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Phase Diagrams for Melonic Tensor / Disordered Models Fidel I. - - PowerPoint PPT Presentation

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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

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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

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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/4GN [ Bekenstein - 1972; Bardeen, Carter, Hawking - 1973; . . . ]

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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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/4GN [ 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

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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/4GN [ 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

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Guiding principles

Existence of parameter N Loss of time-reversal invariance / Unitarity problems

  • Thermodynamical irreversibility (limit N → ∞)

Chaotic dynamics Fβ(t) ∼ ˆ O(0) ˆ O(t) ˆ O(0) ˆ O(t)β,con. ∝ eλLt Lyapunov exponent saturates bound for black holes λL ≤ 2π β

[ Maldacena, Shenker, Stanford - 2015 ] Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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SYK model [ Sachdev, Ye - 1993; Kitaev - 2015 ]

N Majorana fermions ψ1, . . . , ψN in 0 + 1 dim. with Hamiltonian H =

  • i<j<k<l

Jijkl ψi ψj ψk ψl Quenched disorder · ≡

  • dJijkl p(Jijkl) · Jijkl

with σ2(Jijkl) ∝ J2 Some nice features Approximate conformal symmetry in IR = ⇒ NAdS2/NCFT1 Analytical treatment for N → ∞ Explicit numerics for small N (|H| = 2N/2) Saturates bound for λL

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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SYK model [ Sachdev, Ye - 1993; Kitaev - 2015 ]

N Majorana fermions ψ1, . . . , ψN in 0 + 1 dim. with Hamiltonian H =

  • i<j<k<l

Jijkl ψi ψj ψk ψl Quenched disorder · ≡

  • dJijkl p(Jijkl) · Jijkl

with σ2(Jijkl) ∝ J2 Some nice features Approximate conformal symmetry in IR = ⇒ NAdS2/NCFT1 Analytical treatment for N → ∞ Explicit numerics for small N (|H| = 2N/2) Saturates bound for λL Not a proper Quantum Field Theory :-(

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Vector and matrix models: an overview

Large D vector models Large N matrix models Field content φµ with µ = 1, . . . , D X a

b with a, b = 1, . . . , N

Symmetry O(D) U(N)2 or U(N) Interactions

  • φ 2k for k = 1, . . .

Tr(XX †XX † · · · ), . . .

  • Diag. scaling

DV −P+ϕ = D1−ℓ NV −P+f = N2−2g Leading cacti diagrams (auxiliary tree-level) planar diagrams Applications

  • cond. mat. ph., CFT,

higher spin gravity

  • nucl. ph., QCD,

string theory

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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New large N limit [ Ferrari - 2017; Ferrari, Rivasseau, Valette - 2017 ]

O(d) × U(n)2 model for a vector of complex matrices Interaction vertices are VB = Tr

  • Xµ1X †

µ2 · · · Xµ2s−1X † µ2s

  • Usual scaling

S = nd

  • 1

2Tr

  • XµX †

µ

  • +

B tBVB(Xµ)

  • Fidel I. Schaposnik Massolo - IHES

Phase Diagrams for Melonic Tensor / Disordered Models

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New large N limit [ Ferrari - 2017; Ferrari, Rivasseau, Valette - 2017 ]

O(d) × U(n)2 model for a vector of complex matrices Interaction vertices are VB = Tr

  • Xµ1X †

µ2 · · · Xµ2s−1X † µ2s

  • Usual scaling

S = nd

  • 1

2Tr

  • XµX †

µ

  • +

B tBVB(Xµ)

  • Enhance ’t Hooft coupling tB for VB as

tB = λBdE(B) with E(B) ≥ 0 In the n → ∞ limit F =

  • g≥0

n2−2gFg In the d → ∞ limit (g fixed) Fg =

  • k≥0

d1+g−k/2Fg,k

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Quartic models for fermionic matrices

