Integrability and complexity in quantum spin chains

Abstract

There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.

Figures (13)

• Figure 1: Left: The time evolution of the upper bound on Nielsen's complexity for the Hamiltonian evolution of the transverse Ising model at $(h_x,h_z)=(-1.05,0)$ (blue) and the Ising model evaluated at a chaotic parameter point $(h_x,h_z)=(-1.05,0.5)$ (red). The length of the chain is $L=12$. To avoid using candidate minima that are evidently sub-optimal in estimating the complexity \ref{['Cbound']}, at every time step we compare the output of the minimization procedure to the complexity associated to the early time solution $k_n = 0$ and the minimum of the two is chosen. Right: A zoom on the time window inside the plateau region used in the numerics.
• Figure 2: Histograms of the eigenvalues of the Q-matrix for Left column: an integrable Ising spin chain \ref{['eq:Isingspin12']} with $(h_x,h_z)=(-1.05,0)$ and Right column: a chaotic Ising spin chain with $(h_x,h_z)=(-1.05,0.5)$ for varying locality thresholds as Top row:$k_{sp}=k_{op}$ or Bottom row: only varying $k_{op}$ while keeping $k_{sp}=6$ fixed. The length of the chain is set to $L=12$. In the bottom left inset of each plot we zoom in on the size of the kernel of the $Q$-matrix. It can be seen that in the integrable case, the number of zero eigenvalues increases in steps of $2$, in accordance with the form of the conserved charges \ref{['eq: conslawsIsing1']}-\ref{['eq: conslawsIsing4']}.
• Figure 3: Left: The fraction of local generators to the total number of generators, as a function of $k_{op}$ for the two types of locality thresholds $\mathcal{T}_1$ and $\mathcal{T}_2$ we impose in the main analysis. Middle and Right: The distance between the mean of the $Q$-eigenvalue distribution and the right edge of the $Q$-histograms, for, respectively, the chaotic and integrable Ising model as a function of $k_{op}$. The shape of the curves in all three plots is very similar. The results for the chaotic model qualitatively agree with the RMT prediction \ref{['eq: estimate mean Q spectrum']} up to a factor of order one.
• Figure 4: The late-time saturation value of the complexity bound for the integrable $(h_x,h_z)=(-1.05,0)$ and chaotic $(h_x,h_z)=(-1.05,0.5)$ Ising model with $L=12$ sites as a function of a locality threshold specified by $k$. The dashed line corresponds to $k_{sp}=6$ fixed and varying $k_{op}$, while for the solid line we vary both locality degrees $k_{sp}=k_{op}$.
• Figure 5: Histograms of the eigenvalues of the Q-matrix for the $L=12$Left column: integrable XYZ model \ref{['eq:H_XYZ']} and for the Right column: chaotic XYZ model with magnetic field \ref{['eq:H_XYZ_hz']} (right) for varying locality thresholds as Top row:$k_{sp}=k_{op}$ or Bottom row: only varying $k_{op}$ while keeping $k_{sp}=6$ fixed. For the coupling constants we used the numbers $(J_x,J_y,J_z)=(-0.35,0.5,-0.1)$ and on the right plot we chose $h_z=0.8$. In the bottom left corner of each plot we zoom in on the zero eigenvalues. The number of zero $Q$-eigenvalues in the integrable case increases in accordance with the form of the conserved charges.