Lattice nonlinear Schr¨odinger: history and open problems Kun Hao and Vladimir Korepin Dublin Institute for Advanced Studies January 20, 2021 V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 1 / 30 Continuous model The model has many names: Lieb-Liniger, Bose gas with delta interaction, Tonks-Girardeau, nonlinear Sch¨odinger.It has application in quantum optics. It was build in optical lattice by N. J. van Druten https://arxiv.org/pdf/0709.1899.pdf The Hamiltonian is: Z y y y H = dx(@x @x + κψ ); (x); y(y) = iδ(x y) f g − It is integrable both in classical and quantum cases. It has infinitely many conservation laws and Lax representation https://en.wikipedia.org/wiki/Lax_pair. @t Ln = Mn+1(λ)Ln(λ) − Ln(λ)Mn(λ) An integer n is a discrete space variable x = n∆ and ∆ is lattice spacing. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 2 / 30 R matrix We shall discuss the quantum case. Lax representation follows from Yang-Baxter equation. https://arxiv.org/pdf/0910.0295.pdf O O R(λ, µ) Ln(λ) Ln(µ) = Ln(µ) Ln(λ) R(λ, µ) The R(λ, µ) solves the Yang-Baxter equation : C.N. Yang Phys. Rev. Lett. 19, (1967) 1312-1314 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1312 Yang found the following solution: R(λ, µ) = i1 + (µ − λ)Π This is the same R matrix, which describes spin 1=2 XXX Heisenberg chain. Here Π is permutation. C.N. Yang Phys. Rev. 168, (1968) 1920 https://journals.aps.org/pr/abstract/10.1103/PhysRev.168.1920 Note Note that for λ − µ = i the R(λ, µ) = i (1 − Π) is degenerate. It is 1D projector. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 3 / 30 QISM formulation The model was embedded into quantum inverse scattering method https://en.wikipedia.org/wiki/Quantum_inverse_scattering_method. We shall use algebraic Bethe Ansatz https://arxiv.org/abs/hep-th/9605187 The orthogonality and completeness of the Bethe Ansatz eigenstates was proved by Teunis C. Dorlas in 1993. An approximate L operator on a dense lattice was discovered by Faddeev , Sklyanin and Takhtajan in 1979 iλ∆ y 1 2 ipκχn 2 Ln(λ) = − − iλ∆ + O(∆ ); ipκχn 1 + 2 An integer n is a discrete space variable x = n∆ and ∆ is lattice spacing. The χn is the quantum field: y [χn; χm] = ∆ δnm χ = ∆ in the limit ∆ 0. The λ is the spectral parameter! and κ is a coupling constant. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 4 / 30 Lattice nonlinear Schr¨odingerequation Exact lattice Lax operator (all orders in ∆) was constructed by A.G. Izergin and V. E. Korepin in Doklady Akademii Nauk, 1981 https://arxiv.org/pdf/0910.0295.pdf see also Nuclear Physics B 205 [FS5], 401, 1982 iλ∆ κ y y ! 1 2 + 2 χj χj ipκχj %j Lj (λ) = − −iλ∆ κ y : ipκ%j χj 1 + 2 + 2 χj χj y κ y 1 [χ ; χ ] = ∆δ and % = (1 + χ χ ) 2 ; j l j;l j 4 j j here κ > 0, and ∆ > 0. The same R matrix. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 5 / 30 Equivalence of NLS and XXX chain with negative spin We can rewrite the L operator as XXX Heisenberg chain Zeitschrift f¨urPhysik 49, (1928) 619-636 https://link.springer.com/article/10.1007/BF01328601: XXX z k k Lj = −σ Lj = iλ + Sj ⊗ σ + y − z κ y Sj = ipκχ %j ; Sj = ipκ%j χj ; Sj = (1 + χ χj ): − j 2 j representation of SU(2) algebra with negative spin s = 2 . − κ∆ The σ are Pauli matrices. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 6 / 30 Antipode and quantum determinant The monodromy matrix and transfer matrix: A(λ) B(λ) T0(λ) L0L(λ) ::: L01(λ) = ; τ(λ) = tr0 T0(λ): ≡ C(λ) D(λ) N N R(λ, µ)(T0(λ) T0(µ)) = (T0(µ) T0(λ)) R(λ, µ) The antipode of quantum monodromy matrix is: −1 −L y t y T (λ) = dq (λ)σ T (λ + i)σ 2 dq(λ) = ∆ (λ ν)(λ ν + i)=4 ν = 2i=∆ − − − This is similar to Cramer's formula http://pi.math.cornell.edu/~andreim/Lec17.pdf The difference is a shift of the spectral parameter by i. The denominator is the quantum determinant: L detqT (λ) = A(λ)D(λ + i) B(λ)C(λ + i) = dq (λ) This was discovered in 1981− in https://arxiv.org/pdf/0910.0295.pdf Remark When λ − µ = i the R matrix turns into one dimensional projector. