J. De Nardis

Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions

We study quenches in integrable spin-1/2 chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench-action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench-action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit.

Complete Generalized Gibbs Ensembles in an Interacting Theory

In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady-state predictions. Here we solve this long-standing problem for the case of the spin-1/2 Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behavior after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasilocal charges. As a test, we reproduce the exact postquench steady state of the Néel quench problem obtained previously by means of the Quench Action method.

Solution for an interaction quench in the Lieb-Liniger Bose gas

We study a quench protocol where the ground state of a free many-particle bosonic theory in one dimension is let unitarily evolve in time under the integrable Lieb-Liniger Hamiltonian of

Quench action approach for releasing the Neel state into the spin-1/2 XXZ chain

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A Gaudin-like determinant for overlaps of Néel and XXZ Bethe states

-interacting repulsive bosons. By using a recently proposed variational method, we here obtain the exact nonthermal steady state of the system in the thermodynamic limit and discuss some of its main physical properties. Besides being a rare case of a thermodynamically exact solution to a truly interacting quench situation, this interestingly represents an example where a standard implementation of the generalized Gibbs ensemble fails.

Néel-XXZ state overlaps: odd particle numbers and Lieb–Liniger scaling limit

The steady state after a quantum quench from the Neel state to the anisotropic Heisenberg model for spin chains is investigated. Two methods that aim to describe the postquench non-thermal equilibrium, the generalized Gibbs ensemble and the quench action approach, are discussed and contrasted. Using the recent implementation of the quench action approach for this Neel-to-XXZ quench, we obtain an exact description of the steady state in terms of Bethe root densities, for which we give explicit analytical expressions. Furthermore, by developing a systematic small-quench expansion around the antiferromagnetic Ising limit, we analytically investigate the differences between the predictions of the two methods in terms of densities and postquench equilibrium expectation values of local physical observables. Finally, we discuss the details of the quench action solution for the quench to the isotropic Heisenberg spin chain. For this case we validate the underlying assumptions of the quench action approach by studying the large-system-size behavior of the overlaps between Bethe states and the Neel state.

Relaxation dynamics of local observables in integrable systems

We derive a determinant expression for overlaps of Bethe states of the XXZ spin chain with the Neel state, the ground state of the system in the antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic system size. Interestingly, it is remarkably similar to the well-known Gaudin formula for the norm of Bethe states, and to another recently-derived overlap formula appearing in the Lieb-Liniger model.

Analytical expression for a post-quench time evolution of the one-body density matrix of one-dimensional hard-core bosons

We specialize a recently-proposed determinant formula (Brockmann, De Nardis, Wouters and Caux 2014 J. Phys. A: Math. Theor. 47 145003) for the overlap of the zero-momentum N?el state with Bethe states of the spin-1/2 XXZ chain to the case of an odd number of downturned spins, showing that it is still of ?Gaudin-like? form, similar to the case of an even number of down spins. We generalize this result to the overlap of q-raised N?el states with parity-invariant Bethe states lying in a nonzero magnetization sector. The generalized determinant expression can then be used to derive the corresponding determinants and their prefactors in the scaling limit to the Lieb?Liniger (LL) Bose gas. The odd number of down spins directly translates to an odd number of bosons. We furthermore give a proof that the N?el state has no overlap with non-parity-invariant Bethe states. This is based on a determinant expression for overlaps with general Bethe states that was obtained in the context of the XXZ chain with open boundary conditions (Pozsgay 2013 arXiv:1309.4593, Kozlowski and Pozsgay 2012 J. Stat. Mech. P05021, Tsuchiya 1998 J. Math. Phys. 39 5946). The statement that overlaps with non-parity-invariant Bethe states vanish is still valid in the scaling limit to LL which means that the Bose?Einstein condensate state (De Nardis, Wouters, Brockmann and Caux 2014 Phys. Rev. A 89 033601) has zero overlap with non-parity-invariant LL Bethe states.

Separation of Time Scales in a Quantum Newton’s Cradle

We show, using the quench action approach (Caux and Essler 2013 Phys. Rev. Lett. 110 257203), that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient s 0 ψ (λ). This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of s 0 ψ (λ) in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density-density correlation in the interacting Lieb-Liniger gas after a quench from a Bose-Einstein condensate. This shows an approach to equilibrium with power law t −3 which turns out to be independent of the post-quench interaction and of the considered observable.

Metastable Criticality and the Super Tonks-Girardeau Gas

We apply the logic of the quench action to give an exact analytical expression for the time evolution of the one-body density matrix after an interaction quench in the Lieb-Liniger model from the ground state of the free theory (BEC state) to the infinitely repulsive regime. In this limit there exists a mapping between the bosonic wavefuntions and the free fermionic ones but this does not help the computation of the one-body density matrix which is sensitive to particle statistics. The final expression, given in terms of the difference of two Fredholm Pfaffians, can be numerically evaluated and is valid in the thermodynamic limit and for all times after the quench.