Metastable minima of the Heisenberg spin glass in a random magnetic field

We have studied zero-temperature metastable minima in classical $m$-vector component spin glasses in the presence of $m$-component random fields for two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case where one averages the number of minima before taking the log and the quenched case where one averages the complexity itself, both for fields above and below the de Almeida-Thouless (AT) field, which is finite for $m>2$. We have done numerical quenches starting from a random initial state (infinite temperature state) by putting spins parallel to their local fields until there is no further decrease of the energy and found that in zero field it always produces minima that have zero overlap with each other. For the $m=2$ and $m=3$ cases in the SK model the final energy reached in the quench is very close to the energy $E_c$ at which the overlap of the states would acquire replica symmetry-breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states $\rho \left(\lambda \right)$ of the Hessian matrix in the annealed approximation. Despite the fact that in the presence of a random field there are no continuous symmetries, the spectrum extends down to zero with the usual $\sqrt{\lambda }$ form for the density of states for fields below the AT field. However, when the random field is larger than the AT field, there is a gap in the spectrum, which closes up as the AT field is approached. The VB model behaves differently and seems rather similar to studies of the three-dimensional Heisenberg spin glass in a random vector field.

Figure 1

The overlap distribution $P\left(q\right)$ for the VB model ($\sigma =0,z=6$, $h_r=0.6$) for the minima generated by the prescription described in the text. $P\left(q\right)$ seems to be approaching a $\delta$ function as $N$ tends to infinity.

Figure 2

The average energy per site and spin component for the $XY$ SK spin glass model ($m=2$) with $h_r=0$ plotted against $1/N^{2/3}$ in order to estimate the infinite system value of the energy obtained from a quench from infinite temperature. For $m=2$, $E_c=-0.866$ [9].

Figure 3

The average energy per site and spin component for the Heisenberg SK spin glass ($m=3$) with $h_r=0$ plotted against $1/N^{2/3}$ in order to estimate the infinite system value of the energy obtained from a quench from infinite temperature. For $m=3$, $E_c=-0.914$ [9].

Figure 4

The eigenvalue spectrum of the Hessian at zero temperature for the vector spin glass with $m=3$ in the SK limit. The various lines correspond to different values of $h_r$, the external random field.

Figure 5

The magnified view of the same figure as Fig. 4 but for the small eigenvalues.

Figure 6

The inverse of the spin glass susceptibility ${\chi }_{SG}^{-1}$ versus $h_r^2$ for a range of system sizes of the Heisenberg SK model. The analytic curve is the result of Eq. (29). For $h_r\le 1$, one expects that ${\chi }_{SG}^{-1}=0$, but finite-size effects make it nonzero.

Figure 7

The inverse of the spin glass susceptibility ${\chi }_{SG}^{-1}$ versus $h_r^2$ for a range of system sizes for the VB model with $z=6$.

Figure 8

The averaged density of states of the Hessian matrix of the metastable states obtained after a quench to $T=0$ starting from spins with random orientations, i.e., $T=\infty$ for the SK model ($\sigma =0,z=N-1$ of the diluted model). Data shown here for the special case of $h_r=0.8$, for which the system is in the spin glass phase, just below $h_{AT}=1$. The analytical curve is that calculated from Eqs. (31) and (32) for metastable states at the top of the band within the annealed approximation. The numerical results are strikingly similar to the analytical results, despite the fact that they refer to Hessians for quite different situations!