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 7 / 30 XXX chain with spin s = 1 − PL The Hamiltonian is: = k=1 Hk;k+1 The density of the HamiltonianH is Hk;k+1 = ( Jk;k+1) (Jk;k+1 + 1) + (2): − − − Here (x) = d ln Γ(x)=dx and ~ ~ Jk;k+1(Jk;k+1 + 1) = 2Sk Sk+1 ⊗ Can be solved by Vieta's formulas. QCD describes deep inelastic scattering. The scattering of a lepton on a baryon is a sum of Feynman diagrams. In leading logarithmic approximation ladder diagrams dominate. Quarks exchange gluons. The Hamiltonian describing interactions of the gluons is this spin chain. L. Lipatov https://arxiv.org/pdf/hep-th/9311037.pdf L. Alvarez-Gaume, https://arxiv.org/pdf/0804.1464.pdf Our paper: https://arxiv.org/pdf/1909.00800.pdf This is a part of larger trend: spin chains are related to 4D Yang-Mills N. A. Nekrasov, S. L. Shatashvili https://arxiv.org/pdf/0908.4052.pdf V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 8 / 30 Bethe equations An example is XXX chain with spin s = 1. The Bethe equations are https://en.wikipedia.org/wiki/Bethe_ansatz− : L N L N λk + is Y λk λj + i s=−1 λk i Y λk λj + i = − − = − λ is λ λ i λ + i λ λ i k j=1 k j −−−! k j=1 k j − j6=k − − j6=k − − k = 1; ; N ··· These are periodic boundary conditions. The energy is: N N X 1 d λj + i X 2 E ln = 2− ; ≡ i dλj λj i λ + 1 j=1 − j=1 j V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 9 / 30 Analysis of Bethe equations Theorem 1. All solutions of Bethe equations for s = −1 are real numbers. L N λk − i Y λk − λj + i = ; k = 1; ··· ; N: λk + i λk − λj − i j=1; j6=k Proof: Let us use the following properties: λ − i λ − i LHS: ≤ 1; when Imλ ≥ 0; ≥ 1; when Imλ ≤ 0; λ + i λ + i λ + i λ + i RHS: ≥ 1; when Imλ ≥ 0; ≤ 1; when Imλ ≤ 0; λ − i λ − i If we denote the one with maximal imaginary part as λmax 2 fλj g, then Im λmax ≥ Im λj ; j = 1; ··· ; N: For λk = λmax L N λmax − i Y λmax − λj + i = ≥ 1: λmax + i λmax − λj − i j=1 Due to LHS, this results in: Im λj ≤ Im λmax ≤ 0 Similarly, one has 0 ≤ Im λmin ≤ Im λj ! so Im λj = 0 V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 10 / 30 Solution exists and unique The logarithm of Bethe equations for the model s = 1, − N X 2πnk = θ(λk λj ) + L θ (λk ) ; − j=1 Here nk are different integer (or half integer) numbers: Pauli principle http://insti.physics.sunysb.edu/~korepin/PDF_files/Pauli.pdf iκ + λ θ(λ) = θ( λ) = i ln ; π < θ(λ) < π; Im λ = 0 − − iκ λ − − 2κ θ0(λ µ) = K(λ, µ) = ; K(λ) = K(λ, 0): − κ2 + (λ µ)2 − V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 11 / 30 Yang's action is convex Theorem 2. The solutions of the logarithmic form Bethe equations exist. Logarithmic Bethe equations are the extremums of the Yang action: N N N X 1 X X S = L θ (λ ) + θ (λ − λ ) − 2π n λ ; 1 k 2 1 k j k k k=1 k;j k=1 R λ θ1(λ) = 0 θ(µ)dµ. Bethe equations: @S=@λj = 0. N @2S X = δjl [LK(λj ) + K(λj ; λm)] − K(λj ; λl ) @λj @λl m=1 Consider some real vector vj . The quadratic form is positive: 2 N N X @ S X 2 X 2 vj vl = LK(λj )vj + K(λj ; λl )(vj − vl ) ≥ 0 @λj @λl j;l j=1 j>l The K(λj ) are positive. The action is convex: it has unique minimum. Solution of Bethe equation exists and unique. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 12 / 30 Side remark The same second derivative appears later in the theory. The square of the norm of the Bethe wave function is a determinant: @2S ΦN ΦN = det N h j i @λj @λl M. Gaudin conjectured in 1972. V.E. Korepin proved in 1982 http://insti.physics.sunysb.edu/~korepin/PDF_files/norm.PDF N @2S X = δjl [LK(λj ) + K(λj ; λm)] K(λj ; λl ) @λ @λ − j l m=1 V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 13 / 30 Open problem: Is to analyze the integral equation in the limit of κ 0 The K(λ, µ) 2πδ(λ µ): the integral cancel the! LHS ...? ! − The thermodynamic limit at zero temperature For positive κ all λj has to be different: Pauli principle in the momentum space is valid.