Resurgence, Phase Transitions and Large N Gerald Dunne University - - PowerPoint PPT Presentation

Gerald Dunne University of Connecticut

review: arXiv:1601.03414 ; winter school lectures

Yukawa Institute/RIKEN iTHEMS Conference 2020 , September 2020

Resurgence, Phase Transitions and Large N

Physical Motivation

Temperature (MeV) Baryon Doping – B (MeV)

50 100 150 200 250 300 200 400 600 800 1000 1200 1400 1600

Quark-Gluon Plasma

Color Superconductor Hadron Gas

Physical Motivation: Quantum Physics in Extreme Conditions

• QCD phase diagram
• non-equilibrium physics at strong-coupling
• (quantum) phase transitions in cold atom systems
• quantum systems in extreme background fields
• transition to hydrodynamics
• quantum gravity

extreme systems are extremely difficult to analyze quantitatively

• perturbation theory is of limited use
• non-perturbative semi-classical methods: “instantons"
• non-perturbative numerical methods: Monte Carlo
• asymptotics

extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; …

Physical Motivation: Quantum Physics in Extreme Conditions

• QCD phase diagram
• non-equilibrium physics at strong-coupling
• (quantum) phase transitions in cold atom systems
• quantum systems in extreme background fields
• transition to hydrodynamics
• quantum gravity

extreme systems are extremely difficult to analyze quantitatively

• perturbation theory is of limited use
• non-perturbative semi-classical methods: “instantons"
• non-perturbative numerical methods: Monte Carlo
• asymptotics

“resurgence”: new form of asymptotics that unifies these approaches

extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; …

Physical Motivation: Quantum Physics in Extreme Conditions

• QCD phase diagram
• non-equilibrium physics at strong-coupling
• (quantum) phase transitions in cold atom systems
• quantum systems in extreme background fields
• transition to hydrodynamics
• quantum gravity

technical problem: what does a quantum path integral really mean? extreme systems are extremely difficult to analyze quantitatively

• perturbation theory is of limited use
• non-perturbative semi-classical methods: “instantons"
• non-perturbative numerical methods: Monte Carlo
• asymptotics

“resurgence”: new form of asymptotics that unifies these approaches

extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; …

The Feynman Path Integral

• stationary phase approximation: classical physics
• quantum perturbation theory: fluctuations about trivial saddle point
• other saddle points: non-perturbative physics
• resurgence: saddle points are related by analytic continuation, so

perturbative and non-perturbative physics are unified

Z Dx(t) exp  i ~ S[x(t)]

• Z

DA(xµ) exp  i g2 S [A(xµ)]

• QFT:

QM:

Stokes and the Airy Function: “Stokes Phenomenon”

Ai(z) = 1 2π Z +∞

−∞

ei( 1

3 x3+z x)dx ∼

8 > > < > > :

e− 2

3 z3/2

2√π z1/4

, z → +∞

sin( 2

3 (−z)3/2+ π 4 )

√π (−z)1/4

, z → −∞

z Ai(z)

Stokes and the Airy Function: “Stokes Phenomenon”

Ai(z) = 1 2π Z +∞

−∞

ei( 1

3 x3+z x)dx ∼

8 > > < > > :

e− 2

3 z3/2

2√π z1/4

, z → +∞

sin( 2

3 (−z)3/2+ π 4 )

√π (−z)1/4

, z → −∞

z Ai(z)

• integral cannot be evaluated without

contour deformation

• “Stokes transition” at z=0
• fluctuation expansions about saddles

must be divergent, and must be related

• underlies optics and WKB analysis
• 4
• 2

2 4

• 1.0
• 0.5

0.5 1.0

x

Analytic Continuation of Path Integrals

Z Dx(t) exp  i ~ S[x(t)]

• Z

Dx(t) exp  −1 ~ S[x(t)]

• since we need complex analysis and contour deformation to

make sense of oscillatory integrals, it is natural to explore similar methods for (infinite dimensional) path integrals

idea: seek a computationally viable constructive definition

• f the path integral as a resurgent trans-series

Resurgent Trans-Series

resurgence: “new” idea in mathematics

Dingle 1960s, Ecalle, 1980s; Stokes 1850

perturbative series “trans-series”

f(~) = X

k

X

p

X

l

c[kpl] e− k

~ ~p (ln ~)l

f(~) = X

p

c[p] ~p

• unifies perturbative and non-perturbative physics
• trans-series is well-defined under analytic continuation
• expansions about different saddles are related
• exponentially improved asymptotics

mathematics: physics:

“resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the origin. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities”

• J. Ecalle, 1980

Resurgent Functions

conjecture: this structure occurs for all “natural” problems

Resurgence in Exponential Integrals

steepest descent integral through saddle point “n”: all fluctuations beyond the Gaussian approximation:

T (n)(~) ∼

∞

X

r=0

T (n)

r

~r I(n)(~) = Z

Cn

dx e

i ~ f(x) =

1 p 1/~ e

i ~ fn T (n)(~)

Resurgence in Exponential Integrals

steepest descent integral through saddle point “n”: all fluctuations beyond the Gaussian approximation:

T (n)(~) ∼

∞

X

r=0

T (n)

r

~r I(n)(~) = Z

Cn

dx e

i ~ f(x) =

1 p 1/~ e

i ~ fn T (n)(~)

(Fnm ≡ fm − fn)

universal large orders of fluctuation coefficients: straightforward complex analysis implies:

T (n)

r

∼(r − 1)! 2π i X

m

(±1) (Fnm)r " T (m) + Fnm (r − 1) T (m)

1

+ (Fnm)2 (r − 1)(r − 2) T (m)

2

+ . . #

Resurgence in Exponential Integrals

steepest descent integral through saddle point “n”: all fluctuations beyond the Gaussian approximation:

T (n)(~) ∼

∞

X

r=0

T (n)

r

~r

fluctuations about different saddles are quantitatively related

I(n)(~) = Z

Cn

dx e

i ~ f(x) =

1 p 1/~ e

i ~ fn T (n)(~)

(Fnm ≡ fm − fn)

universal large orders of fluctuation coefficients: straightforward complex analysis implies:

T (n)

r

∼(r − 1)! 2π i X

m

(±1) (Fnm)r " T (m) + Fnm (r − 1) T (m)

1

+ (Fnm)2 (r − 1)(r − 2) T (m)

2

+ . . #

Resurgence in Exponential Integrals

canonical example: Airy function: 2 saddle points

T ±

r = (±1)r Γ

• r + 1

6

• Γ
• r + 5

6

• (2π)

4

3

r r! = ⇢ 1, ± 5 48, 385 4608, ± 85085 663552, . . .

Resurgence in Exponential Integrals

canonical example: Airy function: 2 saddle points large orders of fluctuation coefficients: generic “large-order/low-order” resurgence relation

T +

r ∼ (r − 1)!

(2π) 4

3

r 1 − ✓4 3 ◆ 5 48 1 (r − 1) + ✓4 3 ◆2 385 4608 1 (r − 1)(r − 2) − . . . !

T ±

r = (±1)r Γ

• r + 1

6

• Γ
• r + 5

6

• (2π)

4

3

r r! = ⇢ 1, ± 5 48, 385 4608, ± 85085 663552, . . .

Resurgence in Exponential Integrals

canonical example: Airy function: 2 saddle points large orders of fluctuation coefficients: generic “large-order/low-order” resurgence relation

T +

r ∼ (r − 1)!

(2π) 4

3

r 1 − ✓4 3 ◆ 5 48 1 (r − 1) + ✓4 3 ◆2 385 4608 1 (r − 1)(r − 2) − . . . !

amazing fact: this large-order/low-order behavior has been found in matrix models, QM, QFT, string theory, … the only natural way to explain this is via analytic continuation of path integrals

T ±

r = (±1)r Γ

• r + 1

6

• Γ
• r + 5

6

• (2π)

4

3

r r! = ⇢ 1, ± 5 48, 385 4608, ± 85085 663552, . . .

Decoding a Path Integral as a Trans-Series

• expansions along different axes must be quantitatively related
• expansions about different saddles must be quantitatively related

Z DA e

i ~ S[A] =

X

thimbles

e

i ~ S[Athimble] × (fluctuations) × (logs)

logarithms

perturbation theory works, but it is generically divergent and there is a lot of interesting physics behind this this is actually a very good thing !

