Large N fermionic tensor models in d = 2

Transcription

1 Large N fermionic tensor models in d = 2 Sylvain Carrozza Quantum spacetime and the Renormalization roup 2018 Bad Honnef Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

2 Context early 10s: universal large N expansion for tensor models Main feature: dominated by melon diagrams. [urau, Bonzom, Rivasseau,...] Primary applications: Random geometry in d 3 Tensor Field Theory roup Field Theory (renormalization) [Talk by T. Koslowski] [Talk by J. Ben eloun] [Talk by D. Oriti] Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

3 Context early 10s: universal large N expansion for tensor models Main feature: dominated by melon diagrams. [urau, Bonzom, Rivasseau,...] Primary applications: Random geometry in d 3 Tensor Field Theory roup Field Theory (renormalization) [Talk by T. Koslowski] [Talk by J. Ben eloun] [Talk by D. Oriti] 16 17: applications to holography [Witten; Klebanov, Tarnopolsky;... ] and more generally to local QFTs on fixed space-time backgrounds. Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

4 Context early 10s: universal large N expansion for tensor models Main feature: dominated by melon diagrams. [urau, Bonzom, Rivasseau,...] Primary applications: Random geometry in d 3 Tensor Field Theory roup Field Theory (renormalization) [Talk by T. Koslowski] [Talk by J. Ben eloun] [Talk by D. Oriti] 16 17: applications to holography [Witten; Klebanov, Tarnopolsky;... ] and more generally to local QFTs on fixed space-time backgrounds. In this talk: large N QFT in d = 2 ariv: JHEP 2018 with D. Benedetti (Paris Sud), R. urau (-Polytechnique) and A. Sfondrini (ETH). Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

5 Outline 1 SYK, tensor quantum mechanics and holography 2 Tensor field theory in d=2 Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

6 SYK, tensor quantum mechanics and holography 1 SYK, tensor quantum mechanics and holography 2 Tensor field theory in d=2 Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

7 SYK-like tensor models Sachdev-Ye-Kitaev models = disordered systems of N Majorana fermions [Sachdev, Ye, eorge, Parcollet 90s...; Kitaev 15, Maldacena, Stanford, Polchinski, Rosenhaus...] H int J i1 i 2 i 3 i 4 ψ i1 ψ i2 ψ i3 ψ i4, Ji1 i 2 i 3 i 4 0, Ji 2 1 i 2 i 3 i 4 J2 N 3 Many interesting properties: solvable at large N emergent conformal symmetry at strong coupling maximal quantum chaos holography in low dimension: NAdS 2 /NCFT 1 Same melonic large N limit as tensor models [Witten 16] SYK-like quantum-mechanical models: same qualitative properties at large N and strong coupling; no disorder. New class of QFTs with solvable large N limits. Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

8 O(N) 3 tensor models (d = 0) Statistical model for T i1 i 2 i 3 transforming under O(N) 3 as [urau, Bonzom, Rivasseau,... 10s; Tanasa, SC 15] T i1 i 2 i 3 O (1) i 1 j 1 O (2) i 2 j 2 O (3) i 3 j 3 T j1 j 2 j 3 Invariant action: S(T ) = 1 2 T i 1 i 2 i 3 T i1 i 2 i 3 = λ1 N 3/2 T i 6 i 2 i 3 T i1 i 4 i 3 T i6 i 4 i 5 T i1 i 2 i 5 + λ2 N 2 T i 6 i 2 i 3 T i1 i 2 i 3 T i6 i 4 i 5 T i1 i 4 i λ1 N 3/2 + λ2 N 2 + Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

9 O(N) 3 tensor models (d = 0) Statistical model for T i1 i 2 i 3 transforming under O(N) 3 as [urau, Bonzom, Rivasseau,... 10s; Tanasa, SC 15] T i1 i 2 i 3 O (1) i 1 j 1 O (2) i 2 j 2 O (3) i 3 j 3 T j1 j 2 j 3 Invariant action: S(T ) = 1 2 T i 1 i 2 i 3 T i1 i 2 i 3 = λ1 N 3/2 T i 6 i 2 i 3 T i1 i 4 i 3 T i6 i 4 i 5 T i1 i 2 i 5 + λ2 N 2 T i 6 i 2 i 3 T i1 i 2 i 3 T i6 i 4 i 5 T i1 i 4 i λ1 N 3/2 + λ2 N 2 + Large N expansion indexed by urau degree ω F N := ln [dt ]e S(T ) = N 3 ω F ω and dominated by melon diagrams (ω=0). ω N/2 Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

10 Klebanov Tarnopolsky model Tensor quantum mechanics of N 3 Majorana fermions: [Klebanov, Tarnopolsky 16...] ( i S = dt 2 ψ i 1 i 2 i 3 tψ i1 i 2 i 3 + λ ) 4N ψ 3/2 i 1 i 2 i 3 ψ i4 i 5 i 3 ψ i4 i 2 i 6 ψ i1 i 5 i 6 Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

