Quantum phase transition and Resurgence:
Lessons from 3d N=4 SQED
Takuya Yoda
Department of Physics, Kyoto University[arXiv: 2103.13654]
Collaborators: Toshiaki FujimoriA,D, Masazumi HondaB, Syo KamataC, Takahiro MisumiD,E, Norisuke SakaiD

Hiyoshi Phys. Keio U.A, YITPB, NCBJC, RECNS KeioD, Akita->Kindai U.E

Jun./16/2021 Seminar @Kyoto U. Particle Physics GroupResurgence theory[J. Ecalle, 81]• One of the approaches to non-perturbative physics

Lectures and reviews, e.g.
[M. Marino, 12][D. Dorigoni, 19][I. Aniceto, G. Basar, R. Schiappa, 19]• Decodes non-perturbative information from perturbation theory

Perturbation theory

Non-perturbative effects
e.g. phase transitions

Borel resummation

Lefschetz thimble analysis

Obstructions to controlling
asymptotic series?

Contributions from
non-trivial saddles?

Phase transitions and resurgence
Common story:
1st order phase transition = Change of dominant saddles

This is also within the scope of resurgence theory

Recent works:•0-dim Gross-Neveu, Nambu-Jona-Lasinio like model[T. Kanazawa, Y. Tanizaki, 15]•2dim Yang-Mills on lattice (reduced to Gross-Witten-Wadia)[G. Dunne et al., 16, 17, 18]etc.

Is resurgence theory applicable to2nd order phase transitions or more realistic QFTs?

Brief summaryModel•[Russo, Tierz, 17]

SUSY gauge theory +• Fayet-Illiopoulos parameter

hypermultiplets with charge 1, flavor mass

2nd order quantum phase transition at the large-flavor limit

Result: resurgence is applicable!

[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]

2nd order phase transition = Simultaneous Stokes and Anti-Stokes phenomena
Change of the form of asymptotic expansion
and change of dominant saddles• The order of the phase transition is determined by the collision angle of saddles• Such information is encoded in a perturbative series

Contents

 Motivations and Brief summary(3)• 2nd order phase transition in SQED3 (review)(4)• Lefschetz thimble analysis(9)• Borel resummation(11)• Lessons from SQED3(3)• Conclusion and future works(1)

Total: 31

Contents

 Motivations and Brief summary(3)

these, let me• 2nd order phase transition in SQED3 Before(review)(4)provide• Lefschetz thimble analysis

lightning introduction to resurgence(9)• Borel resummation(11)• Lessons from SQED3(3)• Conclusion and future works(1)

Total: 31

Asymptotic series and resurgence structure

Resurgence structure[J. Ecalle, 81][M. Marino, 12][Cherman, Dorigoni, Unsal, 14][Cherman, Koroteev, Unsal, 14][D. Dorigoni, 19][I. Aniceto, G. Basar, R. Schiappa, 19]

Resurgence theory:the perturbative part knows the non-perturbative parts

Example: 0dim Sine-Gordon model
Partition function:

Perturbative partNon-perturbative part

Observation 1

Lefschetz thimble

Observation 2
Borel resummation“Ambiguity”
RecallObservation 3

Resurgence theory[J. Ecalle, 81]• One of the approaches to non-perturbative physics

Lectures and reviews, e.g.
[M. Marino, 12][D. Dorigoni, 19][I. Aniceto, G. Basar, R. Schiappa, 19]• Decodes non-perturbative information from perturbation theory

Perturbation theory

Non-perturbative effects
e.g. phase transitions

Borel resummation

Lefschetz thimble analysis

Obstructions to controlling
asymptotic series?

Contributions fromnon-trivial saddles?

