Competing orders at higher-order Van Hove points

Van Hove points are special points in the energy dispersion, where the density of states exhibits analytic singularities. When a Van Hove point is close to the Fermi level, tendencies towards density wave orders, Pomeranchuk orders, and superconductivity can all be enhanced, often in more than one channel, leading to a competition between different orders and unconventional ground states. Here we consider the effects from higher-order Van Hove points, around which the dispersion is flatter than near a conventional Van Hove point, and the density of states has a power-law divergence. We argue that such points are present in intercalated graphene and other materials. We use an effective low-energy model for electrons near higher-order Van Hove points and analyze the competition between different ordering tendencies using an unbiased renormalization-group approach. For purely repulsive interactions, we find that two key competitors are ferromagnetism and chiral superconductivity. For a small attractive interaction, we find an unconventional spin Pomeranchuk order, in wich the spin oder parameter winds around the Fermi surface. The supermetal state, predicted for a single higher-order Van Hove point, is an unstable fixed point in our case.

Figure 1

Lattice and tight-binding model. Left panel: Lattice in real space with neighboring vectors ${\vec{a}}_n,{\vec{b}}_n,{\vec{c}}_n$. The nearest-neighbor vectors on the honeycomb lattice are ${\vec{a}}_1=(\sqrt{3},1)/2, {\vec{a}}_2=(-\sqrt{3},1)/2, {\vec{a}}_3=(0,-1)$; the second-nearest-neighbor vectors are ${\vec{b}}_1=(\sqrt{3},0),{\vec{b}}_2=(\sqrt{3},3)/2,{\vec{b}}_3=(-\sqrt{3},3)/2, {\vec{b}}_4=-{\vec{b}}_1,{\vec{b}}_5=-{\vec{b}}_2,{\vec{b}}_6=-{\vec{b}}_3$; and the third-nearest-neighbor vectors are ${\vec{c}}_1=-2{\vec{a}}_1,{\vec{c}}_2=-2{\vec{a}}_2,{\vec{c}}_3=-2{\vec{a}}_3$. Middle panel: Energy dispersion for $t_1=1,t_2=0.1,t_3=0.2$. The valence and conduction band touch at the Dirac points. The Van Hove points appear at the $M$ points, which are marked by black dots for the conduction band. Right panel: Energy contours for the same hopping amplitudes and the high-symmetry points $K_1=2\pi /3(1/\sqrt{3},1),K_2=2\pi /3(-1/\sqrt{3},1)$ and $M_1=\pi (0,2/3), M_2=\pi (-1/\sqrt{3},1/3), M_3=-\pi (1/\sqrt{3},1/3)$. The Fermi level at Van Hove filling is given by the red line. At this filling, the system undergoes a Lifshitz transition from a closed to an open Fermi surface. For a small variation of the filling, the Fermi surface is either closed as demonstrated by the nearby dashed line, or changes to open Fermi-surface pockets given by the dotted lines around the $K$ points.

Figure 2

Band flattening at Van Hove point. Left panel: Dispersion along $K_1-M_1-K_2$ for fixed $t_2=0.1t_1$ and varying $t_3=0$ (orange, dotted), $t_3=0.1t_1$ (green, dashed), and $t_3=0.2t_2$ (blue, solid). The last value of $t_3$ leads to a HOVH singularity. Right panel: Corresponding Fermi surface.

Figure 3

DOS and saddle point. Left panel: DOS of the band dispersion for hopping parameters $t_1=1,t_2=1/10,t_3=\frac{1}{4}(1-2/10)$ (solid line) and DOS for hopping parameters $t_1=1,t_2=t_3=0$ for comparison (dashed line). Right panel: Corresponding higher-order saddle point at $M_1$.

Figure 4

Three-patch model and interaction couplings. Graphic representation of the four interaction couplings $g_i, i\in \{1,...,4\}$ representing the scattering processes between the three $M_p$ points for the Van Hove doped dispersion. Solid, dashed, and dotted lines represent electrons near the three $M_p$ points.

Figure 5

Ladder series for vertices. Graphic representation of the ladder series for the spin (top row), charge (middle rows), and pairing (bottom row) vertex. States close to the three $M_p$ points are represented by solid, dashed, and dotted lines. The couplings are colored according to Fig. 4.

Figure 6

RG flow equations. Diagrams representing the different RG flow in Eqs. (40, 41, 42, 43). Note that the two internal lines correspond to the same $M$ point in each diagram. We show the flow equations for three $M_p$ points represented by solid, dashed, and dotted lines.

Figure 7

Flow to strong coupling. Integration of the flow in Eqs. (40, 41, 42, 43) for four sets of bare couplings and $d_0=0.25$. The bare values are $g_1^0=g_2^0=g_4^0=0.5, g_3^0=0$ (top left), $-g_1^0=g_2^0=g_4^0=0.5, g_3^0=0$ (top right), $g_2^0=g_3^0=g_4^0=0.5, g_1^0=0$ (bottom left), and $g_2^0=-g_3^0=g_4^0=0.5, g_1^0=0$ (bottom right). $g_1$, dashed purple; $g_2$, dotted blue; $g_3$, solid green; $g_4$, solid orange.

