Bose-Luttinger liquids

We study systems of bosons whose low-energy excitations are located along a spherical submanifold of momentum space. We argue for the existence of gapless phases which we dub “Bose-Luttinger liquids,” which in some respects can be regarded as bosonic versions of Fermi liquids, while in other respects they exhibit striking differences. These phases have bosonic analogues of Fermi surfaces, and like Fermi liquids they possess a large number of emergent conservation laws. Unlike Fermi liquids, however, these phases lack quasiparticles, possess different RG flows, and have correlation functions controlled by a continuously varying exponent $\eta$, which characterizes the anomalous dimension of the bosonic field. We show that when $\eta >1$, these phases are stable with respect to all symmetric perturbations. These theories may be of relevance to several physical situations, including frustrated quantum magnets, rotons in superfluid He, and superconductors with finite-momentum pairing. As a concrete application, we show that coupling a Bose-Luttinger liquid to a conventional Fermi liquid produces a resistivity scaling with temperature as $T^{\eta }$. We argue that this may provide an explanation for the non-Fermi liquid resistivity observed in the paramagnetic phase of MnSi.

Figure 1

An illustration of a dispersion possessing minima at the two Bose points $\pm k_B$. The IR theory contains only modes within $\pm \mathrm{\Lambda }$ of each Bose point.

Figure 2

How the low-energy annulus in momentum space is broken up into patches. Each patch is labeled by an angle $\gamma$, with corresponding unit vector $\mathit{\gamma }$.

Figure 3

The geometry of the scattering processes contributing to ${\mathrm{\Sigma }}^{\prime \prime }$. A portion of the Fermi surface is drawn in gray, with the tip of the vector $\mathbf{K}$ lying just outside the Fermi surface. The purple sphere has radius $k_B$ and is centered on the tip of $\mathbf{K}$. The two vertical purple circles are separated by a distance of $k=\left|\mathbf{K}\right|-k_F$.

Figure 4

The 1-loop diagrams contributing to ${\mathrm{\Sigma }}_{\alpha {\alpha }^{\prime }}(\mathbf{k},0)$ for $\mathbf{k}>0$.

Figure 5

The diagrams which contribute to the part of ${\mathrm{\Sigma }}_{\chi \overline{\chi }}(k,0)$ odd in $k$. Solid lines represent ${\mathcal{G}}_{\chi \chi }$, dashed lines represent ${\mathcal{G}}_{\overline{\chi }\overline{\chi }}$, and mixed dashed-solid lines represent ${\mathcal{G}}_{\overline{\chi }\chi },{\mathcal{G}}_{\chi \overline{\chi }}$.