Comparison of Gabay–Toulouse and de Almeida–Thouless instabilities for the spin-glass XY model in a field on sparse random graphs

Vector spin glasses are known to show two different kinds of phase transitions in the presence of an external field: the so-called de Almeida–Thouless and Gabay–Toulouse lines. While the former has been studied to some extent on several topologies (fully connected, random graphs, finite-dimensional lattices, chains with long-range interactions), the latter has been studied only in fully connected models, which however are known to show some unphysical behaviors (e.g., the divergence of these critical lines in the zero-temperature limit). Here we compute analytically both these critical lines for XY spin glasses on random regular graphs. We discuss the different nature of these phase transitions and the dependence of the critical behavior on the field distribution. We also study the crossover between the two different critical behaviors, by suitably tuning the field distribution.

Figure 1

Stability parameter ${\lambda }_{\text{BP}}$ for the spin-glass XY model on a $C=3$ RRG at zero field. Data are collected during cooling and heating numerical experiments with 300 iterations for temperature, and averaged over the last 150 iterations. The black dot marks the exact value for the critical temperature.

Figure 2

Stability parameter ${\lambda }_{\text{BP}}$ for the spin-glass XY model on a $C=3$ RRG at zero field. All the points reported have been measured in the stationary regime. The full green line refers to the analytic evaluation of ${\lambda }_{\text{BP}}$ on the paramagnetic solution. The inset shows the power-law behavior below the critical point, ${\lambda }_{\text{BP}}\propto {\tau }^{\alpha }$, with $\alpha =1.6\left(1\right)$.

Figure 3

Stability parameter ${\lambda }_{\text{BP}}$ for the spin-glass XY model on a $C=3$ RRG with a uniform external field of intensity $H$. The two panels show data with different ranges of fields. The lower one makes evident the leftward shift of the curves when increasing the field strength $H$.

Figure 4

Critical lines in a uniform field for a spin-glass XY model on a random $C$-regular graph. The corresponding line in the fully connected model (SK limit) is given by the black curve. The inset shows evidence for the $H_c\left(T\right)\propto {\tau }^{1/2}$ behavior, typical of the GT transition.

Figure 5

Stability parameter ${\lambda }_{\text{BP}}$ for the spin-glass XY model on a $C=3$ RRG with a randomly oriented external field of fixed intensity $H$. At variance with the uniform-field case, the curve ${\lambda }_{\text{BP}}\left(T\right)$ mainly moves downward when increasing $H$, while smoothing away the zero-field singularity.

Figure 6

Critical lines in a field of random direction for a spin-glass XY model on a random $C$-regular graph. The corresponding line in the fully connected model (SK limit) is given by the black curve. The inset shows evidence for the $H_c\left(T\right)\propto {\tau }^{3/2}$ behavior, typical of the dAT transition.

Figure 7

GT and dAT critical lines computed in the XY model with $J_{ij}=\pm J$ on a $C=3$ random regular graph.

Figure 8

Probability distribution of $cos{\vartheta }_i$ over the BP fixed-point population ${\mathbb{P}}^{*}\left[\eta \right]$ for several points along the dAT line (left panel) and the GT line (right panel). For the definition of ${\vartheta }_i$ see Eq. (11) and the main text. Here $C=3$ and $J=1$.

Figure 9

Joint probability distribution of $m_i$ and $cos{\vartheta }_i$ for the same points of Fig. 8, better highlighting the different spin symmetries broken on dAT and GT lines, respectively. Here $C=3$ and $J=1$.

Figure 10

Critical lines in the $(T,H)$ plane for the spin-glass XY model on a $C=3$ RRG with field directions $\varphi =2\pi \kappa /Q$ with $\kappa \in \{0,1,\cdots ,Q^{\prime }-1\}$ uniformly. The choice $Q^{\prime }=1$ corresponds to the GT line, while $Q^{\prime }=Q=64$ gives back the dAT line. Data in the right panel seem to suggest a dAT-like critical behavior for any $Q^{\prime }>1$ (dashed lines have slopes $1/2$ and $3/2$, respectively).

Figure 11

Critical lines in the $(T,H)$ plane for the spin-glass XY model on a $C=3$ RRG with field directions $\varphi =2\pi \kappa /Q$ with $\kappa$ randomly extracted according to $\mathbb{P}[\kappa =n]=(1-w+w/Q){\delta }_{n,0}+w/Q{\sum }_{i=1}^{Q-1}{\delta }_{n,i}$. For $w=0$ and $w=1$ we recover the GT and the dAT lines, respectively. Data in the right panel seem to suggest a dAT-like critical behavior for any $w>0$ (dashed lines have slopes $1/2$ and $3/2$, respectively).

Figure 12

The GT line computed via the replica approach in the unit-norm frame.

Figure 13

The dAT line computed via the replica approach in the unit-norm frame, obtained when using respectively a Gaussian distribution for the field components (purple curve) and a randomly oriented field with constant intensity (green curve).

Figure 14

Convergence to zero of the critical temperature along the dAT line when increasing the field variance for the Gaussian case (upper panel) or the field strength for the randomly oriented case with fixed $H$ (lower panel). The linear trend for large values of ${\sigma }_H$ and $H$ confirms the analytic results (A42) and (A47), respectively. Axis scales refer to the $m=2$-norm choice for the spins, while error bars are due to the numeric precision used in the computation.