Fermionic matrices in 0 + 1 dimensions (ψ†

µ)a b = (ψb µa)†

with

  • ψa

µb, (ψ† ν)c d

  • = 1

nd δµνδa

dδc b

Desired features U(n) × O(d) invariance Single trace Hamiltonian Quadratic mass term mTr

  • ψ†

µψµ

  • Quartic interactions

Tr

  • ψµψ†

νψµψ† ν

  • = −Tr
  • ψ†

νψµψ† νψµ

  • + n

d , etc. (Combinations ψµψµ and ψ†

µψ† µ are suppressed)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Inequivalent interactions

Crossing interactions = ⇒ E(B) = 1/2 λTr(ψ†

µψνψ† µψν) λ′Tr(ψ† µψ† νψµψν) ξTr(ψ† µψνψµψν) ξ∗Tr(ψµψ† νψ† µψ† ν)

Non-crossing interactions = ⇒ E(B) = 0 κTr(ψ†

µψµψ† νψν) κ′Tr(ψµψ† µψνψ† ν) ˜

κTr(ψ†

µψµψνψ† ν) ˜

κ∗Tr(ψµψ†

µψ† νψν)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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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(ψµψµ) = Tr(ψ†

µψ† µ) = 0

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Melon trees

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Two basic models

Charge preserving model with symmetry O(d) × U(n)2 HQ = nd Tr

  • m ψ†

µψµ + 1 2λ

√ dψµψ†

νψµψ† ν

  • λ′Tr(ψ†

µψ† νψµψν) interaction renormalizes λ → λ + 2λ′

Charge violating model with symmetry O(d) × U(n) HQ = nd Tr

  • m ψ†

µψµ + 1 2

√ d

  • ξψ†

µψνψµψν + ξ∗ψ† µψ† νψ† µψν

  • ,

, . . . Melonic-dominated models = ⇒ Physics similar to SYK

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Disordered model formulation

ψa

µb has d × n2 fermionic degrees of freedom

  • Hilbert space is 2dn2 dimensional :-(

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Disordered model formulation

ψa

µb has d × n2 fermionic degrees of freedom

  • Hilbert space is 2dn2 dimensional :-(

Equivalent disordered models with N Dirac fermions

  • χi, χ†

j

  • = δi

j

˜ HQ = mχ†

i χi + λij kl

N3/2 χ†

i χ† j χkχl

˜ HQ = mχ†

i χi +

ξi

jkl

N3/2 χ†

i χjχkχl + ξijk l

N3/2 χ†

i χ† j χ† kχl

Hilbert space is 2N dimensional :-)

WARNING: Equivalence is partial and only to leading large N (= n2d) order! Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Procedure

Euclidean two-point function G(t) =

  • Tr T
  • ψµ(t)ψ†

µ

  • β

Fermionic perturbation theory m ≫ λ = ⇒

  • Exp. around decoupled fermionic oscillators

T ≫ λ = ⇒ Non-standard (SYK-like) perturbation theory G0(t) = em(β−t) emβ + 1 =

  • 1

2sign(t)

m → 0, then T → 0 e−mtΘ(t) T → 0, then m → 0 Feynman diagram structure = ⇒ Schwinger-Dyson equations

= + + · · · Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Schwinger-Dyson equations

Expanding G(t) in Matsubara-Fourier modes G(t) = 1 β

  • k

Gke−iωkt , ωk = 2π β k with k ∈ Z + 1

2

The Schwinger-Dyson equations are G −1

k

= m−iωk+Σk

  • ΣQ(t)

= λ2G(t)2G(−t) ΣQ(t) = − 1

4|ξ|2G(t)

  • G(t)2 + 3G(−t)2

Now Define Seff with Schwinger-Dyson equations as saddle-points Relate its on-shell value to the free energy F = − 1

β log Tr e−βH

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Charge preserving model: Phase diagram structure

Strong coupling regime High T pert. regime T ≫ 1,

S n2d = log 2 ≃ 0.69

High m pert. regime

P e r t u r b a t i v e r e g i m e

m = 0 T → 0

S n2d = Cat π + log 2 4

≃ 0.46 (SYK) G(t) = e−mtΘ(t)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Building the phase diagram

