Asymptotic behavior of series of multiple integrals. K. K. Kozlowski - - PowerPoint PPT Presentation

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion

Asymptotic behavior of series of multiple integrals.

• K. K. Kozlowski

CNRS, Institut de Mathématiques de Bourgogne

3, September 2012

In collaboration with : N. Kitanine, J.-M. Maillet, N.A. Slavnov and V. Terras .

• K. K. Kozlowski, "Riemann–Hilbert approach to the time-dependent generalized sine kernel" , math-ph:

10115897.

• K. K. Kozlowski & V. Terras, "Long-time and large-distance asymptotic behavior of the current-current correlators

in the non-linear Schrödinger model" , J. Stat. Mech.: Th. and Exp., P09013, (2011).

• K. K. Kozlowski, "Large-distance and long-time asymptotic behavior of the reduced density matrix in the

non-linear Schrödinger model" , math-ph:11011626.

Quantum integrable systems and geometry – Olhao 2012

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion

Outline

1

Motivations, predictions, setting The setting and some prequisits Multiple integrals from integrable models

2

Typical results for the asymptotic behavior

3

Asymptotic analysis of multidimensional deformations of Fredholm series Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

4

Conclusion

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

What one would like to know?

⊛ H Hamiltonian in volume L, O(α)

x

• loc. Op.

⊛ Hilbert space H = V1 ⊗ · · · ⊗ VL . i) Find the Eigenstates and Eigenvectors of H| Ψβ = Eβ| Ψβ ; ii) Compute in closed form and characterize the correlation functions Ψγ |O(α1)

1

. . . O(αx)

x

| Ψβ ;

Characterize intrinsic & response properties of a system. Appear in perturbative expansions: H ֒→ H + Hpert .

⊛ Program i) − ii) especially interesting when L → +∞.

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

Predictions for the long-distance behavior of the correlators

⊛ T=0 Model becomes critical if gapless spectrum =⇒ algebraic decay OxO1T=0 ≡ G.S. |OxO1| G.S. G.S. | G.S. ≃ O12

0 +

C1 xα1 + C2 xα2 cos (2xpF) + . . . ...

• Prediction of critical exponents αi by approximate methods

Correspondence with a Conformal Field Theory (’70 Polyakov , ’84 Cardy ) Correspondence with Luttinger liquid (’75 Luther, Peschel , ’81 Haldane )

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

Quantum one-dimensional integrable models

⊛ Models where one can characterize spectrum & eigenvectors. ◮ In some cases, also exist various types of representations for the correlation functions.

• Analysis of expressions from quantum integrability allows to

i) test validity and regime of CFT/Luttinger liquid predictions. ii) Yield value/interpretation of the amplitudes Ci . iii) Go beyond limit of existing predictions Inculde the effects of time, short-distance...

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

The non-linear Schrödinger model

⊛ The non-linear Schrödinger model H =

L

• ∂yΨ† (y) ∂yΨ (y) + cΨ† (y) Ψ† (y) Ψ (y)Ψ (y) − hΨ† (y) Ψ (y)
• dy

L: length of circle, c > 0 coupling constant (repulsive regime), h > 0 chemical potential. Equivalent to a collection of N-body Hamiltonians HN on F

• 0 < x1 < · · · < xN < L
• HN = −

N

• p=1

∂2 ∂x2

k

with b.c.

• ∂

∂xj+1 − ∂ ∂xj − c

• · χN(x1, . . . , xN)|xj+1=xj = 0

⊛ Eigenstates constructible by Bethe Ansatz ( ’63 Lieb-Linniger ) ϕ

• x1, . . . , xN | ℓ1, . . . , ℓN
• ℓa ∈ Z

⊛ Complete set parameterized by all choices of integers ℓ1 < · · · < ℓN (’90 Dorlas) ⊛ Pertinent for physics after thermodynamic limit N, L → +∞ with N/L → D

