Failure of the local GGE for integrable models with bound states Garry Goldstein and Natan Andrei Department of Physics, Rutgers University and Piscataway, New Jersey 08854 In this work we study the applicability of the local GGE to integrable one dimensional systems with bound states. We find that the GGE, when defined using only local conserved quantities, fails to describe the long time dynamics for most initial states including eigenstates. We present our calculations studying the attractive Lieb-Liniger gas and the XXZ magnet, though similar results may be obtained for other models. I. INTRODUCTION Recent years have witnessed spectacular advances in the theory of unitary nonequilibrium dynamics, particu- larly in systems of optically trapped atomic gases. Key to this advance is the extremely weak coupling to the en- vironment which allows for essentially Hamiltonian dy- namics. These experimental advances have spurred many theoretical questions: does a steady state emerge, how do local observables equilibrate, is there any principle which allows us to relate the steady state to the initial condi- tions? One of the most surprising recent experimental [1], and theoretical [2] results is that there is a relation between the initial state and the long time steady state for inte- grable models. It was shown that after a quench inte- grable models retain memory of their initial state and do not appear to relax to thermodynamic equilibrium. This was ascribed to the fact that integrable models possess an infinite family of local conserved charges in involu- tion, {I i }, which include the Hamiltonian H, typically identified with I 2 : [H, I i ]=[I i ,I i ′ ]=0,H = I 2 (1) These conserved quantities in turn imply that there is a complete system of eigenstates for the Hamiltonian which may be parametrized by sets of rapidities {k} and which simultaneously diagonalize all charges. To understand the equilibration of this system it was recently proposed that it is insufficient to consider only thermal ensembles but it is also necessary to include these nontrivial con- served quantities. It was proposed [3] that the system relaxes to a state given by the generalized Gibbs ensem- ble GGE with its density matrix being given by ρ GGE = 1 Z exp  −  α i I i  (2) where the I i are the local conserved quantities; the α i are the generalized inverse temperatures and Z is a nor- malization constant insuring Tr [ρ GGE ]=1. The α i are chosen in such a way as to insure that the con- served quantities I i remain constant, namely, 〈I i 〉 f inal ≡ Tr {ρ GGE I i } = 〈I i (t = 0)〉≡〈I i 〉 initial = I 0 i . Moreover it was proposed that expectation values of local opera- tors and of correlation functions of an integrable model may be computed at long times by taking their expecta- tion value with respect to the GGE density matrix, e.g. 〈Θ(t →∞)〉 = Tr [ρ GGE Θ]. Recent numeric and the- oretical works have, however, put this assumption into question [4]. Here we would like to show that the GGE hypothesis, based on local conserved quantities, fails in general for the class of integrable models possessing bound states, or string eigenstates. Bound states in integrable models are described, in the thermodynamic limit, by rapidi- ties forming n-strings, [5]: k j α = k α + iμ(n − 2j ),j = 0, 1, 2,...n, with n an arbitrary integer and μ a coupling constant in the Hamiltonian. We will show that for such models the GGE hypothesis fails to reproduce the long time dynamics for most states and in particular for eigen- states of the Hamiltonian. We will focus in detail on the attractive Lieb-Liniger model, and repeat our arguments more briefly for the XXZ model. Our results are also applicable to other models with bound states. The Lieb Liniger hamiltonian is given by: H LL = ∞ ˆ −∞ dx  ∂ x b † (x) ∂ x b (x)+ c ( b † (x) b (x) ) 2  , (3) Here b † (x) is the bosonic creation operator at the point x and c is the coupling constant. The eigenstates of the Hamiltonian are parametrized by rapidities |{k}〉. In the basis of the Bethe rapidities, |{k}〉, the local con- served quantities I i are diagonalized and take the form, I i |{k}〉 = ∑ k i |{k}〉. It was pointed out recently [6] that when not acting on eigenstates (or on a finite linear com- bination of them) the charges may generate divergences in the form of powers and derivatives of Dirac-deltas. Great care must then be taken to define their action. Here we assume an appropriate renormalization scheme has been implemented [7]. We consider the attractive Hamiltonian with the cou- pling constant taken to be negative c< 0. In this case bound states are formed and the rapidities fall into n- string configurations k j = k 0 + ic 2 (n − 2j ), with strings of arbitrary length length n =1, 2, 3 ··· The contribution an n-string centered at k 0 to the conserved charge I i is arXiv:1405.4224v3 [cond-mat.quant-gas] 27 Mar 2016