Surface Functional Renormalization Group for Layered Quantum Materials

Abstract

We present an extension to the two-dimensional functional renormalization group to efficiently treat interactions on the surface or at interfaces of three-dimensional systems. As an application, we consider a semi-infinite stack of two-dimensional square lattices, including a Hubbard interaction on the surface layer and an alternating interlayer coupling. We investigate how strongly correlated states of the decoupled two-dimensional Hubbard model on the surface evolve under inclusion of such an SSH-like interlayer coupling. For large parts of the phase diagram as a function of the interlayer hopping parameters, the physics of the two-dimensional system prevails, with antiferromagnetic, superconducting d-wave, and ferromagnetic correlations taking center stage. However, for intermediate interlayer couplings the superconducting state at intermediate interaction strengths separates into two regimes by a small region of incommensurate spin-density-wave and spin-bond order, enabling the potential realization of chiral spin-bond order.

Figures (5)

• Figure 1: Three-dimensional Hubbard-SSH model. Each of the layers has nearest- and next-nearest-neighbor hopping terms $t$ and $t'$, respectively (black and gray). The coupling of subsequent layers is given by two alternating hopping amplitudes $v$ (orange) and $w$ (light blue), as in an SSH chain. We treat the renormalization of the two-particle interaction only in the outermost layer, where we include an on-site Hubbard-$U$ term (blue).
• Figure 2: Diagrammatic representation of the flow equation of the four-point vertex. The diagrams are grouped into channels corresponding to distinct transfer momenta, with the particle-particle channel $\color{p-channel}P^\Lambda$ in light blue, the direct particle-hole channel ${\color{d-channel}D^\Lambda}$ in purple, and the crossed particle-hole channel ${\color{c-channel}C^\Lambda}$ in orange. Slashed propagator pairs denote a scale derivative ($\mathrm{d}/\mathrm{d}\Lambda$) of the corresponding propagator pair. The vertices conserve electron spin along the gray lines connecting two in-/out-going electron lines.
• Figure 3: Non-interacting spectral density for three cases of $t'\in\{0,-0.25,-0.5\}$. The chemical potential is fixed at $\mu=-4t'$ such that the system is at Van Hove filling in the layer-decoupled limit ($\omega$ is measured with respect to $\mu$). For each panel, the four lines represent four different choices of the out-of-plane hopping parameters $v$ and $w$: The gray line corresponds to the layer-decoupled system ($v=w=0$), the orange line to $v=0.9$ and $w=0.4$, the dashed cyan line to $v=0.4$ and $w=0.9$, and the purple line to an almost isotropic three-dimensional system with $v=w=0.9$.
• Figure 4: Critical scale $\Lambda_\mathrm{C}$ (approximately onset temperature of the corresponding order) and type of leading instability for $t'\in\{0,-0.25,-0.5\}$ and $U\in\{3,5\}$. The out-of-plane hopping amplitudes $v$ and $w$ are varied from zero to one on a regular $11\times11$ grid. The red color-map corresponds to spin-density-wave order (SDW) and the blue color-map to superconducting order (SC). Literature results for the two-dimensional Hubbard model are reproduced for $v=0$, where the surface is decoupled from the bulk.
• Figure 5: Magnetic susceptibilities $\chi_f(\boldsymbol{q})$ generated from the vertex at the end of the flow. The Brillouin zone is indicated as a square, with incommensurate peaks (magenta) close to $\boldsymbol{q}=(\pi,\pi)$ for all cases. In the momentum contraction of fermion bilinears with the vertex, both a constant (a,c) and a nontrivial (b,d) formfactor are used. For the topological case $(v,w)=(0.2,0.3)$ in (a,b), the susceptibility is slightly stronger for the nontrivial formfactor (b). For the topologically trivial case $(v,w)=(0.3,0.2)$ in (c,d) the opposite applies and the susceptibility is slightly stronger for the constant formfactor (c). Having a dominant instability with non-trivial formfactor (cf. (b)) can lead to chiral superpositions of degenerate wave-vectors $\boldsymbol{q}$ in the resulting order parameter.