Quasiparticle explanation of the weak-thermalization regime under quench in a nonintegrable quantum spin chain

The eigenstate thermalization hypothesis provides one picture of thermalization in a quantum system by looking at individual eigenstates. However, it is also important to consider how local observables reach equilibrium values dynamically. Quench protocol is one of the settings to study such questions. A recent numerical study [Bañuls, Cirac, and Hastings, Phys. Rev. Lett. 106, 050405 (2007)] of a nonintegrable quantum Ising model with longitudinal field under such a quench setting found different behaviors for different initial quantum states. One particular case called the “weak-thermalization” regime showed apparently persistent oscillations of some observables. Here we provide an explanation of such oscillations. We note that the corresponding initial state has low energy density relative to the ground state of the model. We then use perturbation theory near the ground state and identify the oscillation frequency as essentially a quasiparticle gap. With this quasiparticle picture, we can then address the long-time behavior of the oscillations. Upon making additional approximations which intuitively should only make thermalization weaker, we argue that the oscillations nevertheless decay in the long-time limit. As part of our arguments, we also consider a quench from a BEC to a hard-core boson model in one dimension. We find that the expectation value of a single-boson creation operator oscillates but decays exponentially in time, while a pair-boson creation operator has oscillations with a $t^{-3/2}$ decay in time. We also study dependence of the decay time on the density of bosons in the low-density regime and use this to estimate decay time for oscillations in the original spin model.

Figure 1

(a)–(c) ED calculations of the evolution of some observables for several system sizes compared to infinite-MPS results from Ref. [37] marked as $L=\infty$ (we are grateful to the authors of Ref. [37] for sharing their data with us). For smaller $L$, visible deviations from the $L=\infty$ results appear at smaller $t$, which we associate with recurrence phenomenon in finite systems. Close agreement of our ED results with the infinite-MPS results over a large time window allows us to identify the frequency of the oscillations from finite-size spectra, which we argue is essentially the quasiparticle energy at zero momentum.

Figure 2

Weights of the initial state $|Z+\rangle$ on the eigenstates $|E\rangle$ for our largest ED study with $L=18$. The figure only shows the weights on states with momentum quantum number $K=0$, since the Hamiltonian is translationally invariant and the initial state has this quantum number (also, only states that are invariant under inversion have nonzero weights). We see that even for our largest size, the majority of the weight is still on the ground state and the first excited state, which is expected since $\langle Z+\left|H\right|Z+\rangle =-1.5L=-27$ is close to $E_1$ for this size. Insets: The system size dependence of the weights on the ground state $|E_0\rangle$ and the first excited state $|E_1\rangle$ from $L=12$ to $L=18$. In the thermodynamic limit, the weights on these two states should go to zero, since $|Z+\rangle$ has finite energy density above the ground state. Nevertheless, $E_1-E_0$ is defined in the thermodynamic limit and has a meaning of the quasiparticle gap, controlling oscillations of the observables as in Fig. 1 over extended time interval.

Figure 3

The energy difference between the first excited state and the ground state as a function of $g$. This gap can also be understood as quasiparticle excitation energy. The ED result is from system size $L=17$ (the ED gap estimates are essentially already converged for $L\gtrsim 10$). We also show perturbative SW result at order $O\left(g^2\right)$, Eq. (6). The exact and perturbative calculations agree well at small $g$ and deviate more at larger $g$, but the qualitative picture of the quasiparticle is robust since the gap does not close over the range of $g$ shown.

Figure 4

Properties of the SW-rotated initial state $|{\psi }_{\mathrm{ini}}^{\prime }\rangle =e^{iS}|{\psi }_{\mathrm{ini}}\rangle$, with $iS=i{S}^{\left[1\right]}+i{S}^{\left[2\right]}$ calculated to second order in $g$ (see Appendix pp1 for details). The figure shows expectation values of the particle density $b_j^{†}{b}_j\equiv (1-{\sigma }_j^z)/2$ and BEC order parameter $b_j^{†}\equiv ({\sigma }_j^x-i{\sigma }_j^y)/2$. The particle density $\rho \approx 0.05$ is very low and close to the estimate based on the average energy density in the initial state and the quasiparticle gap.

