# Taming pseudo-fermion functional renormalization for quantum spins: Finite-
temperatures and the Popov-Fedotov trick

Benedikt Schneider  Department of Physics and Arnold Sommerfeld Center for
Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstr. 37,
80333 Munich, Germany  Munich Center for Quantum Science and Technology
(MCQST), 80799 Munich, Germany  Dominik Kiese  Institute for Theoretical
Physics, University of Cologne, 50937 Cologne, Germany  Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York,
NY 10010, USA  Björn Sbierski  Department of Physics and Arnold Sommerfeld
Center for Theoretical Physics, Ludwig-Maximilians-Universität München,
Theresienstr. 37, 80333 Munich, Germany  Munich Center for Quantum Science and
Technology (MCQST), 80799 Munich, Germany

(August 28, 2024)

###### Abstract

The pseudo-fermion 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 pseudo-fermions have so far
omitted the Popov-Fedotov projection. Instead, restrictions to zero
temperature were made and 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 pseudo-fermion
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.

##  I Introduction

Many quantum systems of current interest, ranging from frustrated magnets [1]
to Rydberg atom arrays [2] can be described by Hamiltonians consisting of spin
$S=1/2$ operators $\mathbf{S}=(S^{x},S^{y},S^{z})$ fulfilling the standard
$\mathfrak{su}(2)$ spin algebra [1]. In theoretical treatments, it is often
useful to switch to an auxiliary particle representation of the spin operator.
An established representation in terms of spinful fermions annihilated by
$f_{\alpha=\uparrow,\downarrow}$ goes back to Abrikosov [3],

$\displaystyle
S^{\mu}\xrightarrow{}\bar{S}^{\mu}=\frac{1}{2}\sum_{\alpha,\alpha^{\prime}}f^{\dagger}_{\alpha}\sigma^{\mu}_{\alpha\alpha^{\prime}}f_{\alpha^{\prime}}.$
(1)

Here, $\sigma^{\mu}$ are Pauli matrices $(\mu=x,y,z)$ and an overbar indicates
an operator in the fermionic Hilbert space 111We skip the overbar for
fermionic operators like $f_{\alpha}$ or
$n_{\alpha}=f_{\alpha}^{\dagger}f_{\alpha}$ where no confusion is possible..

The pseudo-fermion (pf) representation (1) allows for a variety of
applications. For example, it is one of the pillars of the theory of spin-
fractionalization and spin-liquids [5] where pf mean-field states are used to
describe highly entangled paramagnetic ground states of frustrated spin
systems. On the other hand, the pf representation has been used extensively
for numerical methods as it allows to transfer the well-developed diagrammatic
toolbox for interacting fermions in equilibrium [6] to quantum spins. These
tools are based on the Wick theorem and perturbation theory. Two popular
examples for more advanced methods are the diagrammatic Monte Carlo [7, 8,
9](pf-diagMC) and the functional renormalization group (pf-fRG) [10] in vertex
expansion. Whereas the first method samples diagrams of a perturbative series
in $J/T$ ($J$ is the exchange coupling and $T$ the temperature) [11], the
second rests on a hierarchy of flow equations for pf vertex functions which
flow under the variation of the regularized bare propagator [12, 13]. Despite
the necessary truncation of this hierarchy, resulting end-of-flow correlation
functions contain infinite-order re-summations of certain diagram classes.

Crucially, there is a well-known problem that appears whenever the pf
representation (1) is used: While the left hand side acts in the two-
dimensional spin Hilbert space spanned by
$\\{\ket{\uparrow},\ket{\downarrow}\\}$, the basis of the right hand side’s
Hilbert space is extended,
$\\{\ket{\uparrow},\ket{\downarrow},\ket{\uparrow\downarrow},\ket{0}\\}$ and
the pf spin operator is $\bar{S}^{\mu}=\mathrm{diag}(S^{\mu},0,0)$. It
faithfully represents a spin $S=1/2$ operator only on the physical subspace
$\\{\ket{\uparrow},\ket{\downarrow}\\}$ while it acts as a $S=0$ operator on
the empty and doubly-occupied subspaces. As the mentioned fermionic methods
are applied in thermal equilibrium, any occupation of the unphysical $S=0$
subspaces will lead to differences between the physical $S^{\mu}$ correlation
functions and the pf ones (using $S^{\mu}\xrightarrow{}\bar{S}^{\mu}$) as well
as between the associated partition functions $Z$ and $\bar{Z}$.

Fortunately, this problem can be circumvented with a trick found by Popov and
Fedotov (PF) in the late 1980s [14, 15]. We will review the details of the PF
trick in Sec. III. In its most simple incarnation it amounts to the addition
of an imaginary valued chemical potential term to the pf Hamiltonian,
$\bar{H}\rightarrow\bar{H}+\bar{H}_{\text{PF}}$, where

$\displaystyle\bar{H}_{\text{PF}}=\frac{i\pi
T}{2}(n_{\downarrow}+n_{\uparrow}-1)$ (2)

and $n_{\alpha}=f^{\dagger}_{\alpha}f_{\alpha}$ is the pf number operator.

