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)