Perturbation Theory

unstable

• L. Euler, De seriebus divergentibus, Opera Omnia, I, 14, 585–617, 1760.

∞

X

n=0

(−1)n n! xn = ???

The Struggle to Make Sense of Divergent Series

The Struggle to Make Sense of Divergent Series factorial: convergent for all x > 0 “Borel summation” f(x) =

∞

X

n=0

(−1)nn! xn = Z ∞ dt e−t 1 1 + x t

The Struggle to Make Sense of Divergent Series factorial: convergent for all x > 0 “Borel summation” f(x) =

∞

X

n=0

(−1)nn! xn = Z ∞ dt e−t 1 1 + x t nonperturbative imaginary part ! f(−x) =

∞

X

n=0

n! xn = Z ∞ dt e−t 1 1 − x t Im[f(−x)] ∼ e−1/x

QM Perturbation Theory: Zeeman & Stark Effects Stark : divergent, non-alternating, asymptotic series Zeeman : divergent, alternating, asymptotic series physics: magnetic field causes energy level shifts (real)

• electric field causes energy level shifts (real)
• and ionization (imaginary, exponentially small)

physics:

an ∼ (−1)n(2n)! an ∼ (2n)!

QM Perturbation Theory: Zeeman & Stark Effects Stark : divergent, non-alternating, asymptotic series Zeeman : divergent, alternating, asymptotic series physics: magnetic field causes energy level shifts (real)

• electric field causes energy level shifts (real)
• and ionization (imaginary, exponentially small)

physics:

an ∼ (−1)n(2n)! an ∼ (2n)!

appears nicely consistent with Borel summation approach …

but not so fast … the story becomes even more interesting …

Instantons and Non-Perturbative Physics

• exponentially small non-perturbative splitting due to tunneling
• Yang-Mills theory and QCD have aspects of both systems
• physics of optical lattices and condensates

(phase transitions) (band structure)

Instantons and Non-Perturbative Physics

• exponentially small non-perturbative splitting due to tunneling
• Yang-Mills theory and QCD have aspects of both systems
• physics of optical lattices and condensates

surprise: perturbation theory is non-alternating divergent ! but these systems are stable ??? (phase transitions) (band structure)

unphysical imaginary parts exactly cancel !

• E. B. Bogomolny, 1980; J. Zinn-Justin et al, 1980

A Brilliant Resolution: “BZJ Cancelation Mechanism”

non-perturbative instanton & anti-instanton interaction: perturbation theory + Borel:

− → −i exp  −2 SI ~

• −

→ +i exp  −2 SI ~

• separately, each of the perturbative and non-perturbative computations

is inconsistent; but combined as a trans-series they are consistent

unphysical imaginary parts exactly cancel !

“Resurgence”: cancelations occur to all orders; the trans-series expression for the energy is real & well-defined tip-of-the-iceberg: perturbative/non-perturbative relations

• E. B. Bogomolny, 1980; J. Zinn-Justin et al, 1980

A Brilliant Resolution: “BZJ Cancelation Mechanism”

non-perturbative instanton & anti-instanton interaction: perturbation theory + Borel:

− → −i exp  −2 SI ~

• −

→ +i exp  −2 SI ~

• separately, each of the perturbative and non-perturbative computations

is inconsistent; but combined as a trans-series they are consistent

Resurgence in Quantum Mechanical Instanton Models

E±(~, N) = Epert(~, N) ± ~ √ 2π 1 N! ✓32 ~ ◆N+ 1

2

exp  −8 ~

• Pinst(~, N) + . . .

trans-series for energy, including non-perturbative splitting:

Resurgence in Quantum Mechanical Instanton Models

E±(~, N) = Epert(~, N) ± ~ √ 2π 1 N! ✓32 ~ ◆N+ 1

2

exp  −8 ~

• Pinst(~, N) + . . .

trans-series for energy, including non-perturbative splitting:

Pinst(~, N) = ∂Epert(~, N) ∂N exp " S Z ~ d~ ~3 ∂Epert(~, N) ∂N − ~ +

• N + 1

2

• ~2

S !#

perturbation theory encodes everything … to all orders fluctuations about first non-trivial saddle:

