Phase Diagrams for Melonic Tensor / Disordered Models Fidel I. Schaposnik Massolo Institut des Hautes ´ Etudes Scientifiques Based on 1707.03431 and 1810.xxxxx In collaboration with T. Azeyanagi and F. Ferrari Critical Phenomena in Statistical Mechanics and Quantum Field Theory Princeton Center for Theoretical Science - October 5, 2018
Reverse engineering black holes Desirable features of a black hole model A) Macroscopic space-time description [Schwarzschild - 1916; Kerr - 1963; . . . ] Definition of the horizon [ Finkelstein - 1958; . . . ] Description of the interior [ Kruskal - 1960; Penrose, Hawking - 1965, 1970; . . . ] Entropy S = A / 4 G N [ Bekenstein - 1972; Bardeen, Carter, Hawking - 1973; . . . ] Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Reverse engineering black holes Desirable features of a black hole model A) Macroscopic space-time description [Schwarzschild - 1916; Kerr - 1963; . . . ] Definition of the horizon [ Finkelstein - 1958; . . . ] Description of the interior [ Kruskal - 1960; Penrose, Hawking - 1965, 1970; . . . ] Entropy S = A / 4 G N [ Bekenstein - 1972; Bardeen, Carter, Hawking - 1973; . . . ] B) Consequences of the existence of the horizon Loss of time-reversal invariance Chaotic dynamics Unitarity problems / Information loss paradox Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Reverse engineering black holes Desirable features of a black hole model A) Macroscopic space-time description [Schwarzschild - 1916; Kerr - 1963; . . . ] Definition of the horizon [ Finkelstein - 1958; . . . ] Description of the interior [ Kruskal - 1960; Penrose, Hawking - 1965, 1970; . . . ] Entropy S = A / 4 G N [ Bekenstein - 1972; Bardeen, Carter, Hawking - 1973; . . . ] B) Consequences of the existence of the horizon Loss of time-reversal invariance Chaotic dynamics Unitarity problems / Information loss paradox Can we study black holes starting from (B) and getting to (A) through holography? Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Guiding principles Existence of parameter N Loss of time-reversal invariance / Unitarity problems � � Thermodynamical irreversibility (limit N → ∞ ) Chaotic dynamics F β ( t ) ∼ � ˆ O (0) ˆ O ( t ) ˆ O (0) ˆ O ( t ) � β, con . ∝ e λ L t Lyapunov exponent saturates bound for black holes λ L ≤ 2 π [ Maldacena, Shenker, Stanford - 2015 ] β Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
SYK model [ Sachdev, Ye - 1993; Kitaev - 2015 ] N Majorana fermions ψ 1 , . . . , ψ N in 0 + 1 dim. with Hamiltonian � H = J ijkl ψ i ψ j ψ k ψ l i < j < k < l Quenched disorder � σ 2 ( J ijkl ) ∝ J 2 � · � ≡ dJ ijkl p ( J ijkl ) � · � J ijkl with Some nice features Approximate conformal symmetry in IR = ⇒ NAdS 2 /NCFT 1 Analytical treatment for N → ∞ Explicit numerics for small N ( |H| = 2 N / 2 ) Saturates bound for λ L Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
SYK model [ Sachdev, Ye - 1993; Kitaev - 2015 ] N Majorana fermions ψ 1 , . . . , ψ N in 0 + 1 dim. with Hamiltonian � H = J ijkl ψ i ψ j ψ k ψ l i < j < k < l Quenched disorder � σ 2 ( J ijkl ) ∝ J 2 � · � ≡ dJ ijkl p ( J ijkl ) � · � J ijkl with Some nice features Approximate conformal symmetry in IR = ⇒ NAdS 2 /NCFT 1 Analytical treatment for N → ∞ Explicit numerics for small N ( |H| = 2 N / 2 ) Saturates bound for λ L Not a proper Quantum Field Theory :-( Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Vector and matrix models: an overview Large D vector models Large N matrix models X a Field content φ µ with µ = 1 , . . . , D b with a , b = 1 , . . . , N U ( N ) 2 or U ( N ) Symmetry O ( D ) φ 2 � k for k = 1 , . . . Tr( XX † XX † · · · ) , . . . � � Interactions D V − P + ϕ = D 1 − ℓ N V − P + f = N 2 − 2 g Diag. scaling cacti diagrams Leading planar diagrams (auxiliary tree-level) cond. mat. ph., CFT, nucl. ph., QCD, Applications higher spin gravity string theory Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