Whereas the PF trick is routinely and straightforwardly employed in pf-diagMC
calculations [7, 8, 9], this is not the case in the pf-fRG literature where
only very limited attention has been paid to the subject [10, 16, 17]. The
justifying narrative for this omission in state-of-the-art pf-fRG [18, 19, 20,
21, 22, 23, 24, 25, 26] is that unphysical $S=0$ states would only occur at
energies above the ground state energy and consequently PF projection would be
unnecessary if calculations are restricted to $T=0$. However, this state of
affairs comes with a number of problems:

(i) In the following Sec. II, we show that even simple and generic frustrated
spin systems like the antiferromagnetic (AFM) Heisenberg trimer have a pf
ground state with sizable occupation of unphysical $S=0$ states if PF
projection is omitted. Even if these clusters are usually not the focus of pf-
fRG applications, they are basic building blocks of highly relevant lattices
like triangular or Kagome. For AFM Heisenberg Hamiltonians on these lattices
it is thus questionable if ground states in pf representation without PF
projection are indeed in the physical subspace. As these systems have been
studied amply with pf-fRG [18, 27, 28, 29, 25, 30], this also questions the
quantitative accuracy of these state-of-the-art pf-fRG results.

(ii) Even for systems where the pf ground state is faithful, the unavoidable
truncation of the hierarchy of fRG flow equations is an uncontrolled
approximation at $T=0$, where, in the absence of magnetic fields $h$, neither
$J/T$ nor $J/h$ can be used to perturbatively justify the truncation. Existing
arguments appealing to the correctness of commonly employed pf-fRG truncations
in the large $S$ or large $N$ limit [23, 31] [in the sense of generalizing
$SU(2)$ to $SU(N)$] are not rigorously helpful in the case $S=1/2$, $N=2$ most
relevant in applications. At best, it is the presence of the Matsubara cutoff
scale $\Lambda$ which perturbatively controls the pf-fRG: As $\Lambda$ flows
from infinity to zero, only ordering tendencies extracted at $\Lambda\gtrsim
J$ tend to be reliable, a point of view not emphasized in the literature
before 222Details will be covered in a forthcoming publication. However, as
the physical model is only recovered as $\Lambda\rightarrow 0$, quantitatively
reliable results for observables cannot be guaranteed by this argument.
Another practical difficulty of the pf-fRG at $T=0$ is the continuous nature
of Matsubara frequencies, requiring major numerical efforts for a stable and
reliable solution of the flow equations [33], in particular if multiloop
schemes are involved [26, 25].

In this work, we tame problem (i) and (ii) of the pf-fRG by implementing the
PF trick in the pf-fRG framework. We term the resulting method ppf-fRG. We
show that this is possible with only minor technical modifications (see Sec.
IV) so that most methodological achievements from the last decade [23, 34, 35,
36, 25, 26, 37] can be seamlessly adapted. As a result, the ppf-fRG leverages
the pf-fRG to finite temperatures $T$ where quantum and thermal fluctuations
compete, and the numerical implementation simplifies by the discrete nature of
Matsubara frequencies. In addition, we show that the ppf-fRG is now
perturbatively justified for small $J/T$ and equipped with a method-intrinsic
gauge for the quality of the results.

In Sec. V we benchmark the ppf-fRG using small spin clusters. Although exact
results are obtained trivially for these systems, diagrammatic approaches are
invoked already at their full complexity, making spin clusters a valuable
testbed. We show that the ppf-fRG indeed yields quantitatively reliable
results at large and moderate $T$ but cannot be trusted for $T$ smaller than
about a third of $J$, at least for the models studied here. Surprisingly, we
find that this situation cannot be improved by considering solutions of the
parquet approximation. The latter is known to be in equivalence with the loop-
converged limit of the multiloop fRG [38, 39, 40] in which the two-particle
vertex is correctly obtained up to errors of order $J^{4}$, i.e., one order
higher than in one-loop fRG.

By applying the ppf-fRG to a translation invariant three-dimensional
Heisenberg magnet on the cubic lattice, we show that the study of finite-
temperature transitions into symmetry-broken (magnetic) phases becomes
possible. In Sec. VI, we conclude and also relate our results to the recently
developed pseudo-Majorana fRG (pm-fRG) [41] where by construction no
unphysical sectors exist in the fermionic Hilbert space.

##  II Spin constraint violation at $T\geq 0$

(a)

(b)

Figure 1: Spin projection
$C_{i}(T)=4\langle\bar{S}^{z}_{i}\bar{S}^{z}_{i}\rangle$ over temperature
calculated via exact diagonalization of the pf Hamiltonian $\bar{H}$ 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 $T\rightarrow 0$ limit of
$C_{i}(T)$ 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.

In this section we substantiate our claim that there exist simple AFM spin
clusters with a ground state of the pf Hamiltonian $\bar{H}$ that – without PF
projection – partially resides outside of the physical $S=1/2$ sector. We use
exact diagonalization (ED) of the pf Hamiltonian $\bar{H}$ corresponding to
the spin Hamiltonian $H$ according to Eq. (1). Our examples serve to
demonstrate the need for PF projection along with the pf representation [cf.
(i) in Sec. I], but the small system sizes allow to disregard problems related
to the solution of $\bar{H}$, i.e. the truncation of the pf-fRG [cf. (ii)].

