Prethermalization from a low-density Holstein-Primakoff expansion

We consider the nonequilibrium dynamics arising after a quench of the transverse magnetic field of a quantum Ising chain, together with the sudden switch-on of a long-range interaction term. The dynamics after the quantum quench is mapped onto a fully connected model of hard-core bosons, after a suitable combination of a Holstein-Primakoff transformation and of a low-density expansion in the quasiparticles injected by the quench. This mapping holds for a broad class of initial states and for quenches which do not cross the critical point of the transverse field Ising model. We then study the algebraic relaxation in time of a number of observables towards a metastable, prethermal state, which becomes the asymptotic steady state in the thermodynamic limit.

Figure 1

Postquench population of the Ising quasiparticle modes ${\widehat{n}}_k$ [cf. Eq. (A14)] at time $t=0^{+}$ as a function of the final transverse field $g$ and the momentum $k$ for four different choices of the prequench field, from left to right $g_0=8$, 2, 1, and 0.5. The dashed lines correspond to $g=g_0$, i.e., the nonquench case, in which the initial fermionic populations vanish. Ising modes become significantly populated only for quenches, which cross from the ferromagnetic $\left(g<1\right)$ to the paramagnetic $\left(g>1\right)$ phase or vice versa, or start or end at the critical point $g=1$. Note that these plots are invariant under the exchange $g_0\leftrightarrow g$, since ${\widehat{n}}_k$ has the same property [see Eq. (6)].

Figure 2

Temporal evolution of $〈{\widehat{n}}_k\left(t\right)〉$ for $k=\pi /2$, and a quench from $g_0=8$, to $g=3$ and $\lambda =1$, for various system sizes $N$: from top to bottom the solid lines correspond to $N=80,120$, and 160. The dashed lines of the respective colors indicate the initial $\left(t=0\right)$ value of each curve. These curves demonstrate that the population $〈{\widehat{n}}_k\left(t\right)〉$ relaxes towards a prethermal value which is different from the initial one. The plateaus last until a recurrence time $t_R$, approximately equal to $N/2$. $t_R$ marks the reappearance of oscillations and it is due to the finite size of the chain.

Figure 3

(a) Temporal evolution of the difference $\mathrm{\Delta }{E}_I\left(t\right)=E_I\left(t\right)-E_{qs}$ between the Ising energy, $E_I$ [see Eq. (32)], and its stationary value, $E_{qs}$ after a quench from $g_0=9$ to $g=4$ with $\lambda =0.1$, for three different system sizes $N=60$ (black), 80 (red), and 120 (blue). (b) Same quantities as in panel (a) at fixed $N=80$ and for three different interaction strengths $\lambda =0.1$ (black), 0.2 (red), and 0.4 (blue). The insets of the various panels show the evolution of the same quantities as those displayed in the corresponding main plots, after a rescaling by a factor $1/\left(\lambda N\right)$, which makes the curves on each panel collapse on a single master curve. (c) and (d) Evolution of the difference $\mathrm{\Delta }{N}_p\left(t\right)=N_p\left(t\right)-N_{p,qs}$ [see Eq. (31)] for exactly the same parameters as those used in (b) and (a), respectively. The insets are again rescaled by $1/\left(\lambda N\right)$ and display collapse on a single master curve.

Figure 4

Temporal evolution of the Ising energy $E_I\left(t\right)$ [Eq. (32)], after a quench from $g_0=8$ to $g=1.5,\lambda =0.1$, and $N=20$, with $t\simeq {10}^4$. Black lines represent data calculated from the exact diagonalization of the original fermionic Hamiltonian (12), whereas the red ones refer to the bosonic approximation, Eq. (30). This shows that at very long times (see the time scales on the horizontal axis) the latter starts to deviate from the actual evolution, signaling a breakdown of the low-density approximation, upon which it is based. However, this breakdown occurs at increasingly longer times as the postquench value of $g$ departs from the critical point in the paramagnetic phase, as exemplified by the curves in the inset, which display the same two quantities but for $g=3.5$, which are practically indistinguishable.

Figure 5

(a) Evolution of the bosonic populations $〈\widehat{N}\left(t\right)〉$ (black, solid lines) and squared populations $〈{\widehat{N}}^2\left(t\right)〉$ (red, dashed lines) for a critical quench from $g_0=8,g=1$, with $\lambda =0.2$ and $N=50$. From the topmost $\left(k=\pi /50\right)$ to the lowermost curve $\left(k=49\pi /50\right)$, the momentum is monotonic decreasing in steps of $N/2=25$ values. This plot reaches $t=50\approx 2t_R$ (with $t_R\approx N/2=25)$ and therefore shows that the prethermal regime is still well-described by the free-boson approximation, despite the presence of modes with a large population close to 0.5. (b) Evolution of the corresponding relative error (36) for momenta $k=\pi /50$ (black solid line), $29\pi /50$ (blue short-dashed line, in the middle), and $43\pi /50$ (red long-dashed line, at the bottom).