Krylov complexity in ergodically constrained nonintegrable transverse-field Ising model

TL;DR

We address whether ergodicity can be suppressed in a nonintegrable transverse-field Ising model without disorder by introducing spatial inhomogeneity in couplings. We employ a tunable inhomogeneity parameter $\mathcal{J}_r$ to drive a crossover from chaotic to constrained dynamics, using diagnostics including $C(d,t)$ OTOCs, level spacing statistics, the spectral form factor, Krylov space growth, and entanglement of eigenstates. We find that increasing $\mathcal{J}_r$ yields a transition from Wigner–Dyson to Poisson spectral statistics, delayed $t_{Th}$ in the SFF, reduced long-time OTOC saturation, and suppressed operator spreading in Krylov space; off-diagonal Hamiltonian elements in the $H_0$ basis decay as $W_{\mathrm{off}} \propto \mathcal{J}_r^{-\alpha}$ with $\alpha \approx 1.83$, consistent with an emergent effective integrability via a Schrieffer–Wolff-like transformation. Eigenstate entanglement also decreases and ground-state entanglement remains small for $\mathcal{J}_r > 1$. Collectively, the results show a minimal, disorder-free route to tunable ergodicity and constrained quantum dynamics in an interacting spin chain.

Abstract

The nonintegrable transverse-field Ising model is a common platform for studying ergodic quantum dynamics. In this work, we introduce a simple variant of the model in which this ergodic behaviour is suppressed by introducing a spatial inhomogeneity in the interaction strengths. For this we partition the chain into two equal segments within which the spins interact with different coupling strengths. The ratio of these couplings defines an inhomogeneity parameter, whose variation away from unity leads to constrained dynamics. We characterize this crossover using multiple diagnostics, such as the long-time saturation of out-of-time-ordered correlators, level-spacing statistics, and the spectral form factor. We further examine the consequences for operator growth in Krylov space and for entanglement generation in the system's eigenstates. Together, these results demonstrate that introducing a macroscopic inhomogeneity in coupling strengths provides a minimal, disorder-free route to breaking ergodicity in this specific model of interacting spins.

Figures (14)

• Figure 1: Schematic of an inhomogeneous spin chain with an odd number of spins, chosen so that the number of nearest-neighbor interaction terms can be equally divided. The spins interact along the $z$ direction with open boundary conditions. Half of the interaction terms have strength $J_1$ and the other half have strength $J_2$, creating a nonuniform interaction pattern. The chain is subjected to a transverse field $h_x$ and a longitudinal field $h_z$.
• Figure 2: 3D heat map of matrix elements, $H_{pq}$ for the Hamiltonian under different values of inhomogeneity parameter $\mathcal{J}_r$, for $N = 7$, $h_x = 1.05$ and $h_z = 0.5$. $(a)$ and $(c)$ are representations in the general computational basis, whereas $(b)$ and $(d)$ follow the basis of the dominant Hamiltonian term $H_0$. Note the suppression in the off-diagonal elements between $\mathcal{J}_r = 1.1$ and $10$.
• Figure 3: Logarithmic variation of normalised off-diagonal Hamiltonian components in the $H_0$ basis, $\sum_{i\neq j}|H_{ij}|$, with the homogeneity parameter for different values of $N$. Fitting a straight line, of slope $m$ between the plot of the logarithmic quantities indicates that $\sum_{i\neq j}|H_{ij}| \propto \mathcal{J}_r^{-1.821}$ for $N = 9$. The inset shows variation of the fitted slope $m$ for different values of $1/N$. This curve indicates the extrapolated value of slope to be $-1.83$ for the large $N$ limit.
• Figure 4: $(a)$ Variation of the OTOC (Eq. \ref{['OTOC_Working']}) in the short time limit for different inhomogeneity parameter $\mathcal{J}_r$. $h_x = 1.05$, $h_z = 0.5$ for $J_1 = 1$, $N = 7$ and $d = 6$ with open boundary conditions. The dotted lines indicate the fitted curve ($\propto t^{26}/13!$) for the corresponding $\mathcal{J}_r$. In $(b)$ we show the variation of parameter $\alpha$ which is found to be dependent on $\mathcal{J}_r$ as $5.39 \times \mathcal{J}_r^4$.
• Figure 5: Variation of the average spectral parameter $r$ for different values of $\mathcal{J}_r$, with system size appearing as a parameter. For all cases, $h_x = 1.05$ and $h_z = 0.5$ with open boundary conditions. The level spacing includes the entire spectrum as a particular symmetry segment does not exist.