11 Klebanov Tarnopolsky model Tensor quantum mechanics of N 3 Majorana fermions: [Klebanov, Tarnopolsky 16...] ( i S = dt 2 ψ i 1 i 2 i 3 tψ i1 i 2 i 3 + λ ) 4N ψ 3/2 i 1 i 2 i 3 ψ i4 i 5 i 3 ψ i4 i 2 i 6 ψ i1 i 5 i 6 Melonic dominance at large N closed Schwinger-Dyson equation: 1 = 1 free + Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

12 Klebanov Tarnopolsky model Tensor quantum mechanics of N 3 Majorana fermions: [Klebanov, Tarnopolsky 16...] ( i S = dt 2 ψ i 1 i 2 i 3 tψ i1 i 2 i 3 + λ ) 4N ψ 3/2 i 1 i 2 i 3 ψ i4 i 5 i 3 ψ i4 i 2 i 6 ψ i1 i 5 i 6 Melonic dominance at large N closed Schwinger-Dyson equation: 1 = 1 free + Strong coupling neglect free : dt (t, t )Σ(t, t ) = δ(t t ), Σ(t, t ) = λ 2 [(t, t )] 3 emergent reparametrization symmetry t f (t), (t, t ) [f (t)f (t )] 1/4 (f (t), f (t )) Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

13 Symmetry breaking 1 = 1 free + Strong coupling conformal solution: (t 2, t 1) sgn(t2 t1) t 2 t 1 1/2 Spontaneous breaking to SL(2, R) oldstone modes φ Diff(R)/SL(2, R)? free term explicit breaking non-zero Schwarzian effective action S eff [φ] 1 dt {φ, t} λ ({φ, t} is the Schwarzian derivative) [Maldacena, Stanford 16] Effective action common to SYK and tensor quantum mechanics. Additional soft contributions are present in tensor models e.g. O(N) 3 sigma model. [Minwalla et al. 17; Benedetti, urau 18] Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

14 Duality with Jackiw-Teitelboim gravity Near-horizon geometry of near-extremal Reissner Nordström BHs AdS 2 S 2 Near-horizon dynamics captured by dilaton Jackiw-Teitelboim gravity S φ 0 d 2 x h R h + 2φ 0 K (1) M M + d 2 x h φ (R h + 2) + 2 φ b K (2) M M Euclidean solutions = cut-outs from the hyperbolic plane ρ τ ds 2 = dρ 2 + sinh 2 ρ dτ 2 Non-trivial zero modes of (1) Diff(R)/SL(2, R) symmetry. Action (2) explicitly breaks the symmetry Schwarzian effective action! [Maldacena, Stanford, Yang 16] Nice review: ábor Sárosi ariv: Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

15 Tensor field theory in d=2 1 SYK, tensor quantum mechanics and holography 2 Tensor field theory in d=2 Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

16 Motivations Unlike SYK, tensor models naturally fit in the framework of local quantum field theory. Natural research programme: Investigate the properties of melonic large N QFTs in d 2. Bosonic: [iombi, Klebanov, Tarnopolsky 17] Fermionic: [Prakash, Sinah 17] [Benedetti, SC, urau, Sfondrini 17] Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

17 Action Tensorial ross-neveu model, with e.g. O(N) Majorana fermions. S N = 1 2 d 2 x ψ i1 i 2 i 3 / ψ i1 i 2 i 3 λ0 4N 3 d 2 x S S =S,V,P λ 1 4N 2 d 2 x =S,V,P λ 2 4N 3/2 d 2 x Spinorial contractions with: 1l (S), γ µ (V) or γ 5 (P) insertions. Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

18 Action Tensorial ross-neveu model, with e.g. O(N) Majorana fermions. S N = 1 2 d 2 x ψ i1 i 2 i 3 / ψ i1 i 2 i 3 λ0 4N 3 d 2 x 1 λ 0 > 0, λ S 1 > 0, λ 2 = 0: S S =S,V,P λ 1 4N 2 d 2 x =S,V,P λ 2 4N 3/2 d 2 x Spinorial contractions with: 1l (S), γ µ (V) or γ 5 (P) insertions. Tadpole diagrams non-perturbative generation of mass in the IR Asymptotic freedom. 2 λ 0 = λ S 1 = 0 and λ 2 0: Melonic Schwinger-Dyson equation Free part still relevant at strong coupling no IR conformal regime Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

19 S λ 0 + λ 1 Spontaneous generation of mass and asymptotic freedom S Σ = + Large-N Schwinger-Dyson equations: 1 (x, x ) = 1 0 (x, x ) Σ(x, x ), Σ(x, x ) = (λ S 0 + λ S 1 )Tr[(x, x)]δ(x, x ) ap equation: m = 2m(λ S 0 + λ S 1 ) Non-perturbative solution: q Λ d 2 q 1 (2π) 2 q 2 + m 2 m(λs 0 + λ S 1 ) 1 π ln Λ m ( m = Λ exp π ) λ S 0 + λs 1 Callan-Symanzik asymptotic freedom: β S 1 = 1/π(λ S 1 ) 2 < 0 Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

