Fermi liquid instabilities in the spin channel

We study the Fermi-surface instabilities of the Pomeranchuk type [Sov. Phys. JETP 8, 361 (1959)] in the spin-triplet channel with high orbital partial waves $\left[F_l^a(l>0)\right]$. The ordered phases are classified into two classes, dubbed the $\alpha$ and $\beta$ phases by analogy to the superfluid ${}^3\mathrm{He}$ A and B phases. The Fermi surfaces in the $\alpha$ phases exhibit spontaneous anisotropic distortions, while those in the $\beta$ phases remain circular or spherical with topologically nontrivial spin configurations in momentum space. In the $\alpha$ phase, the Goldstone modes in the density channel exhibit anisotropic overdamping. The Goldstone modes in the spin channel have a nearly isotropic underdamped dispersion relation at small propagating wave vectors. Due to the coupling to the Goldstone modes, the spin-wave spectrum develops resonance peaks in both the $\alpha$ and $\beta$ phases, which can be detected in inelastic neutron-scattering experiments. In the $p$-wave channel $\beta$ phase, a chiral ground-state inhomogeneity is spontaneously generated due to a Lifshitz-like instability in the originally nonchiral systems. Possible experiments to detect these phases are discussed.

Figure 1

(Color online) The $\alpha$ phases in the $F_1^a$ and $F_2^a$ channels. The Fermi surfaces exhibit the $p$- and $d$-wave distortions, respectively.

Figure 2

(Color online) The $\beta$ phases in the $F_1^a$ channel. Spin configurations exhibit the vortex structures in the momentum space with winding number $w=\pm 1$, which correspond to Rashba and Dresselhaus SO couplings, respectively.

Figure 3

(Color online) The $\beta$ phases in the $F_2^a$ channel. Spin configurations exhibit the vortex structure with winding number $w=\pm 2$. These two configurations can be transformed to each other by performing a rotation around the $x$ axis with the angle of $\pi$.

Figure 4

(Color online) The spin configurations on both Fermi surfaces in the $\beta$ phase ($F_l^a$ channel) map to a large circle on an $S^2$ sphere with the winding number $l$.

Figure 5

The half-quantum vortex with the combined spin-orbit distortion for the $\alpha$ phase at $l=1$. The triad denotes the direction in the spin space.

Figure 6

The $p$-wave counterpart of paramagnon modes in 2D. Taking into account the spin degrees of freedom, there are three (six) transverse triplet modes in 2D (3D) and three longitudinal modes, respectively.

Figure 7

The azimuthal angle $\varphi$ for the propagation wave vector $\vec{q}$ for the Goldstone modes in the 2D alpha phase. The spin-down and -up Fermi surfaces can overlap each other by a shift at the wave vector $\vec{K}=2\overline{n}∕v_F$.

Figure 8

(Color online) The three Goldstone modes of $O_z$, $O_{x^{\prime }}$, $O_{y^{\prime }}$ of the 2D $\beta$ phase can be viewed as a relative spin-orbit rotation for this phase with angular momentum $l=1$ and winding number $w=1$. Here, $\varphi$ is the azimuthal angle of the propagation direction $\vec{q}$.

Figure 9

The linear dispersion relation of Goldstone modes of the 2D $\beta$ phases $(l⩾2)$ at small momentum $\vec{q}$. For $q>q^{\prime }$, the Goldstone modes enter the particle-hole continuum, and are damped.

Figure 10

(Color online) The dispersion relation of Goldstone modes for the $\beta$ phase with $l=1$. The longitudinal modes $O_{y^{\prime }}$ have linear dispersion relation, while the two transverse modes $O_{z\pm i{y}^{\prime }}$ are unstable toward the Lifshitz-like instability at small momentum $\vec{q}$.

Figure 11

Longitudinal and transverse twists for the Lifshitz-like instability in the $\beta$ phase with $l=1$. (a) The longitudinal twist with ${\widehat{n}}_2$ precessing around the ${\widehat{e}}_1$ axis, as described in Eq. (8.20). (b) The transverse twist with the triad formed by ${\widehat{n}}_1$, ${\widehat{n}}_2$, and ${\widehat{n}}_3={\widehat{n}}_1\times {\widehat{n}}_2$ precessing around the ${\widehat{e}}_x$ axis, as described in Eq. (8.24).

Figure 12

The $\alpha$ phases in a $\vec{B}$ field $(l=1,2)$. The larger Fermi surface has spin parallel to the $\vec{B}$ field. The Fermi-surface distortion is not pinned by the $\vec{B}$ field.

Figure 13

(Color online) The spin configuration on the Fermi surfaces in the $\beta$ phase ($F_l^a$ channel). With the $\vec{B}$ field, the large circle splits into two small circles perpendicular to the $\vec{B}$ field with the development of a finite magnetization. The large (small) Fermi surface polarizes parallel (antiparallel) to the $\vec{B}$ field.

Figure 14

(Color online) The order parameters in the 3D $\beta$ phase at $l=1$. (a) Configuration for $B^{\prime }>B$ as defined in Eq. (9.5). (b) Configuration for $B_c>B>B^{\prime }$.