The most elementary example is the AFM Heisenberg trimer with $N=3$ spins
coupled all-to-all ($J=1$), see (b) in Fig. 1,

$\displaystyle
H_{\mathrm{trimer}}=J(\mathbf{S}_{1}\cdot\mathbf{S}_{2}+\mathbf{S}_{2}\cdot\mathbf{S}_{3}+\mathbf{S}_{3}\cdot\mathbf{S}_{1}).$
(3)

Its degenerate physical ground state manifold is described by two of the spins
forming a singlet and the third spin being in any other state. By using the pf
representation (1), $H_{\mathrm{trimer}}\rightarrow\bar{H}_{\mathrm{trimer}}$,
the state with the third spin in an unphysical $S=0$ configuration has the
same energy ($E=-3J/4$) as the physical ground state and is therefore equally
populated at $T=0$. To quantify this further we consider

$C_{i}(T)=4\langle\bar{S}^{z}_{i}\bar{S}^{z}_{i}\rangle=\langle\bar{P_{i}}\rangle,$
(4)

first studied in Ref. [25] in the context of pf-fRG. On the right side, the
projector to the local $S=1/2$ sector is

$\bar{P_{i}}=n_{i\uparrow}+n_{i\downarrow}-2n_{i\uparrow}n_{i\downarrow}.$ (5)

Only if the (local) pf configuration is entirely in the physical subspace,
$\bar{S}^{z}_{i}$ would be a faithful $S=1/2$ operator squaring to $1/4$ and
$C_{i}(T)$ would be unity. Thus $C_{i}(T)$ can be interpreted as a measure for
the suppression of unphysical states and we refer to it as spin projection.
Returning to the trimer, we obtain $C_{i}(T=0)=4/5<1$ signaling the presence
of an admixture of unphysical $S=0$ states to the ground state manifold. We
plot $C_{i}(T)$ for $\bar{H}_{\mathrm{trimer}}$ over a range of temperatures,
see Fig. 1. The spin projection decreases with increasing $T$ reaching
$C_{i}(T\rightarrow\infty)=1/2$, indicating that in the $T=\infty$ state on
the pf Hilbert space the occupation of unphysical states equals the occupation
of physical states.

Besides the trimer we found numerous other AFM Heisenberg spin clusters with
$N=5$ where unphysical states poison the pf ground state. This behavior can be
generally observed when a physical system cannot lower its ground state energy
by adding one more spin and is thus paradigmatic for frustrated systems. In
that case, preparing the additional site in a physical $S=1/2$ or an
unphysical $S=0$ state gives the same ground state energy. In Fig. 1, we show
results for the bow tie (c), centered square (d) and centered tetrahedron (e),
which are qualitatively similar to the trimer case (b). Since these shapes
(including the trimer) are basic building blocks of the triangular, Kagome,
face-centered cubic and centered pyrochlore lattice [42], respectively, it is
questionable if ground states of AFM Heisenberg pf Hamiltonians $\bar{H}$ on
these lattices reside entirely in the physical subspace. Finally, from the
clusters considered in Fig. 1, only the $N=2$ dimer (a) has a pf ground state
in the physical subspace.

##  III Popov-Fedotov trick: A review

As discussed in Sec. I, the spin operator $\bar{S}^{\mu}$ in pf representation
(1) acts like a spin-$1/2$ operator on the subspace
$\\{\ket{\uparrow},\ket{\downarrow}\\}$ but like a spin-$0$ operator on the
subspaces spanned by $\ket{\uparrow\downarrow}$ and $\ket{0}$, respectively.
Thermal occupation of the latter two sectors of the fermionic Hilbert space
will thus compromise the equivalence between the spin and pf partition
functions, $Z\neq\bar{Z}\equiv\mathrm{tr}\,e^{-\beta\bar{H}}$, respectively. A
straightforward projection to the physical subspace using
$\bar{P}=\Pi_{i=1,...,N}\bar{P_{i}}$ with local projectors $\bar{P}_{i}$ from
Eq. (5) becomes unpractical for a large number of spins $N$.

A more feasible projection scheme was originally proposed by Popov and Fedotov
[14] and later generalized by Prokof’ev and Svistunov [15]. It amounts to
replacing $\bar{H}\rightarrow\bar{H}+\bar{H}_{\text{PF}}^{\phi_{1,2}}$ where

$\displaystyle\bar{H}_{\text{PF}}^{\phi_{1,2}}=iT\left[n_{\downarrow}n_{\uparrow}\phi_{1}+(n_{\uparrow}-1)(n_{\downarrow}-1)\phi_{2}\right]$
(6)

with $\phi_{1,2}\in\mathbb{R}$ and the constraint
$e^{i\phi_{1}}+e^{i\phi_{2}}=0$. Now, the partition function in the pf Hilbert
space can be split into a sum over the purely physical part of the Hilbert
space (equivalent to $Z$ since $\bar{H}_{\text{PF}}^{\phi_{1,2}}=0$ in the
$S=1/2$ sector) and a part containing the expectation values of a product
state with $\mathcal{N}=1,2,...,N$ unphysical contributions,

$\displaystyle\bar{Z}=\mathrm{tr}(e^{-\beta\bar{H}-\beta\bar{H}_{\text{PF}}^{\phi_{1,2}}})$
$\displaystyle=Z+\sum_{\mathcal{N}=1}^{N}\sum_{\xi_{\mathcal{N}}}(Z_{\xi_{\mathcal{N}}}\prod_{j\in\xi_{\mathcal{N}}}F_{j}^{\phi_{1,2}}).$
(7)

