Lattice Nonlinear Schrödinger

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 † † † H = dx(∂x ψ ∂x ψ + κψ ψ ψψ),

ψ(x), ψ†(y) = iδ(x y) { } − 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∆ † 1 2 i√κχn 2 Ln(λ) = − − iλ∆ + O(∆ ), i√κχn 1 + 2 An integer n is a discrete space variable x = n∆ and ∆ is lattice spacing. The χn is the quantum ﬁeld:

† [χn, χm] = ∆ δnm χ = ψ∆ 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 † † ! 1 2 + 2 χj χj i√κχj %j Lj (λ) = − −iλ∆ κ † . i√κ%j χj 1 + 2 + 2 χj χj

† κ † 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 + † − z κ † Sj = i√κχ %j , Sj = i√κ%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 = ψ( Jk,k+1) ψ(Jk,k+1 + 1) + ψ(2). − − − Here ψ(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 ∈ {λj }, 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 | 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. A.G. Izergin and V.E. Korepin; Letters in Mathematical Physics 1982: http://insti.physics.sunysb.edu/~korepin/PDF_files/Pauli.pdf In the limit L and N , the λj are condensed into Fermi sphere [ q, q]. → ∞ → ∞ 1 − The distribution function ρp(λj ) = satisfy: L(λj+1−λj ) Z q 2πρp(λ) = K(λ, µ)ρp(µ)dµ + K(λ) −q 2κ Z q N K(λ, µ) = , K(λ) = K(λ, 0), ρ (λ)dλ = D = κ2 + (λ µ)2 p L − −q For κ = 0 Fermi sphere collapse: the ground state is Bose-Einstein condensate.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 14 / 30 The thermodynamic limit at zero temperature

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 14 / 30 Collapse of the Fermi sphere in the continuous case

In the continuous case of nonlinear Schr¨odingerthe integral equation is: Z q 2πρp(λ) = K(λ, µ)ρp(µ)dµ + 1. −q In the limit κ 0 the K(λ, µ) 2πδ(λ µ).The integral cancel the LHS. The limit was studied→ by → − S. Prolhac https://arxiv.org/pdf/1610.08912.pdf G. Lang https://arxiv.org/pdf/1907.04410.pdf C. Tracy, H. Widom https://arxiv.org/pdf/1609.07793.pdf The decomposition is in √κ and log κ. Coeﬃcients are objects of number theory. In the lattice case the limit is an open problem.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 15 / 30 Energy of elementary excitation

Return to the lattice case: special case XXX with spin s = 1. Considering the grand canonical ensemble, the energy spectrum becomes −

N X 2 Eh = ( − h). λ2 + 1 − j=1 j The h is the chemical potential. In thermodynamic limit the energy of elementary excitation ε(λ) satisﬁes the linear integral equation

1 Z +q 2 ε(λ) K(λ, µ)ε(µ)dµ = 2− h ε0(λ), − 2π −q λ + 1 − ≡ ε(q) = ε( q) = 0 −

Remark The elementary excitation has a topological charge: it does not ﬁt into periodical boundary conditions, we have to change the boundary conditions into anti-periodic.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 16 / 30 Construction of elementary excitation

hole

particle particle λ q q −

Figure 1: The energy of the elementary excitation as a function of λ. For −q < λ < q elementary excitation is a hole, but it is the particle for other values of λ.

In the inﬁnite volume limit any energy level is a scattering state of several elementary excitations with diﬀerent momenta.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 17 / 30 Momentum of the elementary excitation.

The momentum of the particle k(λp) is Z q i + λ k(λp) = p0(λp) + θ(λp µ)ρp(µ)dµ, θ(λ) = p0(λ) = i ln . − i λ −q − The momentum kh(λh) of elementary hole excitation is Z q kh(λh) = p0(λh) θ(λh µ)ρp(µ)dµ. − − −q − where q < λh < q. At zero temperature all the observables are described by a linear integral− equation.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 18 / 30 Remark Many body scattering matrix is a product of pairwise scattering matrices. This can be used as a deﬁnition of complete integrability in many body quantum mechanics.

