Operator Algebras and the Renormalization Group in Quantum Field
A Numerical Perturbative Renormalization Approach to ... · • Many other examples in condensed...
Transcript of A Numerical Perturbative Renormalization Approach to ... · • Many other examples in condensed...
A Numerical Perturbative Renormalization Approach to Condensed Matter Field Theory
Kun Chen
Rutgers University
Supported by Simons Foundation
2019.07.22 NYC
Acknowledgement
Kristjan Haule Gabriel Kotliar
Large Parameter in Feynman diagrams
• Large parameters from UV:
e.g. a 𝜙4 model with a small mass 𝑚 at 𝑑 = 4: 𝑆𝑏𝑎𝑟𝑒 = 𝑑𝒌
2𝜋 𝑑 𝑘2𝜙−𝒌𝜙𝒌 + 𝑢
𝑑𝒌𝑑𝒌′𝑑𝒒
2𝜋 𝑑 𝜙𝒌 𝜙𝒌′−𝒒 𝜙𝒌′+𝒒 𝜙𝒌
= −1
2× 3 +⋯Γ4
𝑢
𝒌 + 𝜦
𝒌
= න𝑑𝒌
2𝜋 𝑑
1
𝒌2 𝒌 + 𝚲 2 = # lnΛ𝑈𝑉Λ
Λ𝑈𝑉
Λ𝐼𝑅
Λ
new physics
UV “divergence”: Our ignorance beyond the UV scale
arbitrary intermediate scale
bare theory
• Vertex Problem (curse of dimensionality):
• The UV “divergence” has to be resumed (e.g., with certain skeleton
diagrammatic technique)
• However, Γ4 (𝑘1, 𝑘2, 𝑘3) can be difficult to store and calculate!
Relevance of Vertex Functions
• Scaling analysis:
• Assume UV and IR scales to be well-separated, the system must be scale invariant!
•
•
• The dominating vertex functions can be parameterized with the relevant/marginal
couplings only (up to certain non-universal corrections)!
• Three challenges:
a) How to find all relevant/marginal couplings?
b) With known relevant/marginal couplings, how to perturbatively recover physical
observables?
c) How to non-perturbatively calculate relevant/marginal couplings?
Dimensionless Γ𝑛 = Ψ𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙𝑘𝑖
Λ;𝑢𝑗
Λ𝛿𝑗
+ regular corrections
Relevant couplings: 𝛿𝑗 > 0; Marginal: 𝛿𝑗 = 0; Irrelevant: 𝛿𝑗 < 0.
Λ𝑈𝑉
Λ𝐼𝑅
Λ
new physics
running scale
bare theory
Step 1.
Find Relevant/Marginal Vertex Functions
in Different Field Theories
Example 1: Critical Field Theory
• Critical scalar field theory at 𝒅 = 𝟒:
• Relevant/marginal vertices: Γ2,𝑟𝑒𝑙Λ 𝑘 = # + #𝑘2 and Γ4,𝑟𝑒𝑙
Λ = #
𝑆Λ = 𝑑𝒌
2𝜋 𝑑 (𝑘2 + Λ2)𝜙−𝒌𝜙𝒌 + 𝑢
𝑑𝒌𝑑𝒌′𝑑𝒒
2𝜋 𝑑 𝜙𝒌 𝜙𝒌′−𝒒 𝜙𝒌′+𝒒 𝜙𝒌
𝒌 + 𝒒
𝒌
~ lnΛ𝑈𝑉Λ
Γ4Λ ≡
𝒌𝟏 𝒌𝟐
𝒌𝟑 𝒌𝟒
𝑅 = Γ4Λ(𝑘1 = 0, 𝑘2 = 0, 𝑘3 = 0)
=𝒌𝟏
𝒌𝟏 − 𝒒 𝒌𝟐 + 𝒒
𝒌𝟐
𝒌
𝒌𝒌𝟏
𝒌𝟏 − 𝒒 𝒌𝟐 + 𝒒
𝒌𝟐
Example 2: Fermi Liquid Theory
• A spinless Fermi liquid at 𝒅 = 𝟐:
• Relevant/marginal vertices: Γ2,𝑟𝑒𝑙 𝑘, 𝜔𝑛 = # + #𝜔𝑛 + #(𝒌 − 𝒌𝑭)
• Γ4,𝑟𝑒𝑙 = particle−hole channel effective interaction near the Fermi surface
𝒌 + 𝒒, 𝜔 + 𝛺
𝒌,𝜔
Γ4 ≡
𝒌𝟏, 𝜔1 𝒌𝟐, 𝜔2
𝑅 = Γ𝑝ℎ(𝑘1 = 𝑘𝐹𝒆1, 𝑘2 = 𝑘𝐹𝒆2, 𝒒, 𝛺) − exchange
𝒌𝟏, 𝜔1𝒌𝟐, 𝜔2
𝒒,𝛺
𝒒, 𝛺
𝒌 + 𝒒, 𝜔 + 𝛺
𝒌,𝜔𝒌𝟏 = 𝑘𝐹𝒆1,𝜔1 → 0
𝒒,𝛺
=𝒌𝟐 = 𝑘𝐹𝒆2,𝜔2 → 0
= 0~න 𝑑𝒌𝒌 ∙ 𝒒
𝑖𝛺 − 𝒌 ∙ 𝒒Γ𝐿Γ𝑅 + reg.
