Total correlations of the diagonal ensemble as a generic indicator for ergodicity breaking in quantum systems

The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics in systems without degeneracies. In analogy to the time average of a classical phase-space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general $\alpha$-Renyi entropies and focus on the cases $\alpha =1$ and $\alpha =2$ with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a subextensive behavior when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many-body localized phase in systems with disorder or quasiperiodicity.

Figure 1

The von Neumann total correlations of the diagonal ensemble starting with the Neel state for an XXZ chain with defect of strength $\epsilon$ placed at center of the chain [Eq. (9) with parameters $J_x=J_y=1$, and $J_z=0.5]$. When the defect strength is zero or very strong the model is integrable, which is reflected in a linear scaling of the total correlations, and when it is comparable with the interaction energy it shows a logarithmic growth indicative of ergodic dynamics. Inset: Total correlations for an XXZ chain with next-nearest-neighbour interaction [Eq. (10) with parameters $J_x=J_y=1, J_z=0.5$, and $J_x^{\prime }=J_y^{\prime }=1$, compared to the same model with $J_x^{\prime }=J_y^{\prime }=0]$. The model is nonintegrable and thus the scaling of the total correlations is logarithmic in the system size.

Figure 2

The 2-Renyi total correlations of the diagonal ensemble starting with the Neel state for an XXZ chain with defect of strength $\epsilon$ placed at center of the chain [Eq. (9) with parameters $J_x=J_y=1$, and $J_z=0.5]$. When the defect strength is zero or very strong $T_2$ scales linearly with system size and when it is comparable with the interaction energy it scales logarithmically. Inset. Total correlations for an XXZ chain with next-nearest-neighbour interaction [Eq. (10) with parameters $J_x=J_y=1, J_z=0.5$, and $J_x^{\prime }=J_y^{\prime }=1$, compared to the same model with $J_x^{\prime }=J_y^{\prime }=0]$. The scaling of the total correlations is logarithmic in the system size.

Figure 3

The von Neumann total correlations of the diagonal ensemble, starting with a Neel state, for the Heisenberg model with random fields [whose Hamiltonian is Eq. (11) with $h_i\in [-h,h]$ and $J_x=J_y=J_z=1]$, rescaled with the system size. The markers on the top axis denote the positions of the local peak $h^{*}\left(N\right)$. The curves show a system-size-dependent peak (see Fig. 7) and collapse for $h\gtrsim 2.5$. Inset: System size scaling of the total correlations for three example values of $h$, showing a logarithmic scaling deep in the delocalized phase and a linear scaling for disorder values near the transition (which is at $h_c\approx 3.7$) and in the localized phase.

Figure 4

The von Neumann total correlations of the diagonal ensemble, starting with a Neel state, for the Aubry-André model [whose Hamiltonian is Eq. (11) with the cosine $h_i$ fields and $J_x=J_y=J_z=1]$, rescaled with the system size. The curves show a system-size-dependent peak and collapse for $h\gtrsim 3.5$. Inset: System size scaling of the total correlations for three example values of $h$, showing a logarithmic scaling deep in the delocalized phase and a linear scaling in the localized phase.

Figure 5

The 2-Renyi total correlations of the diagonal ensemble, starting with a Neel state, for the Heisenberg model with random fields [whose Hamiltonian is Eq. (11) with $h_i\in [-h,h]$ and $J_x=J_y=J_z=1]$, rescaled with the system size. The markers on the top axis denote the positions of the local peak $h^{*}\left(N\right)$, excluding the case $N=8$, where no local maximum can be discerned. The curves show a system-size-dependent peak (see Fig. 7) and collapse for $h\gtrsim 2$. Inset: System size scaling of the total correlations for three example values of $h$, showing a logarithmic scaling deep in the delocalized phase and a linear scaling for disorder values near the transition (which is at $h_c\approx 3.7$) and in the localized phase.

Figure 6

The 2-Renyi total correlations of the diagonal ensemble, starting with a Neel state, for the Aubry-André model [whose Hamiltonian is Eq. (11) with the cosine $h_i$ fields and $J_x=J_y=J_z=1]$, rescaled with the system size. The curves show a system-size-dependent peak (see Fig. 7) and collapse for $h\gtrsim 3$. Inset: System size scaling of the total correlations for three example values of $h$, showing a logarithmic scaling deep in the delocalized phase and a linear scaling in the localized phase.

Figure 7

System size scaling of the peak of the total correlations in the Heisenberg model with random fields, for both $T_1$ and $T_2$. The peak is extracted from a polynomial interpolation of each of the curves in Figs. 3 and 5, in which it is denoted with a marker on the top axis. We perform a linear fit in $1/N$ and obtain the infinite size extrapolation given in the text.