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 Motivations, predictions, setting 1 The setting and some prequisits Multiple integrals from integrable models Typical results for the asymptotic behavior 2 Asymptotic analysis of multidimensional deformations of Fredholm series 3 Computing asymptotics is about rewriting Sketch of the asymptotic analysis The large-distance and long-time asymptotics Conclusion 4 K. K. Kozlowski Asymptotic behavior of series of multiple integrals.

Motivations, predictions, setting Typical results for the asymptotic behavior The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion What one would like to know? O ( α ) ⊛ H Hamiltonian in volume L , loc. Op. x ⊛ Hilbert space H = V 1 ⊗ · · · ⊗ V L . i) Find the Eigenstates and Eigenvectors of H| Ψ β � = E β | Ψ β � ; ii) Compute in closed form and characterize the correlation functions � Ψ γ |O ( α 1 ) . . . O ( α x ) | Ψ β � ; x 1 Characterize intrinsic & response properties of a system. Appear in perturbative expansions: H ֒ → H + H pert . ⊛ 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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion Predictions for the long-distance behavior of the correlators ⊛ T=0 Model becomes critical if gapless spectrum = ⇒ algebraic decay �O x O 1 � T = 0 ≡ � G . S . |O x O 1 | G . S . � C 1 C 2 ≃ �O 1 � 2 0 + x α 1 + x α 2 cos ( 2 xp F ) + . . . ... � G . S . | G . S . � • Prediction of critical exponents α i 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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion 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 C i . 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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion The non-linear Schrödinger model ⊛ The non-linear Schrödinger model � L � � ∂ y Ψ † ( y ) ∂ y Ψ ( y ) + c Ψ † ( y ) Ψ † ( y ) Ψ ( y )Ψ ( y ) − h Ψ † ( y ) Ψ ( y ) d y H = 0 L : length of circle, c > 0 coupling constant (repulsive regime), h > 0 chemical potential. � � � Equivalent to a collection of N -body Hamiltonians H N on F 0 < x 1 < · · · < x N < L � � ∂ 2 � N ∂ − ∂ with b . c . H N = − − c · χ N ( x 1 , . . . , x N ) | x j + 1 = x j = 0 ∂ x 2 ∂ x j + 1 ∂ x j p = 1 k ⊛ Eigenstates constructible by Bethe Ansatz ( ’63 Lieb-Linniger ) � � ϕ x 1 , . . . , x N | ℓ 1 , . . . , ℓ N ℓ a ∈ Z ⊛ Complete set parameterized by all choices of integers ℓ 1 < · · · < ℓ N ( ’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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion An Operator of interest ⊛ Define j ( x , 0 ) density operator at x � L � � � � · d N − 1 y � { m a } N 1 | j ( x , 0 ) | { ℓ a } N x , y 1 , . . . , y N − 1 | { m a } N x , y 1 , . . . , y N − 1 | { ℓ a } N 1 � = ϕ · ϕ 1 1 0 and its time evolution � � �� � { m a } N 1 | j ( x , t ) | { ℓ a } N · � { m a } N 1 | j ( x , 0 ) | { ℓ a } N 1 � = exp it E { m a } N 1 − E { ℓ a } N 1 � 1 � In field theoretic framework and j ( x , t ) = e iHt j ( x , 0 ) e − iHt j ( x , 0 ) = Ψ † ( x )Ψ( x ) ⊛ 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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion A correlator of interest Correlation functions at T = 0 K ≡ 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 ∂ 2 ∂ 2 with L N ; β ( x , t ) ∂β 2 L N ; β ( x , t ) | β = 0 = � G . S . | j ( x , t ) · j ( 0 , 0 ) | G . S . � ∂ x 2 ⊛ What do we want Obtain reasonable representation for L N ; β ( x , t ) Show excistence of thermodynamic limit L N ; β → L β Extract the asymptotics of L β ( x , t ) at x , t → + ∞ with x / t fixed � 1 st 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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion Multiple integral representations at the free fermion point • Fredholm determinant of pure sine kernel for L β ( x , 0 ) sin x   � q S x ( λ 1 , λ 1 ) . . . S x ( λ 1 , λ n ) 2 [ λ − µ ] � S x ( λ, µ ) = e β − 1 � �   1      · d n λ with   I + S x =  . . .  det det n  n ! π λ − µ S x ( λ n , λ 1 ) . . . S x ( λ n , λ n ) n ≥ 0 − q 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 The setting and some prequisits Asymptotic analysis of multidimensional deformations of Fredholm series Multiple integrals from integrable models Conclusion 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 ) q � + ∞ � � � • ∂ 2 I n ( x ; λ 1 , . . . , λ n ) · d n λ L β ( x ) β = 0 = ∂β 2 n = 1 − q I n ( x ; λ 1 , . . . , λ n ) = 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.