Scattering matrix

The scattering matrix of two elementary excitation is a transition coeﬃcient:

S = exp iφ(λp, λh) , {− } the scattering phase satisﬁes the integral equation: 1 Z +q φ(λp, λh) K(λp, ν)φ(ν, λ λh)dν = θ(λp λh). − 2π −q − −

iκ + λ θ(λ) = θ( λ) = i ln ; π < θ(λ) < π, Im λ = 0 − − iκ λ − −

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 19 / 30 Scattering matrix

The scattering matrix of two elementary excitation is a transition coeﬃcient:

S = exp iφ(λp, λh) , {− } the scattering phase satisﬁes the integral equation: 1 Z +q φ(λp, λh) K(λp, ν)φ(ν, λ λh)dν = θ(λp λh). − 2π −q − −

iκ + λ θ(λ) = θ( λ) = i ln ; π < θ(λ) < π, Im λ = 0 − − iκ λ − −

Remark Many body scattering matrix is a product of pairwise scattering matrices. This can be used as a deﬁnition of complete integrability in many body quantum mechanics.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 19 / 30 Thermodynamics

The Yang-Yang equation describes thermodynamics Z +∞ 2 T −ε(µ)/T ε(λ) = 2− h K(λ, µ) ln(1 + e )dµ, λ + 1 − − 2π −∞

Z ∞ ρh(λ) ε(λ)/T N = e , D = = ρp(λ)dλ. ρp(λ) L −∞ The rigorous proof was given by Tony Dorlas. Remark Stable excitation exists even at positive temperature, because of the conservation laws. The ε(λ) is the energy of this excitaiton.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 20 / 30 Thermodynamics

The free energy is: LT Z +∞ = Nh K(µ) ln(1 + exp( ε(µ)/T ))dµ. F − 2π −∞ − The pressure is: ∂ T Z +∞ = F = K(µ) ln(1 + e−ε(µ)/T )dµ. P − ∂L T 2π −∞ Thermal entropy is: ∂ L Z +∞ ε(µ) S = = K(µ) ln(1 + e−ε(µ)/T ) + dµ. F ε(µ)/T −∂T 2π −∞ T (e + 1)

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 21 / 30 The Renyi entropy is deﬁned as trρα S = R 1 − α with α > 0. It also scales logarithmically

(1 + α−1) log x S → 6 as in XX spin chain https://arxiv.org/pdf/quant-ph/0304108.pdf

Entanglement entropy

At zero temperature the ground state gs is unique. The entropy of the ground state is zero. Let us consider a block of| xisequential lattice cites. We interpret the rest of the lattice as an environment. We trace away the environment: this will give us the density matrix of the block ρ = trE ( gs gs ). The entropy of the block is a complicated function of x, but for large| xihit scales| logarithmically 1 tr(ρ log ρ) log(x) as x − → 3 → ∞ similar to the continuous case https://arxiv.org/pdf/cond-mat/0311056.pdf Open problem is to study time evolution of entanglement entropy.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 22 / 30 Entanglement entropy

(1 + α−1) log x S → 6 as in XX spin chain https://arxiv.org/pdf/quant-ph/0304108.pdf

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 22 / 30 Open problem Can we describe correlation functions of the XXX with negative spin by number theory?

Open problems

Can we calculate correlation functions? First at zero temperature, time independent in the infinite volume. At spin 1/2 correlation functions in XXX chain can be expressed as polynomials [with rational coefficients] of the values of Riemann zeta function with odd arguments H.E. Boos, V.E. Korepin https://arxiv.org/pdf/hep-th/0104008.pdf T. Miwa, F. Smirnov https://arxiv.org/pdf/1802.08491.pdf The values of Riemann zeta function with odd arguments are celebrated object of number theory. They are conjectured to be transcendental numbers, algebraically independent over the field of rational numbers, see wikipedia: Apery’s theorem.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 23 / 30 Open problems

Open problem Can we describe correlation functions of the XXX with negative spin by number theory?

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 23 / 30 Open problem Does an additional symmetry arise in XXX with negative spin in the limit of inﬁnitely long lattice?

Extra symmetry in the inﬁnite volume is an open problem.

At spin 1/2 the XXX chain gains an additional symmetry in the thermodynamic limit. It is the Yangian symmetry (inﬁnite dimensional quantum group): https://arxiv.org/abs/hep-th/9211133v2

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 24 / 30 Extra symmetry in the inﬁnite volume is an open problem.

At spin 1/2 the XXX chain gains an additional symmetry in the thermodynamic limit. It is the Yangian symmetry (inﬁnite dimensional quantum group): https://arxiv.org/abs/hep-th/9211133v2

Open problem Does an additional symmetry arise in XXX with negative spin in the limit of inﬁnitely long lattice?