Figure 8

Phase diagrams. Bare interactions $g_2^0=g_4^0$ are held fixed and $g_1$ and $g_3$ are varied, $d_0=0.25$ (top) and $d_0=1$ (bottom). Bare values are $g_2^0=g_4^0=0.1$ (left) or $g_2^0=0, g_4^0=0.1$ (right, top), and $g_2^0=-0.1, g_4^0=0.1$ (right, bottom). The coloring encodes the scale where the couplings diverge, i.e., where correlations grow strong. If it is too large, e.g., in the red regime, there is no instability or it only occurs at the lowest scales. This can be used to estimate phase boundaries. FM, ferromagnet; $d\mathrm{FM}, d-\mathrm{wave}$ spin Pomeranchuk; $s\mathrm{POM}, s-\mathrm{wave}$ charge Pomeranchuk; $d\mathrm{POM}, d-\mathrm{wave}$ charge Pomeranchuk; $s\mathrm{SC}, s-\mathrm{wave}$ superconductivity; $d\mathrm{SC}, d-\mathrm{wave}$ superconductivity.

Figure 9

$d-\mathrm{wave}$ spin Pomeranchuk order. Left panel: The order parameter of the $d$-wave spin Pomeranchuk order winds twice around the Fermi surface. Right panel: Diagonalization leads to a Zeeman-like term without net magnetization which splits the Fermi surface.

Figure 10

Temperature derivative of loop corrections. We plot $d/dt{\chi }_{\mathrm{pp}}^X$ and $d/dt{\chi }_{\mathrm{ph}}^X$ [see Eqs. (17) and (18) with $t=ln\mathrm{\Lambda }/T$ for the microscopic model Eq. (1)]. From left to right and top to bottom, we show $d/dt{\chi }_{\mathrm{pp}}^0, d/dt{\chi }_{\mathrm{ph}}^M, d/dt{\chi }_{\mathrm{pp}}^M$, and $d/dt{\chi }_{\mathrm{ph}}^0$. As an example for the situation with a HOVH singularity we chose $t_3=t_1/4,t_2=0$ (solid, purple). For comparison, we also show the case with a logarithmic DOS for $t3=0.01t_1,t_2=0$ (dashed, yellow). When plotted as a function of the logarithmic temperature $t, {\chi }_i^X\propto {(\mathrm{\Lambda }/T)}^{\kappa }$ leads to an exponential growth as observable in ${\chi }_{\mathrm{pp}}^0$ and ${\chi }_{\mathrm{ph}}^0$ (solid, purple). The logarithmic singularity ${\chi }_i^X\propto {ln}^2\mathrm{\Lambda }/T$ leads to a linear behavior in the derivative as observable for ${\chi }_{\mathrm{pp}}^0$ and ${\chi }_{\mathrm{ph}}^M$ (dashed, yellow). The linear growth in ${\chi }_{\mathrm{ph}}^M$ is cut for small enough temperatures $t\gtrsim 5$ due to imperfect nesting $t_3=0.01t_1$.

Figure 11

Loop derivatives ${\partial }_t{\chi }_{\mathrm{pp}/\mathrm{ph}}^X$ as a function of $t_3$ for a fixed (arbitrary) temperature $t=ln\mathrm{\Lambda }/T=5.8$.

Figure 12

Exemplary diagrams at one- and two-loop level if $g_1=g_2=g_3=0$. Diagrams can be classified into three channels, often denoted as crossed particle-hole, particle-particle, and direct particle-hole channel. On the one-loop level (first line), we have contributions from the crossed particle-hole and the particle-particle channel. Diagrams in the direct particle-hole channel cancel against each other. On the two-loop level, we can distinguish pure diagrams, which consist only of contributions belonging to the same channel, and mixed diagrams, which contain subdiagrams from different channels. The second line contains pure diagrams in the crossed particle-hole and the particle-particle channel. The last line shows examples of mixed diagrams: a mixed crossed and direct particle-hole (left) and a mixed particle-particle and particle-hole diagram (right). The RG approximates the mixed diagrams as the product of its subdiagrams.

Figure 13

Susceptibility exponents for the stable fixed trajectories as a function of $d_0$. They correspond to I–VIII from left to right, from top to bottom. $s$-wave spin, solid orange; $d$-wave spin, dashed light orange; $s$-wave charge, solid purple; $d$-wave charge, dotted light purple; $s$-wave SC, solid blue; $d$-wave SC, dot-dashed light blue. Fixed trajectory VII has two regimes of stability; for clarity we connect the exponents in both regimes by thin lines.