Anisotropic long-range spin systems

We consider anisotropic long-range interacting spin systems in $d$ dimensions. The interaction between the spins decays with the distance as a power law with different exponents in different directions: We consider an exponent $d_1+{\sigma }_1$ in $d_1$ directions and another exponent $d_2+{\sigma }_2$ in the remaining $d_2\equiv d-d_1$ ones. We introduce a low energy effective action with nonanalytic power of the momenta. As a function of the two exponents ${\sigma }_1$ and ${\sigma }_2$ we show the system to have three different regimes at criticality, two where it is actually anisotropic and one where the isotropy is finally restored. We determine the phase diagram and provide estimates of the critical exponents as a function of the parameters of the system, in particular considering the case where one of the two $\sigma$'s is fixed and the other varying. A discussion of the physical relevance of our results is also presented.

Figure 1

In panel (a) we plot the parameter space of a anisotropic LR spin system for $d_1=d_2=1$. In the cyan shaded area fluctuations are unimportant and the universal quantities are correctly reproduced by the mean-field approximation. The solid curves are the boundary regions ${\sigma }_1^{*}$ and ${\sigma }_2^{*}$ where the nonanalytic kinetic term becomes irrelevant. We show results for $N=1,2,3$, respectively, in red, blue, green. The dashed lines are the mean-field results for the boundary curves. In panel (b) we show the parameter space of the model in $d_1=2$ and $d_2=1$. In the light cyan shaded area fluctuations are unimportant and the universal quantities are again correctly reproduced by mean-field approximation. The solid curves are the boundary regions ${\sigma }_1^{*}$ and ${\sigma }_2^{*}$. In the inset we show the boundaries in an enlarged scale.

Figure 2

In the left panel (a) we plot for ${\sigma }_1>{\sigma }_1^{*}$ the anomalous dimension ${\eta }_1$ as a function of ${\sigma }_2$ in $d_1=d_2=1$ for field component numbers $N=1,2,3$, respectively, in red, blue, and green from the top. In the right panel (b) the anomalous dimension in $d_1=1\vee 2,d_2=2\vee 1$, again in the case ${\sigma }_1>{\sigma }_1^{*}$, for the field component numbers $N=1,2,3$, respectively, in red, blue, and green is shown (the solid $\vee$ dashed lines are for $d_1=2\vee 1$ and $d_2=1\vee 2)$. In this case the lack of analytic term in the ${\mathbb{R}}^{d_2}$ subspace produces two different results for the isotropic limit ${\sigma }_2\to {\sigma }_2^{*}$ between the two cases $d_2=2,1$ solid and dashed lines, respectively. The gray dashed lines represent the correct values of the anomalous dimension in the isotropic SR Ising model in the considered approximation for $d=2$, panel (a), and $d=3$, panel (b).

Figure 3

Anomalous dimension ${\eta }_1$ as a function of ${\sigma }_1$.

Figure 4

Anomalous dimension ${\eta }_2$ for $d_1=1$ and general $d_2$ when ${\sigma }_1=1$ and ${\sigma }_2>{\sigma }_2^{*}$ for field component numbers $N=2,3,4$, respectively, in blue, green, and red from the top. These results may be used for studying phase transitions in quantum LR spin systems.

Figure 5

The phase space of a LR anisotropic spin system with dimension $d_1=1$ as a function of $d_2$ with ${\sigma }_2>{\sigma }_2^{*}$ for general ${\sigma }_1$. The cyan shaded region represents the mean-field validity region while in the white region $\mathrm{WF}$ type universality is found. The gray dashed line is the mean-field threshold above which SR behavior is recovered. The solid colored lines represent the dressed threshold values for the $N=1,2,3$ cases in red, blue, and green, respectively. The gray area is the region where we expect the critical behavior to disappear and only a single phase is found.

Figure 6

In panel (a) the inverse of the correlation length exponents for the critical point of an anisotropic spin system for dimensions $d_1=d_2=1$ are reported. The two exponents are shown for three values of the number of components $N=1,2,3$ in panels (a), (b), and (c), respectively. For different values of ${\sigma }_2$ we report the behavior of the inverse exponents as a function of ${\sigma }_1$.

Figure 7

Plot of the inverse of the correlation length exponents for the critical point of an anisotropic spin system with dimensions $d_1=2$ and $d_2=1$. The two exponents are shown for three values of the components number $N=1,2,3$ in panels (a), (b), and (c), respectively. For different values of ${\sigma }_2$ we report the behavior of the inverse exponents as a function of ${\sigma }_1$.

Figure 8

The mean-field parameter space of a LR anisotropic spin system with dimensions $d_1=d_2=1$, panel (a), and $d_1=2,d_2=1$, panel (b).