundary condition of type φ 1 ( x , y ) φ 1 ( x , y ) = ηδ ( x ) (Dirac delta on line x = 0) in Einstein equations generated by given Action Previous works with discontinuous BCs in (super)gravity φ 1 φ 1 φ 1 = θ ( x ) and m φ = 0 analytical Janus solutions [D. Bak, M. Gutperle, S. Hirano] φ 1 = θ ( x ) and m φ = 0 at T > 0 numerical and analytical Janus black φ 1 φ 1 holes in d = 2 + 1 [D. Bak, M. Gutperle, R. A. Janik] φ 1 = δ ( x ) and m 2 φ 1 φ 1 φ = − 2 with SUSY, analytical and scale invariant [E. D’Hoker et al. ] → various non-trivial p -forms → hard to generalise to black hole case ( T > 0) → not very useful in AdS/CMT Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 6 / 18

First approach and ”emergent” Supergravity “Emergent” supergravity η z 2 linearised analysis gives φ lin = π ( x 2 + z 2 ) → suggests conformal symmetry along defect line ( SO (2 , 2)) new coordinates r 2 = x 2 + z 2 , tan α = x / z ( φ lin ( α ) = η π cos 2 ( α )) full solution with this symmetry cannot be found! dynamical generation of source φ 1 φ 1 φ 1 ∼ δ ( x ) + 1 / | x | + ... ! a way out – modification of the scalar potential V ( φ ) Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 7 / 18

First approach and ”emergent” Supergravity “Emergent” supergravity Supersymmetric potential 1 SO (2 , 2) symmetry fixes uniquely V ( φ ) √ V ( φ ) = − 6 cosh( φ/ 3) 2 the same potential arises from reduction & truncation of D=11 SUGRA on AdS 4 × S 7 ! [M. Cvetic et al.] 3 with such potential φ 1 ( x , y ) φ 1 ( x , y ) φ 1 ( x , y ) = ηδ ( x ) & SO (2 , 2) symmetry can be both fulfilled Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 8 / 18

Full Solutions T = 0 (no horizon) √ we take the supersymmetric potential V ( φ ) = − 6 cosh( φ/ 3) in action metric ansatz: � p 2 + dr 2 − dt 2 + dy 2 � d α 2 1 ds 2 = A ( α ) 2 r 2 solving both using numerics (pseudospectral collocation method on Chebyschev grid) and perturbative expansion in parameter η = φ (0) Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 9 / 18

Full Solutions T = 0 (no horizon) 1.2 1.2 1.0 1.0 0.8 0.8 ϕ ( α ) A ( α ) 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 α α Metric and scalar field for φ (0) = 1 . 2 Points → numerical solution with N = 47 spectral grid Lines → fourth order perturbative solution Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 10 / 18

Full Solutions T > 0 case problem no longer 1-dimensional – we only replace x coordinate with α = tan( x / z ) we use the most general metric ansatz: � − (1 − z ) G ( z ) H 1 ( α, z ) dt 2 + H 2 ( α, z ) dz 2 1 ds 2 = z 2 (1 − z ) G ( z ) � S 1 ( α, z )( d α + F ( α, z ) dz ) 2 + S 2 ( α, z ) dy 2 + with G ( z ) = 1 + z + z 2 . DeTurk method stands for gauge-fixing [M. Headrick, et al.] numerical method was based on spectral collocation method on Chebyschev grid [P. Grandclement and J. Novak] Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 11 / 18

Full Solutions T > 0 case Scalar field (right) and metric component F ( α, z ) (left) for φ (0) = 1 . 0. Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 12 / 18

Full Solutions An observable: Entanglement Entropy Holographic entanglement entropy [S. Ryu, T. Takayanagi] – EE of some region is proportional to the area of minimal a surface whose boundary is boundary of that region. For strip of width 2 L around the defect the generic form of EE should be: S = 1 ǫ − B L A strip for which we calculated entanglement entropy, with a sketch of minimal surface used in calculation. Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 13 / 18

Full Solutions An observable: Entanglement Entropy 0.0 0.6 0.4 - 0.1 0.2 S ( L ) - 0.2 S ( L ) 0.0 - 0.2 - 0.3 - 0.4 - 0.6 - 0.4 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 L L Left: EE of pure AdS (line) and defect geometry φ (0) = 2. Right: EE difference between pure AdS and: standard AdS-black hole (red points), defected black hole (blue dots). Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 14 / 18

Conclusions and further directions Conclusions We examined a novel setup in numerical GR, and developed methods to handle it It turned out that conformal defect exists only in Supergravity (scalar √ potential is fixed to be V ( φ ) = − 6 cosh( φ/ 3)) In the theory with defect, entanglement entropy of a strip is lower than in theory without it Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 15 / 18

Conclusions and further directions Further directions Construction of holographic lattice from such local defects Introduction of nonzero chemical potential (gauge field in bulk) Computation of various quantities – i.e. optical conductivity or heat transport Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 16 / 18

Conclusions and further directions Thank you for your attention Piotr Witkowski (WFAIS UJ) Conformal defects in SUGRA 17 / 18