QFT Dynamics from CFT Data Zuhair U. Khandker University of - PowerPoint PPT Presentation

QFT Dynamics from CFT Data Zuhair U. Khandker University of Illinois, Urbana-Champaign Boston University with N. Anand, V. Genest, E. Katz, C. Hussong, M. Walters Non-Perturbative Methods in Quantum Field Theory, ICTP, Sep 4 th 2019

Preface This talk: A new numerical method (“conformal truncation”) to study real-time, infinite-volume dynamics of strongly-coupled QFTs

Basic Strategy QFT

Basic Strategy Write QFT as deformation of UV CFT. Use CFT data to organize QFT calculation. CFT (UV) λ i O (relevant) X + i QFT QFT (IR)

Basic Strategy Free Fields Minimal / Integrable e.g. Perturbative Supersymmetric CFT (UV) Bootstrap-able λ i O (relevant) X + i QFT (IR)

Basic Strategy Goal: Extract QFT dynamics from CFT data Input UV CFT Data: CFT (UV) Δ ’s + OPE coefficients λ i O (relevant) X + i Output IR QFT Observables: QFT (IR) • Spectrum • Correlation Functions (real-time, infinite-volume)

Novel Feature of Conformal Truncation Formulated so that entire computation takes place in real time and infinite volume, allowing access to dynamics No Wick rotation, no lattice, no compactification

Novel Feature of Conformal Truncation Formulated so that entire computation takes place in real time and infinite volume, allowing access to dynamics No Wick rotation, no lattice, no compactification Conformal truncation is a specific implementation of Hamiltonian truncation.

Hamiltonian Truncation 1. Identify a basis of QFT states (infinite) | b 1 i , | b 2 i , | b 3 i , . . . 2. Write Hamiltonian in chosen basis   H 11 H 12 · · · H 21 H 22 H = · · ·     . . . . . . 3. Truncate in some way . . . evals + evecs 4. Diagonalize numerically m 2 m 1 5. Look for convergence w/ truncation level

Hamiltonian Truncation Heart of any truncation scheme. How to discretize QFT??? 1. Identify a basis of QFT states (infinite) | b 1 i , | b 2 i , | b 3 i , . . . 2. Write Hamiltonian in chosen basis   H 11 H 12 · · · H 21 H 22 H = · · ·     . . . . . . 3. Truncate in some way . . . evals + evecs 4. Diagonalize numerically m 2 m 1 5. Look for convergence w/ truncation level

Conformal Truncation Basis Use UV CFT operators O ∆ ( x µ ) to construct basis | b 1 i , | b 2 i , | b 3 i , . . . CFT (UV) λ i O (relevant) X + i QFT (IR)

Conformal Truncation Basis Use UV CFT operators O ∆ ( x µ ) to construct basis | b 1 i , | b 2 i , | b 3 i , . . . Think: [ H, ~ P ] = 0 . Z d d x e − iP · x O ∆ ( x ) | 0 i | ∆ , ~ P, P 2 i = O ∆ ( x ) � ! P 2 Λ 2 0 P 2 P 2 P 2 · · · k max 1 2 | ∆ , ~ P, P 2 � ! k i ( k = 1 , . . . , k max ) Final basis states Note: Still real time and infinite volume

Truncation Parameters: ∆ max , k max Z d d x e − iP · x O ∆ ( x ) | 0 i | ∆ , ~ P, P 2 i = O ∆ ( x ) � ! P 2 Λ 2 0 P 2 P 2 P 2 · · · k max 1 2 | ∆ , ~ P, P 2 � ! k i ( k = 1 , . . . , k max ) ∆ max k max

Why Truncate in ? ∆ max Holographic Intuition: AdS d +1 CFT d O ∆ ( x ) Φ ( x, z ) M 2 AdS ∼ ∆ 2 ← → Large ∆ operators = heavy objects in AdS (expect to decouple)

Why Truncate in ? ∆ max Experimental Evidence: (1+1)d λφ 4 -theory B 1 ! µ 2 small parameter: i ( ∆ max ) = A + ( ∆ max ) # ( ∆ max ) #

