Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime, we find the strong-coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at noninteger dimensions. Apart from the well-known weak-coupling and the now well-established isotropic strong-coupling behavior, we find an anisotropic strong-coupling fixed point for nonlinear couplings of opposite signs at noninteger dimensions.

Figure 1

Typical RG trajectories in the $({\widehat{g}}_{\kappa }^{\prime },{\gamma }_{\kappa })$ plane for $d<2$ (here $d_{\parallel }=d_{\perp }=0.8$). All weak-coupling fixed points EW (with ${\gamma }_{*}=1$), EWU (with ${\gamma }_{*}=0$), and EWA [with ${\gamma }_{*}=(4-d_{\parallel }d)/(d{d}_{\perp }-4)$] are unstable in the $\widehat{g}$ direction and we find two locally fully stable strong-coupling fixed points SC (with ${\gamma }_{*}=1$) and A (with ${\gamma }_{*}<0$). Moreover, there is another U fixed point with ${\gamma }_{*}=0$ which is attractive along the $\gamma =0$ axis and repulsive in the $\gamma$ direction.

Figure 2

RG trajectories in the $({\widehat{g}}_{\kappa }^{\prime },{\gamma }_{\kappa })$ plane at $d=2$ with an increasing difference between the sector dimensions (from up to down). (a) represents a typical flow for $d_{\parallel }<-1+\sqrt{5}$ (here $d_{\parallel }=d_{\perp }=1$, which corresponds to Wolf's result [8]). Both the anisotropic fixed point EWA and the isotropic SC fixed point are fully attractive. The uniaxial fixed point EWU is attractive in the $\widehat{g}$ direction but unstable in the anisotropy direction. (b) Typical flow for $-1+\sqrt{5}<d_{\parallel }<1+1/\sqrt{5}$ (here $d_{\parallel }=1.3$ and $d_{\perp }=0.7$). EWU is now repulsive in the $\widehat{g}$ direction. (c) Typical flow for $d_{\parallel }>1+1/\sqrt{5}$ (here $d_{\parallel }=1.5$ and $d_{\perp }=0.5$). EWA becomes repulsive in the $\widehat{g}$ direction because the anisotropic fixed point A crosses it. Note that in all cases the isotropic fixed point SC is not affected by the weak-coupling stability changes and is always fully attractive.

Figure 3

RG trajectories in the $({\widehat{g}}_{\kappa }^{\prime },{\gamma }_{\kappa })$ plane for $d>2$ but with small $d-2$. The difference between the sector dimensions increases along the panels. The three weak-coupling fixed points EW, EWU, and EWA are attractive in the $\widehat{g}$ direction in all panels. (a) Typical flow for $d_{\parallel }\lesssim 1.2$ (here $d_{\parallel }=d_{\perp }=1.1$). One unstable transition fixed point T (${\gamma }_{*}=1$) is present. (b) Typical flow for $1.2\lesssim d_{\parallel }\lesssim 1.4$ (here $d_{\parallel }=1.31$ and $d_{\perp }=0.702$). Two unstable transition fixed points T and TU (${\gamma }_{*}=0$) are present. (c) Typical flow for $1.4\lesssim d_{\parallel }$ (here $d_{\parallel }=1.7$ and $d_{\perp }=0.5$). Three unstable transition fixed points T, TU, and TA (${\gamma }_{*}<0$) are present.

Figure 4

RG trajectories in the $({\widehat{g}}_{\kappa }^{\prime },{\gamma }_{\kappa })$ plane. Typical flow for $d$ approaching 3 with $d_{\perp }\sim 0.5$ (here $d_{\parallel }=2.5$ and $d_{\perp }=0.5$). The unstable transition fixed points T (${\gamma }_{*}=1$), TA ($0<{\gamma }_{*}<1$), and TU (${\gamma }_{*}=0$) are present. The ordering of the three transition fixed points along the $\gamma$ axis is no longer the same as the one of the weak-coupling solutions (EWA, EW, and EWU). It is useful to compare with Fig. 3. Note that we do not find a splitting in $d=3$ which corresponds to scenario C in TF [29] where the ordering becomes EWA, EW, and EWU and TA, T, and TU.

Figure 5

RG trajectories in the $({\widehat{g}}_{\kappa },{\gamma }_{\kappa })$ plane for $d=3$ and a decreasing difference between both sector dimensions (from up to down). (a) The fixed point A is enclosed by the $\gamma =0$ axis and the separatrix (highlighted in blue) and shows a spiral flow (here $d_{\parallel }=2.04855$ and $d_{\perp }=0.95145$). (b) The separatrix has formed a closed loop around A. Inside this loop, the flow is directed towards A, outside and for $\gamma <0$ the couplings always flow to the weak-coupling regime (here $d_{\parallel }=2.047$ and $d_{\perp }=0.953$). (c) The enclosed area around A decreases in size (here $d_{\parallel }=2.035$ and $d_{\perp }=0.965$). (d) The size of the attractive zone around A shrank down to zero. The TA fixed point has approached TU at $\gamma =0$ and merges with it. (here $d_{\parallel }=2$ and $d_{\perp }=1$). (e) The end point of the separatrix where A was located approaches the TU fixed point and merges with it. TU and U approach each other (here $d_{\parallel }=1.73$ and $d_{\perp }=1.27$). (f) TU and U have annihilated, so the flow in the uniaxial case is always driven to the weak-coupling EWU. (here $d_{\parallel }=d_{\perp }=1.5$).

Figure 6

RG trajectories in the $({\widehat{g}}_{\kappa }^{\prime },{\gamma }_{\kappa })$ plane for $d_{\parallel }=2$ and $d_{\perp }=1$ in the NLO approximation, to be compared with Fig. 5. The strong-coupling SC fixed point is locally fully attractive, but neither the A nor the U fixed points are present. Note that the position of the separatrix (highlighted in blue) is only roughly estimated.

Figure 7

Coupling constant ${\widehat{g}}_{*}^{\prime }$ at the anisotropic A fixed point close to $d=2$ for different values of the sector dimension $d_{\parallel }$. The ${\mathrm{LPA}}^{\prime }$ results are obtained from a numerical solution of the NPRG flow equations and perturbative results (labeled as pert.) from the one-loop expression Eq. (D3). For $d<2$ and small $d_{\parallel }$ the A fixed point is weak coupling and is well described by a perturbative treatment. For larger values of $d_{\parallel }$ the A fixed point becomes strong coupling and the perturbative treatment is no longer justified. The sector dimension $d_{\parallel }=1+1/\sqrt{5}\approx 1.45$ stands as the boundary beyond which the one-loop treatment for A breaks down, separating the weak-coupling regime (white) from the strong-coupling regime (shaded in gray).