New large N limit [ Ferrari - 2017; Ferrari, Rivasseau, Valette - 2017 ] O ( d ) × U ( n ) 2 model for a vector of complex matrices � � X µ 1 X † µ 2 · · · X µ 2 s − 1 X † Interaction vertices are V B = Tr µ 2 s � � 1 X µ X † � � + � Usual scaling S = nd 2 Tr B t B V B ( X µ ) µ Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
New large N limit [ Ferrari - 2017; Ferrari, Rivasseau, Valette - 2017 ] O ( d ) × U ( n ) 2 model for a vector of complex matrices � � X µ 1 X † µ 2 · · · X µ 2 s − 1 X † Interaction vertices are V B = Tr µ 2 s � � 1 X µ X † � � + � Usual scaling S = nd 2 Tr B t B V B ( X µ ) µ Enhance ’t Hooft coupling t B for V B as t B = λ B d E ( B ) with E ( B ) ≥ 0 In the n → ∞ limit � n 2 − 2 g F g F = g ≥ 0 In the d → ∞ limit ( g fixed) � d 1+ g − k / 2 F g , k F g = k ≥ 0 Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Quartic models for fermionic matrices Fermionic matrices in 0 + 1 dimensions = 1 � � ( ψ † µ ) a b = ( ψ b µ a ) † ψ a µ b , ( ψ † ν ) c nd δ µν δ a d δ c with d b Desired features U ( n ) × O ( d ) invariance Single trace Hamiltonian � � ψ † Quadratic mass term m Tr µ ψ µ Quartic interactions + n � � � � ψ µ ψ † ν ψ µ ψ † ψ † ν ψ µ ψ † Tr = − Tr ν ψ µ , etc. ν d (Combinations ψ µ ψ µ and ψ † µ ψ † µ are suppressed) Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Inequivalent interactions Crossing interactions = ⇒ E ( B ) = 1 / 2 λ Tr( ψ † µ ψ ν ψ † µ ψ ν ) λ ′ Tr( ψ † µ ψ † ν ψ µ ψ ν ) ξ Tr( ψ † µ ψ ν ψ µ ψ ν ) ξ ∗ Tr( ψ µ ψ † ν ψ † µ ψ † ν ) Non-crossing interactions = ⇒ E ( B ) = 0 κ Tr( ψ † µ ψ µ ψ † ν ψ ν ) κ ′ Tr( ψ µ ψ † µ ψ ν ψ † κ Tr( ψ † µ ψ µ ψ ν ψ † κ ∗ Tr( ψ µ ψ † µ ψ † ν ) ˜ ν ) ˜ ν ψ ν ) Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Diagramatics Leading order diagrams are generated by melonic moves λ ′ λ ′ λ ′ λ ′ λ λ ξ ∗ ξ ∗ ξ ∗ ξ ∗ ξ ξ ξ ξ κ ∗ κ κ, ˜ ˜ κ ′ κ ∗ ˜ κ, ˜ Mixed structures ( λ, ξ ), . . . are avoided if we require � Tr( ψ µ ψ µ ) � = � Tr( ψ † µ ψ † µ ) � = 0 Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Melon trees Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Two basic models Charge preserving model with symmetry O ( d ) × U ( n ) 2 √ � � m ψ † µ ψ µ + 1 d ψ µ ψ † ν ψ µ ψ † H Q = nd Tr 2 λ ν λ ′ Tr( ψ † µ ψ † ν ψ µ ψ ν ) interaction renormalizes λ �→ λ + 2 λ ′ Charge violating model with symmetry O ( d ) × U ( n ) √ � � �� m ψ † µ ψ µ + 1 ξψ † µ ψ ν ψ µ ψ ν + ξ ∗ ψ † µ ψ † ν ψ † H � Q = nd Tr d µ ψ ν 2 , , . . . Melonic-dominated models = ⇒ Physics similar to SYK Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Disordered model formulation µ b has d × n 2 fermionic degrees of freedom ψ a � � Hilbert space is 2 dn 2 dimensional :-( Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Disordered model formulation µ b has d × n 2 fermionic degrees of freedom ψ a � � Hilbert space is 2 dn 2 dimensional :-( � � χ i , χ † = δ i Equivalent disordered models with N Dirac fermions j j i χ i + λ ij H Q = m χ † ˜ N 3 / 2 χ † i χ † kl j χ k χ l ξ i i χ j χ k χ l + ξ ijk i χ i + jkl H � Q = m χ † ˜ N 3 / 2 χ † N 3 / 2 χ † i χ † j χ † k χ l l Hilbert space is 2 N dimensional :-) WARNING: Equivalence is partial and only to leading large N (= n 2 d ) order! Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Procedure Euclidean two-point function � � �� ψ µ ( t ) ψ † G ( t ) = Tr T µ β Fermionic perturbation theory m ≫ λ = ⇒ Exp. around decoupled fermionic oscillators T ≫ λ = ⇒ Non-standard (SYK-like) perturbation theory � G 0 ( t ) = e m ( β − t ) 1 2 sign ( t ) m → 0 , then T → 0 e m β + 1 = e − mt Θ( t ) T → 0 , then m → 0 Feynman diagram structure = ⇒ Schwinger-Dyson equations = + + · · · Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
Schwinger-Dyson equations Expanding G ( t ) in Matsubara-Fourier modes G ( t ) = 1 ω k = 2 π � G k e − i ω k t k ∈ Z + 1 , β k with 2 β k The Schwinger-Dyson equations are � = λ 2 G ( t ) 2 G ( − t ) Σ Q ( t ) G − 1 = m − i ω k +Σ k G ( t ) 2 + 3 G ( − t ) 2 � k = − 1 4 | ξ | 2 G ( t ) � Σ � Q ( t ) Now Define S eff with Schwinger-Dyson equations as saddle-points Relate its on-shell value to the free energy F = − 1 β log Tr e − β H Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models
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