Functional renormalization group approach to $\text{SU}\left(N\right)$ Heisenberg models: Real-space renormalization group at arbitrary $N$

The pseudofermion functional renormalization group (pf-FRG) is one of the few numerical approaches that has been demonstrated to quantitatively determine the ordering tendencies of frustrated quantum magnets in two and three spatial dimensions. The approach, however, relies on a number of presumptions and approximations, in particular the choice of pseudofermion decomposition and the truncation of an infinite number of flow equations to a finite set. Here we generalize the pf-FRG approach to $\mathrm{SU}(N)$-spin systems with arbitrary $N$ and demonstrate that the scheme becomes exact in the large-$N$ limit. Numerically solving the generalized real-space renormalization group equations for arbitrary $N$, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible $\mathrm{SU}(N)$ models. In a case study of the square-lattice $\mathrm{SU}(N)$ Heisenberg antiferromagnet, we explicitly demonstrate that the generalized pf-FRG approach is capable of identifying the instability indicating the transition into a staggered flux spin liquid ground state in these models for large, but finite, values of $N$. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for $\mathrm{SU}(N)$ spin models that allows us to explicitly study the large-$N$ limit and access the low-temperature spin liquid phase.

Figure 1

$\text{SU}\left(N\right)$ spin model for the staggered flux spin liquid. The original spins are decomposed into pseudofermions and recombine into a finite, nonlocal order parameter on the bonds that generates a $\pi$-flux on every plaquette.

Figure 2

Generalized flow equations for $\mathrm{SU}(N)$-symmetric spin systems of spin length $S$. The prefactors indicated here are only schematic and to leading order in $S$ and $N$. The exact prefactors are given in Fig. 6 in the Appendix. Slashed propagator lines should be understood as the single-scale propagator. A pair of slashed propagators in the flow equation for the two-particle vertex should be understood as a pair of single-scale and full propagators where the two permutations of the propagators are summed over. The Fock diagram and the particle-hole ladder contribution, which both describe quantum fluctuations, are enhanced as $N$ is increased. Similarly, as the classical limit is approached for increasing spin length $S$ the Hartree and random-phase approximation (RPA) channels are boosted.

Figure 3

Susceptibility for various $N$ calculated by pf-FRG with Katanin truncation. For finite $N$ the flow equations are solved numerically. In the limit $N\to \infty$ the flow equation is solved analytically.

Figure 4

Real-space correlations for (a) $N=2$ and (b) $N=1000$ plotted just above the respective critical cutoff. Correlations are given with respect to a single reference site (grey). Ferromagnetic correlations are represented by blue circles, AFM correlations by red ones. The size of the circle represents the correlation's magnitude relative to the reference site. Correlations for $N=2$ clearly reveal Néel order while for $N=1000$ no correlations are visible.

Figure 5

Susceptibility for various $N$ without Katanin truncation calculated by pf-FRG. Finite-$N$ results are obtained numerically, the $N\to \infty$ limit is exact.

Figure 6

Generalized flow equations for the $\text{SU}\left(N\right)$-symmetric Heisenberg model. The spin dependency has been calculated explicitly by splitting the two-particle vertex into the two symmetry-allowed terms, a spin contribution (dashed lines) and a density contribution (solid lines).