Here, $\xi_{\mathcal{N}}$ counts different configurations of sites with
unphysical spin and $Z_{\xi_{\mathcal{N}}}$ is the physical partition function
of the subsystem with the $\mathcal{N}$ unphysical sites removed. Since
$\bar{H}$ acts trivially on unphysical states, we can compute the local trace
$F_{j}^{\phi_{1,2}}$ over the unphysical states at site $j$ explicitly,

$\displaystyle F_{j}^{\phi_{1,2}}$
$\displaystyle=\bra{0}_{j}e^{-\beta\bar{H}_{\text{PF}}^{\phi_{1,2}}}\ket{0}_{j}+\bra{\uparrow\downarrow}_{j}e^{-\beta\bar{H}_{\text{PF}}^{\phi_{1,2}}}\ket{\uparrow\downarrow}_{j}$
(8) $\displaystyle=e^{i\phi_{1}}+e^{i\phi_{2}}=0$ (9)

and Eq. (7) yields $\bar{Z}=Z$. A similar analytic argument shows the
faithfulness of spin correlation functions computed using the pf
representation with the PF term (6), in particular $C_{i}(T)=1$ for all
temperatures.

In the rest of the paper, we make the choice
$\phi_{1}=-\phi_{2}=\frac{\pi}{2}$. This reduces Eq. (6) to Eq. (2), taking
the simple non-interacting form of a potential which is, however, imaginary.

We emphasize that the PF trick applies also in the limit $T\rightarrow 0$ as
easily seen from the fact that in the partition function (for which the PF
term is rigorously defined) the Hamiltonian is multiplied by $\beta=1/T$ and
the PF contribution amounts to phase factors independent of $T$.

##  IV Symmetries, correlation functions and the fRG

In this technical section we review the symmetries of the generic pf
Hamiltonian $\bar{H}$ following Buessen et al. [35]. We then discuss the
necessary changes enforced by the addition of the PF potential term (2). The
resulting modifications in the parametrization of correlation- (and vertex-)
functions are minor and can easily be implemented into established numerical
codes solving the pf-fRG flow equations [26, 43]. We call the resulting pf-fRG
formalism including the PF potential ppf-fRG and restrict our focus to one-
loop evaluation of the flow equations (including Katanin truncation) and an
iterative solution of the parquet approximation, ppf-PA.

###  IV.1 Symmetries of the pf Hamiltonian

In analogy to the pf-fRG literature, we restrict our discussion to spin
Hamiltonians with two-spin interaction across bonds $(i,j)$ and disregard
magnetic fields,

$\displaystyle
H=\sum_{(i,j)}\sum_{\mu,\nu=x,y,z}J_{(i,j)}^{\mu\nu}S^{\mu}_{i}S^{\nu}_{j}.$
(10)

Using Eq. (1), $S_{i}^{\mu}\xrightarrow{}\bar{S}^{\mu}_{i}$, and the PF
potential term $\bar{H}_{\text{PF}}$ from Eq. (2), we consider the pf
Hamiltonian

$\bar{H}+\bar{H}_{\text{PF}}=\sum_{(i,j)}\sum_{\mu,\nu}J_{(i,j)}^{\mu\nu}\bar{S}^{\mu}_{i}\bar{S}^{\nu}_{j}+\frac{i\pi}{2\beta}\sum_{j}(n_{\uparrow
j}+n_{\downarrow j}-1).$ (11)

For $\bar{H}$ alone, Buessen et al. [35] have discussed the following
properties: $\bar{H}$ is hermitian (H) and symmetric with respect to local
$\text{U}(1)$, local particle-hole (lPH) and (anti-unitary) time-reversal (TR)
transformation. These symmetries act on
$\mathcal{F}_{i\alpha}\equiv(f_{i\alpha}^{\dagger},f_{i\alpha})^{T}$ as
follows,

$\displaystyle\mathcal{F}_{i\alpha}\xrightarrow{\text{H}}\left(\begin{array}[]{c}f_{i\alpha}\\\
f_{i\alpha}^{\dagger}\end{array}\right)\forall i,$
$\displaystyle\mathcal{F}_{i\alpha}\xrightarrow{\text{U(1)}}\left(\begin{array}[]{c}e^{i\theta_{i}}f_{i\alpha}^{\dagger}\\\
e^{-i\theta_{i}}f_{i\alpha}\end{array}\right),$ (16)
$\displaystyle\mathcal{F}_{i\alpha}\xrightarrow{\text{lPH}}\left(\begin{array}[]{c}\alpha
f_{i\bar{\alpha}}\\\ \alpha f_{i\bar{\alpha}}^{\dagger}\end{array}\right),$
$\displaystyle\mathcal{F}_{i\alpha}\xrightarrow{\text{TR}}\left(\begin{array}[]{c}e^{i\pi\alpha/2}f_{i\bar{\alpha}}^{\dagger}\\\
e^{-i\pi\alpha/2}f_{i\bar{\alpha}}\end{array}\right)\forall i.$ (21)

Note, that H and TR also involve a complex conjugation when acting on complex
numbers. We denote the spin index by
$\alpha=\\{\uparrow,\downarrow\\}=\\{+1,-1\\}$ and $\bar{\alpha}$ indicates a
spin-flip, $\bar{\alpha}=-\alpha$.