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

An Operator of interest

⊛ Define j(x, 0) density operator at x {ma}N

1 |j(x, 0)| {ℓa}N 1 = L

• ϕ
• x, y1, . . . , yN−1 | {ma}N

1

• · ϕ
• x, y1, . . . , yN−1 | {ℓa}N

1

• · dN−1y

and its time evolution {ma}N

1 |j(x, t)| {ℓa}N 1 = exp

• it
• E{ma }N

1 − E{ℓa}N 1

• · {ma}N

1 |j(x, 0)| {ℓa}N 1

In field theoretic framework j(x, 0) = Ψ†(x)Ψ(x) and j(x, t) = eiHtj(x, 0)e−iHt ⊛ Eigenvectors highly intricate expression. Long series of works to compute & prove determinant rep. Lenard, Izergin, Korepin, Kitanine, Maillet, Terras, Korepin, Slavnov, Dorlas, Oota, K.

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

A correlator of interest

Correlation functions at T = 0K ≡ expectation values in the ground state Current-current correlation function G.S. |j(x, t) · j(0, 0)| G.S. computed from its generating function LN;β(x, t) with ∂2 ∂x2 ∂2 ∂β2 LN;β(x, t)|β=0 = G.S. |j(x, t) · j(0, 0)| G.S. ⊛ What do we want Obtain reasonable representation for LN;β(x, t) Show excistence of thermodynamic limit LN;β → Lβ Extract the asymptotics of Lβ(x, t) at x, t → +∞ with x/t fixed 1st results for c = ∞ (free fermions) =⇒ Toeplitz or Fredholm det. rep. 45 years of efforts: Kaufman, Onsager, Lieb, Mattis, Schulz, Lenard, McCoy, Wu, Korepin, Slavnov . . .

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

Multiple integral representations at the free fermion point

• Fredholm determinant of pure sine kernel for Lβ(x, 0)

det

• I + Sx
• =
• n≥0

1 n!

q

• −q

detn        Sx (λ1, λ1) . . . Sx (λ1, λn) . . . Sx (λn, λ1) . . . Sx (λn, λn)        · dnλ with Sx (λ, µ) = eβ − 1 π sin x 2 [λ − µ] λ − µ

The number of integrals varies from 0 to +∞ extract x → +∞ behavior. Tour de force direct analysis (’79 Tracy, Vaidya ); Sine kernel related to Painlevé V (’80 Jimbo, Miwa, Mori, Sato ); RHP setting for integrable integral operators (’90 Its, Izergin, Korepin, Slavnov ).

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

Correlation functions out of the free fermion point

⋆ Algebraic version of Bethe Ansatz (’79 Faddeev, Takhtadjan, Sklyanin )

• First series of multiple integrals at T 0 and h 0 ( ’84 Izergin-Korepin )
• ∂2

∂β2

• Lβ (x)
• β=0 =

+∞

• n=1

q

• −q

In (x; λ1, . . . , λn) · dnλ In (x; λ1, . . . , λn) = combinatorial sums & non − linear integral equations

• Explicit multiple integral representations at T=0 (’92 -’96 Kyoto Group )
• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

More recent results

• Series of mult. int. for σz

1σz m+1(t) at T = 0 ( ’04 Kitanine, Maillet, Slavnov, Terras )

σz

1σz m+1(t) = +∞

• n=1

q

• −q
• Γ

F (n)

(m,t)

µ1, . . . , µn z1, . . . , zn

• · dnµdnz
• New form of series of multiple integrals (’06-’08 KKMST ) Lβ(x, 0) =

+∞

• n=0

1 n!I(x)

n

[Fn]

• Long-distance asymptotics
• j(x, 0)j(0, 0)
• from first principles (’08 KKMST )
• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

Time and space dependent generating function: non-free fermion case

’11 K., Terras ⊛ Series of multiple integrals Lβ(x, t) =

+∞

• n=0

1 n!I(x,t)

n

[Fn] ⊛ I(x)

n

• scillatory integral in 3n variables

I(x,t)

n

[Fn] =

q

• −q

dnλ (2iπ)n

• Γ

dnz (2iπ)n

• C

dny (2iπ)n · detn eix[yℓ−λℓ− t

x (y2 ℓ −λ2 ℓ )]

(zℓ − λℓ) (zℓ − λj)

• ·

n

• a=1

f(ya) ya − za · Fn {λk }n

1

{yk}n

1

• .

