Random-Field Spin Models beyond 1 Loop: A Mechanism for Decreasing the Lower Critical Dimension

The functional renormalization group for the random-field and random-anisotropy $O(N)$ sigma models is studied to 2 loop. The ferromagnetic-disordered (F-D) transition fixed point is found to next order in $d=4+ϵ$ for $N>N_c$ ($N_c=2.8347408$ for random field, $N_c=9.44121$ for random anisotropy). For $N<N_c$ the lower critical dimension $d=d_{\mathrm{lc}}$ plunges below $d_{\mathrm{lc}}=4$: we find two fixed points, one describing the quasiordered phase, the other is novel and describes the F-D transition. $d_{\mathrm{lc}}$ can be obtained in an ($N_c-N$) expansion. The theory is also analyzed at large $N$ and a glassy regime is found.

Figure 1

Phase diagram. $\mathrm{D}=\mathrm{\text{disordered}}$; $\mathrm{F}=\mathrm{\text{ferromagnetic}}$; $\mathrm{QLRO}=\mathrm{\text{quasi-long-range order}}$.

Figure 2

Parametric plot for solutions of (5) for $N<N_c$ (solid circles) for RFs, equivalent to (6) (solid line, parabola) and flow (arrows). $f$ parametrizes disorder, and only $f\ge 0$ is physical. Compare with Fig. 1, right panel.

Figure 3

Flow for $R^{\prime \prime \prime \prime }(0)/|ϵ|$ for $d=4-ϵ>4$ as a function of $N$. The two branches behave as $1/N$ and $4/N$ at large $N$.