S n2d

m mc ≃ 0.304 SYK-like solution Perturbative solution T = 0.05

F n2d

m

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Phase diagram: (m, T) plane

High entropy phase Tc = 0.06872 Low entropy phase mc = 0.3451 Supercritical phase λ = 1

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Possible interpretation

Stringy description of gravitational collapse n n m O(d)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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4-point functions

  • T Tr
  • ψµ(t1)ψ†

µ(t2)

  • Tr
  • ψν(t3)ψ†

ν(t4)

  • β = n2G(t1, t2)G(t3, t4) + 1

d F(t1, t2, t3, t4)

Leading diagrams for F t1 t2 t4 t3 + t1 t2 t4 t3 + t1 t2 t4 t3 + t1 t2 t4 t3 + · · · F =

∞

  • n=0

K n ⋆F0 = (1−K)−1 ⋆F0 with F0 = −G(t1, t4)G(t3, t2) Rungs

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Lyapunov exponents

λL 2π/β

β SYK at q → ∞ SYK for βJ ≫ 1 m = 0 m = 0.1 m = 0.2 m = 0.24 m = 0.34 m = 0.4 m = 0.5

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Critical behavior of the Lyapunov exponent

λ 2π/β

m α− = 0.311 α+ = 0.401

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Charge violating model: Phase diagram structure

Strong coupling regime High T pert. regime T ≫ 1,

S n2d = log 2 ≃ 0.69

High m pert. regime

P e r t u r b a t i v e r e g i m e

m = 0 T → 0

S n2d = Cat π + log 2 4

≃ 0.46 (SYK) G(t) = e−mtΘ(t)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Charge violating model

Quantum phase transition |λ| = 1 m

S n2d

T = 0 T = 0.01 T = 0.05 T = 0.1

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Lyapunov exponents

β

λL 2π/β

m = 0 m = 0.05 m = 0.1 m = 0.2 m = 0.3 m = 0.4 m = 0.5 m = 0.6

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Finite N picture

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Finite N picture

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Finite N spectrum

N = 4

|λ| = 1

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Finite N spectrum

N = 5

|λ| = 1

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Finite N spectrum

N = 6

|λ| = 1

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Finite N spectrum

N = 10

|λ| = 1 Quantum phase transition

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Quantum critical mass: HQ

mc(N) ≃ 0.232 + 1.03

N − 2.510 N2

λ = 1

− →

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Quantum critical mass: HQ

λ = 1 mc(N) ≃ 0.371 − 0.399

N T = 0 T = 0.01 T = 0.05 T = 0.1

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Outlook

1 Many generalizations: q-body interactions, bosonic models,

supersymmetry, . . .

2 Effective description `

a la Landau

3 Holographic picture Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Outlook

1 Many generalizations: q-body interactions, bosonic models,

supersymmetry, . . .

2 Effective description `

a la Landau

3 Holographic picture

Takeaway: quantum black hole playground in a computer :-)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Thanks!

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Generalizations and extensions

q-body interacting generalization of the fermionic model I Σ(t) = λ2G(t)2G(−t) = ⇒ Σ(t) = (−1)

q 2 λ2G(t) q 2 G(−t) q 2 −1

q = 4 q = 6 q = 8

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Bosonic model phase diagram structure

HB = nd Tr m2 2 XµXµ + λ3 4 √ dXµXνXµXν

  • Classical regime

Perturbative regime

Strong coupling regime x−1 = m2 − λ6

β2 x3

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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Phase diagram for a bosonic model

Unstable region One solution (S2 < 0) Quantum corrections S2 = 0 Two solutions (F1 < F2)

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models

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(q1, q2) = (4, 8) domain wall

m = 0; T = 10−4 λ8 = 100 λ8 = 200 ∆ log ω

Fidel I. Schaposnik Massolo - IHES Phase Diagrams for Melonic Tensor / Disordered Models