Figure 5

Comparisons of the dynamics from ED and second-order SW for two different parameters for system size $L=13$. (a) $g=-0.5$. The perturbative calculation is still quantitatively accurate at short time. The difference between ED and SW mostly comes from the difference between the exact and perturbative quasiparticle energy gap, resulting in the different oscillation frequencies. (b) $g=-1.05$. The quantitative comparison becomes inaccurate at this parameter. For example, the oscillation frequency in the SW calculation is almost half of the ED calculation. Nevertheless, the SW calculation still captures the qualitative behavior observed in the ED calculation.

Figure 6

ED results for $\mathrm{\Phi }\left(t\right)\equiv \langle b_j^{†}\left(t\right)\rangle$ for the hard-core boson model, Eq. (10), and the product BEC state, Eq. (14), with average boson density $\rho ={\left|\beta \right|}^2=0.25$, for system sizes $L=14,16,18,20$. The parameters of the dynamical Hamiltonian are $J_{\mathrm{eff}}=1$ and $W_{\mathrm{eff}}=5J_{\mathrm{eff}}$. Note that the main panel shows $ln\left|\mathrm{\Phi }\right(t\left)\right|$ vs $J_{\text{eff}}t$, and the results strongly suggest exponential decay in $t$ until recurrence time: as the system size increases, the time interval over which the exponential decay is observed also increases. Inset: linear scale for the observable. The recurrence times where the smaller-size results peel off from the largest-size results are roughly in agreement with those in Fig. 7.

Figure 7

ED results for $\mathrm{{\rm Y}}\left(t\right)\equiv \langle b_j^{†}\left(t\right)b_{j+1}^{†}\left(t\right)\rangle$ for the same systems as in Fig. 6 with average density $\rho =0.25$. This observable can be also calculated analytically using Jordan-Wigner transformation (checked against ED for small $L$), allowing us to study much larger sizes and times, as illustrated here for $L=500$. Note that the main panel shows log-log plot, and the long-time behavior in the largest system clearly has a power-law envelope with decay $t^{-3/2}$, in agreement with the analytical calculation in the text. Inset: linear scale for the observable. By comparing data for $L=14,\cdots ,20$ and much larger $L=500$, we can also clearly see where the recurrences appear for the smaller sizes.

Figure 8

Evolution of the absolute value of the BEC order parameter $\left|\mathrm{\Phi }\right(t\left)\right|$ in the same setting as in Fig. 6 but for different average boson densities $\rho$ in the initial state and showing only the largest ED size $L=20$ ($\rho =0.25$ data the same as in Fig. 6). The measurements are normalized by their initial value in order to compare the decay rates. We clearly see that the decay rate increases as the density increases. The exponential decay ends at roughly $J_{\text{eff}}t\approx 5$, where the finite-size recurrence effect shows up, indicated by the broken line (see also Fig. 6). Inset: density dependence of the inverse lifetime $1/J_{\text{eff}}\tau$. The circle symbols are obtained from fitting the exponential decay regime $J_{\text{eff}}t\in [1.5,4.5]$ to a function $A{e}^{-t/\tau }$, while the solid line is calculated from the conjectured Eq. (23). The inverse lifetime decreases as density decreases, with a ${\rho }^{2}ln\left(\frac{1}{\rho }\right)$ dependence at small density.

Figure 9

Evolution of the real part of the BEC order parameter $\mathrm{Re}\left[\mathrm{\Phi }\right(t\left)\right]$ for different average boson densities $\rho$ and showing only the largest ED size $L=20$; the systems are the same as in Fig. 8. Inset: density dependence of the frequency $\omega$ obtained from fitting the exponential decay regime $J_{\text{eff}}t\in [1.5,4.5]$ to a function $A{e}^{-t/\tau }cos(\omega t-\alpha )$, with $\tau$ determined from Fig. 8. We see that the frequency decreases towards $W_{\mathrm{eff}}-2J_{\mathrm{eff}}=3$ as the density decreases to zero. The solid line indicates our conjectured dependence of the oscillation frequency on density, Eq. (24).

