One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O($N$) spin model near lower critical dimension

We consider the random-field $\mathrm{O}(N$) spin model with long-range exchange interactions which decay with distance $r$ between spins as $r^{-d-\sigma }$ and/or random fields which correlate with distance $r$ as $r^{-d+\rho }$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\to d-\sigma$ dimensional reduction at the leading order of the $d-2\sigma$ expansion holds only for $N>2(4+3\sqrt{3})\simeq 18.3923\dots$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\to d-\sigma -\rho$ dimensional reduction does not hold within the present framework, as far as $N$ is finite.

Figure 1

Exponent ${\alpha }^{*}=\alpha {(N,\widehat{\epsilon })}^{*}$ characterizing the nonanalyticity ${(1-z)}^{{\alpha }^{*}}$ of the LRF TT FP (20) for three values of $\widehat{\epsilon }$. The ordinate is ${\alpha }^{*}$, and the abscissa is $N$. (i) For $\widehat{\epsilon }=16/15$, the value $N$ of the lower boundary above which the LRF TT FP are singly unstable with respect to the cuspless perturbation is $N=18$. $\alpha {(N\searrow 18,\widehat{\epsilon })}^{*}=3/2$. (ii) For $\widehat{\epsilon }=2(3+\sqrt{23})/13$, $N=16$. $\alpha {(N\searrow 16,\widehat{\epsilon })}^{*}=1$. (iii) For $\widehat{\epsilon }=2(3+\sqrt{15})/5$, $N=8$. $\alpha {(N\searrow 8,\widehat{\epsilon })}^{*}=1$.

Figure 2

The regions where various FPs are singly unstable. The ordinate is $\widehat{\epsilon }[=\epsilon /(\epsilon -\rho )]$, and the abscissa is $N$. The broken line (black) denotes the lower boundary above which the SR TT and the LRF TT FPs are singly unstable against the cuspless perturbation. The border line (black line) between the SR TT and LRF TT FPs is given by $\widehat{\epsilon }=(N-2)/(N-3)$. The solid line (red line) denotes the lower boundary above which the SR TT and LRF TT FPs are singly unstable against the cuspy perturbation.

Figure 3

Exponent ${\alpha }^{*}=\alpha {\left(N\right)}^{*}$ characterizing the subcuspy singularity ${(1-z)}^{{\alpha }_{*}}$ of the LRE TT FP (70). The ordinate is ${\alpha }^{*}$, and the abscissa is $N$. $\alpha {(N\searrow 18)}^{*}=3/2$.

Figure 4

Exponent ${\alpha }^{*}=\alpha {(N,\tilde{\epsilon })}^{*}$ characterizing the subcuspy singularity ${(1-z)}^{{\alpha }^{*}}$ of the LREF TT FP (72) for three values of $\tilde{\epsilon }$. The ordinate is ${\alpha }^{*}$, and the abscissa is $N$. (i) For $\tilde{\epsilon }=18/17$, the value $N$ of the lower boundary above which the LREF TT FP are singly unstable with respect to the cuspless perturbation is $N=18$. $\alpha {(N\searrow 18,\tilde{\epsilon })}^{*}=3/2$. (ii) For $\tilde{\epsilon }=2(4+\sqrt{23})/15$, $N=16$. $\alpha {(N\searrow 16,\tilde{\epsilon })}^{*}=1$. (iii) For $\tilde{\epsilon }=2(4+\sqrt{15})/7$, $N=8$. $\alpha {(N\searrow 8,\tilde{\epsilon })}^{*}=1$.

Figure 5

The regions where various FPs are singly unstable. The ordinate is $\tilde{\epsilon }[=(\epsilon +\rho )/\epsilon ]$, and the abscissa is $N$. The broken line (black) denotes the lower boundary above which the LRE TT and the LREF TT FPs are singly unstable against the cuspless perturbation. The border line (black line) between the LRE TT and LREF TT FPs is given by $\widehat{\epsilon }=N/(N-1)$. The solid line (red line) denotes the lower boundary above which the LRE TT and LREF TT FPs are singly unstable against the cuspy perturbation.

Figure 6

The graphs of Eq. (108) in $\sigma \in [1.80,2.00]$. Here we have put $\rho =2-\sigma$ in Eq. (108). Then the reduced variable $\tilde{\epsilon }$ takes $\tilde{\epsilon }=(\epsilon +2-\sigma )/\epsilon$. The ordinate is ${\left(\epsilon {\nu }_{\mathrm{LREF}}\right)}^{-1}$, and the abscissa is $\sigma$. Here we have set $\epsilon =0.01$. Then $\tilde{\epsilon }=(2.01-\sigma )/0.01$.