20 λ 1 S + λ 2 melonic Schwinger-Dyson equation Y 1 = 1 free + Y Same melonic-like equation as in d = 1. Main difference: free cannot be neglected in the IR limit. Is there a non-trivial IR fixed point nonetheless? Strategy: 1 Ansatz: Ĝ(p) 1 i/p + m. 2 Solve gap equation and find conditions to cancel m. 3 Impose stability of λ S 0 = λs 1 = 0 and other conditions under radiative correction. 4 Impose that the remaining coupling(s) have 0 beta function. Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

21 Results 1 Ansatz: Ĝ(p) 1 i/p + m. 2 Solve gap equation and find conditions to cancel m: (λ S 2 ) 2 + (λ P 2 ) 2 4λ V 2 λ P 2 = 0 3 Impose stability of λ S 0 = λ S 1 = 0 and other conditions under radiative correction: (λ S,1 2 ) 2 = (λ P,1 2 ) 2 = λ 2 0 and λ V,1 2 = 0, or (λ S,1 2 ) 2 = (λ P,1 2 ) 2 = 0 and λ V,1 2 = λ 0. Note: satisfied by models invariant under continuous chiral symmetry. 4 Impose that the remaining coupling(s) have 0 beta function: non-trivial wave-function renormalization β λ = 3 π 2 λ3 (perturbative calculation) Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

22 Weakly interacting fixed point in d = 2 ε Very interesting feature in d = 2 ε: β β = 3 π 2 (λ)3 β (ε) = ελ+ 3 π 2 (λ)3 UV ε IR λ weakly interacting IR fixed point λ 2 ε, analogous to Wilson-Fisher in bosonic ϕ 4 4 ε. Conjecture: becomes strongly coupled and governs the conformal regime of SYK in the limit ε 1. ε-expansion for SYK Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

23 Conclusion and outlook Recent developments in tensor models beyond the intended applications to random geometry / discrete quantum gravity: tensor quantum mechanics reproducing salient features of the SYK many-body system unanticipated connections to other quantum gravity approaches: AdS 2 /CFT 1, black holes, higher spins... Motivates a more systematic investigation of such large N QFTs higher dimensions (anti)symmetrized tensors [Klebanov, Tarnopolsky 17; Benedetti, SC, urau, Sfondrini 17] multi-matrix models [Ferrari 17;...] condensed matter propagators [Ben eloun, Rivasseau 17] hierarchy of Schwinger-Dyson equations [Perez-Sanchez, Wulkenhaar 17] Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

24 Conclusion and outlook Recent developments in tensor models beyond the intended applications to random geometry / discrete quantum gravity: tensor quantum mechanics reproducing salient features of the SYK many-body system unanticipated connections to other quantum gravity approaches: AdS 2 /CFT 1, black holes, higher spins... Motivates a more systematic investigation of such large N QFTs higher dimensions (anti)symmetrized tensors [Klebanov, Tarnopolsky 17; Benedetti, SC, urau, Sfondrini 17] multi-matrix models [Ferrari 17;...] condensed matter propagators [Ben eloun, Rivasseau 17] hierarchy of Schwinger-Dyson equations [Perez-Sanchez, Wulkenhaar 17] Controversial statement: holography in dimensions is very interesting. Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

25 Conclusion and outlook Recent developments in tensor models beyond the intended applications to random geometry / discrete quantum gravity: tensor quantum mechanics reproducing salient features of the SYK many-body system unanticipated connections to other quantum gravity approaches: AdS 2 /CFT 1, black holes, higher spins... Motivates a more systematic investigation of such large N QFTs higher dimensions (anti)symmetrized tensors [Klebanov, Tarnopolsky 17; Benedetti, SC, urau, Sfondrini 17] multi-matrix models [Ferrari 17;...] condensed matter propagators [Ben eloun, Rivasseau 17] hierarchy of Schwinger-Dyson equations [Perez-Sanchez, Wulkenhaar 17] Controversial statement: holography in dimensions is very interesting. Provocative question: convergence between tensor models for discrete quantum gravity and holographic tensor models? Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17

26 Conclusion and outlook Recent developments in tensor models beyond the intended applications to random geometry / discrete quantum gravity: tensor quantum mechanics reproducing salient features of the SYK many-body system unanticipated connections to other quantum gravity approaches: AdS 2 /CFT 1, black holes, higher spins... Motivates a more systematic investigation of such large N QFTs higher dimensions (anti)symmetrized tensors [Klebanov, Tarnopolsky 17; Benedetti, SC, urau, Sfondrini 17] multi-matrix models [Ferrari 17;...] condensed matter propagators [Ben eloun, Rivasseau 17] hierarchy of Schwinger-Dyson equations [Perez-Sanchez, Wulkenhaar 17] Controversial statement: holography in dimensions is very interesting. Provocative question: convergence between tensor models for discrete quantum gravity and holographic tensor models? Thank you for your attention! Sylvain Carrozza Large N fermionic tensor models Bad Honnef 20/6/ / 17