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 24 / 30 The non-uniqueness of quantization in integrable models was ﬁrst found in massive Thiring model: Commun. Math. Phys. 76, (1980) 165-176 http://insti.physics.sunysb.edu/~korepin/PDF_files/NewEff.pdf

Side remark

M. J. Ablowitz and J. F. Ladik in J. Math. Phys. 17, (1976) 1011 constructed a diﬀerent integrable discretization of nonlinear Schoedinger. The L-operator is diﬀerent. The equations of motion are simple. i ∂ ψ(n, t) = (1 + 4ψ(n, t)ψ†(n, t))(ψ(n + 1, t) + ψ(n − 1, t)) 2 ∂t

i ∂ − ψ†(n, t) = (1 + 4ψ(n, t)ψ†(n, t))(ψ†(n + 1, t) + ψ†(n − 1, t)). 2 ∂t In classical case Tim Hoffmann proved that the L operator of Ablowiz-Ladik model is gauge equivalent to the one, which we presenting here. Physics Letters A 265 (2000) 62-67. http://insti.physics.sunysb.edu/~korepin/PDF_files/Hoff.pdf Also Hamiltonian structures are different. The gauge matrix depends on dynamical variable ψ. The R matrices are different. Non-uniqueness of quantization.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 25 / 30 Side remark

The non-uniqueness of quantization in integrable models was ﬁrst found in massive Thiring model: Commun. Math. Phys. 76, (1980) 165-176 http://insti.physics.sunysb.edu/~korepin/PDF_files/NewEff.pdf

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 25 / 30 Double discrete version

In classical case Tim Hoﬀmann constructed an integrable double discrete version of nonlinear Schoedinger and related it to geometry: Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow. Discrete Diﬀerential Geometry, (2008), Vol. 38, pp 95-115. https://link.springer.com/chapter/10.1007%2F978-3-7643-8621-4_5.

Figure 2: An oval curve under the Hashimoto ﬂow and the discrete evolution.

Tropical geometry https://en.wikipedia.org/wiki/Tropical_geometry

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 26 / 30 Open problem Quantization of the double discrete nonlinear Schr¨odinger.

Open problem

The ﬁeld was developed by Alexander Bobenko and Yuri Suris. The book: Discrete Diﬀerential Geometry, Integrable Structure https://books.google.com/books/about/Discrete_Differential_Geometry.html? id=H1u1OanYfigC It has applications.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 27 / 30 Open problem

Open problem Quantization of the double discrete nonlinear Schr¨odinger.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 27 / 30 Open problem Construction of quantum completely integrable cellular automata.

Review on quantum CA is https://arxiv.org/pdf/1904.13318.pdf The book by Gerard ’t Hooft : https://www.springer.com/gp/book/9783319412849 A special case was analytically solved case by means of MPS: https://arxiv.org/abs/2012.12256 MPS is related to algebraic Bethe Ansatz https://arxiv.org/pdf/1201.5627.pdf https://arxiv.org/pdf/1201.5636.pdf Can we relate this version of quantum CA to Yang-Baxter?

Cellular automata

In classical case one can also discretize the value of the unknown function [the ﬁeld]. This leads us to cellular automata https://en.wikipedia.org/wiki/Cellular_automaton Mathematicians worked on classical completely integrable cellular automata [CA] https://scholar.google.com/scholar?q=completely+integrable+cellular+ automata&hl=en&as_sdt=0&as_vis=1&oi=scholart Has Lax representation.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 28 / 30 Cellular automata

Open problem Construction of quantum completely integrable cellular automata.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 28 / 30 Related models

It would be important to study related models, with similar problems: The sine-Gordon model : A.G. Izergin, V.E. Korepin: Nuclear Physics B 205 [FS5], 401, 1982 V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993.

Fermi Hubbard in one space dimension: F. Essler, H. Frahm, F. Goehmann, A. Kluemper, V.E. Korepin The One-Dimensional Hubbard Model, Cambridge University Press, 2005.

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 29 / 30 References

T.C. Dorlas Rigorous Bethe Ansatz for the Nonlinear Schroedinger Model In: Fannes M., Maes C., Verbeure A. (eds) On Three Levels. NATO ASI Series (Series B: Physics), vol 324. Springer, Boston, MA 1994 V.O.Tarasov, L.A.Takhtajan and L.D.Faddeev, “Local hamiltonians for integrable quantum models on a lattice”, Theor. Math. Phys. 57 (1983) 163-181. L.A.Takhtajan and L.D.Faddeev, “Hamiltonian Methods in the Theory of Solitons”, Springer. 2007

C. N. Yang Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett 19 (1967) 1312-1314 A. G. Izergin and V. E. Korepin, Doklady Akademii Nauk vol 259, (l981), page 76.

V. E. Korepin and A.G. Izergin, “Lattice model connected with nonlinear Schrodinger equation”, Sov. Phys. Doklady, 26 (1981) 653-654; “Lattice versions of quantum ﬁeld theory models in two dimensions”, Nucl. Phys. B 205 (1982) 401-413. V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993. F. Essler, H. Frahm, F. Goehmann, A. Kluemper, V. Korepin, The One-Dimensional Hubbard Model, Cambridge University Press, 2005. C. Yang and C. Yang, J.Math.Phys. 10 (1969) 1115–1122.

Gerard ’t Hooft “The Cellular Automaton Interpretation of Quantum Mechanics’, Springer. 2016

V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 30 / 30