Bosonic model of quantum holography

We analyze a model of qubits that we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model we consider is known as the quantum $q$-spin model because it features $q$-local interactions between qubits. It was previously studied as a model of a quantum spin glass, and while we find that the model is glassy for $q=2, q=3$, and likely $q=4$, we also find evidence for previously unexpected SYK-like behavior for the quenched free energy down to the lowest temperatures for $q\ge 5$. This SYK-like physics includes power-law correlation functions and an extensive low-temperature entropy, so we refer to the model as Spin SYK. The model is generic in that it includes all possible $q$-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. Furthermore, we discuss a generalized family of models that includes Spin SYK and which provably exhibit SYK-like physics in the solvable limit of large local Hilbert space dimension. We also comment on the implications of a bosonic system with SYK-like properties for the study of holography, Hamiltonian complexity, and related topics.

Figure 1

The strategy we follow in this paper is in gray. The top flowchart of blue boxes is the general procedure proposed in [34] in which we construct the replicated action, carry out the minimax procedure, and then use the quenched saddle to determine the free energy. While this procedure should be possible to carry out, here we adopt an alternate procedure in which we try to determine $q_{\text{EA}}$ using an extrapolation from finite-size numerics (orange box). When $q_{\text{EA}}=0$, we can restrict to single-replica saddles, in which case our results indicate that the only possibility is the SYK-like saddle.

Figure 2

Solution to the EOM with $q=2, \beta =4, J=1, N_{\tau }=80$, and $N_s=1000$ samples. The red dashed line is the initialization (large $q$), the black line shows the final converged result, and the blue shaded region gives $3/\sqrt{N_s}$ around the black line. The result is consistent with the annealed saddle in which $G$ approaches unity at long time.

Figure 3

Solution to the EOM with $q=3, \beta =4, J=1, N_{\tau }=80$, and $N_s=1000$ samples. The red dashed line is the initialization ($J=0$), the black line shows the final converged result, and the blue shaded region gives $3/\sqrt{N_s}$ around the black line. The result is consistent with the annealed saddle in which $G$ approaches unity at long time.

Figure 4

Solution to the EOM with $q=3, \beta =4, J=1, N_{\tau }=300$, and $N_s=10000$ samples. Compared to the previous figures, we have now dramatically increased the number of time points and the number of samples. The red dashed line is the initialization (large $q$), the black line shows the final converged result, and the blue shaded region gives $3/\sqrt{N_s}$ around the black line. The blue dashed line is the SYK conformal result (capped at 3). We see quite good agreement with the large-$q$ result at short times and the SYK result at long times. Note how the solution goes well below the annealed value at large $\tau$.

Figure 5

Solution to the EOM with $q=4, \beta =1, J=1, N_{\tau }=200$, and $N_s=10000$ samples. The red dashed line is the initialization (large $q$), the black line shows the final converged result, and the blue shaded region gives $3/\sqrt{N_s}$ around the black line. The blue dashed line is the SYK conformal result (capped at 3). This result is roughly consistent with the large-$q$ form for all $\tau$. We again see good agreement with the large-$q$ result at short times and the SYK result at long times.

Figure 6

Solution to the EOM with $q=5, \beta =0.1, J=1, N_{\tau }=200$, and $N_s=10000$ samples. The red dashed line is the initialization (large $q$), the black line shows the final converged result, and the blue shaded region gives $3/\sqrt{N_s}$ around the black line. The blue dashed line is the SYK conformal result (capped at 3). The blue dashed line is the SYK conformal result (capped at 3). We have significantly reduced $\beta$ relative to the previous figures to demonstrate the good agreement with the large-$q$ result.

Figure 7

The Edwards-Anderson order parameter in the ground state (log scale) for $q=3$ as a function of system size. Each point is averaged over 2000 samples. The growth with system size and the macroscopic value indicate glassiness. The red line is a linear fit to $log{q}_{\text{EA}}$.

Figure 8

A histogram of 2000 samples of the finite-size energy gap between the ground state and first excited state for $q=3$ and $n=14$. The exponential distribution of the gap and the corresponding lack of level repulsion are consistent with the two lowest-lying states being quantum TAP states.

Figure 9

The Edwards-Anderson order parameter in the ground state (log scale) for $q=4$ as a function of system size. Each point is averaged over 500 samples. The slow decrease with system size is consistent with the absence of glassiness. We also see a very clear even-odd effect consistent with the antiunitary time-reversal symmetry. The red line is a linear fit to $log{q}_{\text{EA}}$.

Figure 10

A histogram of 500 samples of the finite-size energy gap between the ground state and first excited state for $q=4$ and $n=15$. The clear level repulsion is consistent with a nonglassy low-energy state.

Figure 11

The energy per spin plotted vs $1/N$ for $q=4$. The red line is a linear extrapolation to $1/N=0$ and the blue star is the large-$q$ prediction. Each point is 500 samples.

Figure 12

The Edwards-Anderson order parameter in the ground state (log scale) for $q=5$ as a function of system size. Each point is averaged over 2000 samples. The rapid decay with system size strongly indicates the absence of glassiness. The red line is a linear fit to $log{q}_{\text{EA}}$. One possible reason the finite-size scaling is exponential in system size is that, like the [64] SYK model, the Spin SYK has an exponentially large ground-state space and the single spin operators behave like random matrices on this space.

Figure 13

A histogram of 2000 samples of the finite-size energy gap between the ground state and first excited state for $q=5$ and $n=14$. The clear level repulsion is consistent with a nonglassy low-energy state.

Figure 14

The energy per spin plotted vs $1/N$ for $q=5$. The red line is a linear extrapolation to $1/N=0$ and the blue star is the large-$q$ prediction. Each point is 2000 samples.

Figure 15

The Edwards-Anderson order parameter in the ground state (log scale) for $q=3$ as a function of system size from $N=13$ to 17. Each point is averaged over 100 samples. The approximate saturation with system size indicates a glassy ground state at large $N$. The red line is a linear fit to $log{q}_{\text{EA}}$.

Figure 16

The Edwards-Anderson order parameter in the ground state (log scale) for $q=4$ as a function of system size from $N=13$ to 17. Each point is averaged over 100 samples. The approximate saturation with system size indicates a glassy ground state at large $N$. The red line is a linear fit to $log{q}_{\text{EA}}$.

Figure 17

The Edwards-Anderson order parameter in the ground state (log scale) for $q=5$ as a function of system size from $N=13$ to 17. Each point is averaged over 100 samples. The rapid decrease with system size indicates a nonglassy ground state at large $N$. The red line is a linear fit to $log{q}_{\text{EA}}$.

Figure 18

The Edwards-Anderson order parameter in the ground state (log scale) for $q=6$ as a function of system size from $N=13$ to 17. Each point is averaged over 100 samples. The rapid decrease with system size indicates a nonglassy ground state at large $N$. The red line is a linear fit to $log{q}_{\text{EA}}$.

Figure 19

The Edwards-Anderson order parameter in the ground state (log scale) for $q=7$ as a function of system size from $N=13$ to 17. Each point is averaged over 100 samples. The rapid decrease with system size indicates a nonglassy ground state at large $N$. The red line is a linear fit to $log{q}_{\text{EA}}$.