Quantum phase transition in 3dim SQED
13and Lefschetz thimble analysis (work in progress)
GTakuya Yoda

Collaborators: Toshiaki FujimoriA,D, Masazumi HondaB, Syo KamataC, Tatsuhiro MisumiD,E,F, Norisuke SakaiD
Hiyoshi Phys. Keio U.A, YITPB, NCBJC, RECNS Keio U.D, Akita U.E, RIKEN iTHEMSF, Kyoto U.G

Abstract• Quantum phase transition in 3d N=4 SQED is understoodfrom the viewpoint of Lefschetz thimble analysis• Resurgence structure for large-flavor expansion is discussed• A class of Borel singularities found in this model should appear
universally in large-flavor gauge theories

3. Quantum phase transition and Lefschetz thimbles
Results• The quantum phase transition is understood as
infinite number of saddles
w/ stokes phenomenon

trivial saddle only
w/o stokes phenomenon

Thimble structure for λ<λcarg N=-0.025, λ=0.4, m=1

arg N=+0.025, λ=0.4, m=1

1. Resurgence and aim of this work
Resurgence

In a wide range of QFTs, it is expected that
Perturbation theory

Non-perturbative effects

logbranch cuts

Phase transitions

thimbledual thimble

Borel resummationAmbiguities caused byBorel singularities

Resurgence
Lefschetz thimble analysis
structure
Ambiguities caused by
Cancelstokes phenomena

Borel resummation

thimbletrivial saddle

Thimble structure for λ>λc

arg N=+0.025, λ=1.2, m=1

arg N=-0.025, λ=1.2, m=1• A resummation technique to handle
divergent formal series• Singularities or branch cuts hinder
the resum, yielding ambiguities• Considering the Borel resum ofpartition functions, singularities areassociated with saddles

Lefschetz thimble• “Steepest descents” in configuration
space• Changing the phase of parameters,
dominant saddles may jump,
yielding ambiguities• The jumps may cause “discontinuity”
of free energy

4. Borel transformation in large-flavor expansion
Results• A class of the Borel singularities should appear universally inlarge-flavor gauge theories

Borel plane structure
λ<λc

Aim of this work

λ=0.4, m=1
26 terms, Padé (13,13)

To describe a phase transition in QFT in terms of resurgenceλ>λc

λ=1.2, m=1
26 terms, Padé (13,13)λ>>λc

λ=8.0, m=120 terms, Padé (10,10)?

2. SQED3 and its quantum phase transition
SetupsModel:Toy model
3d N=4 U(1) SUSY gauge theory+ 2N hypermultiplets with charge 1Parameters: FI parameter η and flavor mass m

Free energy• Partition function on S3 can becomputed exactly by the SUSY
localization method [1,2,3]• The result depends only on η, m
• In the limit N→∞ w/ λ=fixed, thesecond derivative of the freeenergy jumps [4]References[1] Anton Kapustin, Brian Willett, and Itamar Yaakov. Exact Results for Wilson Loops in Superconformal Chern-Simons Theories withMatter. JHEP, 03:089, 2010.
[2] N. Hama, K. Hosomichi, and S. Lee, “Notes on SUSY Gauge Theories on Three-Sphere,” JHEP 1103 (2011) 127.
[3] Daniel L. Jafferis. The Exact Superconformal R-Symmetry Extremizes Z. JHEP, 05:159, 2012.
[4] J. G. Russo and M. Tierz, “Quantum phase transition in many-flavor supersymmetric QED3,” Phys. Rev. D 95 no. 3, (2017) 031901.

• Singularities found for λ>>λc are captured bya toy model:• There are infinite number of Borel singularities
associated with saddles on infinite number ofRiemann surfaces• Such a structure should appear at least in SUSY
observables of large-flavor gauge theories w/
 3dim N=2 SUSY on S3, rank(G)=1

 2dim N=(2,2) SUSY on S2, rank(G)=1

5. Conclusion and future works• A quantum phase transition in 3d N=4 SQED is discussed
in terms of resurgence the phase transition is interpreted as a stokes phenomenon A class of Borel singularities found in this model should appear
universally in large-flavor gauge theories• Description in terms of resurgence is not yet completed Borel singularities should be found on the positive real axis