Quantum Chaos in Many-Body Systems Without a Classical Analogue

Abstract

In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.

Figures (19)

• Figure 1: Heatmap of the values of the matrix that represents the PXP Hamiltonian for $L=10$, in the zero-momentum, inversion-symmetric, no adjacent excitations sector. The dimension of the reduced Hilbert space in this case is $\mathcal{D} = 14$, and therefore the Hamiltonian is a matrix of dimension $14 \times 14$.
• Figure 2: Strong violation of the ETH revealed by the eigenstate expectation values $\langle O^Z \rangle$, with $O^Z = (1/L) \sum_{j=1}^L Z_j$, plotted as a function of energy. While the majority of the points are concentrated around the canonical ensemble prediction, the band of special eigenstates is also clearly visible. For these eigenstates, $\langle O^Z \rangle$ strongly deviates from the canonical prediction at the corresponding energy, see Section \ref{['sec:canonical']} for the calculation of the canonical curve. The system contains $L=28$ atoms in the zero-momentum, inversion-symmetric, no adjacent excitations sector. We see a total of 15, $L/2 + 1$ for $L=28$, special eigenstates (one of them shown with an arrow), at the top of the tower-like structures.
• Figure 3: Standard deviation $\sigma$ of the EEVs of the $O^Z = (1/L) \sum_{j=1}^L Z_j$ operator for various system sizes $L$, as a function of the dimension of the Hilbert space $\mathcal{D}$ in the zero-momentum, inversion symmetric, no adjacent excitations sector. The regression is performed only on the last 3 points.
• Figure 4: Overlap of all the different energy eigenstates $\ket{\psi}$ of the PXP Hamiltonian, with the $\ket{\mathbb{Z}_2}$ product state. Data are shown for $L=28$ sites in the zero-momentum, inversion symmetric, no adjacent excitations sector. The states with high overlap with the $\ket{\mathbb{Z}_2}$ state, lying on top of the "towers" observed, are identified with the states at the top of the ETH-breaking "towers" in Fig. \ref{['fig:eev']}, one of which is shown with an arrow.
• Figure 5: Level spacing statistics, using the quantity $r_n = \min\{\delta_n, \delta_{n+1}\} / {\max\{\delta_n, \delta_{n+1}\}}$ defined in OH2007. We compare the theoretical prediction (solid curve), surmised in Atas:2013wy, with the numerical calculation (histogram) for random matrices of size $\mathcal{D} = 6,000$. For the Poisson case, we generate a diagonal matrix with real random entries, while for the GOE case, we generate a random matrix $M$ of size $6,000 \times 6,000$ with real entries, and then symmetrize it using $(M + M^T)/2$.