𝐿 = σ𝒌 𝜓𝒌 ,𝜏+ 𝜕
𝜕𝜏+ 𝒌𝟐 − 𝜇 𝜓𝒌,𝜏 +
1
2𝑉σ𝒒𝒌𝒌′
8𝜋
𝑞2+𝜆𝜓𝒌,𝜏+ 𝜓𝒌−𝒒,𝜏
+ 𝜓𝒌′+𝒒,𝜏𝜓𝒌′,𝜏
Other Examples
• Superconductivity:
pp channel effective interaction
• 2D/3D dilute Bose gas:
pp channel (B. Svistunov, E. Babaev, and N. Prokof'ev, Superfluid states of matter)
• Conformal field theory (e.g., many common critical theories, like the Ising criticality):
• 2-point and 3-point correlations are fixed up to a few universal constants and critical exponents.
• All high order correlations can be constructed using operator product expansion.
• Conformal bootstrap.
• Many other examples in condensed matter field theory books…
K. Wilson, On products of quantum field operators at short distances. Cornell Report (1964)A. Polyakov, Conformal symmetry of critical fluctuations JETP let. 12, 381 (1970).
e.g., S. Sachdev, Quantum Phase Transitions, (2011)
Step 2.
Q: With known relevant/marginal couplings, how to
perturbatively calculate physical observables?
A: Perturbative Renormalization
Renormalized Expansion
• Represent any physical observables with the relevant/marginal couplings:
= −1
2× 3 +⋯Γ4
𝑢
𝜉2𝜉1
≡Γ4
𝒌1 𝒌2
𝒌3 𝒌4
𝑅
= 𝑅Γ4 −1
2× 3 +⋯𝑅 𝑅
𝑅 𝑅 𝑅 𝑅−
Bare expansion
Renormalized expansion
2
13
BPHZ Renormalization Scheme
• Zimmermann’s forest formula:
Identify all 1PI 4-vertex subgraphs, then subtract the “divergent” pieces one by one (smaller vertices first).
N. N. Bogoliubov and O. S. Parasiuk, Acta. Math. 97, 227 (1957). K. Hepp, Comm. Math. Phys. 2, 301 (1966). W. Zimmermann, Math. Phys. 11,1 (1968) . W. Zimmermann, Math. Phys. 15, 208 (1969).
𝑅
𝑅
𝑅
(1 − 𝑃 )(1 − 𝑃 )
=𝑅
𝑅
𝑅
𝑅
𝑅
𝑅
𝑅
𝑅
𝑅− + −
2
1
3
2
13
• Greatly simplifies when interactions are point-wise(all boxes are simple constants), e.g. QED, 𝝓𝟒
theory, …
• Too expansive for generic condensed matter field theories with complicated interactions, e.g. Fermi
liquid theory, superconductivity, …
𝑅
𝑅
𝑅
Dyson-Schwinger Equations
• Dyson-Schwinger Equations: A set of EXACT equations connecting different 1PI vertex functions
Γ4Γ4 = −1
2 Γ4
Γ4 × 3× 3 +1
2𝑢−1
6Γ6
Γ6 =Γ4
Γ4
× 15 + ⋯
• The key ideas:
1. DSEs can recursively generate all Feynman diagrams.
2. If one can renormalize DSEs, then the renormalized DSEs can be a generator for all
renormalized diagrams.
𝑛 + 1 loops
𝑛 loops
DSE Renormalization Scheme
• Derivation of the recursive rules:
Γ4𝑅= +1
2 Γ4
Γ4 × 3× 3 −1
2𝑢+⋯
=𝑢 𝑢
≡Γ4
Γ4𝑟𝑒𝑓
𝒌1 𝒌2
𝒌3 𝒌4
𝑅Γ4Γ4= +
1
2 Γ4
Γ4 × 3× 3 −1
2𝑢+⋯(1)
(2)
(3)=(1)-(2) Γ4𝑅= −1
2 Γ4
Γ4 × 3× 3 +1
2−⋯
Γ4
• The key insight: two dual equations
𝜉2 + 𝜉3 +⋯𝜉1 += 1 2Γ4 𝑅 2𝑅Γ4 3
𝜉2 − 𝜉3 +⋯𝜉1 −= 1 2𝑅 2𝑅 3𝑢
𝑖contain 𝑖 number of
renormalized couplings
DSE Renormalization Scheme
Γ4𝑅= +1
2 Γ4
Γ4 × 3× 3 −1
2𝑢+⋯
Γ4𝑅= −1
2 Γ4
Γ4 × 3× 3 +1
2−⋯Γ4 𝜉2 + 𝜉3 +⋯𝜉1 += 1 2Γ4 𝑅 2𝑅Γ4 3
𝜉2 − 𝜉3 +⋯𝜉1 −= 1 2𝑅 2𝑅 3𝑢
2 𝑅 𝑅= −1
2× 3
3 2= −1
2𝑅 × 3 𝑅+
1
22 × 3 +
𝑅
𝑅
𝑅 × 3
1 𝑅=
Bottom-Up Recursive DiagMC
• Renormalized DSEs leads to a bottom-up recursive DiagMC.
