Pseudo-fermion functional renormalization group with magnetic fields

The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry-breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at $T=0$ in the physical limit of vanishing renormalization group parameter $\mathrm{\Lambda }$. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb, and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three-dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.

Figure 1

(a), (c) Magnetization $M^{\mathrm{\Lambda }}=\left|{\mathit{M}}_i^{\mathrm{\Lambda }}\right|$ as a function of $\mathrm{\Lambda }$ for the (a) ferromagnetic and (c) antiferromagnetic square lattice Heisenberg model. Full (dotted) lines correspond to PFFRG (mean-field) results. Different colors correspond to different strengths of seed fields ($\delta /\left|J\right|=0$, 0.01, 0.02, 0.1 and 0.5) [see legend in (b) which applies to all subfigures]. Seed fields are oriented parallel or antiparallel to the $z$ axis according to the insets. (b), (d) Longitudinal order-parameter susceptibilities ${\chi }^{zz,\mathrm{\Lambda }}\left(\mathit{q}\right)$ as a function of $\mathrm{\Lambda }$ for the (b) ferromagnetic and (d) antiferromagnetic square lattice Heisenberg model where $\mathit{q}=\mathit{0}$ and $\mathit{q}=(\pi ,\pi )$, respectively. Full lines are PFFRG results derived from Eq. (43) while dashed lines are derived from ${\chi }^{\mathrm{\Lambda }}=\partial M^{\mathrm{\Lambda }}/\partial \delta$. In the latter approach, $\delta$ derivatives are approximated by the variation of $M^{\mathrm{\Lambda }}$ when $\delta$ is increased by $10\%$ from the value stated in the legend.

Figure 2

(a) Néel order on the honeycomb lattice. (b) Three-sublattice ${120}^{\circ }$ Néel order. Different sublattices are distinguished by different colors of the spins. (c) Magnetization $M^{\mathrm{\Lambda }\to 0}$ in the small-$\mathrm{\Lambda }$ limit as a function of maximum correlation distance $L$, for the ferromagnetic Heisenberg model on the square lattice as well as for the antiferromagnetic Heisenberg models on the square, honeycomb, and triangular lattices. Here, $L$ is defined as the maximal distance of spin-spin correlations (in units of the lattice constant), while longer-distance spin-spin correlations are treated as zero. The magnetizations shown are obtained at cutoffs $\frac{\mathrm{\Lambda }}{J}=0.02$, except for the square lattice ferromagnet where $\frac{\mathrm{\Lambda }}{J}=0.01$ is used. Seed fields are of size $\frac{\delta }{J}=0.01$ on the square lattice, and of size $\frac{\delta }{J}=0.02$ on the honeycomb and triangular lattices.

Figure 3

(a), (b) Two choices of seed fields $\delta {\mathit{n}}_i$ deviating from a ${120}^{\circ }$ Néel pattern on the triangular lattice. Red dots indicate vanishing amplitudes $\delta$ on this sublattice. On the other two sublattices with identical $\delta$, the arrows depict the directions of ${\mathit{n}}_i$ which either enclose (a) angles of ${120}^{\circ }$ or (b) angles of ${90}^{\circ }$. (c) $M_{{120}^{\circ }}^{\mathrm{\Lambda }}$ defined in Eq. (46) as a function of $\mathrm{\Lambda }$ for the two seed fields of the triangular lattice Heisenberg antiferromagnet in (a) and (b) and for the perfect ${120}^{\circ }$ seed field in Fig. 2. (d) ${\mathrm{\Delta }}_M^{\mathrm{\Lambda }}$ defined in Eq. (47) as a function of $\mathrm{\Lambda }$ for the same seed fields as considered in (c). The inset in (d) shows the magnetization $M^{\mathrm{\Lambda }}$ of the triangular lattice Heisenberg antiferromagnet as a function of $\mathrm{\Lambda }$ for the ideal seed fields illustrated in Fig. 2.

Figure 4

Blue squares show the magnetization $M^{z,\mathrm{\Lambda }\to 0}$ from PFFRG of the antiferromagnetic Heisenberg model on the square lattice as a function of the homogeneous field strength $h$, linearly extrapolated to zero seed fields, $\delta \to 0$. For comparison, red diamonds are QMC results from Ref. [49]. The dashed black line is the linear classical magnetization curve. The inset illustrates the magnetization process schematically, where red and blue arrows depict spins on the two sublattices in a field $\mathit{h}$ that increases from bottom to top.

Figure 5

Magnetization $M$ of a free spin in a magnetic field $h$ either as a function of the temperature $T$ or as a function of the sharp frequency cutoff parameter $\mathrm{\Lambda }$, both represented by the variable $u\in \{T,\mathrm{\Lambda }\}$. The explicit functions are given in Eqs. (A3) and (A4).

Figure 6

Susceptibilities ${\chi }^{\mu \mu }$ for $\mu =x,z$ of a free spin in a magnetic field $h$ along the $z$ axis. Shown is either the dependence on the temperature $T$ or on the sharp frequency cutoff $\mathrm{\Lambda }$, both represented by the variable $u\in \{T,\mathrm{\Lambda }\}$. The explicit functions are given by Eqs. (A5, A6, A7, A8).