Out-of-equilibrium dynamics of the XY spin chain from form factor expansion

Dear Editor and Referees,

We appreciate the very positive reports about our manuscript. Please find below our answers to the referees with the corresponding changes.

Report1: We thank the referee for their very positive report and recommendation to publish our manuscript in its present form.

Report2: We thank the referee for their very positive report, constructive comments, and spotting a number of typos (which we have corrected). We indeed consider even lattice lengths L and now state this explicitly above equation (1). Our replies to the referee’s points (1) and (2) are as follows:

(1) "Coherent state" usually refers to states of the form $e^{b^\dagger}|0\rangle$ with $b^\dagger$ bosonic creation operators. We refer to states (10) as coherent states as they can be viewed as coherent states of bosonic zero-momentum pairs of fermions. We have added a comment below eqn (10) to make this clear. The states defined in this way are indeed a specific class.

(2) As we show in Section 2.4 the state of the system initialised in a ground state of H(h,\gamma) following and then driven out of equilibrium by any variation of the magnetic field and/or the anisotropy can always be written as a coherent state at all times. So their physical relevance is not merely that they relax to GGE: they exactly describe the out-of-equilibrium time evolution of the system at all times in such protocols. Also, the actual experimental realizability of these coherent states is thus equivalent to that of quantum quenches.

Report3: We thank the referee for their positive report and helpful comments. Our replies to the referee’s points (1)-(3) are as follows:

(1) In contrast to the references quoted by the referee we are concerned with a homogeneous system here. Hence the local physics at late times is described by a generalised Gibbs ensemble, which describes an equilibrium state in the sense of Jaynes (under the constraints that the values of the extensive number of conserved quantities are fixed by the choice of initial state). In this view “equilibrium” refers to the fact that local physical observables cease to be time dependent.

In order to avoid any confusion we have added a short comment on p.6 and added a references to the work of Jaynes, and Rigol et al:
E. T. Jaynes, Information Theory and Statistical Mechanics. II, Phys. Rev. 108, 171 (1957)

Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons Marcos Rigol,1 Vanja Dunjko,2,3 Vladimir Yurovsky,4 and Maxim Olshanii2

(2) The referee states that "Normally, by the doubling trick, in the doubled model, order parameter correlation functions may be evaluated using the standard techniques from U(1) invariant models.” Our understanding of the situation is at variance with this statement. We are of course well aware that the doubling technique is useful in the continuum limit, but as far as we know the latter is in fact crucial, because Jordan-Wigner string operators reduce to bosonic vertex operators and become simple. On the lattice they are as far as we know difficult to deal with, see e.g. (3.22) in JB Zuber, C Itzykson - Physical Review D15, 2875 (1977). We are not aware of works in the literature that use the doubling trick on the lattice to obtain the kind of results the referee alludes to and we would be grateful for any specific references.

(3) The proposed expectation value would require more work. However we could compute the expectation value of $e^{i\sum_{j>J}\sigma^z_j(t)}e^{-i\sum_{j>J}\sigma^z_j(0)}$ exactly along the same lines as for the dynamical two-point function of \sigma^x. Indeed, the form factor of $e^{i\sum^z_{j>J}\sigma^z_j(0)}$ is a determinant similarly to \sigma^x, so exactly the same calculations would apply.

Best regards,
The authors