Local integrals of motion encoded in a few eigenstates

Abstract

Many properties of a quantum system can be obtained from just a single eigenstate of its Hamiltonian. For example, a single eigenstate can be used to determine whether a system is integrable or chaotic and, in the latter case, to establish its thermal properties. Focusing on the XXZ model, we show that the local integrals of motion, which lie at the heart of integrability, can also be estimated from a small number of eigenstates. Moreover, as the system size increases, fewer eigenstates are required, so that in the thermodynamic limit, the integrals of motion can be obtained from a vanishingly small fraction of all eigenstates. Interestingly, this property does not extend to integrals of motion arising solely from Hilbert space fragmentation, as found in the folded XXZ model, where the majority of eigenstates has to be used. This represents one of the few fundamental differences known between integrability and Hilbert space fragmentation.

Figures (7)

• Figure 1: Numerical results for the largest singular value obtained for the set of imaginary operators that are even under the spin-flip transformation and supported on up to $M=4$ sites. This set $\{A^1,\ldots, A^{D_O} \}$ contains $D_O=9$ operators including those in Eqs. \ref{['s11']}-\ref{['s12']}. Continuous lines in (a) show the projection $\langle |V_{11}| \rangle$ averaged over different choices of $N_s$ eigenstates for $L=14,\ldots, 24$. The number of these choices is $10^4$, also in Figs. \ref{['fig2']}-\ref{['fig4']}. The shaded areas indicate the standard deviation of $|V_{11}|$ for the smallest ($L=14$) and largest ($L=24$) systems, using consistent color coding. The edge of the former is marked with a thin black curve for visibility. Additionally, the horizontal dashed line marks the analytical prediction, ${\cal V}_{11}$, for the energy current, ${\cal Q}_1$, see Eq. (\ref{['encu']}). (b) The same resuts as in (a), but for the projection $V_{21}$. The projection $V_{31}$ is indistinguishable from $V_{21}$ on the presented scale. (c) Continuous lines show the projections $\langle |V_{s1}| \rangle$ on other operators $(s\ge 4)$ for $L=24$. (d) The largest singular value, $\langle \lambda^2_1\rangle/N_S$, see Eq. (\ref{['svd']}). Strict LIOMs correspond to $\lambda^2/Z=1$ obtained from all eigenstates (when $N_S=Z$).
• Figure 2: The same as in Fig. \ref{['fig1']}, but for a larger support $M=6$, for which the set $\{A^1,\ldots, A^{D_O}\}$ contains $D_O=155$ operators. The approximate LIOMs corresponding to two largest singular values, $Q^{'1}$ and $Q^{'2}$, are rotated according to the procedure described in the main text, see Eq. \ref{['rot']}. (a) and (b) show the projections of $Q^1$ on the operators from Eqs. \ref{['s11']} and \ref{['s112']}, respectively. The projections on operators from Eqs. \ref{['s112']}-\ref{['s12']} are mutually indistinguishable on the presented scale. (c) The projections of $Q^1$ on other operators with $s\ge 4$. Dashed lines in (a) and (b) mark the analytical predictions, as explained below Eq. \ref{['s12']}.
• Figure 3: (a) The same as in Figs. \ref{['fig2']}(a) and \ref{['fig2']}(b) but for $Q^2$ instead of $Q^1$. This panel shows the projections of $Q^2$ on the operators from Eqs. \ref{['s21']}-\ref{['s22']}. The largest projection is on $A^1$ from Eq.\ref{['s21']}, while the projections on other operators are mutually indistinguishable on the scale shown in (a). The horizontal dashed lines mark the analytical predictions, see text below Eq. \ref{['s22']}. The number of eigenstates $N^*$ is rescaled by $\mathrm{rank}({\cal R}) = 57$. (b) The numerical results for $Q^2$ constructed from the set of real operators that are even under the spin-flip transformation and are supported on up to $M=5$ sites. In this sector, $Q^1$ represents the normalized Hamiltonian. This plot shows the projections of $Q^2$ on the operators from Eqs. \ref{['s31']}-\ref{['s32']}. The largest projection is on $A^1$ from Eq. \ref{['s31']} and the projections on other operators are indistinguishable on the scale shown in (b). The number of eigenstates $N^*$ is rescaled by $\mathrm{rank}(R) = 31$.
• Figure 4: Results for the sector of imaginary operators that are odd under the spin-flip transformation. We consider the set of operators with the support $M=6$, for which $\mathrm{rank}(R)= 66$. This symmetry sector includes the spin-current operator but does not contain any LIOMs. (a) The projections of $Q^1$ onto the operators defined in Eqs. \ref{['s41']}-\ref{['s42']}. The shaded areas indicate the standard deviations for the largest ($L=24$) systems, using consistent color coding. (b) The largest singular value, $\langle \lambda_1^2 \rangle/N_S$, obtained for $N_{S} = 30\, \mathrm{rank}(R)$. The extrapolation to $L\to\infty$ indicates that it increases with increasing support in the thermodynamic limit, and suggest that it corresponds to a QLIOM.
• Figure 5: Results for the folded XXZ model from Eq. \ref{['folded']}. The set of $A^s$ is constructed only from $S^z_l$ and $I_l$ acting on up to $M=4$ consecutive sites, for which $\mathrm{rank}(R)=7$. All results are averaged over $10^4$ different choices of $N_S$ eigenstates, also in Fig. \ref{['fig6']}. (a) The largest singular value, $\langle\lambda_1^2\rangle/\tilde{N_s}$, for $g=0$. (b) Top curves correspond to the largest projection of $Q^1$ onto the operator $A^1$ from Eq. \ref{['s5']}, while bottom curves correspond the next largest projection. The shaded areas indicate the standard deviations of the projections for the smallest ($L=12$) and largest ($L=18$) systems, using consistent color coding. The edge of the former is marked with a thin black curve for visibility. Panels (c) and (d) show the same results as panels (a) and (b), respectively, but for the perturbed model with $g=2.0$.