The PF-term $\bar{H}_{\text{PF}}$ is invariant under the local U(1) symmetry,
but changes its sign under hermitian conjugation, global particle-hole (PH)
and time-reversal symmetry,

$\displaystyle\bar{H}+\bar{H}_{\text{PF}}\xrightarrow{\text{H},\text{PH},\text{TR}}\bar{H}-\bar{H}_{\text{PF}}.$
(22)

Therefore, the full Hamiltonian $\bar{H}+\bar{H}_{\text{PF}}$ is only
symmetric under pairwise combinations of H, PH and TR symmetry. The lPH
symmetry ceases to be useful in the presence of $\bar{H}_{\text{PF}}$.

Depending on the model-specific $J_{(i,j)}^{\mu\nu}$, we can further find
lattice (L) or spin rotation (S) symmetries of $\bar{H}$,

$\displaystyle
J^{\mu\nu}_{(i,j)}\xrightarrow{L}J^{\mu\nu}_{(i^{\prime},j^{\prime})},$
$\displaystyle
J^{\mu\nu}_{(i,j)}\xrightarrow{S}J^{\mu^{\prime}\nu^{\prime}}_{(i,j)}.$ (23)

These symmetries are not broken by presence of $\bar{H}_{\text{PF}}$.

###  IV.2 Symmetries of correlation functions

The symmetries of the pf Hamiltonian impose symmetries on the correlation
functions, which constitute the basic starting point for the fRG treatment.
The single-particle correlation function (or propagator) is

$G\left(1^{\prime};1\right)=\int_{0}^{\beta}d\tau^{\prime}d\tau
e^{i\tau^{\prime}\omega^{\prime}-i\tau\omega}\left\langle\mathcal{T}_{\tau}f_{i^{\prime}\alpha^{\prime}}^{\dagger}(\tau^{\prime})f_{i\alpha}(\tau)\right\rangle,$
(24)

where we used the imaginary time-ordering and operators in the Heisenberg
picture. The two-particle correlation function reads

$G\left(1^{\prime},2^{\prime};1,2\right)=\\\
\int_{0}^{\beta}d\tau_{1^{\prime}}d\tau_{2^{\prime}}d\tau_{1}d\tau_{2}e^{i\left(\tau_{1^{\prime}}\omega_{1^{\prime}}+\tau_{2^{\prime}}\omega_{2^{\prime}}-\tau_{1}\omega_{1}-\tau_{2}\omega_{2}\right)}\\\
\times\left\langle\mathcal{T}_{\tau}f_{i_{1^{\prime}}\alpha_{1^{\prime}}}^{\dagger}(\tau_{1^{\prime}})f_{i_{2^{\prime}}\alpha_{2^{\prime}}}^{\dagger}(\tau_{2^{\prime}})f_{i_{1}\alpha_{1}}(\tau_{1})f_{i_{2}\alpha_{2}}(\tau_{2})\right\rangle.$
(25)

On the left-hand side of the above equations, we use multi-indices,

$\displaystyle 1$
$\displaystyle\equiv\left(i_{1},\phantom{-}\omega_{1},\alpha_{1}\right),$
$\displaystyle-1$
$\displaystyle\equiv\left(i_{1},-\omega_{1},\alpha_{1}\right),$
$\displaystyle\bar{1}$
$\displaystyle\equiv\left(i_{1},\phantom{-}\omega_{1},\bar{\alpha}_{1}\right).$

Following Ref. [35], we summarize the symmetry constraints on the correlation
functions in Tab. 1. For the two particle correlation function, there is the
additional crossing symmetry (X) related to anticommuting two fermionic
creation or annihilation operators.

$\displaystyle
G\left(1^{\prime},2^{\prime};1,2\right)=-G\left(1^{\prime},2^{\prime};2,1\right)$
(26) $\displaystyle=\>$ $\displaystyle
G\left(2^{\prime},1^{\prime};2,1\right)=-G\left(2^{\prime},1^{\prime};1,2\right).$

Table 1: Symmetries of $\bar{H}$ acting on $\bar{H}_{\text{PF}}$ and
correlation functions: Hermitian (H), local $\text{U}(1)$, global particle-
hole (PH) and time-reversal (TR) symmetry.

| $\bar{H}_{\text{PF}}$ | $G\left(1^{\prime};1\right)$ | $G\left(1^{\prime},2^{\prime};1,2\right)$
---|---|---|---
H | $-\bar{H}_{\text{PF}}$ | $\phantom{-\alpha^{\prime}\alpha}G(-1;-1^{\prime})^{*}$ | $\phantom{\alpha_{1}\alpha_{2}\alpha_{1}^{\prime}\alpha_{2}^{\prime}}G(-1,-2;-1^{\prime},-2^{\prime})^{*}$
PH | $-\bar{H}_{\text{PF}}$ | $-\alpha^{\prime}\alpha G(-\bar{1};-\bar{1}^{\prime})\phantom{{}^{*}}$ | $\alpha_{1}\alpha_{2}\alpha_{1}^{\prime}\alpha_{2}^{\prime}G(-\bar{1},-\bar{2};-\bar{1}^{\prime},-\bar{2}^{\prime})\phantom{{}^{*}}$
TR | $-\bar{H}_{\text{PF}}$ | $\phantom{-}\alpha^{\prime}\alpha G(-\bar{1}^{\prime};-\bar{1})^{*}$ | $\alpha_{1}\alpha_{2}\alpha_{1}^{\prime}\alpha_{2}^{\prime}G(-\bar{1}^{\prime},-\bar{2}^{\prime};-\bar{1},-\bar{2})^{*}$
U(1) | $\phantom{-}\bar{H}_{\text{PF}}$ | $e^{i(\theta_{i_{1}^{\prime}}-\theta_{i_{1}})}G\left(1^{\prime};1\right)$ | $e^{i(\theta_{i_{1}^{\prime}}+\theta_{i_{2}^{\prime}}-\theta_{i_{1}}-\theta_{i_{2}})}G\left(1^{\prime},2^{\prime};1,2\right)$

