Taming pseudofermion functional renormalization for quantum spins: Finite temperatures and the Popov-Fedotov trick

The pseudofermion representation for $S=1/2$ quantum spins introduces unphysical states in the Hilbert space, which can be projected out using the Popov-Fedotov trick. However, state-of-the-art implementation of the functional renormalization group method for pseudofermions have so far omitted the Popov-Fedotov projection. Instead, restrictions to zero temperature were made and the absence of unphysical contributions to the ground state was assumed. We question this belief by exact diagonalization of several small-system counterexamples where unphysical states do contribute to the ground state. We then introduce Popov-Fedotov projection to pseudofermion functional renormalization, enabling finite-temperature computations with only minor technical modifications to the method. At large and intermediate temperatures, our results are perturbatively controlled and we confirm their accuracy in benchmark calculations. At lower temperatures, the accuracy degrades due to truncation errors in the hierarchy of flow equations. Interestingly, these problems cannot be alleviated by switching to the parquet approximation. We introduce the spin projection as a method-intrinsic quality check. We also show that finite-temperature magnetic-ordering transitions can be studied via finite-size scaling.

Figure 1

Spin projection $C_i\left(T\right)=4\langle {\overline{S}}_i^z{\overline{S}}_i^z\rangle$ over temperature calculated via exact diagonalization of the pf Hamiltonian $\overline{H}$ for Heisenberg spin-clusters of $N=2,3,5$ sites as shown in the legend. A bond represents an anti-ferromagnetic Heisenberg coupling $J=1$. For the bow tie and the centered square, with inequivalent sites, the site-label $i$ refers to the center spin. For all clusters except the dimer, the pf ground state manifold contains unphysical states and the $T\to 0$ limit of $C_i\left(T\right)$ reduces from the physical value of unity (dashed line) to $\frac{4}{5}$ for the trimer, $\frac{7}{8}$ for the bow tie, $\frac{2}{3}$ for the centered square and $\frac{13}{15}$ for the centered tetrahedron, respectively.

Figure 2

Heisenberg dimer: Local and non-local susceptibilities (top panel: squares and crosses, respectively) and spin projection $C_i\left(T\right)=4\langle {\overline{S}}_i^z{\overline{S}}_i^z\rangle$ (bottom panel) computed with PF projection using the fRG (ppf-fRG) and the parquet approximation (ppf-PA). For comparison, we show the exact results (solid lines), second-order perturbation theory in $J$ (dotted lines), pf-fRG results without PF projection and data from the pseudo-Majorana fRG (pm-fRG) building on a fermionic spin representation without unphysical states.

Figure 3

Heisenberg trimer: The top panel shows the static susceptibilities that diverge as $T\to 0$ and have been multiplied by temperature to obtain finite values at $T=0$. The bottom panel shows the spin projection $C_i\left(T\right)$. The symbols follow the convention of Fig. 2.

Figure 4

ppf-fRG: Correlation length $\xi$, Néel susceptibility ${\chi }_N$ and spin projection $C$ for the nearest neighbor AFM Heisenberg model on the cubic lattice. We find a line crossing at $T_c^{\mathrm{fRG}}\simeq 0.61$. The corresponding scaling collapse can be found in Fig. 5.

Figure 5

ppf-fRG: Collapse plot for the data of Fig. 4, assuming the mean-field $\nu =0.5$ (points) or exact $\nu \simeq 0.71$ (triangles) correlation length critical exponents.

Figure 6

Error of the pf self energy in the Heisenberg dimer with $J=1$: (a) In ppf-fRG, the error scales with $\frac{J^3}{T^2}$. For small temperatures, this scaling breaks down because $J/T$ ceases to be a small parameter. (b) In ppf-PA, the observed error scaling depends on the the frequency grid, three choices are indicated by different colors. For larger frequency grids the self-energy error approaches the expected scaling with $\frac{J^3}{T^4}$ more closely.

Figure 7

Correlation length, Néel susceptibility, spin projection and scaling collapse from the calculations on the nearest neighbor AFM Heisenberg model on the cubic lattice. For panel (a) and (c) the flow equations were truncated via the RPA and integrated out numerically. We find $T_c\simeq 1.498$ and a critical exponent consistent with the mean-field value $\nu =0.5$. The spin projection deviates strongly from the exact result and is greater than 1. For panel (b) and (d) one-loop fRG without Katanin truncation was used. We find $T_c\simeq 1.29$ and a critical exponent also consistent with $\nu =0.5$. The spin projection exceeds the exact result of $C\left(T\right)=1$.

Figure 8

ppf-PA: Correlation length $\xi$, Néel susceptibility ${\chi }_N$ and spin projection $C$ for the nearest neighbor AFM Heisenberg model on the cubic lattice. We find a smeared-out line crossing at $T_c^{\mathrm{ppf}-\mathrm{PA}}\simeq 0.855$. A corresponding scaling collapse consistent with $\nu =0.71$ or $\nu =0.5$ could not be found. For $C\left(T\right)$ all five lines lie on top of each other.