Nematic fluctuations balancing the zoo of phases in half-filled quantum Hall systems

Half-filled Landau levels form a zoo of strongly correlated phases. These include non-Fermi-liquids (NFLs), fractional quantum Hall (FQH) states, nematic phases, and FQH nematic phases. This diversity begs the following question: what keeps the balance between the seemingly unrelated phases? The answer is elusive because the Halperin-Lee-Read description that offers a natural departure point is inherently strongly coupled. However, the observed nematic phases suggest that nematic fluctuations play an important role. To study this possibility, we apply a recently formulated controlled double-expansion approach in large-$N$ composite fermion flavors and small $\epsilon$ nonanalytic bosonic action to the case with both gauge and nematic boson fluctuations. In the vicinity of a nematic quantum critical line, we find that depending on the amount of screening of the gauge- and nematic-mediated interactions controlled by $\epsilon$'s, the renormalization-group flow points to all four mentioned correlated phases. When pairing preempts the nematic phase, NFL behavior is possible at temperatures above the pairing transition. We conclude by discussing measurements at low tilt angles, which could reveal the stabilization of the FQH phase by nematic fluctuations.

Figure 1

Phase diagram in the vicinity of a quantum nematic transition occurring at critical isotropic pressure $p_c\left(\delta \right)$ (red line) as a function of a measure $\delta$ of dominance of nematic-mediated over gauge-mediated bare interactions. The hatched region is nematically ordered. Along the dashed red line, the nematic transition is preempted by a paired (composite) fermion state (orange). At ambient pressure ($p_0$), a non-Fermi-liquid (blue) and gapless nematic (gray on right) are also present. A continuous pairing transition occurs on the black line.

Figure 2

Model and RG scheme. (a) Patch pair $j$, with corresponding Fermi momenta $\pm k_j$. (b) One-loop diagrams contributing to the flow of coupling constants, with a fermion propagator (full line), a nematic propagator (dashed), a gauge field propagator (dotted), and four-fermion interaction (dot).

Figure 3

RG flows at the NQCL projected onto planes in space of fermion-gauge ($g$), fermion-nematic ($\eta$), and pairing interaction ($v$) couplings. Fixed points, stable (large dot), or unstable (small dot) in the plane are marked. (a,b) In the orange region, the (composite) fermions pair with vanishing bare attraction, i.e., a gapped FQH is formed. (a) The value $g_{*}/{\eta }_{*}=0.7$ is representative for all ${\epsilon }_g<{\epsilon }_{\eta }$. In the dotted orange area, the NFL energy scale is higher than the pairing gap scale. (b) The choice $g_{*}/{\eta }_{*}=1.44$ is representative for all ${\epsilon }_g>{\epsilon }_{\eta }$. The $(g_{*},0)$ point is isotropic NFL (blue). At the boundary between the regions, the pairing energy scale becomes zero. This phase diagram is robust away from the nematic quantum critical line (see Fig. 8). (c) Pairing instability for initial repulsion ($v>0$) or attraction ($v<0$), with $g_{*}/{\eta }_{*}=1.44$ as in (b) and the choice ${\eta }_0=0.67{\eta }_{*}$. The stable NFL fixed point $(g,\eta ,v)=(g_{*},0,v_{*})$ is given by $v_{*}=\sqrt{g_{*}}=0.8$ [see Eq. (C3)]. Flows starting in the orange region show pairing instability since $v\to -\infty$ at a finite RG scale.

Figure 4

Phases realized for different bare fermion-gauge ($g_0$) and fermion-nematic (${\eta }_0$) couplings when the system is in the vicinity of the nematic quantum transition. (a) In the disordered phase with positive nematic mass $r_0>0$, either a NFL (blue) or a paired state (orange). (b) In the nematically ordered phase with negative nematic mass $r_0<0$, either an anisotropic NFL (shaded blue) or a gapped paired nematic state (shaded orange).

Figure 5

Pairing the scale factor $z_P$ [Eq. (C10)] dependence on the parameters $a/k^2$ and $b/k^2$ [Eq. (C6)]. The curves from bottom to top are for $a/k^2={10}^{-6},{10}^{-5},\cdots ,{10}^{-1}$.

Figure 6

Line of continuous transition to the paired state in the plane of bare couplings $(g_0,{\eta }_0)$ for ${\epsilon }_g>{\epsilon }_{\eta }$. (a) On the NQCL, with $g_{*}=1.3,{\eta }_{*}=0.9$. Blue dots are obtained from numerical solutions to RG flow by having the $v\left(l\right)$ coupling diverge near the largest available scale $l$. The boundary of the blue shading is the approximate expression Eq. (D3), and the upper boundary of the red shading is Eq. (14); see the discussion in Appendix pp4. Inset: zoom-in at small bare couplings. (b) Away from the NQCL, $R_0=0.05$, and $g_{*}=1.3,{\eta }_{*}=0.8$. Red dots are obtained from numerical solutions to RG flow by having the $v\left(l\right)$ coupling diverge near the largest available scale $l$. The red line is obtained by equating $l_T^{\prime }$ [the scale at which the pairing function $f\left(l\right)$ starts repressing the pairing tendency], obtained numerically from Eq. (F3), to the approximation $l_P=\pi /\left(2\sqrt{{\eta }_0-g_0}\right)$ (see the discussion in Appendix pp4). The boundary of the blue and yellow shadings uses the two approximate expressions for $l_T^{\prime }$ in Eq. (F4), respectively. Note that blue corresponds to $l_T$ with $R_0=0$, while yellow matches the slope near the origin better.

Figure 7

The NFL energy scale $exp(-l_F)$ calculated using Eq. (E3) along two lines in the $(g_0,{\eta }_0)$ plane (inset). The bottom line in the inset is the pairing transition line, along which the NFL scale is the top (blue) curve. The top line in the inset gives the bottom (red) line for the NFL scale. The ratio $g_{*}/{\eta }_{*}=1.22$.

Figure 8

Numerical RG flows projected onto the fermion-boson coupling plane $(g,\eta )$ in the vicinity of the NQCL. Colored regions show $(g_0,{\eta }_0)$ values, which give states corresponding to the top (bottom) label when $r_0>0$ ($r_0<0$). In the orange area there is infinitesimal pairing instability. (a) Case ${\epsilon }_g<{\epsilon }_{\eta }$ corresponds to Fig. 3, with $R_0/g_{*}=0.06$, $g_{*}/{\eta }_{*}=8/13$. (b) Case $g_{*}>{\eta }_{*}$ corresponds to Fig. 3, with $R_0/g_{*}=0.008$, $g_{*}/{\eta }_{*}=13/8$. Here the topology of RG flows is the same as at the NQCL, except that the unstable fixed point $(0,{\eta }_{*})$ is replaced by $(0,\infty )$ affecting flow at $g=0$. (c,d,e,f) Flows of $g$, $\eta$, $R$, and pairing $v$ in (c,e) correspond to initial values marked in panel (a); same for panels (d,f) marked in (b). The horizontal axis is $lnl$.