Contents

 Motivations and Brief summary(3)• 2nd order phase transition in SQED3 (review)(4)• Lefschetz thimble analysis(9)• Borel resummation(11)• Lessons from SQED3(3)• Conclusion and future works(1)

Total: 31

SQED3Model:SUSY gauge theory with•hypermultiplets (charge 1)• Fayet-Illiopoulos parameter• flavor masses

Partition function:
Exactly computed on 𝑆 3 by SUSY localization technique[Pestun, 12][A. Kapustin, B. Willett, I. Yaakov, 10][N. Hama, K. Hosomichi, S. Lee, 11][D. L. Jafferis, 12]: Coulomb branch parameteri.e. constant configuration of the scalar belonging to the vector multiplet inLanguage

Saddles’t Hooft like limit:[Russo, Tierz, 17]“Action”:

Saddles:

Something must
happen at

Saddles[Russo, Tierz, 17]

𝜆 < 𝜆𝑐𝜆 ≥ 𝜆𝑐Saddles[Russo, Tierz, 17]

𝜆 < 𝜆𝑐Smallest
real part𝜆 ≥ 𝜆𝑐Saddles[Russo, Tierz, 17]

𝜆 ≥ 𝜆𝑐𝜆 < 𝜆𝑐Smallest
real part2nd order phase transition
If the saddles of smallest real part contribute,[Russo, Tierz, 17]2nd order phase transition
Questions:• Smallest real part does not necessarily mean that such saddles contribute to thepath integral. Can we justify it in more precise way?
• Can we interpret the 2nd order phase transition from the viewpoint ofresurgence, and draw lessons for generic QFTs?

We will answer to these questions, Yes!

Contents

 Motivations and Brief summary
 2nd order phase transition in SQED3 (review)• Lefschetz thimble analysis• Borel resummation• Conclusion and future works

(3)

(4)
(9)
(11)
(3)
(1)

Total: 31

Lefschetz thimbles and dual-thimbles
Lefschetz thimbles = “Steepest descents” in configuration space= “Steepest ascents” in configuration space [E. Witten, 11]Dual-thimble[M. Cristoforetti et al., 12,13,14][H. Fujii et al., 13][G. Aarts, 13][A. Alexandru et al, 16](thimble decomposition)intersection number

Stokes and anti-Stokes phenomena

Stokes phenomenon: Change of an intersection number, which occurs at

Anti-Stokes phenomenon : Change of dominant saddles, which occurs at

1st order phase transition

What we do
Recall our goal:• Smallest real part does not necessarily mean that such saddles contribute to thepath integral. Can we justify it in more precise way?
• Can we interpret the 2nd order phase transition from the viewpoint ofresurgence, and draw lessons for generic QFTs?

Sub-questions in this part:• Is the 2nd order phase transition justified by thimble decomposition?
• Is the 2nd order phase transition interpreted as (anti-)Stokes phenomena?

Application to the “path integral”[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]Partition function:

Thimble/dual-thimble equations:Lefschetz thimble structure[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]

𝜆 < 𝜆𝑐𝜆 ≥ 𝜆𝑐arg 𝑁 = 0, 𝜆 = 0.4, 𝑚 = 1

arg 𝑁 = 0, 𝜆 = 1.2, 𝑚 = 1

branch cut

thimblethimbleTrivial saddle

Thimble decompositionis ambiguous
Vary the phase ofLefschetz thimble structure (subcritical)
𝜆 < 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = +0.025

thimbleDual thimbleLefschetz thimble structure (subcritical)
𝜆 < 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = +0.025

thimbleDual thimbleLefschetz thimble structure (subcritical)
𝜆 < 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = +0.025

thimbleDual thimbleLefschetz thimble structure (subcritical)
𝜆 < 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = +0.025

thimbleDual thimble• No Stokes phenomenon• Only the trivial saddle

contributes to the path integral

Lefschetz thimble structure (supercritical)
𝜆 ≥ 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = +0.025

Lefschetz thimble structure (supercritical)
𝜆 ≥ 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = +0.025

