Fixed-point structure and effective fractional dimensionality for O($N$) models with long-range interactions

We study, by renormalization group methods, $\mathrm{O}\left(N\right)$ models with interactions decaying as power law with exponent $d+\sigma$. When only the long-range momentum term $p^{\sigma }$ is considered in the propagator, the critical exponents can be computed from those of the corresponding short-range $\mathrm{O}\left(N\right)$ models at an effective fractional dimension $D_{\mathrm{eff}}$. Neglecting wave function renormalization effects the result for the effective dimension is $D_{\mathrm{eff}}=\frac{2d}{\sigma }$, which turns to be exact in the spherical model limit $(N\to \infty )$. Introducing a running wave function renormalization term the effective dimension becomes instead $D_{\mathrm{eff}}=\frac{(2-{\eta }_{\mathrm{SR}})d}{\sigma }$. The latter result coincides with the one found using standard scaling arguments. Explicit results in two and three dimensions are given for the exponent $\nu$. We propose an improved method to describe the full theory space of the models where both short- and long-range propagator terms are present and no a priori choice among the two in the renormalization group flow is done. The eigenvalue spectrum of the full theory for all possible fixed points is drawn and a full description of the fixed-point structure is given, including multicritical long-range universality classes. The effective dimension is shown to be only approximate, and the resulting error is estimated.

Figure 1

Comparison between the LPA fixed points landscapes of a LR model in dimension $d$ and a SR model in dimension $D_{\mathrm{eff}}$. Each value of $\mathrm{\Sigma }\equiv {\overline{U}}_{*}^{\prime }\left(0\right)$ for which we have a spike in the above figure is the derivative at the origin of a well-defined fixed-point effective potential: thus every spike is the signature of a different universality class. Solid lines represent spike plots of LR models in dimension $d$ with power-law exponent $\sigma$, while dashed lines represent spike plots of SR models in dimension $D=D_{\mathrm{eff}}=2d/\sigma$. The plot is for $d=2$, in the cases (a) $N=1$, (b) $N=2$, and (c) $N=5$, with $\sigma =1.25,1.75,1.9$, from bottom to top.

Figure 2

$y_t=1/{\nu }_{\mathrm{LR}}$ exponent as a function of $\sigma$ in $d=2$ for some values of $N$ (from top: $N=1,2,3,4,5,10,100$). The dashed line is the analytical result obtained for the spherical model $N=\infty$. Inset: $y_t=1/{\nu }_{\mathrm{LR}}$ vs $\sigma$ for the $d=2$ LR Ising model compared with MC data of [14] (red circles) and of [18] (blue squares). The three continuous lines represent the estimates made using (8) with the numerical values of ${\nu }_{\mathrm{SR}}\left(D_{\mathrm{eff}}^{\prime }\right)$ and ${\eta }_{\mathrm{SR}}\left(D_{\mathrm{eff}}^{\prime }\right)$ taken from recent high-precision estimates in fractal dimensions [31] (top magenta line with square markers), from [27, 28] where the $\mathrm{O}\left(N\right)$ model definition for ${\eta }_{\mathrm{SR}}$ is used (purple bottom line with round markers). This definition is obtained taking the $N\to 1$ limit of the anomalous dimension of the Goldstone modes propagator. We also considered the results from [22], where the Ising definition of ${\eta }_{\mathrm{SR}}$ is used instead.

Figure 3

$y_t=1/{\nu }_{\mathrm{LR}}$ exponent as a function of $\sigma$ in $d=3$ for some values of $N$ (from top: $N=1,2,3,4,5,10,100$). As in Fig. 2, the dashed line is the analytical result obtained for the spherical model.