Hamiltonian Matrix Elements CFT Spectrum − → basis OPE Coe ffi cients − → H matrix elements Z H QF T = H CF T + � x O rel ( ~ x ) d ~ Z d d x d d x 0 e i ( P · x � P 0 · x 0 ) h O ( x ) O rel (0) O 0 ( x 0 ) i h ∆ , P | � H | ∆ 0 , P 0 i = � ( ~ P � ~ P 0 ) H matrix element Fourier transform of CFT 3PF Quantization scheme: Lightcone

Technology CFT Spectrum − → basis OPE Coe ffi cients − → H matrix elements 1. How to enumerate all primary operators in a CFT (even just free CFT)? 2. How to efficiently compute OPE coefficients (even just free CFT)? 3. How to Fourier transform general-spin CFT 3PFs? specifically, Wightman functions

Conformal Truncation Deliverables - Spectrum: bound states, onset of critical behavior, etc. - Real-time, infinite-volume correlation functions: e.g., ρ O ( µ ) K¨ all´ en-Lehmann spectral density d d p Z Z (2 π ) d e − ip · x θ ( p 0 )(2 π ) δ ( p 2 � µ 2 ) dµ 2 ρ O ( µ ) h O ( x ) O (0) i = Z µ 2 dµ 0 2 ρ O ( µ 0 ) I O ( µ ) ≡ 0

Conformal Truncation Deliverables Z µ 2 dµ 0 2 ρ O ( µ 0 ) I O ( µ ) ≡ 0 ρ O ( µ ) K¨ all´ en-Lehmann spectral density I O ( µ ) Encodes RG µ IR UV

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� ● ● λ � π = ���� ● 0.10 ● ● ● 0.05 ● ● ● 0.00 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ ● � +- ���������� �������� ������� 0.15 Δ ��� = �� ● λ � π = ���� ● 0.10 ● ● 0.05 ● ● ● ● 0.00 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� ● λ � π = ���� ● 0.10 ● ● 0.05 ● ● ● ● 0.00 ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ ● � +- ���������� �������� ������� 0.15 Δ ��� = �� λ ● � π = ���� 0.10 ● ● 0.05 ● ● ● ● ● 0.00 ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ ● � +- ���������� �������� ������� 0.15 Δ ��� = �� λ ● � π = ���� 0.10 ● ● 0.05 ● ● ● ●● ● 0.00 ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ ● � π = ���� 0.10 ● ● 0.05 ● ● ● ●● ● 0.00 ● ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ � π = ���� ● 0.10 ● ● 0.05 ● ● ●● ● ● 0.00 ● ● ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ � π = ���� 0.10 ● ● 0.05 ● ● ●●● ● ● 0.00 ● ● ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ � π = ���� 0.10 ● 0.05 ● ● ● ●●● ● ● 0.00 ● ● ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ � π = ���� 0.10 ● 0.05 ● ● ● ● ● ● ● 0.00 ●● ● ● ● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ � π = ���� 0.10 ● 0.05 ● ● ● ●● ● ● 0.00 ● ● ● ●● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Example: (1+1)d λφ 4 -theory µ : Spectral Density vs. ¯ T µ λ � +- ���������� �������� ������� 0.15 Δ ��� = �� λ � π = ���� 0.10 ● 0.05 ● CFT! ● ● ●● ● ● 0.00 ● ● ● ●● 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2

Convergence ∆ max (@ fixed λ ) 0.15 � +- ���������� �������� ������� ▲ Δ ��� = �� ▲ ■ ◆ Δ ��� = �� ■ Δ ��� = �� 0.10 ◆ ● ● Δ ��� = �� ■ ● 0.05 ◆ ■ ▲ ● ■ ◆ ● ■■■ ■ ■ ▲ ●●● ● ◆ ◆ ◆ ▲▲ ▲ 0.00 ◆ ● ◆ ■ ◆ ■ ■ ● ● ● ● 0 2 4 6 8 10 12 14 μ � / � �

Example: (1+1)d λφ 4 -theory φ 2n : Spectral Density vs. ¯ λ ϕ � � ���������� �������� ������� Δ ��� = �� ● 0.35 ● 0.30 λ � π = ���� ● 0.25 ● ● ϕ � 0.20 ● ● 0.15 ■ ϕ � ● 0.10 ● ◆ ϕ � ● 0.05 ● 0.00 ◆ ■ ◆ ■ ◆◆ ◆ ◆ ■■ ■ ■ ◆ ■ ◆ ■ ◆ ■ ◆ ■ 0 2 4 6 8 λ ≡ λ μ � / � � ¯ m 2