Lefschetz thimble structure (supercritical)
𝜆 ≥ 𝜆𝑐arg 𝑁 = −0.015

arg 𝑁 = +0.015

Lefschetz thimble structure (supercritical)
𝜆 ≥ 𝜆𝑐arg 𝑁 = −0.015

arg 𝑁 = +0.015

Lefschetz thimble structure (supercritical)
𝜆 ≥ 𝜆𝑐arg 𝑁 = −0.025

arg 𝑁 = −0.015

arg 𝑁 = 0・・・

• Infinite number of Stokes phenomena occur around• Infinite number of saddles

contribute to the path integral

Stokes and anti-Stokes pheno. at the critical pt.
𝜆 < 𝜆𝑐𝜆 ≥ 𝜆𝑐Stokes and anti-Stokes pheno. at the critical pt.
𝜆 < 𝜆𝑐𝜆 ≥ 𝜆𝑐Saddles collide at the critical pointAt the critical point,• we have seen a Stokes phenomenon• anti-Stokes phenomenon occurs at the same timePhase transition and Lefschetz thimble structure
Summary
𝜆 < 𝜆𝑐𝜆 ≥ 𝜆𝑐• No Stokes phenomenon• Infinite number of Stokes phenomena• Only the trivial saddle
contributes • Infinite number of saddles
contribute(Two of whichsurvive the large-flavor limit)

branch cut

thimblethimbleTrivial saddle• Stokes and anti-Stokes phenomena occur at the same time

The phase transition is interpreted from the view point of thimbles

Contents

 Motivations and Brief summary(3)

 2nd order phase transition in SQED3 (review)(4) Lefschetz thimble analysis(9)• Borel resummation(11)• Lessons from SQED3(3)• Conclusion and future works(1)

Total: 31

Asymptotic series and resurgence structure

Resurgence structure[J. Ecalle, 81][M. Marino, 12][Cherman, Dorigoni, Unsal, 14][Cherman, Koroteev, Unsal, 14][D. Dorigoni, 19][I. Aniceto, G. Basar, R. Schiappa, 19]

Resurgence theory:the perturbative part knows the non-perturbative parts

Borel resummation
Resuming a asymptotic series[J. Ecalle, 81]

Asymptotic series

Lectures and reviews, e.g.
[M. Marino, 12][D. Dorigoni, 19][I. Aniceto, G. Basar, R. Schiappa, 19]

Borel resummation
Borel transformation

Borel transformation may have Borel singularities

Borel singularities mean
non-perturbative corrections!

Non-trivial saddle and Borel singularities[Lipatov, 77]

Non-trivial saddles are encoded in an asymptotic series

What we doLarge-flavor expansion around the trivial saddle:
𝜆 < 𝜆𝑐asymptotic series!

Borel resummation:

Sub-questions in this part:• Is the Borel plane structure consistent with the Lefschetz thimble structure?

• Can we decode the phase transition from the perturbative series?

But please wait (1/2): Borel-Padé approximation
Exact quantities:
asymptotic series

Borel transformation

Approximate these from finite number of inputs
Borel-Padé approximation

But please wait (2/2): Larger 𝜃 = arg 𝑁𝑓
Stokes phenomena occur on different Riemann sheetsBut please wait (2/2): Larger 𝜃 = arg 𝑁𝑓
Stokes phenomena occur on different Riemann sheets

Borel plane structure
𝜆 < 𝜆𝑐𝜆 = 0.4, 𝑚 = 1, Padé (25,25)[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]

𝜆 ≥ 𝜆𝑐𝜆 = 1.4, 𝑚 = 1, Padé (25,25)

Poles of the Padé approximant are consumed for branch cuts…

Improvement: Padé-Uniformized approximation[Costin, Dunne, 20]

Uniformize the Borel 𝑡-plane by a map:

Branch cut singularity at 𝑡 = 𝑠 is eliminated

Perform the standard Padé approximation on the 𝑢-planeBorel plane structure (improved)[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]< 𝜆𝜆𝑐𝑐𝜆𝜆 <𝜆 = 0.4, 𝑚 = 1, Padé-Uniformized (25,25)

