Confined Quasiparticle Dynamics in Long-Range Interacting Quantum Spin Chains

We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law ($1/r^{\alpha }$) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic states in high-energy physics. We show that these quasiparticles have signatures in the dynamics of order parameters following a global quench, and the Fourier spectrum of these order parameters can be exploited as a direct probe of the masses of the confined quasiparticles. We introduce a two-kink model to qualitatively explain the phenomenon of long-range-interaction-induced confinement and to quantitatively predict the masses of the bound quasiparticles. Furthermore, we illustrate that these quasiparticle states can lead to slow thermalization of one-point observables for certain initial states. Our work is readily applicable to current trapped-ion experiments.

Figure 1

(a)–(c) $⟨{\sigma }_j^z{\sigma }_k^z{⟩}_c$ and (d) $S_A(t)$ versus $t$ after a quantum quench with initial state $|{\mathrm{\Psi }}_0⟩$. $L=19$, $k=10$, and $B=0.27$. (a) Short-range interacting case ($\alpha \to \infty$), (b) $\alpha =2.6$, (c) $\alpha =2.3$. The dashed white lines illustrate the maximal velocity $4B$ of freely propagating domain walls in the short-range interacting case [34]. (d) $S_A(t)$ for various $\alpha$.

Figure 2

(a),(b) $⟨{\sigma }^z(t)⟩$ (black line) versus time after quenching to (a) $\alpha =2.3$, $B=0.27$, (b) $\alpha =1.4$, $B=0.35$ for $L=20$. The dashed green lines show $⟨{\sigma }^z(t)⟩$ for the short-range model (same $B$). (c),(d) Fourier spectrum of $⟨{\sigma }^z(t)⟩$ for the long-range case in (a) and (b), respectively. The largest time for the Fourier transform is $t=30$ and 12 for (c) and (d), respectively. The parameters in (b),(d) are accessible in current trapped-ion experiments [15]. The dashed lines show the mesonic masses ($m_i$) and their differences ($m_{ij}\equiv m_j-m_i$) calculated using the two-kink model.

Figure 3

(a) Potential energy as a function of distance $x$ between the two domain walls ($n=|x|$). Red line, $\alpha =1.9$; green line, $\alpha =2.3$. Inset: Typical spin configuration of two-domain-wall states. (b)–(d) Comparison of two-kink model (blue) and ED results (red). (b) Energy spectrum. Parameters: $\alpha =2.3$, $B=0.27$, $L=20$. (c) $m_1$ versus $\alpha$ with parameters $B=0.27$ and $L=22$. Inset: Difference of $m_1$ between the two methods $\mathrm{\Delta }{m}_1$ versus $L$. (d) $m_1$ versus $L$ [same parameters as (b)]. The dashed lines are $m_{1,\infty }$. The inset shows $m_{1,\infty }-m_1$ versus system size. The black line shows the fitting of the two-kink model’s data to $(1/L{)}^{\beta }$, with $\beta =1.315$. ED data have similar scaling with $\beta =1.34$. $m_{1,\infty }$ is chosen as 5.56 (5.62) for the two-kink model (ED).

Figure 4

Strong (a) and weak (b) thermalization for different initial states. (a) $⟨{\sigma }^{\mu }(t)⟩-{⟨{\sigma }^{\mu }⟩}_{\mathrm{th}}$ for initial state $|Y_{+}⟩$. (b) Same as (a) but for initial state $|Z_{+}⟩$. Parameters: $\alpha =2.3$, $B=0.37$, $L=20$.