The functions Fn are symmetric and analytic in (y1, . . . , yn) and (λ1, . . . , λn); satisfy reduction properties Fn and Fn−1 related when yn = λn.

✲ ✬ ✫ ✩ ✪

Γ

✛ ✚ ✘ ✙ ✲

−q q λj x λℓ x R ⋆ Poles zℓ = λℓ & zℓ = λj ⋆ Poles ys = zs.

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion The setting and some prequisits Multiple integrals from integrable models

Some particular cases

Example : F (F,g)

n

λ1, . . . , λn z1, . . . , zn

• =

n

• a=1

F(λa) eg(za) eg(λa) pole combinatorics simplify most integrals separate t = 0 strongly simplifies final formula I(x,0)

n

• F (F,g)

n

• =

q

• −q

detn

• V(x)

F,g(λℓ, λj)

• · dnλ .

⊛ Initial series turns into a Fredholm series L(F,g)

β

(x, t) =

+∞

• n=0

1 n!I(x,0)

n

• F (F,g)

n

• = det
• I + V(x)

F,g

• ⊛ Lβ (x, t) ≡ Multidimensional deformation of Fredholm determinant representation
• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion

Asymptotic behavior of the generating function at t = 0

’08 Kitanine, K., Maillet, Slavnov, Terras Lβ(x, 0) = e

βx

q

• −q

ρ(µ)dµ

2π sinh(2q)ρ(q)x β2Z2(q)

2π2

A βZ(∗) 2iπ

• ·
• 1 + O

ln x x

• ⊛ ρ(λ), Z(λ) solutions to linear integral equations driven by exponent p0 and Fn :

ρ(λ) +

q

• −q

K(λ − µ)ρ(µ)dµ = p′

0(µ)

2π and Z(λ) +

q

• −q

K(λ − µ)Z(µ)dµ = 1 kernel K dictated by Fn A explicit functional built out of Fn ’s. Asymptotics valid under various summability hypothesis. ⋆ Fredholm det. asympt.

• functional of a function of the data

det

• I + V(x)

F,g

• ∼ Fx
• ν, g
• where

ν(λ) = i 2π ln

• 1 + F(λ)
• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

A one-dimensional integral

Consider Ix[f] = δ

0 g(t)f(e−xs)ds

with g, f ∈ C∞([ −2δ ; 2δ ]) ⊛ Ix[f] is linear in f ; ⊛ If P(s) = sn then Ix[P] is a Laplace integral O.K. for asymptotics; Ix[P] = δ g(s)e−nxsds = g(0) − g(δ)e−nxδ nx + 1 x δ g′(s)e−nxs n ds ⊛ By linearity asymptotics computable for any polynomial P(s) =

N

• n=1

ansn; Ix[P] = g(0) x

N

• n=1

an n − g(δ) x

N

• n=1

an n e−nxδ + 1 x δ g′(s)

N

• n=1

ane−nxs n ds

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

A one-dimensional integral

⊛ Recast the answer solely in terms of P

N

• n=1

ansn n =

N

• n=1

an s yn−1dy = s P(y) y dy ≡ H[P](s) Ix[P] = g(0) x H[P](1) − g(δ) x H[P](e−xδ) + 1 x δ g′(s)H[P](e−xs)ds lhs & rhs are linear functionals continuous in f1 = sups{|f(s)| + |f′(s)|}; Polynomals are dense in (C1([ −2δ ; 2δ ]), ·1) Ix[f] = g(0) x H[f](1) − g(δ) x H[f](e−xδ)

• O(x−∞)

+ 1 x δ g′(s)H[f](e−xs)ds

• O(x−1)
• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

Distance and time dependent case

’11 K., Terras K. ⊛ Convergence properties of generating function

• pen ⇒ analysis of γ-regularization:

L(p)

β (x, t) =

• n≥0

1 n!

q

• −q

dnλ (2iπ)n

• Γ

dnz (2iπ)n

• C

dny (2iπ)n ·detn eix[yℓ−λℓ− t

x (y2 ℓ −λ2 ℓ )]

(zℓ − λℓ) (zℓ − λj)

• · ∂

∂γp

• F (γ)

n

• {zk}n

1; {λk}n 1

{yk }n

1

• |γ=0

.

