qft dynamics from cft data -...

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 4th 2019

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

This talk:

Preface

Basic Strategy

QFT

Basic Strategy

QFT

Write QFT as deformation of UV CFT. Use CFT data to organize QFT calculation.

CFT (UV)

QFT (IR)

+X

�iO(relevant)i

Free Fields Minimal / Integrable Perturbative Supersymmetric Bootstrap-able

e.g.

CFT (UV)

QFT (IR)

+X

�iO(relevant)i

Basic Strategy

CFT (UV)

QFT (IR)

+X

�iO(relevant)i

UV CFT Data: Δ’s + OPE coefficients

Input

IR QFT Observables: • Spectrum • Correlation Functions (real-time, infinite-volume)

Output

Goal: Extract QFT dynamics from CFT data

Basic Strategy

Novel Feature of Conformal Truncation

No Wick rotation, no lattice, no compactification

Formulated so that entire computation takes place in real time and infinite volume, allowing access to dynamics

Novel Feature of Conformal Truncation

No Wick rotation, no lattice, no compactification

Formulated so that entire computation takes place in real time and infinite volume, allowing access to dynamics

Conformal truncation is a specific implementation of Hamiltonian truncation.

Hamiltonian Truncation

1. Identify a basis of QFT states

2. Write Hamiltonian in chosen basis

3. Truncate in some way

4. Diagonalize numerically

5. Look for convergence w/ truncation level

m1

m2

|b1i, |b2i, |b3i, . . .

H =

0

B@H11 H12 · · ·H21 H22 · · ·...

......

1

CA

evals + evecs

(infinite)

Hamiltonian Truncation

1. Identify a basis of QFT states

2. Write Hamiltonian in chosen basis

3. Truncate in some way

4. Diagonalize numerically

5. Look for convergence w/ truncation level

m1

m2

|b1i, |b2i, |b3i, . . .

H =

0

B@H11 H12 · · ·H21 H22 · · ·...

......

1

CA

evals + evecs

(infinite)

Heart of any truncation scheme. How to discretize QFT???

Conformal Truncation Basis

Use UV CFT operators O�(xµ) to construct basis |b1i, |b2i, |b3i, . . .

CFT (UV)

QFT (IR)

+X

�iO(relevant)i

Conformal Truncation Basis

Use UV CFT operators O�(xµ) to construct basis |b1i, |b2i, |b3i, . . .

0 ⇤2P 21 P 2

2 P 2kmax

· · · P 2

O�(x) �! |�,

~

P , P

2i =Z

d

d

x e

�iP ·x O�(x)|0i

Final basis states

(k = 1, . . . , kmax

)�! |�, ~P , P 2k i

Think: [H, ~P ] = 0.

Note: Still real time and infinite volume

Truncation Parameters:

0 ⇤2P 21 P 2

2 P 2kmax

· · ·P 2

O�(x) �! |�,

~

P , P

2i =Z

d

d

x e

�iP ·x O�(x)|0i

(k = 1, . . . , kmax

)�! |�, ~P , P 2k i

�max

, kmax

�max

kmax

Why Truncate in ?�max

Holographic Intuition:

CFTd AdSd+1

O�(x) ! �(x, z) M2AdS ⇠ �2

Large � operators = heavy objects in AdS

(expect to decouple)

Why Truncate in ?�max

(1+1)d ��4-theoryExperimental Evidence:

µ2

i (�max

) = A+B

(�max

)#1

(�max

)#small parameter: !

Hamiltonian Matrix Elements

CFT Spectrum �! basis

OPE Coe�cients �! H matrix elements

HQFT = HCFT + �

Zd~xOrel(~x)

h�, P |�H|�0, P 0i = �(~P � ~P 0)

Zddx ddx0 ei(P ·x�P

0·x0) hO(x)Orel(0)O0(x0)i

Fourier transform of CFT 3PFH matrix element

Quantization scheme: Lightcone

Technology

CFT Spectrum �! basis

OPE Coe�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:

hO(x)O(0)i =Z

dµ

2⇢O(µ)

Zd

d

p

(2⇡)de

�ip·x✓(p0)(2⇡)�(p

2 � µ

2)

IO(µ) ⌘Z µ2

0dµ02 ⇢O(µ0)

⇢O(µ)Kallen-Lehmann spectral densitye.g.,

Conformal Truncation Deliverables

⇢O(µ)Kallen-Lehmann spectral density

µ

IO(µ)

UVIR

Encodes RG

IO(µ) ⌘Z µ2

0dµ02 ⇢O(µ0)

Example: (1+1)d ��4-theory

Tµµ : Spectral Density vs. �

� ⌘ �

m2

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CFT!

Convergence�max

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0 2 4 6 8 10 12 140.00

0.05

0.10

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μ� / ��

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(@ fixed �)


� ⌘ �

m2

�2n : Spectral Density vs. �

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0.20

0.25

0.30

0.35

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0.05

0.10

0.15

0.20

0.25

0.30

0.35

μ� / ��

ϕ��

Δ�� = ��λ� π = ��

Universal Behavior!


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■

◆◆ ◆ ◆ ◆◆

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■ ϕ�

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0.0 0.5 1.0 1.5 2.0 2.50.00

0.01

0.02

0.03

0.04

0.05

μ� / ��

ϕ�� Δ�� = ��

λ� π = ��

� ⌘ �

m2

Ising Prediction⇢"

IR Zoom-In

Summary of Conformal Truncation

It’s a Hamiltonian truncation method formulated directly in real time and infinite volume, allowing access to nonperturbative dynamics.

Tries to harness small parameter:1

(�max

)#

Input is CFT data. Output is QFT dynamics.

qft dynamics from cft data -...

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