Scenario for delocalization in translation-invariant systems

We investigate the possibility of many-body localization in translation-invariant Hamiltonian systems, which was recently brought up by several authors. A key feature of many-body localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with strong quenched disorder, and that these mobile spots constitute a possible mechanism for delocalization, albeit possibly only on very long timescales. In some models, this argument for delocalization can be made very explicit in first order of perturbation theory in the hopping. For models where this does not work, we present instead a nonperturbative argument that relies solely on ergodicity inside the resonant spots.

Figure 1

First order hopping in $\mathsf{J}$ for harmonic and anharmonic interactions in $d=1$. On the left, interaction is harmonic: hopping never results in frequency mismatches. On the right, interaction is anahormonic: resonances only occur when two levels are swapped (rightmost interaction).

Figure 2

Order resonances in $\mathsf{J}$ for quenched versus thermal disordered systems. Left panel: quenched disorder Hamiltonian in $d=2$. Resonances form fixed isolated nonpercolating islands. Right panel: translation invariant Hamiltonian in $d=1$, with an extra second-neighbor interaction $b_x^{*}{b}_{x+2}+b_x{b}_{x+2}^{*}$. A bit of trial and error should convince the reader that it is possible to connect $\eta$ to ${\eta }^{\prime }$ through a sequence of resonant transitions. The naive resonant spot in $\eta$ appears thus as part of a larger resonant cluster.

Figure 3

Coexistence of classes where percolation does and does not occur, for $d=1,\mathsf{N}\ge 3$. All configurations, where a single site hosts one particle and all other sites are unoccupied, sit in the same class as ${\eta }_{\left(1\right)}$. There is percolation in this class, and the resonant dynamics restricted to it is in fact ballistic (restriction of $U_{\mathrm{res}}$ is equivalent to the lattice Laplacian). The state ${\eta }_{\left(2\right)}$ is such that neighboring sites always have a difference in occupation number larger than 2. There is not a single resonant spot and ${\eta }_{\left(2\right)}$ is the only configuration in its class.

Figure 4

An ergodic spot $\mathcal{F}$ for the resonant Hamiltonian (20), delimited by the points $a$ and $b$. Let $p\ge 2$ ($p=2$ in the figure). The first site on the left has maximal occupation number $\mathsf{N}$, the next $p$ sites have occupation number $\mathsf{N}-1$, the next $p^2$ sites have occupation number $\mathsf{N}-2,\cdots$, and the last $p^{\mathsf{N}}$ sites are vacant. Therefore the size of the spot $\mathcal{F}$ is $b-a=(p^{\mathsf{N}+1}-1)/(p-1)$.

Figure 5

Validity of (A1): it is impossible to swap occupation numbers between two adjacent sites, if their difference is larger than 1.

Figure 6

The spots $\mathcal{F}={\mathcal{F}}_0,{\mathcal{F}}_1,\cdots ,{\mathcal{F}}_{\mathsf{N}} \left(p=2\right)$. All of what is depicted in the figure takes place inside the volume delimited by $a$ and $b$. (a) We aim to show that the spots ${\mathcal{F}}_0,\cdots ,{\mathcal{F}}_{\mathsf{N}}$ are connected. (b) It is enough to establish that $\mathcal{F}$ and $\mathcal{G}$ are connected, as the procedure can then be iterated. (c) Half of the way from $\mathcal{F}$ to $\mathcal{G}$; the vacancy is transferred from the rightmost atom to leftmost site. Once this is accomplished, it is realized that all moves can be undone, while letting the leftmost site remain vacant. So it is actually seen how to move from $\mathcal{F}$ to $\mathcal{G}$, hence from $\mathcal{F}$ to ${\mathcal{F}}_1,\cdots ,{\mathcal{F}}_{\mathsf{N}}$. The case $p>2$ is analogous.

Figure 7

Ergodic spot for the resonant Hamiltonian (21). (a) Sites in blue have occupation number $\mathsf{N}$, sites in magenta have occupation number $\mathsf{N}-1,\cdots$, sites in yellow are vacant. There are $p=4$ times more sites occupied by $k$ particles than sites occupied by $k+1$ particles. As a consequence, sites with $k+1$ particles can be diluted among sites with $k$ particles. This spot can play, for Hamiltonian (21), the role played by the spot depicted in Fig. 4 for Hamiltonian (20). (b) Example of dilution of the sites with $\mathsf{N}$ and $\mathsf{N}-1$ particles among sites with $\mathsf{N}-2$ particles.