𝜆 ≥ 𝜆𝑐𝜆 = 1.2, 𝑚 = 1, Padé-Uniformized (25,25)

Borel plane structure (improved)[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]< 𝜆𝜆𝑐𝑐𝜆𝜆 <𝜆 = 0.4, 𝑚 = 1, Padé-Uniformized (25,25)

𝜆 ≥ 𝜆𝑐𝜆 = 1.2, 𝑚 = 1, Padé-Uniformized (25,25)• The Stokes phenomena are encoded as Borel non-summability

Borel plane structure (improved)[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]< 𝜆𝜆𝑐𝑐𝜆𝜆 <𝜆 = 0.4, 𝑚 = 1, Padé-Uniformized (25,25)

𝜆 ≥ 𝜆𝑐𝜆 = 1.2, 𝑚 = 1, Padé-Uniformized (25,25)• The collision of saddles are encoded as collision of Borel singularities• The anti-Stokes phenomenon is encoded as Borel singularities along the vertical axis

Analytical study for large 𝜆Lading contribution for large 𝜆[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]

𝜎-plane and Borel 𝑡-plane are directly related via

Contents

 Motivations and Brief summary(3)

 2nd order phase transition in SQED3 (review)(4) Lefschetz thimble analysis(9) Borel resummation(11)• Lessons from SQED3(3)• Conclusion and future works(1)

Total: 31

Collision of saddles
Consider[T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, TY, 21]

If the “action” is holomorphic, and 𝑛 saddles collide as

Then, the “action” value at 𝑚-th saddle isPhase transition is of order

Revisiting the SQED3
𝜆 < 𝜆𝑐Phase transition is of order𝜆 ≥ 𝜆𝑐Revisiting the SQED3
The 2nd order phase transition corresponds to followings
Lefschetz thimble analysis:

Stokes andanti-Stokes phenomena
at the same timei.

Contributing saddles jump asii.

Two saddles collide with an angleiii.

Infinite number of Stokes phenomena associated withoccur

Borel resummation:I.

Two Borel singularities collide and line up along the vertical axis

II.

Two Borel singularities collide with an angle

III.

Large-flavor expansion becomes Borel non-summable

Revisiting the SQED3
The 2nd order phase transition corresponds to followings
Lefschetz thimble analysis:i.

Contributing saddles jump asii.

Two saddles collide with an angleiii.

Infinite number of Stokes phenomena associated withThe order of phasetransition is decodedfrom “scattering angle”

occur

Borel resummation:I.

Two Borel singularities collide and line up along the vertical axis

II.

Two Borel singularities collide with an angle

III.

Large-flavor expansion becomes Borel non-summable

Revisiting the SQED3
The 2nd order phase transition corresponds to followings
Lefschetz thimble analysis:i.

Contributing saddles jump asii.

Two saddles collide with an angleiii.

Infinite number of Stokes phenomena associated with

Borel resummation:

occur

Due to SUSYI.

Two Borel singularities collide and line up along the vertical axis

II.

Two Borel singularities collide with an angle

III.

Large-flavor expansion becomes Borel non-summable

Revisiting the SQED3
The 2nd order phase transition corresponds to followings
Lefschetz thimble analysis:i.

Contributing saddles jump asii.

Two saddles collide with an angleiii.

Infinite number of Stokes phenomena associated withoccur

Borel resummation:I.

Two Borel singularities collide and line up along the vertical axis

II.

Two Borel singularities collide with an angle

III.

Large-flavor expansion becomes Borel non-summable

They can be generalized as long asContents

 Motivations and Brief summary(3)

 2nd order phase transition in SQED3 (review)(4) Lefschetz thimble analysis(9) Borel resummation(11)

 Lessons from SQED3(3)• Conclusion and future works(1)

Total: 31

Conclusion and future works
Question:Is resurgence applicable to 2nd order phase transitions or more realistic QFTs?