###  IV.3 Parameterization of correlators and vertices

The local U(1) symmetry (21) constrains the single-particle correlator to be
site-local and the two-particle correlator to be bi-local in real space. In
addition, imaginary time translation symmetry reduces the number of
independent frequencies by one. The dependence on the spin indices $\alpha$
can be parameterized by an expansion in Pauli matrices $\sigma^{\mu}$ with
$\mu=0,x,y,z$ and $\sigma^{0}$ the identity matrix. These considerations allow
for the parametrization [35]

$\displaystyle
G\left(1^{\prime};1\right)=\delta_{i^{\prime}i}\delta_{\omega^{\prime},\omega}\sum_{\mu=0,x,y,z}G^{\mu}_{i}(\omega)\sigma_{\alpha^{\prime}\alpha}^{\mu},$
(27)

and

$\displaystyle G(1^{\prime},2^{\prime};1,2)=$ (28)
$\displaystyle\times[\big{(}\sum_{\mu,\nu=0,x,y,z}G_{i_{1}i_{2}}^{\mu\nu}$
$\displaystyle(s,t,u)\sigma_{\alpha_{1^{\prime}}\alpha_{1}}^{\mu}\sigma_{\alpha_{2^{\prime}}\alpha_{2}}^{v}\big{)}\delta_{i_{1^{\prime}}i_{1}}\delta_{i_{2^{\prime}}i_{2}}$
$\displaystyle-(1^{\prime}\leftrightarrow
2^{\prime})]\delta_{\omega_{1^{\prime}}+\omega_{2^{\prime}},\,\omega_{1}+\omega_{2}},$

where the bosonic transfer frequencies are

$\displaystyle
s=\omega_{1^{\prime}}+\omega_{2^{\prime}}=\omega_{1}+\omega_{2},$ (29)
$\displaystyle
t=\omega_{1^{\prime}}-\omega_{1}=\omega_{2}-\omega_{2^{\prime}},$ (30)
$\displaystyle
u=\omega_{1^{\prime}}-\omega_{2}=\omega_{1}-\omega_{2^{\prime}}.$ (31)

The complex numbers $G^{\mu}_{i}(\omega)$ and $G_{i_{1}i_{2}}^{\mu\nu}(s,t,u)$
can be further constrained using the relations in the first three lines of
Tab. 1. This is summarized in Tab. 2.

Table 2: Constraints on the parameterized pf correlation functions of Eqns. (27) and (28). The constraints are labeled by the symmetries that have been used. $G_{i}^{\mu}(\omega)=\xi(\mu)G_{i}^{\mu}(\omega)$ | $(\mathrm{H}\circ\mathrm{TR})$
---|---
$G_{i}^{\mu}(\omega)=-G_{i}^{\mu}(\omega)^{*}$ | $(\mathrm{TR}\circ\mathrm{PH})$
$G_{i_{1}i_{2}}^{\mu\nu}(s,t,u)=\xi(\mu)\xi(v)G_{i_{1}i_{2}}^{\mu\nu}(s,-t,u)$ | $(\mathrm{H}\circ\mathrm{TR})$
$G_{i_{1}i_{2}}^{\mu\nu}(s,t,u)=\xi(\mu)\xi(v)G_{i_{2}i_{1}}^{v\mu}(s,t,-u)$ | $(\mathrm{X}\circ\mathrm{H}\circ\mathrm{TR})$
$G_{i_{1}i_{2}}^{\mu\nu}(s,t,u)=\xi(\mu)\xi(v)G_{i_{1}i_{2}}^{\mu\nu}(s,t,u)^{*}$ | $(\mathrm{H}\circ\mathrm{PH})$

In these relations, we have introduced the sign function

$\displaystyle\xi(\mu)=\begin{cases}+1&\text{ if }\mu=0,\\\ -1&\text{
otherwise. }\end{cases}$ (32)

Our results indicate that the propagator takes the simple diagonal form

$G(1^{\prime},1)=\delta_{i^{\prime},i}\delta_{\omega^{\prime},\omega}\delta_{\alpha^{\prime},\alpha}G_{i}(\omega)$
(33)

and we define the real self-energy $\gamma_{i}(\omega)$ via

$G_{i}(\omega)=\frac{1}{i\omega+i\gamma_{i}(\omega)}\equiv-ig_{i}(\omega)\in
i\mathbb{R}.$ (34)

In comparison to the standard pf-fRG without PF potential term [35],
$\gamma_{i}(\omega)$ is no longer anti-symmetric in $\omega$. Likewise, the
two-particle correlator has no symmetry relating $s\leftrightarrow-s$ or
$s\leftrightarrow u$. This reduction of symmetries amounts to a factor
$\approx 4$ in memory and computation time compared to the standard pf-fRG
scheme.

