Variational Ansatz for the Ground State of the Quantum Sherrington-Kirkpatrick Model

We present an Ansatz for the ground states of the quantum Sherrington-Kirkpatrick model, a paradigmatic model for quantum spin glasses. Our Ansatz, based on the concept of generalized coherent states, very well captures the fundamental aspects of the model, including the ground state energy and the position of the spin glass phase transition. It further enables us to study some previously unexplored features, such as the nonvanishing longitudinal field regime and the entanglement structure of the ground states. We find that the ground state entanglement can be captured by a simple ensemble of weighted graph states with normally distributed phase gates, leading to a volume law entanglement, contrasting with predictions based on entanglement monogamy.

Figure 1

Average error in energy density $ϵ=\overline{\mathrm{\Delta }E}/\overline{W}$ as a function of the transverse field for different methods (CS in orange, GCS in blue, and ED in purple) and system sizes $N=8$, 12, 16, 22 (light to dark). $\mathrm{\Delta }E$ is the difference between the variational energy and the exact ground state energy, $W$ is the difference between the highest and lowest energies in the exact spectrum. Inset: difference between CS and GCS energies per site $(E_{\mathrm{CS}}-E_{\mathrm{GCS}})/N$ as a function of the system size $N$ for different values $g/J=0.1$, 1.0, 1.5, 2.0, 3.0 (light to dark). All data are for $h=0$ and averaged over $n_{\text{samples}}=1000$ disorder realizations.

Figure 2

Left: spin glass susceptibility ${\chi }_{\mathrm{sg}}$ as a function of the transverse field $g$ for ED (purple circles) and GCS (blue squares). Right: spin glass susceptibility per site ${\chi }_{\mathrm{sg}}/N$ for GCS (top, blue) and CS (bottom, orange) and system sizes $N=20$, 100, 200 (from light to dark). All data are for $h=0$ and averaged over $n_{\text{samples}}=1000$ disorder realizations.

Figure 3

Average Rényi-2 entanglement entropy as a function of the subsystem size $L$ for the QSK ground state at $g=1J$, $h=0$ (top panel in blue) and for an ensemble of weighted graph states (7) (bottom panel in green). Data are plotted for total system sizes $N=50$, 100, 150, 200 (from light to dark markers). In all cases, including ground state data for other values of the fields $g$ and $h$, the entropy is well fitted by the function (6) (orange dashed lines).

Figure 4

Coefficient $C(N)$ extrapolated from the Rényi entanglement entropy fit as a function of the transverse field $g$ at $h=0$ for different system sizes (different shades of blue).

Figure 5

Spin glass susceptibility per site ${\chi }_{\mathrm{sg}}/N$ (green squares) and entropy coefficient $C$ (blue circles) as functions of $g$ for different values $h/J=0,0.1,\dots ,1$ (dark to light) at $N=150$. We observe that both functions develop a singularity typical of a phase transition only in the $h=0$ limit.