Quantum dynamics of impenetrable $\mathrm{SU}\left(N\right)$ fermions in one-dimensional lattices

We study quantum quench dynamics in the Fermi-Hubbard model, and its $\mathrm{SU}\left(N\right)$ generalizations, in one-dimensional lattices in the limit of infinite onsite repulsion between all flavors. We consider families of initial states with generalized Néel order, namely, initial states in which there is a periodic $N$-spin pattern with consecutive fermions carrying distinct spin flavors. We introduce an exact approach to describe the quantum evolution of those systems, and study two unique transient phenomena that occur during expansion dynamics in finite lattices. The first one is the dynamical emergence of Gaussian one-body correlations during the melting of sharp (generalized) Néel domain walls. Those correlations resemble the ones in the ground state of the $\mathrm{SU}\left(N\right)$ model constrained to the same spin configurations. This is explained using an emergent eigenstate solution to the quantum dynamics. The second phenomenon is the transformation of the quasimomentum distribution of the expanding strongly interacting $\mathrm{SU}\left(N\right)$ gas into the rapidity distribution after long times. Finally, we study equilibration in $\mathrm{SU}\left(N\right)$ gasses and show that observables after equilibration are described by a generalized Gibbs ensemble. Our approach can be used to benchmark analytical and numerical calculations of dynamics of strongly correlated $\mathrm{SU}\left(N\right)$ fermions at large $U$.

Figure 1

Melting of a Néel domain wall in the Fermi-Hubbard model at infinite onsite repulsion. (a) At $t=0$ the system is in a Néel product state with 12 fermions. At $t>0$ the fermions expand into an empty lattice. The length of the arrows is proportional to the magnitude of the $z$ component of the spin at site $l$, $S_l^z\left(t\right)=|n^{\uparrow }(l;t)-n^{\downarrow }(l;t)|/2$, where $n^{\sigma }(l;t)$ are the site occupations of fermions with “flavor” $\sigma$ at time $t$. We show results at (b) $t=4$ and (c) $t=8$. This setup is studied in Sec. 3.

Figure 2

Spin-resolved site occupations $n^{\sigma }(l;t)$ during the “melting” of a domain wall [see Eq. (17)]. (a) [(b)] Initial SU(2) [SU(4)] domain walls with generalized Néel order (the spin flavors are encoded in the patterns and colors used). (c), (e) [(d), (f)] $n^{\sigma }(l;t)$ for $\sigma =1$ and $\sigma =2$ in the SU(2) case [$\sigma =2$ and $\sigma =4$ in the SU(4) case], respectively, at $t=8$. Thick dashed lines in (c)–(f) show the total site occupations $n(l;t)$, while the overlapping thin solid lines show the analytical predictions $n{(l;t)}_{\infty }$ from Eq. (18) using $l_0=N_p+1/2$. (g) [(h)] Integrated number of particles $Q^{\sigma }\left(t\right)={\sum }_{l>l_0}{n}^{\sigma }(l;t)$ that have crossed the initial edge of the domain wall (at $l_0$) for the SU(2) [SU(4)] case. Black dashed lines depict results for spinless fermions ${\sum }_{\sigma }{Q}^{\sigma }\left(t\right)$. Numerical results are shown for $N_p=24$ particles in (a)–(f) and for $N_p=120$ particles in (g) and (h).

Figure 3

Matrix elements $C_{l,j}=\left|C_l(j-l;t)\right|$ of the total one-body correlation matrix for an initial domain wall with $N_p=200$ particles [see Eq. (17)]. Results are shown at times (a) $t/N_p=0.1$ and (b) $t/N_p=0.25$. $l_0=N_p$ is the initial edge of the domain wall.

Figure 4

Total one-body correlation function $|C_l(x;t)|$ for an initial domain wall with $N_p=1200$ particles [see Eq. (17)]. We set the target particle occupation $n(l;t)=n_0$ and use Eq. (18) to find the optimal $l$ at a given time $t$. (a) $n_0=0.5$, which corresponds to $l=1200$ at all times. (b) $n_0=0.2$, which corresponds to $l=1281,1443,1685,1928,2171$ at times $t=50,150,300,450,600$, respectively. Dashed lines are results for $C_l\left(x\right)$ in the ground state of the emergent local Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{SF}}^{\mathrm{LDA}}\left(s\right)$ [see Eq. (21)]. For the latter calculations, we set the particle filling $N_p/L=0.5$ in (a) and $N_p/L=0.2$ in (b), and measure correlations from the center of the lattice ($l=L/2$, $L=2400$).

Figure 5

Total quasimomentum distribution $m(k;t)$ during the melting of a domain wall [see Eq. (17)] with $N_p=400$ particles. (a) $m(k;t)$ for different times. Thick lines are exact results for the dynamics while thin solid lines are results obtained in the ground state of the emergent local Hamiltonian $\widehat{\mathcal{H}}\left(t\right)$ [see Eq. (19)]. (b) Rescaled exact results for the quasimomentum distribution $[m(k;t)-\overline{m}]/t$, where $\overline{m}=N_p/L=0.5$.

