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. �
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m2
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CFT!
Convergence�max
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(@ fixed �)
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Example: (1+1)d ��4-theory
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0.15
0.20
0.25
0.30
0.35
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Universal Behavior!
Example: (1+1)d ��4-theory
●● ● ●●
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●
●
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■
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0.01
0.02
0.03
0.04
0.05
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λ� π = ����
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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.