Figure 10

Numerical results for $R(\ell ,t)$ obtained using the Pfaffian method; the system size is $L=400$, and we consider separations $\ell =25,50,75$, and 100. The initial state is the ground state of Eq. (25) with parameters ${\mathrm{\Delta }}_0=0.60$ and ${\mu }_0=-1.60$, corresponding to particle density $\rho =0.25$. The parameters of the dynamical Hamiltonian are chosen as $J_{\mathrm{eff}}=1$ and $W_{\mathrm{eff}}=5$. We are primarily interested in the regime $t\ll \ell /v$, where $v$ is some information-spreading velocity. In this regime, the observable exhibits exponential decay. The time interval over which we see the exponential decay increases as the separation $\ell$ increases.

Figure 11

$R(\ell ,t)$ as in Fig. 10 but for different average densities $\rho$ obtained by tuning ${\mu }_0$ and ${\mathrm{\Delta }}_0$ according to Eqs. (35) and (36) respectively. We show only the largest separation $\ell =100$; the full system size is $L=400$ and is sufficiently large to reflect the thermodynamic limit in $L$. The values are normalized by the initial value in order to bring out the decay rate, which clearly increases as the density increases. Insets: inverse relaxation time ${\tau }^{-1}$ of the exponential decay and the oscillation frequency $\omega$ as a function of density, obtained from fitting the data in the main panel to form $A{e}^{-2t/\tau }{cos}^2(\omega t-\alpha )+C$. The circle symbols denote the fitted values, while the solid lines denote the conjectured forms Eqs. (23) and (24) for ${\tau }^{-1}$ and $\omega$ respectively (see text for details).

Figure 12

Evolution of $\text{Re}\left[\mathrm{\Phi }\right(t\left)\right]$ for the initial state $|{\psi }_{\text{ini,}}\text{A}\rangle$ and system size $L=20$ and $\left|\text{Re}\right[\mathrm{\Phi }\left(t\right)\left]\right|$ calculated from $\frac{1}{2}\sqrt{R(\ell ,t)}$ for the initial state $|{\psi }_{\text{ini,}}\text{B}\rangle$ with separation $\ell =100$ and system size $L=400$, both with density $\rho =0.05$. The dynamical Hamiltonian has $J_{\text{eff}}=0.44$ and $W_{\text{eff}}=4.52$, chosen to reproduce the spin-flip quasiparticle dispersion and gap $W_{\mathrm{eff}}-2J_{\mathrm{eff}}=\mathrm{\Delta }E=3.64$ in the original spin problem. The relaxation times obtained by fitting the exponential decay regime as in Figs. 9 and 11 are ${\tau }_A\approx 46.5$ and ${\tau }_B\approx 51.0$ respectively. This means that at time $t=18$—the longest time simulated in the original spin problem—the amplitude will be roughly 0.7 of the initial value.

Figure 13

Comparison of the dynamics under the simplified hard-core boson Hamiltonian $H$ in Eq. (10) and under various Hamiltonians with different additional interactions. The initial state is $|{\psi }_{\mathrm{ini},\mathrm{A}}\rangle$. (a)–(c) BEC order parameter (left panels) and pair boson correlation function (right panels) for density $\rho =0.05$ and system size $L=14$. The parameters of $H_{\mathrm{eff}}$ are obtained from second-order SW for the original spin model, and similar parameters are also used in $H, H_{\mathrm{nn}}, H_{\mathrm{nnn}}$: $J_{\mathrm{eff}}=0.88, W_{\mathrm{eff}}=3.68, W_{\mathrm{nn}}=-0.47$, and $W_{\mathrm{nnn}}=2.35$ (in units of $J$). The additional interactions basically introduce a small amplitude oscillation in $\mathrm{\Phi }\left(t\right)$. For the pair-boson correlation function, the decay behavior and the oscillation frequencies are affected more strongly (see text for details). (d) BEC order parameter $\mathrm{\Phi }\left(t\right)$ and pair-boson correlation function $\mathrm{{\rm Y}}\left(t\right)$ for density $\rho =0.25$, system size $L=20$ and parameters $J_{\mathrm{eff}}=1.0, W_{\mathrm{eff}}=1.0, W_{\mathrm{nn}}=-0.5$, and $W_{\mathrm{nnn}}=1.5$ (the results for $H$ are identical as in Figs. 6 and 7). The additional interaction of the JW fermions in $H_{\mathrm{nn}}$ changes the decay in $\mathrm{{\rm Y}}\left(t\right)$ to be faster than the previous power law.