Figure 6

Sudden expansion of $N_p=100$ impenetrable $\mathrm{SU}\left(N\right)$ fermions after turning off a harmonic trap with $R=200$. We take $L=1200$ and $l_0=600$. (a) Total site occupations $n(l;t)$. (b) Total quasimomentum distribution $m(k;t)$.

Figure 7

Sudden expansion from a harmonic trap with $R=200$. Results are shown for impenetrable $\mathrm{SU}\left(N\right)$ fermions in the initial state ($t=0$) and at $t=375$, and for spinless fermions (SF) at $t=375$. Thin solid lines depict the results for the impenetrable $\mathrm{SU}\left(N\right)$ fermions in the ground state of the emergent local Hamiltonian (27). (a) Total quasimomentum distribution function $m(k;t)$. (b) Total one-body correlations $|C_l(x;t)|$ measured from the center of the system (at $l=1200$). The results shown are for $N_p=100$ and $L=2400$.

Figure 8

Total one-body correlations $|C_l(x;t)|$ during the sudden expansion of $N_p=100$ impenetrable $\mathrm{SU}\left(N\right)$ fermions after turning off a harmonic trap with $R=100$. We take $L=2400$ and measure correlations from the center of the system (at $l=1200$). The results from the exact time evolution are shown as symbols, and the results in the ground state of ${\widehat{\mathcal{H}}}_{\mathrm{SF}}^{\prime \mathrm{LDA}}\left(t\right)$ [see Eq. (28)] are shown as straight lines. The chemical potential in ${\widehat{\mathcal{H}}}_{\mathrm{SF}}^{\prime \mathrm{LDA}}\left(t\right)$ is set to match the particle filling $n(l;t)$ at any given time.

Figure 9

Long-time dynamics of $N_p=100$ impenetrable $\mathrm{SU}\left(N\right)$ fermions after turning off a harmonic trap with $R=200$ in a lattice with $L=1200$ sites ($l_0=600$). (a) Total site occupations $n(l;t)$. (b) Total quasimomentum distribution $m(k;t)$, computed following Eq. (25).

Figure 10

Generalized thermalization in a box trap after turning off a harmonic trap. (a) Evolution of the total occupation of three quasimomentum modes ($k=0$, $\pi /5$, and $\pi /3$) plotted vs $t/L$ for three sizes $L$ of the box trap. For $L=1500$ there are $N_p=125$ impenetrable $\mathrm{SU}\left(N\right)$ fermions and $R=250$. The other system sizes have the same ratios $N_p/L$ and $N_p/R$. The horizontal dashed lines are the GGE predictions for $L=1500$. (b) Total quasimomentum distribution function $m(k;t)$ in the initial state ($t=0$), in the time-evolving state after a long equilibration time ($t=50000$), in the GGE, in the grand canonical ensemble (GE), and for the spinless fermions (SF) onto which we map the impenetrable $\mathrm{SU}\left(N\right)$ fermions. The results shown are for the $L=1500$ case in (a). (Inset) Distance $\mathrm{\Delta }m$ [see Eq. (32)] between time-averaged results from the time evolution and the GGE predictions. The time average is computed using 100 times between $t_{\text{min}}=10L$ and $t_{\text{max}}=L^2/14$ (set to avoid including revivals that occur in timescales $\propto L^2$), with a time spacing $dt=(t_{\text{max}}-t_{\text{min}})/100$. Error bars show the standard deviation for the average. The solid line is a power-law fit $\mathrm{\Delta }m\propto L^{-\alpha }$ for $L\ge 720$, which yields $\alpha =1.0\left(1\right)$.

Figure 11

Total one-body correlations $|C_l\left(x\right)|$ after equilibration. Comparison between the time-averaged total one-body correlations $|{\overline{C}}_l\left(x\right)|$ after equilibration [see Eq. (33)], the initial total one-body correlations, and the predictions of the GGE and the grand canonical ensemble (GE). ${\overline{C}}_l\left(x\right)$ is computed using times $t_i$ in the interval $15000\le t_i\le 160000$, with a time spacing $dt=50$. The system studied has $L=1500$, $N_p=125$, and initial $R=250$. (Inset) Same results plotted versus $x^2$.

Figure 12

Spin-resolved quasimomentum distribution function $m^{\sigma }(k;t)$ for impenetrable SU(2) fermions with generalized Néel order. (a) Melting of a domain wall with $N_p=400$ fermions at $t=200$ [see Fig. 5 in Sec. 3]. (b) Sudden expansion of $N_p=100$ fermions from a harmonic trap with $R=200$ at $t=375$ [see Fig. 7 in Sec. 4]. (c) Equilibration of $N_p=100$ particles in a box trap with $L=1200$. Results are shown for $t=50000$ after turning off a harmonic trap with $R=200$ [see Fig. 10 in Sec. 5]. Thin black lines depict the results averaged over spin flavors $\overline{m}(k;t)={\sum }_{\sigma =1}^2{m}^{\sigma }(k;t)/2$.