𝑁 +
𝑖+𝑗+𝑘=𝑁
1
2
𝑖
𝑗
𝑘 × 3𝑖 𝑗= −
𝑖+𝑗=𝑁
1
2× 3 −
𝑖+𝑗=𝑁
1
6 Γ6
𝑖
𝑗
• Build higher order diagrams from lower order vertices.
• Similar efficiency for any representation (space/time, momentum/frequency, or mixed).
• Easy to separate different channels (very important for Fermi liquid/superconductivity).
• If necessary, 𝚪𝟔, 𝚪𝟖, … can also be renormalized without sacrificing the efficiency.
R. Rossi, Determinant diagrammatic Monte Carlo algorithm in the
thermodynamic limit. PRL, 119, 045701 (2017).
Step 3.
Q: How to calculate relevant/marginal couplings?
A: Renormalization Group/Skeleton diagrammatic techniques/DSE
Fermi Liquid
• A spinless Fermi liquid at 𝒅 = 𝟐:
• Effective interaction:
• The IR regulator suppresses the slow modes near the Fermi surface.
• Recover the physical system when 𝜦 → 𝟎.
𝐿 = σ𝒌 𝜓𝒌 ,𝜏+ 𝜕
𝜕𝜏+ 𝒌𝟐 − 𝜇 + 𝑅𝑘
Λ 𝜓𝒌,𝜏 +1
2𝑉σ𝒒𝒌𝒌′
8𝜋
𝑞2+𝜆𝜓𝒌,𝜏+ 𝜓𝒌−𝒒,𝜏
+ 𝜓𝒌′+𝒒,𝜏𝜓𝒌′,𝜏
Γ4Λ ≡
𝒌𝟏, 𝜔1 𝒌𝟐, 𝜔2
𝑅Λ = Γ𝑝ℎΛ (𝑘1 = 𝑘𝐹𝒆1, 𝑘2 = 𝑘𝐹𝒆2, 𝒒, 𝛺) − exchange
𝒒,𝛺
𝐸𝐹
2Λ
𝑘1 𝑘2
𝜃
𝐺Λ 𝒌, 𝜏 = 𝐺 𝒌, 𝜏𝑘 − 𝑘𝐹
2
𝑘 − 𝑘𝐹2 + Λ2
= − × 3 +𝑅Λ
= × 15 ∙ 𝜉 + ⋯𝑅Λ 𝑅Λ
𝑅Λ
Γ4Λ Γ4
Λ Γ6Λ
Γ6Λ
𝑑
𝑑Λ
1
2
∙ 𝜉3 +⋯= 1 2∙ 𝜉2 +Γ4 𝑅 2𝑅ΛΓ4Λ 3
where,
∙ 𝜉 +
Functional RG Approach
The effective interaction remains marginal
until Λ < 𝑞 or Λ < 𝑇1/2
Consistent with theoretical predictions:R. Shankar, Rev. Mod. Phys. 66, 129 (1994)N. Dupuis and G. Y. Chitov, Phys. Rev. B. 54,3040 (1996)G. Hitov, D. Senechal, Phys. Rev. B. 57, 1444 (1998)
• DiagMC samples the diagrams for 𝜷 function,
and at the same time, solve the RG equation!
𝑑 𝑑Λ
2D spinless Fermi liquid, 𝑟𝑠 = 1, 𝑇 = 0.05𝐸𝐹 , 𝜆 = 2
Improved Convergence/Efficiency
Λ → 0: the physical vertex function
Bare coupling expansion Renomalized expansion+RG
Sign Cancellation Between Diagrams
−
+
−
−
+Tw
o-l
oo
p c
hai
n d
iagr
am c
on
trib
uti
on
Bubble diagrams + vertex corrections + ph-ladder-type diagrams
Workflow
Effective field theory
“Guess” the relevant/marginal vertex functions
Renormalized perturbative expansion
Are the corrections
~unity?
Problem Solved!
YES
NO
RG/skeleton diagrams/
DSE/variational approach
Finite number Infinitely many
Integrate out the local degrees
of freedom with
impurity solver or ED𝑈
𝑡
Λ
Vertex function
scale barrier
~Λ−𝛿
~Λ𝜎
on
e-b
od
yfe
w-b
od
ym
any-b
od
y
e.g. Bose/Fermi Hubbard model,Spin models (AFM, spin liquids)