Dimerization tendencies of the pyrochlore Heisenberg antiferromagnet: A functional renormalization group perspective

We investigate the ground-state properties of the spin-$1/2$ pyrochlore Heisenberg antiferromagnet using pseudofermion functional renormalization group techniques. The first part of our analysis is based on an enhanced parton mean-field approach, which takes into account fluctuation effects from renormalized vertex functions. Our implementation of this technique extends earlier approaches and resolves technical difficulties associated with a diagrammatic overcounting. Using various parton ansätze for quantum spin liquids, dimerized and nematic states our results indicate a tendency for lattice symmetry breaking in the ground state. While overall quantum spin liquids seem unfavorable in this system, the recently proposed monopole state still shows the strongest support among all spin liquid ansätze that we have tested, which is further confirmed by our complementary variational Monte Carlo calculations. In the second part of our investigation, we probe lattice symmetry breaking more directly by applying the pseudofermion functional renormalization group to perturbed systems. Our results from this technique confirm that the system's ground state either exhibits broken $C_3$ rotation symmetry, or a combination of inversion and $C_3$ symmetry breaking.

Figure 1

[(a), (b)] PFFRG flow equations for $n$-particle vertex functions up to $n=2$: Regular (slashed) arrows represent the full Green's function $G^{\mathrm{\Lambda }}$ (single-scale propagator $S^{\mathrm{\Lambda }}$). The first equation in (a) couples the self energy ${\mathrm{\Sigma }}^{\mathrm{\Lambda }}$ (circle) to the 2-particle vertex ${\mathrm{\Gamma }}^{\mathrm{\Lambda }}$ (square) and to itself via $S^{\mathrm{\Lambda }}$. In the second equation (b), the 2-particle vertex couples to the 3-particle vertex (hexagon), itself, and the self energy. This procedure leads to an infinite hierarchy of coupled differential equations, which we truncate by neglecting the 3-particle vertex. In (c) the diagrammatic relation between the self energy, the 2-particle vertex, and the spin-spin correlation $\langle {\mathbf{S}}_i{\mathbf{S}}_j\rangle$ is depicted.

Figure 2

Diagrammatic representations of different self-consistent schemes. (a) Bare self-consistent Fock mean-field approach. The dashed line is a bare interaction $\sim J_{ij}$. The thick line is a dressed propagator resulting from Dyson's equation in (b). The thin line in (b) is the free fermionic propagator $G_0\left(\omega \right)=\frac{1}{i\omega }$. In an enhanced parton mean-field scheme, Dyson's equation is evaluated together with the self-energy ${\mathrm{\Sigma }}_{\text{MF}+}$ shown in (c) making use of the renormalized vertex from PFFRG (gray box). Note that in this approximation, the thin line in (b) also contains the self-energy $\gamma \left(\omega \right)$ from PFFRG. (d) Illustration of a contribution from the bare self-consistent Fock mean-field scheme, obtained via iteration, see main text for details. Blue lines in (c) and (d) illustrate a cut for a two-particle decoupling in the crossed particle-hole channel. To avoid overcounting in ${\mathrm{\Sigma }}_{\text{MF}+}$, the enhanced mean-field equation in (c) needs to be evaluated with a two-particle irreducible vertex ${\tilde{\mathrm{\Gamma }}}^{\mathrm{\Lambda }}$ (white square), which follows from the Bethe-Salpeter equation in (e). Note that in this equation the propagator line is dressed with ${\gamma }^{\mathrm{\Lambda }}$ but not with ${\mathrm{\Sigma }}_{\text{MF}+}$.

Figure 3

Investigated hopping models on the pyrochlore lattice: The blue (red) bonds denote positive (negative) real-valued hoppings $\chi$ ($-\chi$). Black arrows depict imaginary hoppings $i\chi$ ($-i\chi$) in (against) the direction of the arrow. Gray bonds carry zero hopping amplitudes. Within each ansatz, all finite hoppings have the same absolute value. (a) The uniform state. (b) The staggered state. (c) The $(\pi /2,\pi /2,0)$ monopole state. (d) The localized tetramer state breaking inversion symmetry. (e) Two infinitely extended lines breaking the $C_3$ rotation symmetry. (f) Two localized dimers breaking inversion and rotation symmetry. (g) One localized trimer breaking inversion and rotation symmetry. (h) One localized fourfold loop breaking inversion and rotation symmetry. (i) The $(\pi /2,-\pi /2,0)$ monopole-antimonopole state.

Figure 4

Self-consistently determined nearest-neighbor hopping amplitudes as a function of the decoupling scale $\overline{\mathrm{\Lambda }}$ for the ansätze given in Fig. 3. While the amplitudes for symmetric quantum spin liquids are comparatively small, by far the largest amplitude results for the dimerized pattern in Fig. 3. The decay of all amplitudes when $\overline{\mathrm{\Lambda }}\lesssim 0.4$ is caused by the finite pseudofermion lifetime due to the imaginary self energy from PFFRG. Note that the data points for the $(\pi /2,\pi /2,0)$ flux monopole state and $(\pi /2,-\pi /2,0)$ flux monopole-antimonopole state lie almost on top of each other, with slightly larger amplitudes for the $(\pi /2,-\pi /2,0)$ state.

Figure 5

Renormalization group flows of response functions ${\chi }_D$ [see Eq. (15)] for the three symmetry breaking perturbations illustrated in Figs. 3, 3, and 3. In the bottom part of the figure, the colored bonds illustrate which weakened (gray) and strengthened (black) bonds are compared to each other (the colors in these illustrations match the ones of the curves). Note that for the dimer pattern in Fig. 3 two different response functions ${\chi }_{\text{D},C_3,i}^{\left(1\right)}$ and ${\chi }_{\text{D},C_3,i}^{\left(2\right)}$ can be defined, depending on the precise choice of weakened and strengthened bonds that are compared (bottom part of the figure).