Resurgent Functions

• ccurs in QM path integrals with an infinite number of saddles

Parametric Resurgence and Phase Transitions

• for a phase transition: large N ``thermodynamic limit’’
• in general, we are interested in many parameters
• multiple parameters: different limits are possible
• “uniform” ’t Hooft limit:
• trans-series transmutes into different form in the large N limit
• hallmark of a phase transition

N → ∞ , ~ → 0 : ~ N = fixed

0.5 1.0 1.5 ℏ

• 1.0
• 0.5

0.5 1.0 1.5 2.0 2.5 u(ℏ)

Phase Transition in the Mathieu Equation Spectrum

• N= band/gap label; =coupling
• phase transition: narrow bands vs. narrow gaps:
• real instantons vs. complex instantons
• phase transition = “instanton condensation”
• universal phase transition

~ N = 8 π ~

Basar, GD, 1501.05671, GD, Unsal, 1603.04924

Resurgence in QFT: Euler-Heisenberg Effective Action

• integral representation = Borel sum
• analogue of Stark effect ionization and Dyson’s argument
• particle production in E field implies series are divergent

Stokes Phase Transition in QFT

• phase transition: tunneling vs. multi-photon “ionization”
• phase transition: real vs. complex instantons
• the same transition as in the Mathieu equation
• non-trivial quantum interference effects for general E(t)

E(t) = E cos(ω t)

• Schwinger effect with monochromatic E field:
• Keldysh adiabaticity parameter:
• WKB:

γ ≡ m c ω e E ΓQED ∼ exp  −π m2 c3 e ~ E g(γ)

• (Keldysh, 1964;

Brezin/Itzykson, 1980; Popov, 1981)

ΓQED ⇠ 8 > > < > > : exp h π m2 c3

e ~ E

i , γ ⌧ 1 (tunneling) e E

m c ω

4mc2/~ω , γ 1 (multiphoton)

Basar, GD, 1501.05671 GD, Dumlu,1004.2509, 1102.2899

Resurgence in QFT: Ultra-Fast Dynamics

• the adiabatic expansion is divergent
• resurgence: expansion can be (Borel) resummed to a universal form
• novel quantum interference effects: complex saddles
• applications in AMO and CM physics and in QFT

time evolution of quantum systems with ultra-fast perturbations

Resurgence in Asymptotically Free Quantum Field Theory

1979

“infrared renormalon puzzle”: the BZJ cancelation appears to fail …

Resurgence in Quantum Field Theory

infrared renormalon puzzle of asymptotically free QFT perturbation theory + Borel: non-perturbative instantons :

UV renormalon poles instanton/anti-instanton poles IR renormalon poles

− → −i exp  −2 SI g2

• −

→ +i exp  − 2 SI g2 β1

Resurgence in Quantum Field Theory

infrared renormalon puzzle of asymptotically free QFT perturbation theory + Borel: non-perturbative instantons :

UV renormalon poles instanton/anti-instanton poles IR renormalon poles

− → −i exp  −2 SI g2

• −

→ +i exp  − 2 SI g2 β1

• UV renormalon poles

instanton/anti-instanton poles IR renormalon poles neutral bion poles

new non-perturbative objects (“neutral bions”) lead to Bogomolny/Zinn-Justin style resurgent cancelation

GD/Unsal, 1210.2423

− → −i exp  − 2 SI g2 β1

Analytic Continuation of Path Integrals: “Lefschetz Thimbles”

Lefschetz thimble = “functional steepest descents contour”

• n a thimble, the path integral becomes

well-defined and computable ! complexified gradient flow:

Z(~) = Z DA exp ✓ i ~ S[A] ◆ = X

thimble

Nth ei φth Z

th

DA × (Jth) × exp ✓ Re  i ~S[A] ◆

?

Analytic Continuation of Path Integrals: “Lefschetz Thimbles”

(2013)

• 4d relativistic Bose gas: complex scalar field theory
• Monte Carlo on thimble softens the sign problem
• results comparable to “worm algorithm”

Fujii et al (2013)

