Numerical treatment of spin systems with unrestricted spin length $S$: A functional renormalization group study

We develop a generalized pseudofermion functional renormalization group (PFFRG) approach that can be applied to arbitrary Heisenberg models with spins ranging from the quantum case $S=1/2$ to the classical limit $S\to \infty$. Within this framework, spins of magnitude $S$ are realized by implementing $M=2S$ copies of spin-1/2 degrees of freedom on each lattice site. We confirm that even without explicitly projecting onto the highest spin sector of the Hilbert space, ground states tend to select the largest possible local spin magnitude. This justifies the average treatment of the pseudofermion constraint in previous spin-1/2 PFFRG studies. We apply this method to the antiferromagnetic $J_1\text{−}{J}_2$ honeycomb Heisenberg model with nearest-neighbor $J_1>0$ and second-neighbor $J_2>0$ interactions. Mapping out the phase diagram in the $J_2/J_1\text{−}S$ plane, we find that upon increasing $S$, quantum fluctuations are rapidly decreasing. In particular, already at $S=1$ we find no indication for a magnetically disordered phase. In the limit $S\to \infty$, the known phase diagram of the classical system is exactly reproduced. More generally, we prove that for $S\to \infty$ the PFFRG approach is identical to the Luttinger-Tisza method.

Figure 1

Illustration of the honeycomb lattice where $J_1$ nearest-neighbor ($J_2$ second-neighbor) interactions are highlighted by red (dashed blue) lines. The two sublattices are indicated by numbers and the nearest-neighbor distance is assumed to be one throughout the paper.

Figure 2

Diagrammatic illustration of the PFFRG flow equations for the self-energy and the two-particle vertex, see also Eqs. (13) and (14). The gray lines crossing two fermion propagators denote the term $P_{i_1{i}_2}^{\mathrm{\Lambda }}({\omega }_1,{\omega }_2)=G_{i_1}^{\mathrm{\Lambda }}\left({\omega }_1\right)S_{i_2}^{\mathrm{\Lambda }}\left({\omega }_2\right)+G_{i_2}^{\mathrm{\Lambda }}\left({\omega }_2\right)S_{i_1}^{\mathrm{\Lambda }}\left({\omega }_1\right)$, while slashes crossing only one line are the single scale propagators. Site indices $i_1,i_2,j$ illustrate the real-space structure of the flow equations. The five terms on the right-hand side of the second equation are the particle-particle channel, the RPA term, two vertex correction terms and the crossed particle hole term in the same order as they appear in Eq. (14).

Figure 3

Index structure of the bare two-particle vertex including the flavor index $\kappa$. The dashed line denotes the exchange couplings $J_{ij}$. Note that the site indices $i$ and flavor indices $\kappa$ do not change along fermion lines.

Figure 4

Flowing PFFRG susceptibility for the nearest-neighbor honeycomb Heisenberg antiferromagnet with onsite level repulsion terms [see Eq. (27)]. In the main plots and in all subsequent figures, the susceptibility ${\chi }^{\mathrm{\Lambda }}\left(\mathbf{k}\right)$ (RG scale $\mathrm{\Lambda }$) is given in units of $1/J_1$ ($J_1$). In the insets, energy scales are in units of $\sqrt{A^2+J_1^2}$ to compensate for energy renormalization effects in $\mathrm{\Lambda }$ (see text for details). (a) Flow behavior of the susceptibility for the nearest-neighbor model at $S=3/2$ and varying negative values of $A$ (due to almost diverging susceptibilities, the RG flow is not shown below the critical $\mathrm{\Lambda }$ scale). Depicted is the maximal susceptibility component in $\mathbf{k}$ space which, here, corresponds to antiferromagnetic Néel order on the honeycomb lattice. (b) Same as in (a) but with $S=1/2$. (c) Susceptibility for $J_2/J_1=0.1,S=1/2$, and positive level repulsion terms $A\ge 0$.

Figure 5

(a) Phase diagram in the $g\text{−}S$ plane obtained via PFFRG. We find a nonmagnetic (NM) intermediate phase at $S=1/2$ and three magnetically ordered phases [antiferromagnetic (AF) state and two spiral phases S1 and S2]. For the susceptibility profiles of the magnetic states in $\mathbf{k}$ space, see Figs. 6–6 respectively. (b) $\mathrm{\Lambda }$ flow of the maximal $\mathbf{k}$ space component of the susceptibility ${\chi }^{\mathrm{\Lambda }}\left(\mathbf{k}\right)$ for $g=0.3$ and increasing values of $S$. While the flow for $S=1/2$ does not show signatures of an instability, for $S\ge 1$, we find a kink in the susceptibility at $\mathrm{\Lambda }\approx 0.2$.

Figure 6

(Top) Susceptibility ${\chi }^{\mathrm{\Lambda }}\left(\mathbf{k}\right)$ in reciprocal space for the three magnetically ordered phases at $S=3/2$. (a) antiferromagnetic state at $g=0$, (b) S1 spiral at $g=0.3$, and (c) S2 spiral at $g=0.9$. All plots correspond to $\mathrm{\Lambda }$ values right above the instability feature during the RG flow. Outer (inner) hexagons indicate the boundaries of the extended (first) Brillouin zone. (Bottom) Below each susceptibility profile we depict the corresponding real space spin patterns, which yield magnetic Bragg peaks in $\mathbf{k}$ space at the marked positions (black dots). Arrows illustrate the unit vectors of the honeycomb lattice and indicate the pitch angles of the spiral state along these directions.

Figure 7

Position of the magnetic wave vectors in reciprocal space at $S=3/2$ for increasing values of $g$: The antiferromagnetic state is characterized by susceptibility peaks at the corners of the extended Brillouin zone. In the S1 and S2 spiral phases, the maxima move along the Brillouin zone edges and along a radial direction, respectively.

Figure 8

Fermionic two-particle vertex in RPA approximation: dashed lines denote bare exchange couplings $J_{i_1{i}_2}$ and arrows illustrate free fermion propagators.

Figure 9

Degenerate spiral magnetic wave vectors of the classical antiferromagnetic $J_1\text{−}{J}_2$ honeycomb Heisenberg model in reciprocal space. At $1/6<g<1/2$, the degenerate states form contours around the corners of the extended Brillouin zone, see red ring for $g=0.3$. For $g>0.5$, the contours are around the corners of the first Brillouin zone, see blue ring for $g=0.9$.