Answer: resurgence is applicable!
2nd order phase transition = simultaneous Stokes and anti-Stokes phenomenon• The order of phase transition is determined by a collision of saddles• It is decoded from a perturbative series
Generalized to other systems

Future works:• Relation to Lee-Yang zeros？

• Expansion with respect to other parameters？
• Physical meaning of the phase transition？

Backups

Borel singularities• The Borel plane structure is consistent with the Lefschetz thimble structure• Still there are artifacts, and missing singularities far from the origin
𝜆 < 𝜆𝑐𝜆 = 0.4, 𝑚 = 1, Padé (25,25)

𝜆 < 𝜆𝑐𝜆 = 0.4, 𝑚 = 1, Padé-Uniformized (25,25)

Borel singularities• The Borel plane structure is consistent with the Lefschetz thimble structure• Still there are artifacts, and missing singularities far from the origin
𝜆 < 𝜆𝑐𝜆 = 0.4, 𝑚 = 1, Padé (25,25)

𝜆 < 𝜆𝑐𝜆 = 0.4, 𝑚 = 1, Padé-Uniformized (25,25)↔ NO stokes pheno.

Borel singularities• The Borel plane structure is consistent with the Lefschetz thimble structure• Still there are artifacts, and missing singularities far from the origin
𝜆 ≥ 𝜆𝑐𝜆 = 1.4, 𝑚 = 1, Padé (25,25)

𝜆 ≥ 𝜆𝑐𝜆 = 1.2, 𝑚 = 1, Padé-Uniformized (25,25)

Borel singularities• The Borel plane structure is consistent with the Lefschetz thimble structure• Still there are artifacts, and missing singularities far from the origin
𝜆 ≥ 𝜆𝑐𝜆 = 1.4, 𝑚 = 1, Padé (25,25)

𝜆 ≥ 𝜆𝑐𝜆 = 1.2, 𝑚 = 1, Padé-Uniformized (25,25)

The order of the phase transition
2nd order phase transition corresponds tocollision of two saddles with the reflection angle 𝜋Τ2
𝜆 < 𝜆𝑐𝜆 = 0.4, 𝑚 = 1, Padé-Uniformized (25,25)

𝜆 ≥ 𝜆𝑐𝜆 = 1.2, 𝑚 = 1, Padé-Uniformized (25,25)

Lefschetz thimble analysis
0dim Sine-Gordon model[Cherman, Dorigoni, Unsal, 14][Cherman, Koroteev, Unsal, 14]

Saddles and Lefschetz thimbles
Trivial saddle and non-trivial saddles

Lefschetz thimble analysis[Cherman, Dorigoni, Unsal, 14]
Around arg 𝑔 = 0,
[Cherman, Koroteev, Unsal, 14]
Stokes phenomenon associated with the trivial saddle

NO Stokes phenomenon associated with the non-trivial saddles

Borel resummation[Cherman, Dorigoni, Unsal, 14][Cherman, Koroteev, Unsal, 14]

Perturbation theory around the trivial saddle diverges

There is a Borel singularity (and a branch cut) around arg 𝑔 = 0

Borel resummation[Cherman, Dorigoni, Unsal, 14][Cherman, Koroteev, Unsal, 14]

Perturbation theory around a non-trivial saddle diverges

There is NO Borel singularity (nor branch cut) around arg 𝑔 = 0

Resurgence structure[Cherman, Dorigoni, Unsal, 14][Cherman, Koroteev, Unsal, 14]

The two types of ambiguities cancel𝜋
and the location of the Borel singularity agrees with 𝑆= 1/22Information of non-trivial saddles isencoded in perturbation theory around the trivial saddle

Large-flavor expansion[Fujimori, Honda, Kamata, Misumi, Sakai, TY, to appear]

Consider the Borel resummation of 1Τ𝑁 expansion
to see how thimbles’ structure is encoded𝜆 < 𝜆𝑐𝜆 ≥ 𝜆𝑐

Does perturbation theory aroundthe trivial saddle know non-trivial
saddles and the phase transition?