• 4
• 2

2 4 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

• 4
• 2

2 4

• 1.0
• 0.5

0.5 1.0

Generalized Thimble Method

idea: flow to an approximate Lefschetz thimble

exact steepest descents contour

Alexandru, Basar, Bedaque et al 2016

Ai(z) = 1 2π Z +∞

−∞

ei( 1

3 x3+z x)dx

• 4
• 2

2 4 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

• 4
• 2

2 4

• 1.0
• 0.5

0.5 1.0

Generalized Thimble Method

idea: flow to an approximate Lefschetz thimble

exact steepest descents contour

• 4
• 2

2 4

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

approximate steepest descents contour

Alexandru, Basar, Bedaque et al 2016

Ai(z) = 1 2π Z +∞

−∞

ei( 1

3 x3+z x)dx

Generalized Thimble Method recall that thimble structure can change as parameters change

• G. Basar

Phase Transitions in QFT: 2d Thirring Model L = ¯ ψa (γν∂ν + m + µ γ0) ψa + g2 2Nf ¯ ψaγνψa ¯ ψbγνψb

• chain of interacting fermions: asymptotically free
• prototype for dense quark matter
• sign problem at nonzero density
• cousin of Hubbard model

Monte Carlo thimble computation

(Alexandru et al, 2016)

Tempered Lefschetz Thimble Method (Fukuma et al, 2017, 2019,…)

• probe all relevant thimbles ???
• sign problem vs. ergodicity
• coupling dynamical variable
• parallelized tempering
• e.g. 2d Hubbard model
• probes multiple thimbles

Tempered Lefschetz Thimble Method (Fukuma et al, 2017, 2019,…)

• probe all relevant thimbles ???
• sign problem vs. ergodicity
• coupling dynamical variable
• parallelized tempering
• e.g. 2d Hubbard model
• probes multiple thimbles

with tempering without tempering

Phase Transitions in 2d Gross-Neveu Model LGross−Neveu = ¯ ψai∂ /ψa + g2 2 ¯ ψaψa 2

• asymptotically free; dynamical mass; chiral symmetry; model for QCD
• large Nf chiral symmetry breaking phase transition
• physics = (relativistic) Peierls dimerization instability in 1+1 dim.

saddles solve inhomogeneous gap equation

σ(x; T, µ) = δ δσ(x; T, µ) ln det (i ∂ / − σ(x; T, µ))

σ(x; T, µ) ⌘ h ¯ ψψi(x; T, µ)

chiral symmetry breaking condensate develops crystalline phases !

(Thies et al) massless massive

Phase Transitions in Gross-Neveu Model

• (divergent) Ginzburg-Landau expansion = mKdV hierarchy
• exact saddles are known
• successive orders of GL expansion “reveal” crystal phase

saddles solve inhomogeneous gap equation

• thermodynamic potential
• all orders gives full crystal phase … but T=0 critical point is difficult

Basar, GD, Thies, 0903.1868

Ψ[σ; T, µ] = X

n

αn(T, µ)fn[σ(x, T, µ)] = α0 + α2 σ2 + α4

• σ4 + (σ0)2

+ . . .

Phase Transitions in Gross-Neveu Model

• low-density expansion at T=0: (non-perturbative trans-series)
• density expansion has non-perturbative terms: “trans-series”
• high-density expansion at T=0: (convergent)
• T=0 quantum phase transition

E(ρ) ∼ π 2 ρ2 ✓ 1 − 1 32(πρ)4 + 3 8192(πρ)8 − . . . ◆ E(ρ) ∼ − 1 4π + 2ρ π + 1 π

∞

X

k=1

e−k/ρ ρk−2 Fk−1(ρ) µcritical = 2 π ↔ ρ = 0

Resurgence and Large N Phase Transitions in Matrix Models

3rd order phase transition in Gross-Witten-Wadia unitary matrix model phase transition in the ``thermodynamic’’ large N limit Z depends on two parameters: ’t Hooft coupling t, and matrix size N

Z(t, N) = Z

U(N)

DU exp N t tr

• U + U †

Z(t, N) = det  Ij−k ✓N t ◆

j,k=1,...N

Gross-Witten, 1980 Wadia, 1980 Marino, 2008

Resurgence in Matrix Models at Large N

“order parameter” ∆(t, N) ⌘ hdet Ui satisfies a Painleve III equation

t2∆00 + t∆0 + N 2∆ t2

• 1 − ∆2

= ∆ 1 − ∆2 ⇣ N 2 − t2 (∆0)2⌘ N appears only as a parameter: perfect for large N asymptotics

∆(t, N) ∼ X

n

an(t) N n + e−N S(t) X

n

bn(t) N n + e−2N S(t) X

n

cn(t) N n + . . .

all physical observables inherit the large N trans-series structure large N instanton contributions: generated from ODE e.g. a0(t) = √ 1 − t

Ahmed & GD, 1710.01812

ODE large N weak coupling trans-series: weak coupling large N action:

∆(t, N) ∼ √ 1 − t

∞

X

n=0

d(0)

n (t)

N 2n − i σweak 2 √ 2πN t e−NSweak(t) (1 − t)1/4

∞

X

n=0

d(1)

n (t)

N n + . . .

