New results on the entanglement entropy of singular regions in CFTs Pablo Bueno QUIST 2019, YITP, Kyoto University June 22, 2019 Pablo Bueno EE of singular regions in CFTs 22/06/2019 1 / 18
Talk based on arXiv:1904.11495 with Horacio Casini and William Witczak-Krempa Pablo Bueno EE of singular regions in CFTs 22/06/2019 1 / 18
Talk based on arXiv:1904.11495 with Horacio Casini and William Witczak-Krempa + some mentions to previous work Phys.Rev. B96 (2017) no.3, 035117 with Lauren Sierens , Rajiv Singh , Rob Myers , Roger Melko Phys.Rev. B93 (2016) 045131 with William Witczak-Krempa JHEP 1512 (2015) 168 JHEP 1508 (2015) 068 with Rob Myers Phys.Rev.Lett. 115 (2015) 021602 with Rob Myers , William Witczak-Krempa Pablo Bueno EE of singular regions in CFTs 22/06/2019 1 / 18
Outline 1 EE of singular regions in CFTs: known facts and conjectures 2 EE of singular regions in CFTs: New results Vertex-induced universal terms Wedge entanglement vs corner entanglement Singular regions and EE divergences Pablo Bueno EE of singular regions in CFTs 22/06/2019 2 / 18
EE of singular regions in CFTs: known facts and conjectures 1. EE of singular regions in CFTs: known facts and conjectures Pablo Bueno EE of singular regions in CFTs 22/06/2019 2 / 18
EE of singular regions in CFTs: known facts and conjectures Entanglement entropy in CFTs Rényi/Entanglement entropy of subregions is intrinsically divergent for QFTs, “area law” divergence built in. Pablo Bueno EE of singular regions in CFTs 22/06/2019 3 / 18
EE of singular regions in CFTs: known facts and conjectures Entanglement entropy in CFTs Rényi/Entanglement entropy of subregions is intrinsically divergent for QFTs, “area law” divergence built in. Luckily, well-defined “uni- versal terms”. [Even for those, some care must be taken when theory contains superselection sectors; see Javier’s talk & Horacio’s last lecture; subtlety ignored here] Pablo Bueno EE of singular regions in CFTs 22/06/2019 3 / 18
EE of singular regions in CFTs: known facts and conjectures Entanglement entropy in CFTs Rényi/Entanglement entropy of subregions is intrinsically divergent for QFTs, “area law” divergence built in. Luckily, well-defined “uni- versal terms”. [Even for those, some care must be taken when theory contains superselection sectors; see Javier’s talk & Horacio’s last lecture; subtlety ignored here] Given smooth spatial entangling region V with characteristic length scale H , � d − 1 b 1 H H d − 2 H d − 4 2 s univ δ + ( − 1) , (odd d ) , S ( d ) n = b d − 2 δ d − 2 + b d − 4 δ d − 4 + · · · + log � H n d − 2 b 2 H 2 � 2 s univ δ 2 + ( − 1) + b 0 , (even d ) . n δ where δ , UV regulator. Pablo Bueno EE of singular regions in CFTs 22/06/2019 3 / 18
EE of singular regions in CFTs: known facts and conjectures Universal terms in d = 3 , 4 Even d : s univ ⇔ logarithmic term, linear combination of local integrals on Σ ≡ ∂V n weighted by theory-dependent “charges”. Odd d : s univ ⇔ constant term, no longer controlled by local integral on Σ ≡ ∂V . n Less robust than logarithmic terms ⇒ May use Mutual Information as a regulator. [Casini; Casini, Huerta, Myers, Yale] Pablo Bueno EE of singular regions in CFTs 22/06/2019 4 / 18
EE of singular regions in CFTs: known facts and conjectures Universal terms in d = 3 , 4 Even d : s univ ⇔ logarithmic term, linear combination of local integrals on Σ ≡ ∂V n weighted by theory-dependent “charges”. d = 4, Σ ⇐ smooth surface [Solodukhin; Fursaev] � H � � � � � = − 1 � k 2 − f c ( n ) s univ f a ( n ) R + f b ( n ) W log n 2 π δ Σ Σ Σ where f a (1) = a , f b (1) = f c (1) = c trace-anomaly coefficients. Geometry and theory dependences factorize term by term. Odd d : s univ ⇔ constant term, no longer controlled by local integral on Σ ≡ ∂V . n Less robust than logarithmic terms ⇒ May use Mutual Information as a regulator. [Casini; Casini, Huerta, Myers, Yale] Pablo Bueno EE of singular regions in CFTs 22/06/2019 4 / 18
