Excited state correlations of the finite Heisenberg chain

We consider short range correlations in excited states of the finite XXZ and XXX Heisenberg spin chains. We conjecture that the known results for the factorized ground state correlations can be applied to the excited states too, if the so-called physical part of the construction is changed appropriately. For the ground state we derive simple algebraic expressions for the physical part; the formulas only use the ground state Bethe roots as an input. We conjecture that the same formulas can be applied to the excited states as well, if the exact Bethe roots of the excited states are used instead. In the XXZ chain the results are expected to be valid for all states (except certain singular cases where regularization is needed), whereas in the XXX case they only apply to singlet states or group invariant operators. Our conjectures are tested against numerical data from exact diagonalization and coordinate Bethe Ansatz calculations, and perfect agreement is found in all cases. In the XXX case we also derive a new result for the nearest-neighbour correlator $\{\sigma_1^z\sigma_2^z\}$, which is valid for non-singlet states as well. Our results build a bridge between the known theory of factorized correlations, and the recently conjectured TBA-like description for the building blocks of the construction.Comment: 27 pages, v2: minor modifications, a table is added (displaying the numerical errors), v3: minor modification

Form factor approach to diagonal finite volume matrix elements in Integrable QFT

We derive an exact formula for finite volume excited state mean values of local operators in 1+1 dimensional Integrable QFT with diagonal scattering. Our result is a non-trivial generalization of the LeClair-Mussardo series, which is a form factor expansion for finite size ground state mean values.Comment: 29 page

Failure of the Generalized Eigenstate Thermalization Hypothesis in integrable models with multiple particle species

It has been recently observed for a particular quantum quench in the XXZ spin chain that local observables do not equilibrate to the predictions of the Generalized Gibbs Ensemble (GGE). In this work we argue that the breakdown of the GGE can be attributed to the failure of the Generalized Eigenstate Thermalization Hypothesis (GETH), which has been the main candidate to explain the validity of the GGE. We provide explicit counterexamples to the GETH and argue that generally it does not hold in models with multiple particle species. Therefore there is no reason to assume that the GGE should describe the long time limit of observables in these integrable models.Comment: 16 pages, 2 figures, v2: minor modification

Mean values of local operators in highly excited Bethe states

We consider expectation values of local operators in (continuum) integrable models in a situation when the mean value is calculated in a single Bethe state with a large number of particles. We develop a form factor expansion for the thermodynamic limit of the mean value, which applies whenever the distribution of Bethe roots is given by smooth density functions. We present three applications of our general result: i) In the framework of integrable Quantum Field Theory (IQFT) we present a derivation of the LeClair-Mussardo formula for finite temperature one-point functions. We also extend the results to boundary operators in Boundary Field Theories. ii) We establish the LeClair-Mussardo formula for the non-relativistic 1D Bose gas in the framework of Algebraic Bethe Ansatz (ABA). This way we obtain an alternative derivation of the results of Kormos et. al. for the (temperature dependent) local correlations using only the concepts of ABA. iii) In IQFT we consider the long-time limit of one-point functions after a certain type of global quench. It is shown that our general results imply the integral series found by Fioretti and Mussardo. We also discuss the generalized Eigenstate Thermalization hypothesis in the context of quantum quenches in integrable models. It is shown that a single mean value always takes the form of a thermodynamic average in a Generalized Gibbs Ensemble, although the relation to the conserved charges is rather indirect.Comment: 32 pages, v2: minor changes, v3: minor correction

Overlaps with arbitrary two-site states in the XXZ spin chain

We present a conjectured exact formula for overlaps between the Bethe states of the spin-1/2 XXZ chain and generic two-site states. The result takes the same form as in the previously known cases: it involves the same ratio of two Gaudin-like determinants, and a product of single-particle overlap functions, which can be fixed using a combination of the Quench Action and Quantum Transfer Matrix methods. Our conjecture is confirmed by numerical data from exact diagonalization. For one-site states the formula is found to be correct even in chains with odd length, where existing methods can not be applied. It is also pointed out, that the ratio of the Gaudin-like determinants plays a crucial role in the overlap sum rule: it guarantees that in the thermodynamic limit there remains no $\mathcal{O}(1)$ piece in the Quench Action.Comment: 22 pages, v2: references adde

LeClair-Mussardo series for two-point functions in Integrable QFT

We develop a well-defined spectral representation for two-point functions in relativistic Integrable QFT in finite density situations, valid for space-like separations. The resulting integral series is based on the infinite volume, zero density form factors of the theory, and certain statistical functions related to the distribution of Bethe roots in the finite density background. Our final formulas are checked by comparing them to previous partial results obtained in a low-temperature expansion. It is also show that in the limit of large separations the new integral series factorizes into the product of two LeClair-Mussardo series for one-point functions, thereby satisfying the clustering requirement for the two-point function.Comment: 27 pages, v2: minor modifications, a note and a reference adde