Resurgence in Matrix Models at Large N ⇒

Sweak(t) = 2√1 − t t − 2 arctanh √ 1 − t

• <latexit sha1_base64="VSyH58Z6rqjPE3aoqM+8M6kVmJ8=">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</latexit>

ODE large N weak coupling trans-series: weak coupling large N action:

∆(t, N) ∼ √ 1 − t

∞

X

n=0

d(0)

n (t)

N 2n − i σweak 2 √ 2πN t e−NSweak(t) (1 − t)1/4

∞

X

n=0

d(1)

n (t)

N n + . . .

∞

X

n=0

d(1)

n (t)

N n = 1 + (3t2 − 12t − 8) 96(1 − t)3/2 1 N + . . .

“one-instanton" fluctuations:

Resurgence in Matrix Models at Large N ⇒

Sweak(t) = 2√1 − t t − 2 arctanh √ 1 − t

• <latexit sha1_base64="VSyH58Z6rqjPE3aoqM+8M6kVmJ8=">ACOHicdVBdSxtBFJ21dpoNdrHvgwNhQg17Abrx0NB7ItvWjQqZEO4O7mbDJmdXWfuKmHZn9UXf4Zv0pc+VKSv/QWdxJRa0QMDh3PO5c49UakJd+/8WZevJydezX/urKw+GZpubqyemLT3AhsiVSl5iwCi0pqbJEkhWeZQUgihafR8MvYP71AY2Wqj2mUYSeBvpaxFEBO6lYPjrpFaBJ+iTAs67T2OYwNiKIZ2nNDRbBOZVlQydeb4Uc+CYIRBHpQhgpjqv+LhUb2B7TWrdb8hj8Bf0A+cHOZsCDqVJjUx2q9dhLxV5gpqEAmvbgZ9RpwBDUigsK2FuMQMxhD62HdWQoO0Uk8NL/sEpPR6nxj1NfKI+nCgsXaURC6ZA3sY28sPuW1c4q3O4XUWU6oxf2iOFecUj5ukfekQUFq5AgI91fuRiAa45c1xVXwt9L+fPkpNkINho7Xzdqu3vTOubZO/ae1VnAtgu2eHrMUE+8a+s5/s1rvyfnh3q/76Iw3nXnL/oP3+w/9IK0t</latexit>

ODE large N weak coupling trans-series: weak coupling large N action:

∆(t, N) ∼ √ 1 − t

∞

X

n=0

d(0)

n (t)

N 2n − i σweak 2 √ 2πN t e−NSweak(t) (1 − t)1/4

∞

X

n=0

d(1)

n (t)

N n + . . .

∞

X

n=0

d(1)

n (t)

N n = 1 + (3t2 − 12t − 8) 96(1 − t)3/2 1 N + . . .

“one-instanton" fluctuations: resurgence: large-order growth of “perturbative coefficients”:

d(0)

n (t) ∼

−1 √ 2(1 − t)3/4π3/2 Γ(2n − 5

2)

(Sweak(t))2n− 5

2

 1 + (3t2 − 12t − 8) 96(1 − t)3/2 Sweak(t) (2n − 7

2) + . . .

