Stability of the quantum Sherrington-Kirkpatrick spin glass model

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida–Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida–Thouless (QuAT) line in the transverse field-longitudinal field plane at $T=0$. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

Figure 1

A sketch of the proposed phase diagram of the quantum SK model in terms of temperature $T$, the standard deviation of the (random) longitudinal field $h$, and the transverse field $h^T$. The phase diagram for a uniform longitudinal field $h$ would look the same. The replica symmetric solution is unstable everywhere below the surface indicated. For $h^T=0$ the boundary where replica symmetry breaking occurs is called the AT line [11]. For $T=0$, I call the boundary in the $h-h^T$ plane the quantum AT (QuAT) line. The phase boundary in the $T-h^T$ plane is drawn precisely in Fig. 2.

Figure 2

The phase diagram in the $h^T-T$ plane, determined from the vanishing of the denominator of Eq. (13) where ${\chi }_{{}_{\mathrm{SG}}}^0$ is given by Eq. (17). The phases are spin glass (SG) and paramagnet (P).

Figure 3

The triangles show results for ${\chi }_{{}_{\mathrm{SG}}}$ from my calculations extrapolated to $\mathrm{\Delta }\tau =0$ at very low temperature ($T/J=0.1$) in the paramagnetic phase. The horizontal axis is ${(J/h^T)}^2$. On the vertical axis I multiply ${\chi }_{{}_{\mathrm{SG}}}$ by ${\left(h^T\right)}^2$ because, for $h^T\to \infty$, perturbation theory gives ${\chi }_{{}_{\mathrm{SG}}}=1/{\left(h^T\right)}^2$. The spin glass susceptibility diverges at $h_c^T/J\simeq 1.51$ as I shall see later, and this is indicated by the dashed vertical line. For the range of data shown there is a gap in the spectrum so my results converge rapidly as $T\to 0$ and I find that $T/J=0.1$ is low enough that the results are almost indistinguishable from those at $T=0$. The line is the result of a 14-term series expansion [27] in powers of ${(J/h^T)}^2$, evaluated for $T$ strictly equal to 0. The agreement is excellent until the critical point is approached where (a) I need to be more careful in extrapolating my results to $T=0$ as I shall see later, and (b) the solid line is the “raw” series from Ref. [27] so it does not show a divergence at $h_c^T$.

Figure 4

The spin glass order parameter $q$ near the quantum critical point at $T=0,h^T=h_c^T$. The linear behavior shows that the order parameter exponent $\beta$ has the value $\beta =1$ in agreement with analytic work [19].

Figure 5

A plot of $1/{\chi }_{{}_{\mathrm{SG}}}$, the inverse of the spin glass susceptibility, near the quantum critical point (QCP) at $T=0,h_c^T\simeq 1.51$. One sees that $1/{\chi }_{{}_{\mathrm{SG}}}$ tends to zero as the QCP is approached from above. Below the QCP the spin glass order parameter becomes nonzero as shown in Fig. 4. Nonetheless, ${\chi }_{{}_{\mathrm{SG}}}$ is negative for $h^T<h_c^T$. However this is impossible since ${\chi }_{{}_{\mathrm{SG}}}$ is a positive quantity, and hence the assumption of replica symmetry, made in the calculation, must be wrong.

Figure 6

A plot of $1/{\chi }_{{}_{\mathrm{SG}}}^2$ against $\delta h^T/|log\delta h^T|$ for $N=256,\beta =20$, where $\delta h^T=h^T-h_c^T$ in which $h_c^T$, the critical value, was taken to be 1.508. According to analytic predictions [24, 25], ${\chi }_{{}_{\mathrm{SG}}}\propto {\left(|ln\delta h^T|/\delta h^T\right)}^{1/2}$, so the plot should be a straight line. Clearly this form works very well.

Figure 7

A plot of $1/{\chi }_{{}_{\mathrm{SG}}}$ against $h$ for $h^T=1.35$ and very low temperature. For $h\lesssim 0.045$, the calculated ${\chi }_{{}_{\mathrm{SG}}}$ is negative indicating that the RS solution is unstable, whereas for $h\gtrsim 0.045,{\chi }_{{}_{\mathrm{SG}}}$ is positive indicating the RS solution is stable. The set of parameters $T=0,h=0.045,h^T=1.35$ lies on the quantum AT (QuAT) line, see Fig. 8.

Figure 8

A plot of the QuAT line near the quantum critical point.

Figure 9

A speculative phase diagram for two dimensions. There is no spin glass phase at finite $T$ but there is a quantum spin glass phase at $T=0$ up to $h_c^T$, the critical value of $h^T$, so there might possibly be a QuAT line in the $h-h^T$ plane at $T=0$. While perhaps unlikely, one cannot rule out this scenario at present.