Finally, the fRG flow equations in vertex expansion are written in terms of
the (one-particle irreducible) vertices defined from the connected correlators
via the tree expansion [12]. According to Eq. (34), we have for the self-
energy
$\Sigma(1^{\prime};1)=\delta_{i^{\prime},i}\delta_{\omega^{\prime},\omega}\delta_{\alpha^{\prime},\alpha}\left\\{-i\gamma_{i}(\omega)\right\\}$.
The two-particle vertex $\Gamma(1^{\prime},2^{\prime};1,2)$ is defined from
the connected part of $-G(1^{\prime},2^{\prime};1,2)$ by amputating external
propagators. As the latter take the simple diagonal form (33), the vertex can
be parameterized in analogy to the correlator, i.e. Eq. (28) with $G$ replaced
by $\Gamma$ and the symmetries listed in Tab. 2 are also applicable to
$\Gamma_{i_{1}i_{2}}^{\mu\nu}(s,t,u)$.

So far, we have made no assumptions on the form of the spin-spin interaction
$J_{(i,j)}^{\mu\nu}$. If present, spin rotation and lattice symmetries can be
used to relate different site and spin indices of $\gamma_{i}(\omega)$ and
$G_{i_{1}i_{2}}^{\mu\nu}$. From now on we focus on the SO(3) symmetric
Heisenberg case,

$J_{(i,j)}^{\mu\nu}=\delta_{\mu,\nu}J_{(i,j)},$ (35)

for $\mu,\nu=x,y,z$ so that $G_{i_{1}i_{2}}^{00}\equiv
G_{i_{1}i_{2}}^{d}\in\mathbb{R}$ and
$G_{i_{1}i_{2}}^{xx}=G_{i_{1}i_{2}}^{yy}=G_{i_{1}i_{2}}^{zz}\equiv
G_{i_{1}i_{2}}^{s}\in\mathbb{R}$ are the only non-vanishing correlation
functions. Analogously, $\Gamma_{i_{1}i_{2}}^{s,d}$ are the only non-vanishing
vertices.

###  IV.4 ppf-fRG

For the (p)pf-fRG, a Matsubara cutoff scheme is applied to the bare propagator
$G^{(0)}_{j}(\omega)=1/i\omega\rightarrow
G^{(0),\Lambda}_{j}(\omega)\equiv\theta^{\Lambda}(\omega)G^{(0)}_{j}(\omega)$
[10, 16] with cutoff function $\theta^{\Lambda}(\omega)$ smoothly
interpolating from unity to zero as the magnitude of $\omega$ drops below the
cutoff scale $\Lambda$. When $\Lambda=0$ the bare propagator,
$G^{(0),\Lambda=0}_{j}(\omega)=1/i\omega$, is recovered and the action
describes the physical system of interest. At $\Lambda=\infty$, however, the
modified propagator vanishes and the vertex functions are trivial and
frequency independent,

$\displaystyle\gamma_{i}^{\Lambda=\infty}(\omega)=$
$\displaystyle\frac{\pi}{2\beta},$ (36)
$\displaystyle\Gamma_{i_{1}i_{2}}^{s,\Lambda=\infty}(s,t,u)=$ $\displaystyle
J_{(i_{1},i_{2})}/4,$ (37)
$\displaystyle\Gamma_{i_{1}i_{2}}^{d,\Lambda=\infty}(s,t,u)=$ $\displaystyle
0.$ (38)

The Wetterich equation [44] describes the flow of all $n-$particle vertex
functions under variation of $\Lambda$ from the trivial starting point
$\Lambda=\infty$ to the physical endpoint $\Lambda=0$. The resulting hierarchy
of flow equations can usually not be solved exactly, but has to be truncated,
with multiple truncation schemes available. Here, we focus on the established
one-loop scheme [12] together with Katanin truncation [45] which partially
considers the effect of the three-particle vertex in the flow of the two-
particle vertex and constitutes the standard choice in the pf-fRG literature
[10]. We refer to App. A for the choice of cutoff function and the flow
equations which do not differ from the standard pf-fRG case. We use discrete
Matsubara grids for all frequency arguments with about $N_{w}=30$ (positive)
frequencies and ensure that our results are converged in $N_{w}$. The ppf-fRG
flows are smooth in $\Lambda$ with features appearing around $\Lambda\sim J$
and a plateau towards $\Lambda\rightarrow 0$ from which we obtain the end-of-
flow results reported in the following. In App. B, we show how imaginary
frequency spin susceptibilities $\chi_{i_{1}i_{2}}(\Omega)$ and equal-time
spin correlation functions like the spin projection $C_{i}(T)$ are computed,
with a technical subtlety appearing for the bubble contribution in the latter
case. In summary, the main difference between the ppf-fRG and the pf-fRG is
the slightly reduced symmetry of the vertex functions discussed in the
previous subsection, the finite initial condition for the self energy and fRG
flows that are smooth and convergent.

