Emergent local integrals of motion without a complete set of localized eigenstates

Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a set of localized eigenstates to be complete. Given a set of localized eigenstates comprising a nonzero fraction $(1-p_f)$ of the full many-body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have nonzero weight in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a “global dressing” whose weight vanishes as $\sim p_f$ as $p_f\to 0$. In this sense, the existence of a nonzero fraction of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability.

Figure 1

Entanglement growth as a function of time for the projected problem. The initial condition is obtained by starting from a randomly chosen unentangled product state and projecting out an energy window containing a fraction $p_f$ of the eigenstates, and the time evolution is with respect to an effective Hamiltonian obtained by performing the same projection on (1). Note that the projection increases the entanglement of the initial condition and turns the Hamiltonian into a nonlocal operator; the time evolution remains logarithmic, suggesting a residual local integrability. The results were obtained by exact diagonalization on a system of $N=14$ spins at $h=4$, with 40 disorder realizations; for each realization the results of $\sim 300$ different initial product states were averaged over.

Figure 2

Quasilocal operators in a projected Hilbert space, not constrained to commute with the Hamiltonian. Panels (a)–(c) show the fidelity $\mathcal{F}$ as a function of $p_f$ (averaged over position) for a sequence of system sizes $N$, and with $N_A=1$ (a), 2 (b), 3(c). Increasing $N_A$ brings the fidelity closer to 1, but the “improvement” grows smaller as system size is increased. Panels (d)–(f) show $|\mathcal{F}-(1-p_f\left)\right|$ as a function of system size $N$ with $N_A=1$ (d), 2 (e), 3 (f). The fact that the points follow a straight line (with negative slope) on these double-logarithmic plots suggests that the quantities are decaying to zero. Therefore at large $N$ for fixed $N_A$ we suspect that $\mathcal{F}\to 1-p_f$, though with the presently available system sizes we cannot conclusively confirm this. Though we cannot establish the precise value of $\mathcal{F}$ as $N\to \infty$, our data clearly show that $\mathcal{F}$ does not vanish in this limit. Data were taken for $h=8$ (though the results are largely independent of $h$) and 200 disorder realizations.

Figure 3

Solid lines show the weight of the integral of motion $\tilde{O}$ as a function of distance for the case where ${\mathcal{D}}_{\mathrm{trunc}}$ does not grow with system size $N$, therefore $p_f\sim 2^{-N}$. The $y$ axis of this plot is on a log scale, and the $x$ axis is rescaled (see Appendix pp2) so that straight lines on such a plot indicate exponential decay, and indeed we observe straight lines. Dashed lines show the corresponding weight for the parent quasilocal operator $O$, which does not commute with the Hamiltonian. Data were taken for $N_A=1,h=8,{\mathcal{D}}_{\mathrm{trunc}}=7$, and 200 disorder realizations. Note that $M_{ij}$ decays exponentially at small distances before saturating to a constant background (indicated by the dashed lines), but this background itself is exponentially small in $N$. In the thermodynamic limit the background disappears, such that the integral of motion $\tilde{O}$ is truly an LIOM, with exponential tails out to long distances; i.e., when the number of states projected out is held constant as we increase system size, we can construct truly localized integrals of motion.

Figure 4

$M_{ij}$ as a function of $c\left(\right|i-j\left|\right)$ (see Appendix pp2) for different sets of operators, system sizes, and projected fractions $p_f$. Solid lines show $M_{ij}$ for time-averaged operators $\tilde{O}$ (which commute with the Hamiltonian) and dashed lines show the corresponding overlaps for the parent projected operators $O$ (which do not commute with the Hamiltonian). The columns of the figure correspond to $N_A=1$ and 3.

Figure 5

The dependence on system size of the global background on a single site, for (a) the case where the number of states projected out is held constant, so $p_f$ vanishes exponentially in the thermodynamic limit (semilog plot) and (b) the case where $p_f$ is independent of system size (log-log plot). In (a) the background decays exponentially with $N$, while in (b) it decays as $1/N$. Data were taken for $h=8,N_A=1$, and 40 disorder realizations.

Figure 6

The equivalent of Fig. 2, but for a system where the middle states have been projected out instead of states at the upper end of the energy spectrum. The qualitative physics is the same: increasing $N_A$ increases $\mathcal{F}$ at finite sizes, with this effect vanishing at the thermodynamic limit. But the size of this effect is is much smaller, which makes it harder to extract the relevant physics from the background.

Figure 7

The equivalent of Fig. 4, but for a system where the middle states have been projected out instead of states at the upper end of the energy spectrum. At finite sizes this choice raises the background, making the exponential decay more difficult to see.

Figure 8

The same data as Fig. 3 but plotted with respect to actual distance (not rescaled). The exponential decay is harder to see but still present, showing that it is not an artifact of our rescaling of the $x$ axis by the chord distance. As in the main text, the dashed lines are the “background” coming from the fact that we do not have access to local operators in this projected Hilbert space. Our data show exponential decay until this background is reached, showing that our integrals of motion are as local as possible.

Figure 9

The same data as Fig. 4 but plotted with respect to actual distance. The exponential decay is harder to see but still present, showing that it is not an artifact of our rescaling of the $x$ axis by the chord distance. The dashed lines represent the background from the nonlocal nature of the projected operators.

Figure 10

The same data as the bottom row in Fig. 2, but plotted against $1/N$. We see that $\mathcal{F}$ does not decay to zero in the thermodynamic limit, and in fact seems to be consistent with the value $(1-p_f)$ in the thermodynamic limit, indicated by the stars on the $y$ axis.

Figure 11

The two-point correlator for non-time-averaged operators. This should be a delta function if $p_f=0$, and the corrections to that give the “background” plotted in the main text.