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Introduction to the Bethe Ansatz Solvable Models

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Introduction to the Bethe Ansatz Solvable Models Universit`adi Firenze Dipartimento di Fisica Dottorato di Ricerca in Fisica Introduction to the Bethe Ansatz Solvable Models Lectures given in 1998-1999 by Anatoli G. Izergina) Lecture notes edited by Filippo Colomob) and Andrei G. Pronkoa) a)Steklov Mathematical Institute (POMI), St. Petersbourg, Russia b) INFN, Sezione di Firenze, and Dipartimento di Fisica, Universit`adi Firenze Florence, November 2000 Preface In memory of Anatoli \Tolia" G. Izergin It is still unbelievable for me to realize that Prof. Anatoli G. Izergin untimely died last December. I met Tolia for the first time in Leningrad on October 1990. We immediately became friends. A collaboration started by which I learned a lot on quantum integrable systems. We worked together for many years with exchange of visits. He visited many times our Department giving also lectures for our Ph.D. courses. My graduated students, young researchers, colleagues and myself enjoyed a lot attending his beautiful lec- tures. Last year, I invited him to give again a short course on integrable systems and Tolia gave 30 hours of lectures on this subject. I would like to thank Filippo Colomo and Andrei Pronko who collected notes of these lectures and carefully edited them. I am sure that this will be very useful for all those wishing to start studying this exciting subject. I hope that these notes can also represent a small monument to the memory of Tolia Izergin. Florence, November 2000 Valerio Tognetti i Contents 1 The classical nonlinear Schr¨odingerequation. 1 2 The quantization. The quantum NS model as the one-dimensional nonrelativistic Bose gas. 2 3 A δ-potential as a boundary condition. 4 4 Bethe eigenfunctions. The coordinate Bethe Ansatz. 5 5 Particles on the whole axis. The spectrum. 9 6 Particles on the whole axis. The S-matrix. 11 7 Periodic boundary conditions. Bethe equations. 13 8 On the solutions of the Bethe equations for the one-dimensional Bose gas. 15 9 Classical inverse scattering method. The Lax representation. 16 10 Why \the inverse scattering method"? . 18 11 The transition matrix and the monodromy matrix. 22 12 The trace identities for the classical nonlinear Schr¨odingerequation. 23 13 Tensor products; notations. 26 14 The classical r-matrix. 28 15 Trace identities and conservation laws. 32 16 On the r-matrix and the M-operator. 33 17 The Lax representation for lattice systems. 36 18 The quantum inverse scattering method. The quantum monodromy matrix and the transfer matrix. 39 19 The quantum R-matrix. 40 20 The Yang-Baxter equation. 45 21 The examples of the R-matrices. 47 22 Some properties of the R-matrix. 50 23 The algebraic Bethe Ansatz. Preliminaries. The generating state. 54 24 The algebraic Bethe Ansatz. Bethe eigenstates and Bethe equations. 58 25 The algebraic Bethe Ansatz. Some remarks. 61 26 The quantum determinant. 64 27 The quantum inverse scattering method for the quantum nonlinear Schr¨odingerequation. The transfer matrix and the trace identities. 65 iii 28 The quantum inverse scattering method for the quantum nonlinear Schr¨odingerequation. The R-matrix. 68 29 The quantum inverse scattering method for the quantum nonlinear Schr¨odingerequation. The algebraic Bethe Ansatz. 69 30 Spin models on a one-dimensional lattice. The fundamental L-operator. 72 31 Fundamental spin models on a one-dimensional lattice. The local trace identities. 73 32 The Heisenberg spin chains. The trace identities. 77 1 33 The algebraic Bethe Ansatz for the Heisenberg spin 2 XXX chain. 80 1 34 The Heisenberg spin 2 XXZ chain. 81 35 Fundamental vertex models of classical statistical physics on a two- dimensional lattice. 83 36 The six vertex model. 88 37 The partition function of the six vertex model with the domain wall boundary conditions. 94 38 The ground state of the nonrelativistic Bose gas in the periodic box. 101 39 The ground state in the thermodynamic limit. The Lieb equation. 103 40 Excitations at zero temperature. 106 41 The thermodynamics of the Bose gas at finite temperature. The thermal equilibrium. 