Odd thermodynamic limit for the Loschmidt echo

Is it possible to readily distinguish a system made by an Avogadro's number of identical elements and one with a single additional one? Usually, the answer to this question is negative but, in this work, we show that in antiferromagnetic quantum spin rings a simple out-of-equilibrium experiment can do so, yielding two qualitatively and quantitatively different outcomes depending on whether the system includes an even or an odd number of elements. We consider a local quantum-quench setup and calculate a generating function of the work done, namely, the Loschmidt echo, showing that it displays different features depending on the presence or absence of topological frustration, which is triggered by the even/oddness in the number of the chain sites. We employ the prototypical quantum Ising chain to illustrate this phenomenology, which we argue being generic for antiferromagnetic spin chains, as it stems primarily from the different low energy spectra of frustrated and nonfrustrated chains. Our results thus prove that these well-known spectral differences lead indeed to distinct observable characteristics and open the way to harvest them in quantum thermodynamics protocols.

Figure 1

Loschmidt echo comparison between frustrated and unfrustrated chains of similar length $N$, fixing the magnetic field and the perturbation parameter, respectively, to $h=0.4, \lambda =0.2$ (left plot) and $h=0.8, \lambda =0.1$ (right plot). The time is rescaled for a better comparison. For even $N$ (unfrustrated systems), due to the negligible hybridization with the first excited states, the LE presents small oscillations around a value near one (left columns). For odd $N$ instead the higher number of hybridized states results in a strong sensitivity of the LE oscillations to the system parameters.

Figure 2

Comparison between the result for frustrated (red squares) and unfrustrated (blue circles) chains of the time-average (left panel) and of the standard deviations (right panel) of the LE for several sets of parameters as a function of the inverse system length. Differently from the frustrated case, the unfrustrated time average is mostly size independent. The standard deviation deviation for the frustrated case is always larger, even a few orders of magnitude, than the one of the unfrustrated case.

Figure 3

Loschmidt echo's comparison between the numerics (dotted red line) and the analytic expression Eq. (5) (blue line) for a spin chain of length $N=201$ and for $\lambda =0.1$. The time is rescaled for a better comparison. The results are in agreement for $h=0.01$, that corresponds to the limit $0<h\ll \lambda \ll 1$ (upper panel, the curves are mostly superimposed). We also find similar results when $h$ and $\lambda$ are comparable, as shown in the middle panel for the case $\lambda =h=0.1$. Finally, in the lower panel the failure of the approximation for $h=0.5$ is shown, where the value of the magnetic field is beyond the assumed range of validity.