⊛ The reasons Lβ(x, t) =

+∞

• p=0

L(p)

β (x, t)

p! formally; x → +∞ asymptotics of L(p)

β

(x, t) are a theorem; x → +∞ asymptotics sum-up termwise.

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

A sketch of the asymptotic analysis

⊛ Steps of the analysis i) Approach F (γ)

n

by density ii) Relate initial object to Fredholm determinant-like object L(p)

β (x, t) =

∂ ∂γp

• det
• I + V(x)

γν, g

• · F (γ)[ν]
• |γ=0
• iii) Construct a Natte series representation for classical det
• I + γV(x)

F,g

• iv) Carry out the < · > averaging lift Natte series up to L(p)

β (x, t)

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

⊛ Lifting of the Natte series ( ’11 K., Terras , ’11 K. ) L(p)

β

(x, t) = ∂ ∂γp ·

• H0(x, t) [γν0] F (γ)

[γν0]

• |γ=0

+

• n≥1
• {ǫt}
• C

∂ ∂γp ·

• Hn (x, t; {zt})
• γνǫt (∗, {zt})
• ×F (γ)

n

{zt}ǫt>0 {zt}ǫt<0 γνǫt (∗; {zt})

|γ=0

dnzt (2iπ)n . where νǫt (λ; {zt}) = β 2iπ Z (λ) −

t

ǫt φ(λ, zt) fixed by solving linear integral equations φ(λ, µ) + q

−q

K(λ − τ)φ(τ, µ) · dτ 2π = ϑ(λ − µ) Hn (x, t; {zt}) ≫ Hn+1 (x, t; {zt})

Yields large-time and space asymptotics of two-point functions =⇒ beyond applicability of existing approximate techniques .

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

Turining the time on

Predictions for the long-distance/long-time behavior at T = 0K restricted to x ≫ vFt:

• j (x, t) j (0, 0)
• ≃ j (0, 0)2 + C′

1

x2 + v2

Ft2

• x2 − v2

Ft22 + C′ 2

cos (2xpF)

• x2 − v2

Fx2Z2 + ...

⇒ Consistency problem with time-dependent asymptotics x2 + v2

Ft2

• x2 − v2

Ft22 (1 + o (1)) = 1

x2 (1 + o (1)) when x ≫ vFt

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics

T=0K leading harmonics in long-time & distance asymptotics

⊛ : Aymptotic regime x → +∞ and x/t fixed. Overall structure of the asymptotic series (space-like regime):

• j (x, t) j (0, 0)
• =

pF π 2 − Z2 2π2 x2 + t2v2

F

• x2 − t2v2

F

2 (1 + o (1)) +

• ℓ+;ℓ−∈Z

ℓ++ℓ−≤0 ∗

eixℓ+pF [−i(x − vFt)]

∆(R)

ℓ+;ℓ−

e−ixℓ−pF [i(x + vFt)]

∆(L)

ℓ+;ℓ−

× e−i(ℓ++ℓ−)[xp(λ0)−tε(λ0)]

• [p′ (λ0)]2

−i[xp′′(λ0) − tε′′(λ0)] |ℓ++ℓ−|2

2

· (2π) |ℓ++ℓ−|

2

• F (j)

ℓ+,ℓ−

• 2

G

• 1 +
• ℓ+ + ℓ−
• (1 + o (1)) .

⋆ λ0 Saddle-point of the oscillating phase: p′(λ0) − tε′(λ0) /x = 0. p dressed momentum & ε dressed energy.

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior Asymptotic analysis of multidimensional deformations of Fredholm series Conclusion

Conslusion and perspectives

Today’s main message

Correlatiors ≡ multidimensional deformations of Fredholm determinants ;

Multidimensional Natte series Mellin-Barnes equivalent for multiple integrals ;

Method aplicable in a broad setting.

Some roads that follow

⊛ Develop techniques for studying convergence of series ; ⊛ Short distance asymptotics in massive integrable QFT ; ⊛ Emptiness formation probability ; ⊛ Develop analog for multidimensional deformations of OP subordinate integrals .

• K. K. Kozlowski

Asymptotic behavior of series of multiple integrals.