Figure 4

The figure shows the eigenvalues $\theta$ of the RG stability matrix in $d=2$ and $N=1$ as a function of $\sigma$ for the SR (red solid lines) and LR (blue dashed lines) fixed points. For $\sigma >{\sigma }_{*}$, only the SR solution exists and two of the three exponents are independent of $\sigma$, where the third exponent describes the flow in the direction of the ${\overline{J}}_{\sigma }$ coupling and is positive, showing that the SR fixed point is attractive in this direction. For $\sigma <{\sigma }_{*}$, the situation changes and the SR fixed point has two repulsive directions (two negative exponents), while a LR fixed point emerges. This LR fixed point is repulsive in one single direction and has nontrivial $\sigma$-dependent exponents (blue dashed lines). Finally, for $\sigma <d/2$, the SR and LR interacting fixed points disappear, leaving only a LR Gaussian fixed-point solution, whose exponents can be obtained using a mean-field approximation on the original LR Ising model (gray dash-dotted lines).

Figure 5

Anomalous dimension vs $\sigma$ in the SR (red solid line) and LR (blue dashed line) cases. In the LR dominant region $\sigma <{\sigma }_{*}$, we have $\eta =2-\sigma$, while for $\sigma >{\sigma }_{*}$, the anomalous dimension keeps its SR value. Inset: The difference between the $\nu$ exponent found using ansatz (9) and the same exponent calculated using the effective dimension relation (5) at the same truncation level. This result shows that effective dimension relations are exact only when no SR term is present in the propagator, but the error committed is very small in all of the $\sigma$ range.

Figure 6

Different results for the effective dimension $D_{\mathrm{eff}}^{\prime }$ of a LR $\mathrm{O}\left(N\right)$ model in $d=2$: from left to right $N=1,2,3,5,100$ using the data from [27, 28]. The $N=100$ case already practically overlaps with $D_{\mathrm{eff}}=2d/\sigma$ valid in the large-$N$ limit. The gray dashed line is the proposal made in [16].

Figure 7

(a) Anomalous dimension ${\eta }_2$ and (b)–(d) fixed-point values $J_{\sigma ,*},{\kappa }_{*},{\lambda }_{*}$, respectively, in the truncation considered in the text for the Ising model in $d=2$. For $\sigma >{\sigma }_{*}\equiv 2-{\eta }_{\mathrm{SR}}$, only the fixed point (purple straight line) is present, characterized by ${\eta }_2={\eta }_{\mathrm{SR}}$ and $J_{\sigma ,*}=0$. At $\sigma ={\sigma }_{*}$, the LR fixed point (blue curves) branches from the SR fixed point and then controls the critical behavior for every $\sigma <{\sigma }_{*}$. Thus, even in the case of both SR and LR terms in the propagator, the anomalous dimension as a function of $\sigma$ is LR (${\eta }_2=2-\sigma$) for $\sigma <{\sigma }_{*}$ and SR for $\sigma >{\sigma }_{*}$.

Figure 8

(a) Anomalous dimension ${\eta }_2$ and (b)–(d) fixed-point values ${\overline{J}}_{\sigma ,*},{\kappa }_{*},{\lambda }_{*}$, respectively, in the truncation (B1) with LR dimensions for the Ising model in $d=2$. With this dimensional choice, we are able to describe only the $\sigma <{\sigma }_{*}$ region, since in the other region the coupling ${\overline{J}}_{2,k}$ is divergent, as can be understood from the ${\overline{J}}_{2,k}$ plot. Also in this case, the anomalous dimension as a function of $\sigma$ is LR (${\eta }_2=2-\sigma$) for $\sigma <{\sigma }_{*}$.

Figure 9

Eigenvalues ($\theta$) of the RG stability matrix as a function of $\sigma$ for the LR (blue dashed lines) fixed point in the case of LR dimensions for the Ising model in $d=2$. In this case, we are able to describe only the LR fixed point present when $\sigma <{\sigma }_{*}$ and, since RG eigenvalues are universal quantities, they agree with the SR dimension ones. We also report the results for the LR Gaussian fixed point (gray dash-dotted lines).