EE of singular regions in CFTs: known facts and conjectures Universal terms in d = 3 , 4 Even d : s univ ⇔ logarithmic term, linear combination of local integrals on Σ ≡ ∂V n weighted by theory-dependent “charges”. d = 4, Σ ⇐ smooth surface [Solodukhin; Fursaev] � H � � � � � = − 1 � k 2 − f c ( n ) s univ f a ( n ) R + f b ( n ) W log n 2 π δ Σ Σ Σ where f a (1) = a , f b (1) = f c (1) = c trace-anomaly coefficients. Geometry and theory dependences factorize term by term. Odd d : s univ ⇔ constant term, no longer controlled by local integral on Σ ≡ ∂V . n Less robust than logarithmic terms ⇒ May use Mutual Information as a regulator. [Casini; Casini, Huerta, Myers, Yale] d = 3, Σ ⇐ smooth curve H S (3) δ − s univ = b 1 n n e.g., Σ = S 1 , then s univ = free energy of CFT on S 3 [Casini, Huerta, Myers; 1 Dowker] , non-local quantity. Geometry and theory dependences entangled. Pablo Bueno EE of singular regions in CFTs 22/06/2019 4 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Situation changes when geometric singularities present on Σ. Consider corner of opening angle Ω on a time slice of a d = 3 CFT, H � H � S corner δ − a (3) = b 1 n (Ω) log + b 0 EE δ Pablo Bueno EE of singular regions in CFTs 22/06/2019 5 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Situation changes when geometric singularities present on Σ. Consider corner of opening angle Ω on a time slice of a d = 3 CFT, H � H � S corner δ − a (3) = b 1 n (Ω) log + b 0 EE δ Logarithmic universal term arises, controlled by a (3) n (Ω). Vast literature, free fields, lattice models, holography, etc. [Many people] Angular and theory dependences do not dis- entangle ( e.g., simple result for holographic theories [Drukker, Gross, Ooguri; Hirata, Takayanagi] vs horrendous expressions for free fields [Casini, Huerta] ). Pablo Bueno EE of singular regions in CFTs 22/06/2019 5 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Still, remarkable amount of universality observed [PB, Myers, Witczak-Krempa] σ = π 2 1 (Ω) = σ (Ω − π ) 2 + . . . , a (3) 24 C T (1) Conjectured to hold ∀ CFTs in d = 3. Pablo Bueno EE of singular regions in CFTs 22/06/2019 6 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Still, remarkable amount of universality observed [PB, Myers, Witczak-Krempa] σ = π 2 1 (Ω) = σ (Ω − π ) 2 + . . . , a (3) 24 C T (1) Conjectured to hold ∀ CFTs in d = 3. Proven! [Faulkner, Leigh, Parrikar] Pablo Bueno EE of singular regions in CFTs 22/06/2019 6 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Still, remarkable amount of universality observed [PB, Myers, Witczak-Krempa] σ = π 2 1 (Ω) = σ (Ω − π ) 2 + . . . , a (3) 24 C T (1) Conjectured to hold ∀ CFTs in d = 3. Proven! [Faulkner, Leigh, Parrikar] Stress tensor charge C T provides natural normalization. Pablo Bueno EE of singular regions in CFTs 22/06/2019 6 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Still, remarkable amount of universality observed [PB, Myers, Witczak-Krempa] σ = π 2 1 (Ω) = σ (Ω − π ) 2 + . . . , a (3) 24 C T (1) Conjectured to hold ∀ CFTs in d = 3. Proven! [Faulkner, Leigh, Parrikar] Stress tensor charge C T provides natural normalization. 1 (Ω) ≥ π 2 C T Universal lower bound ⇔ a (3) log[1 / sin(Ω / 2)] [PB, Witczak-Krempa] 3 Pablo Bueno EE of singular regions in CFTs 22/06/2019 6 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Still, remarkable amount of universality observed [PB, Myers, Witczak-Krempa] σ = π 2 1 (Ω) = σ (Ω − π ) 2 + . . . , a (3) 24 C T (1) Conjectured to hold ∀ CFTs in d = 3. Proven! [Faulkner, Leigh, Parrikar] Stress tensor charge C T provides natural normalization. 1 (Ω) ≥ π 2 C T Universal lower bound ⇔ a (3) log[1 / sin(Ω / 2)] [PB, Witczak-Krempa] 3 Analogous result to (1) for (hyper)-cones in general d . [PB, Myers; Mezei; Miao] Pablo Bueno EE of singular regions in CFTs 22/06/2019 6 / 18
EE of singular regions in CFTs: known facts and conjectures Corner entanglement in d = 3 Still, remarkable amount of universality observed [PB, Myers, Witczak-Krempa] σ = π 2 1 (Ω) = σ (Ω − π ) 2 + . . . , a (3) 24 C T (1) Conjectured to hold ∀ CFTs in d = 3. Proven! [Faulkner, Leigh, Parrikar] Stress tensor charge C T provides natural normalization. 1 (Ω) ≥ π 2 C T Universal lower bound ⇔ a (3) log[1 / sin(Ω / 2)] [PB, Witczak-Krempa] 3 Analogous result to (1) for (hyper)-cones in general d . [PB, Myers; Mezei; Miao] Rényi entropy generalization is trickier... Pablo Bueno EE of singular regions in CFTs 22/06/2019 6 / 18
EE of singular regions in CFTs: known facts and conjectures Cone entanglement in d = 4 Fundamentally different from corner, theory dependence completely disentangled from angular dependence (which is the same for all CFTs) [Klebanov, Nishioka, Pufu, Safdi] H 2 � H � � H � n (Ω) log 2 S (4) cone δ 2 − a (4) = b 2 + b 0 log + c 0 n δ δ 4 f b ( n )cos 2 Ω n (Ω) = 1 a (4) ∀ CFTs sin Ω Pablo Bueno EE of singular regions in CFTs 22/06/2019 7 / 18
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