Nonequilibrium Dynamics of One-Dimensional Hard-Core Anyons Following a Quench: Complete Relaxation of One-Body Observables

We demonstrate the role of interactions in driving the relaxation of an isolated integrable quantum system following a sudden quench. We consider a family of integrable hard-core lattice anyon models that continuously interpolates between noninteracting spinless fermions and strongly interacting hard-core bosons. A generalized Jordan-Wigner transformation maps the entire family to noninteracting fermions. We find that, aside from the singular free-fermion limit, the entire single-particle density matrix and, therefore, all one-body observables relax to the predictions of the generalized Gibbs ensemble (GGE). This demonstrates that, in the presence of interactions, correlations between particles in the many-body wave function provide the effective dissipation required to drive the relaxation of all one-body observables to the GGE. This relaxation does not depend on translational invariance or the tracing out of any spatial domain of the system.

Figure 1

(a) Ground-state momentum distribution of $N=64$ hard-core anyons in a box of $L=128$ sites for various values of the anyon statistical parameter $\theta$. (b) Momentum distribution of the anyons after (symmetric) expansion into a larger box of $L=256$ lattice sites, as predicted by the GGE (see text).

Figure 2

Time evolution of (a)–(d) the normalized distance $\delta \mathcal{M}(t)$ between instantaneous and GGE momentum distributions and (e)–(h) the normalized trace distance $\mathcal{D}\mathbf{(}\sigma (t),{\sigma }_{\mathrm{GGE}}\mathbf{)}$, for $N$ hard-core anyons expanding from a hard-wall box of $2N$ sites to one of $L=4N$ sites. The inset to (e) shows the time evolution of the normalized distance $\delta \mathcal{N}(t)$ between instantaneous and GGE site occupations, which is identical for all statistical parameters $\theta$.

Figure 3

Dependence of $\mathcal{D}(\sigma ,{\sigma }_{\mathrm{GGE}}{)}_{\infty }$ (see text) on (a) the HCA statistical parameter $\theta$ and (b) the system size $L$ (both panels report the same data). The red dashed line indicates a power-law fit (fitted to data points for $L=96,\dots ,512$), which yields $\mathcal{D}(\sigma ,{\sigma }_{\mathrm{GGE}}{)}_{\infty }\propto L^{-0.516\pm 0.001}$ in the limit of HCBs ($\theta =\pi$). Inset: Dependence of the SF trace distance ${\mathcal{D}}^f$ on $L$. The corresponding dashed line indicates a power-law fit (fitted to data points for $L=1024,\dots ,12\hspace{0.17em}288$), which yields ${\mathcal{D}}^f-1/2\propto L^{-0.833\pm 0.004}$.

Figure 4

Dependence of $\mathcal{D}(\sigma ,{\sigma }_{\mathrm{GGE}}{)}_{\infty }$ on $L$, for HCAs undergoing expansion within the Aubry-André model with $\lambda =1$. The black dashed line indicates a power-law fit (fitted to data points for $L=96,\dots ,512$), which yields $\mathcal{D}(\sigma ,{\sigma }_{\mathrm{GGE}}{)}_{\infty }\propto L^{-0.512\pm 0.003}$ for HCBs ($\theta =\pi$). Inset: dependence of the SF trace distance ${\mathcal{D}}^f$ on $L$.