110 iv 1 The classical nonlinear Schr¨odingerequation. The classical nonlinear Schr¨odinger(NS) equation is the following system of equa- tions for the nonrelativistic Bose fields (x; t), +(x; t), (in this Section the cross denotes complex conjugation of classical fields) in two space-time dimensions 2 + i@t = −@x + 2c ; (1:1) + 2 + + + −i@t = −@x + 2c (the \ordering" of , + is not essential, of course, in the classical case). It is a Hamiltonian system. The Hamiltonian H is Z + + + H = dx(@x @x + c ) ; (1:2) the momentum P and the charge Q are given as i Z Z Z P = dx(@ + − +@ ) = i dx(@ +) = −i dx +@ ; 2 x x x x (1:3) Z Q = dx + (the time argument t is fixed to be the same for all fields). At this level, boundary conditions for , + are not fixed, but it is assumed that integration by parts, as in the expressions for P , can be done. Equations of motion (1.1) can be obtained by computing the Poisson brackets with the Hamiltonian. For functionals A, B of fields (x; t), +(x; t) (at fixed time t) the Poisson brackets are defined as Z δA δB δA δB fA; Bg = −i dz · − · ; (1:4) δ (z) δ +(z) δ +(z) δ (z) the definition of the functional derivative is as usual: one represents the first varia- tion of the functional in the form Z δA δA δA = dx δ (x) + δ +(x) ; δ (x) δ +(x) the coefficients of δ , δ + being the corresponding functional derivatives. The canonical Poisson brackets are f (x; t); +(y; t)g = −iδ(x − y); f ±(x; t); ±(y; t)g = 0; − ≡ : (1:5) Equations ± ± @t = f Hg (1:6) give just the equations of motion (1.1). It is easy to calculate also that ± ± ± ± f P g = −@x ; f Qg = ±i ; 1 and to show that Q, P are \commuting" integrals of motion, fH; P g = fH; Qg = 0; fP; Qg = 0: (1:7) We will see later that there exist, in fact, infinitely many commuting integrals of motion for the nonlinear Schr¨odingerequation. 2 The quantization. The quantum NS model as the one-dimensional nonrelativistic Bose gas. In the quantum case, the fields (x; t), +(x; t) are operators, and the cross stands for the hermitean conjugation. The Poisson brackets (1.5), as usual in canonical quantization, are changed for the commutators ( − ≡ ) h i h i (x; t); +(y; t) = δ(x − y) ; ±(x; t); ±(y; t) = 0 (2:1) (remind that under quantization the Poisson bracket between the coordinate q and the momentum p is changed for the commutator of the corresponding operators according to the rule fq; pg = 1 ! [q; p] = ih¯; we puth ¯ = 1). The Hamiltonian H, the momentum P , and the operator Q of the number of particles are given by the same expressions (1.2){(1.3) as in the classical case (however, the normal ordering is now essential!). The equations of motion for operators , + are ± ± −i@t = [H; ] ; (2:2) which gives just the NS equation (1.1) for quantum fields. Also, we have the relations ± ± i@x = [P; ] ; and [Q; ±] = ± ± : Operators P and Q commute with the Hamiltonian and between themselves, [H; P ] = [H; Q] = [P; Q] = 0 : (2:3) We shall see later that also in the quantum case there exist infinitely many integrals of motion (conservation laws). The space where operators , + act is the Fock space. The Fock vacuum j0i is defined as usual by the requirement (x)j0i = 0; 8x ; (2:4) h0j ≡ (j0i)+ ; h0j +(x) = 0 ; h0j0i ≡ 1 : 2 It is an eigenstate of H, P , Q with zero eigenvalues, Hj0i = 0 ;P j0i = 0 ;Qj0i = 0 : (2:5) The other states in the Fock space are linear combinations of the states obtained by acting with operators + on the Fock vacuum: Z N + + + jΨN i = d z χN (fzg) (z1) (z2) : : : (zN )j0i ;N = 0; 1; 2;::: (2:6) N Here, χN (fzg) ≡ χN (z1; : : : ; zN ) is a wave function; d z ≡ dz1dz2 : : : dzN . Due to + the commutativity of operators (zj) (2.1), we can restrict ourselves to the wave functions which are symmetrical under permutations of zj, χN (zS1 ; : : : ; zSN ) = χN (z1; : : : ; zN ); χN (fSzg) = χN (fzg); (2:7) where S is a permutation, S : (1; 2;:::;N) ! (S1;S2;:::;SN ). In other words, one has Bose statistics. Using commutation relations (2.1), it is not difficult to calculate the action of operators Q; P and H on the state (2.6), expressing it in terms of operators acting on the wave function. It is evident that QjΨN i = NjΨN i ; (2:8) i.e., jΨN i is an \N-particle" state. Moreover, integration by parts gives: Z + P jΨN i = i dx@x · jΨN i Z N + + = d z (PN χN ) (z1) : : : (zN )j0i; where N X @ PN = −i @j;@j ≡ ; (2:9) j=1 @zj and Z N + + HjΨN i = d z (HN χN ) (z1) : : : (zN )j0i; with N X 2 X HN = − @j + 2c δ(zj − zk): (2:10) j=1 1≤j<k≤N It follows from the expressions for PN and HN that in the sector containing N particles, our model describes a \gas" of N nonrelativistic bosons on a line (the mass of the boson is equal to 1/2) interacting via the pair potentials, 2cδ(zj −zk), of \zero radia". The fields , + are second quantized fields, describing the gas with any number of particles (N = 0; 1; 2;:::). 3 Due to the commutativity of operators H, P , and Q, one can find their simulta- neous eigenstates: HjΨN i = EN jΨN i ;P jΨN i = PN jΨN i ;QjΨN i = NjΨN i: (2:11) Here, EN , PN , and QN ≡ N are the eigenvalues.
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