Functional renormalization group for three-dimensional quantum magnetism

We formulate a pseudofermion functional renormalization group (PFFRG) scheme to address frustrated quantum magnetism in three dimensions. In a scenario where many numerical approaches fail due to sign problem or small system size, three-dimensional (3D) PFFRG allows for a quantitative investigation of the quantum spin problem and its observables. We illustrate 3D PFFRG for the simple cubic $J_1\text{−}{J}_2\text{−}{J}_3$ quantum Heisenberg antiferromagnet, and benchmark it against other approaches, if available.

Figure 1

(a) Quantum phase diagram of the spin-$\frac{1}{2}$ simple cubic $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg antiferromagnet [see Eq. (1)] as obtained by PFFRG. It features a paramagnetic domain surrounded by commensurate magnetic orders. (Classical magnetic phase boundaries are drawn by black dashed lines for comparison.) (b)–(d) Illustrations of the real space pattern (upper row) and magnetic susceptibility profile (lower row) in units of $1/J_1$ for magnetism at ordering vectors $\mathit{Q}=(\pi ,\pi ,\pi ),(0,\pi ,\pi ),$ and $(0,0,\pi )$ obtained for the parameters $(J_2/J_1,J_3/J_1)=(0,0),(1,0),$ and $(1,1)$, respectively.

Figure 2

(a)–(f) Evolution of magnetic susceptibility for $J_3/J_1=0.3$ from $J_2/J_1=0.50$ to 0.95. As we traverse the paramagnetic region into the $(0,0,\pi )$ antiferromagnet (AF) in close proximity to the $(0,\pi ,\pi )$ AF for lower $J_3/J_1$, the short-range correlations deviate significantly from the characteristic susceptibility profile deep in the ordered regimes [see Figs. 1–1]. We have cut out one octant of the Brillouin zone such that one can also visually track the spin correlations around the $\mathrm{\Gamma }$ point.

Figure 3

(a) PFFRG coupling flow deep in the paramagnetic and different magnetic phases, shown for the parameter values $(J_2/J_1,J_3/J_1)=(0.5,0.25)\left[\mathrm{Paramagnet}\right],(0,0)\left[\right(\pi ,\pi ,\pi \left)\right],(1,0)\left[\right(0,\pi ,\pi \left)\right]$, and $(1,1)\left[\right(0,0,\pi \left)\right]$. In the case of ordering, the flow exhibits a sharp singularity at ${\mathrm{\Lambda }}^{*}\equiv \frac{2}{\pi }{T}_{\text{Néel}}$. (b)–(d) Distinct change in flow behavior from the paramagnet into the magnetic phases highlighted by pairs of filled data point labels in Fig. 1. (e) $T_{\text{Néel}}$ for different cuts of constant $J_3/J_1$ as a function of $J_2/J_1$. The black asterisk highlights the value for the $J_1$ model as obtained by quantum Monte Carlo [17].

Figure 4

(a)–(c) Valence bond crystal candidates in the paramagnetic regime of Fig. 1. Red bonds correspond to strengthened $\left(J_1\to J_1+\delta \right)$ and gray bonds to weakened $\left(J_1\to J_1-\delta \right)$ nearest-neighbor interactions. (d) Columnar and plaquette VBC show nearly identical and weak dimer response strength throughout the paramagnetic regime (variations of the response within the PM phase are smaller than the thickness of the line). (e) The cubic VBC exhibits a similarly weak response strength which, however, varies as a function of $J_2/J_1$ and $J_3/J_1$, shown here for $(J_2/J_1,J_3/J_1)=(0.45,0.15)$ (dotted line) and $(0.70,0.25)$ (dashed line).