• Resurgence in Matrix Models at Large N

⇒

Sweak(t) = 2√1 − t t − 2 arctanh √ 1 − t

• <latexit sha1_base64="VSyH58Z6rqjPE3aoqM+8M6kVmJ8=">ACOHicdVBdSxtBFJ21dpoNdrHvgwNhQg17Abrx0NB7ItvWjQqZEO4O7mbDJmdXWfuKmHZn9UXf4Zv0pc+VKSv/QWdxJRa0QMDh3PO5c49UakJd+/8WZevJydezX/urKw+GZpubqyemLT3AhsiVSl5iwCi0pqbJEkhWeZQUgihafR8MvYP71AY2Wqj2mUYSeBvpaxFEBO6lYPjrpFaBJ+iTAs67T2OYwNiKIZ2nNDRbBOZVlQydeb4Uc+CYIRBHpQhgpjqv+LhUb2B7TWrdb8hj8Bf0A+cHOZsCDqVJjUx2q9dhLxV5gpqEAmvbgZ9RpwBDUigsK2FuMQMxhD62HdWQoO0Uk8NL/sEpPR6nxj1NfKI+nCgsXaURC6ZA3sY28sPuW1c4q3O4XUWU6oxf2iOFecUj5ukfekQUFq5AgI91fuRiAa45c1xVXwt9L+fPkpNkINho7Xzdqu3vTOubZO/ae1VnAtgu2eHrMUE+8a+s5/s1rvyfnh3q/76Iw3nXnL/oP3+w/9IK0t</latexit>

large N strong coupling trans-series: completely different structure strong coupling large N action:

Resurgence in Matrix Models at Large N Sstrong(t) = arccosh (t) − r 1 − 1 t2

∆(t, N) ∼ √ t e−NSstrong(t) √ 2πN (t2 − 1)1/4

∞

X

n=0

Un (q(t)) N n

+ 1 4(t2 − 1) ✓ √ te−NSstrong(t) √ 2πN (t2 − 1)1/4 ◆3

∞

X

n=0

U (1)

n

(q(t)) N n + . . .

large N strong coupling trans-series: completely different structure strong coupling large N action:

Resurgence in Matrix Models at Large N Sstrong(t) = arccosh (t) − r 1 − 1 t2

∆(t, N) ∼ √ t e−NSstrong(t) √ 2πN (t2 − 1)1/4

∞

X

n=0

Un (q(t)) N n

+ 1 4(t2 − 1) ✓ √ te−NSstrong(t) √ 2πN (t2 − 1)1/4 ◆3

∞

X

n=0

U (1)

n

(q(t)) N n + . . .

resurgence: large-order growth of “perturbative coefficients”:

Un (q(t)) ∼ 1 2π (−1)n (n − 1)! (2Sstrong(t))n ✓ 1 + U1 (q(t)) (2Sstrong(t)) (n − 1) + U2 (q(t)) (2Sstrong(t))2 (n − 1)(n − 2) + . . . ◆

Large N Transmutation of Transseries weak-coupling trans-series changes its form across the phase transition into the strong-coupling phase universal transition: cf. Mathieu & Schwinger effect examples immediate vicinity of t=1 is described by Painleve II equation (“double-scaling limit”) physics = instanton condensation physics = Stokes transition between real and complex instantons

Lee-Yang view of Large N Phase Transitions in Matrix Models

Lee-Yang: complex zeros of Z(t, N) pinch the real t axis at the phase transition, in the thermodynamic (large N) limit complex parameters can indicate phase transitions

2 dim Yang-Mills: Douglas-Kazakov Large N Phase Transition e.g., 2d Yang-Mills on sphere “spectral sum” for partition function: large N phase transition at critical area “saddle sum” for partition function: dual descriptions: generalized Poisson duality phase transition = change of saddles phase transition = transmutation of trans-series Z(a, N) = X

~ n

F(~ n) e− 2π2N

a

~ n2

Z(a, N) = X

R

(dim R)2 e− a

2N C2(R)

2d Lattice Ising Model paradigm of phase transitions Kramers-Wannier duality phase transition when expansions about T=0 and T=infinity are both convergent Z(βJ) sinhN/2(2βJ) = Z(β ˜ J) sinhN/2(2β ˜ J) , tanh(βJ) ≡ e−2β ˜

J

k ≡ sinh2(2βJ) = 1

2d Lattice Ising Model paradigm of phase transitions Kramers-Wannier duality phase transition when expansions about T=0 and T=infinity are both convergent Z(βJ) sinhN/2(2βJ) = Z(β ˜ J) sinhN/2(2β ˜ J) , tanh(βJ) ≡ e−2β ˜

