Dimensional reduction and its breakdown in the driven random-field $\mathbf{O}\left(N\right)$ model

The critical behavior of a random-field $\mathbf{O}\left(N\right)$ model driven at a uniform velocity is investigated near three dimensions at zero temperature. From intuitive arguments, we predict that the large-scale behavior of the $D$-dimensional driven random-field $\mathbf{O}\left(N\right)$ model is identical to that of the $(D-1)$-dimensional pure $\mathbf{O}\left(N\right)$ model. This is an analog of the dimensional reduction property of equilibrium cases, which states that the critical exponents of $D$-dimensional random-field models are identical to those of $(D-2)$-dimensional pure models. However, the dimensional reduction property breaks down in low enough dimensions due to the presence of multiple metastable states. By employing the nonperturbative renormalization group approach, we calculate the critical exponents of the driven random-field $\mathbf{O}\left(N\right)$ model in the first order of $\epsilon =D-3$ and determine the range of $N$ in which the dimensional reduction breaks down.

Figure 1

Schematic picture of the RG flow of $R_k^{\prime }\left(1\right)={\eta }_k$ for the $\mathrm{RFO}\left(N\right)\mathrm{M}$. The horizontal axis represents the spatial dimension $D$. “D”, “LRO”, and “QLRO” denote the disordered, long-range order, and quasi-long-range order phases, respectively.

Figure 2

Anomalous dimensions $\eta$ and $\overline{\eta }$ for the $\mathrm{DRFO}\left(N\right)\mathrm{M}$ and $\mathrm{RFO}\left(N\right)\mathrm{M}$ as functions of $N$. The red solid lines represent ${\eta }_{\perp }$ and ${\overline{\eta }}_{\perp }$ for the $\mathrm{DRFO}\left(N\right)\mathrm{M}$ at $D=3+\epsilon$. The green dashed lines represent $\eta$ and $\overline{\eta }$ for the $\mathrm{RFO}\left(N\right)\mathrm{M}$ at $D=4+\epsilon$. $\epsilon$ is set to unity. The blue dotted line represents the dimensional reduction value ${\eta }_{\mathrm{DR}}={(N-2)}^{-1}$. The insets show $\eta /{\eta }_{\mathrm{DR}}$ and $\overline{\eta }/{\eta }_{\mathrm{DR}}$.

Figure 3

Fixed functions ${\delta }_{00}^{*}\left(z\right)$ and ${\delta }_{21}^{*}\left(z\right)$ for $N=3$, 4, and 5.

Figure 4

Graphical representations for the flow equations of ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_1^{\left(1\right)}$.

Figure 5

Graphical representation for the flow equation of ${\mathrm{\Gamma }}_1^{\left(2\right)}$.

Figure 6

Graphical representation for the flow equation of ${\mathrm{\Gamma }}_2$.

Figure 7

Graphical representation for the flow equation of ${\mathrm{\Gamma }}_2^{\left(11\right)}$.