J

k ≡ sinh2(2βJ) = 1 −β F(k) + 1 4 ln k ∼ 1 4 ln k − k 4 + k2 32 − k3 48 + 9k4 1024 + · · · ∼ 1 4 ln 1 k − 1 4k + 1 32k2 − 1 48k3 + 9 1024k4 − · · ·

2d Lattice Ising Model paradigm of phase transitions Kramers-Wannier duality phase transition when expansions about T=0 and T=infinity are both convergent Z(βJ) sinhN/2(2βJ) = Z(β ˜ J) sinhN/2(2β ˜ J) , tanh(βJ) ≡ e−2β ˜

J

k ≡ sinh2(2βJ) = 1 −β F(k) + 1 4 ln k ∼ 1 4 ln k − k 4 + k2 32 − k3 48 + 9k4 1024 + · · · ∼ 1 4 ln 1 k − 1 4k + 1 32k2 − 1 48k3 + 9 1024k4 − · · · resurgence: logarithmic behavior at critical T

Resurgence in 2d Lattice Ising Model diagonal correlation functions: C(s, N) = tau function for Painleve VI equation (Jimbo, Miwa) convergent conformal block expansions at low T and high T: resurgence also for convergent expansions ! C(s, N) = hσ0,0 σN,Ni(s) C(s, N) has a trans-series expansion: convergent about T=0, T= ∞ scaling limit: PVI PIII as N → ∞ & T → Tc ⌧(s) ∼

∞

X

n=−∞

⇢n C(~ ✓, +n) B(~ ✓, +n; s) B(~ ✓, ; s) ∝ sσ2 X

λ,µ∈Y

Bλ,µ(~ ✓, )s|λ|+|µ|

(McCoy et al) (Lisovyy et al, 2012, 2013 …) GD, 1901.02076

Resurgent Extrapolation

• often, asymptotics is the ONLY thing we can do
• question: how much global information can be decoded from a

FINITE number of perturbative coefficients ?

• how much “perturbative” information is required to detect, and

to probe the properties of, a phase transition ?

Temperature (MeV) Baryon Doping – B (MeV)

50 100 150 200 250 300 200 400 600 800 1000 1200 1400 1600

Quark-Gluon Plasma

Color Superconductor H a d r

• n

G a s

Costin, GD: 1904.11593, 2003.07451 2009.01962 Zach Harris poster: Wed/Thurs

Resurgent Extrapolation

• case-study: Painleve I equation

y00(x) = 6 y2(x) − x

• Re[x]

Im[x]

• Painleve I equation has 5 sectors in the complex x plane, separated

by phase transitions

• tritronquée solution: poles only in shaded region
• suppose we expand about x=+infty to finite order N: how

much do these coefficients “know” about the other sectors?

Resurgent Extrapolation

• extrapolate across Stokes transitions, even into the tritronquée

pole region

• resurgent extrapolation can decode global behavior from

surprisingly little input data from some other regime

• Pade-Borel + conformal or uniformizing maps: extreme precision
• 10 terms at x=+infty encode 23 digits of precision at x=0
• 15
• 10
• 5

5 10 15 Re[x]

• 15
• 10
• 5

5 10 15 Im[x]

Costin, GD: 1904.11593, 2009.01962

Resurgent Extrapolation

• resurgent extrapolation can decode global behavior

y(x) ≈ 1 (x − xpole)2 + xpole 10 (x − xpole)2 + 1 6(x − xpole)3 +hpole(x − xpole)4 + x2

pole

300 (x − xpole)6 + . . .

• transmutation of the trans-series:
• near x → +∞

y(x) ∼ − rx 6

∞

X

n=0

an 1 x5n/2 , x → +∞

• into the pole region:

4π 5 ≤ arg(x) ≤ 6π 5

• this phase transition is encoded in (few) fluctuation coeffs at
• along Stokes/anti-Stokes lines: exponentials

x = +∞

• resurgence: new summation & extrapolation methods

Conclusions

• “resurgence” is a new and improved form of asymptotics
• deep connections between perturbative and non-perturbative physics
• recent applications to differential eqs, QM, QFT, string theory, …
• 2-parameter trans-series can describe phase transitions
• outlook: new theoretical approach to quantum systems in extreme

conditions

• outlook: computational definition of real-time path integrals
• outlook: computational access to strongly-coupled systems, phase

transitions, particle production, and far-from-equilibrium physics