###  IV.5 ppf-PA

Recently, a lot of effort has been put into generalizing the one-loop
truncation of general fermionic fRG flows to higher loop orders using
multiloop fRG (mfRG) [38, 39, 40]. Applications include the Anderson impurity
model [46] and the two-dimensional Hubbard model [47, 48]. By construction,
vertices obtained with mfRG in the limit of infinite loops converge to
solutions of the parquet approximation (PA), a complementary diagrammatic
formulation of the many-body problem in which the self-energy and frequency-
dependent contributions to the two-particle vertex are self-consistently
described by the Schwinger-Dyson and three Bethe-Salpeter equations,
respectively. In-depth discussions of their structure for pseudo-fermion
Hamiltonians can, for example, be found in Refs. [25, 26]. In contrast to the
full parquet equations, the PA neglects frequency dependent contributions to
the fully two-particle irreducible vertex $I_{\text{2PI}}$ which thus reduces
to the bare vertex $\Gamma_{0}$, with $\Gamma_{0}\sim J$ for pf systems.
Deviations from exact vertices set in at fourth order in $J/T$, corresponding
to the so-called envelope diagram.

Using the very same initial conditions as in the $\Lambda\to\infty$ limit of
ppf-fRG (36)–(38), we numerically converge the algebraic equations of the PA
using forward iterations combined with a mixed update scheme, which determines
the input for the next iteration as

$\displaystyle x_{\text{new}}=(1-\lambda)x_{\text{old}}+\lambda
f(x_{\text{old}})\,,$ (39)

where $x=(\Sigma,\Gamma)$ and $f$ schematically denotes the Schwinger-Dyson
and Bethe-Salpeter equations. For $J/T\ll 1$, full updates ($\lambda=1$) were
sufficient to meet the convergence criterion
$||f(x_{\text{old}})-x_{\text{old}}||<10^{-6}$ between subsequent iterations
($||.||$ is the maximum norm). For lower temperatures $\lambda$ had to be
reduced from unity in order to obtain converged results. We choose extended
Matsubara grids with up to $48\crossproduct 24^{2}$ frequencies in mixed
bosonic-fermionic frequency notation to parameterize the two-particle vertex
as well as $32$ frequencies for the self-energy. For the vertex, we take into
account the decomposition of each channel into its respective asymptotic
functions $K_{1}$, $K_{2}$ ($\bar{K}_{2}$) and $R$ as detailed in Ref. [49].
We checked convergence with respect to the number of frequencies and, in most
cases (see App. C and D), found no significant changes of our results if more
were included.

##  V Results

###  V.1 Benchmark: Small spin clusters

To benchmark the proposed ppf-fRG and ppf-PA approaches, we consider the AFM
Heisenberg dimer $H_{\mathrm{dimer}}=J\mathbf{S}_{0}\cdot\mathbf{S}_{1}$ with
$J=1$ and focus on the static local and non-local susceptibilities and the
spin projection $C_{i}$ as a function of $T$. Exact ED results of the spin
Hamiltonian are shown as solid black lines in Fig. 2, see Ref. [41] for
closed-form expressions. The (end-of-flow) ppf-fRG results are denoted by
orange symbols. For large and moderate temperatures $T\gtrsim 0.4J$, the exact
susceptibilities are accurately reproduced by the ppf-fRG. For lower $T$, the
susceptibilities become unphysically large in magnitude. This breakdown of
accuracy is also reflected in $C_{i}$ which considerably drops below unity
with decreasing $T$.

The inaccuracies at low $T$ are due to the truncation of the ppf-fRG flow
equations so that self-energy and (two-particle) vertex are only correct up to
order $J^{2}/T$, see App. C for numerical confirmation. However, the advantage
of fRG over naive second-order perturbation theory (SOPT, dotted line in Fig.
2) is the re-summation of certain diagrams to infinite order [16], which is
essential for the detection of magnetic ordering tendencies (see below). In
the dimer case, the re-summation stabilizes the susceptibilities beyond
temperatures where SOPT is applicable.

As the treatment of the full three-particle vertex is prohibitively expensive,
an interesting question is if the above truncation problem can be alleviated
by invoking higher-loop orders beyond the Katanin truncation. This has been
shown to be numerically feasible in the pf-fRG [26, 25]. In fact, as explained
in Sec. IV, our finite-temperature application makes it even possible to
converge the ppf-PA equations equivalent to the loop-converged multi-loop
result with an error of order $J^{4}/T^{3}$ and $J^{5}/T^{4}$ for vertex and
self-energy, respectively. Surprisingly, our ppf-PA results (green symbols in
Fig. 2) show no systematic improvement compared to the ppf-fRG. This
unexpected finding also questions the usefulness of multi-loop extensions in
the context of pf applications.

For completeness, blue symbols in Fig. 2 show results of the standard one-loop
pf-fRG applied to $\bar{H}$ (without PF potential $\bar{H}_{\text{PF}}$) which
at large $T$ compare well with the exact (but unphysical) ED results obtained
from $\bar{H}$ (dashed black line). Like in the ppf-fRG, agreement only holds
for $T\gtrsim 0.4J$, suggesting the truncation error in the fRG flow equations
is largely independent of $\bar{H}_{\text{PF}}$.

Finally, in Fig. 3 we consider the Heisenberg trimer, Eq. (3), where the ppf-
fRG and ppf-PA results are qualitatively similar to the dimer and strengthen
the conclusions given above for the latter case. It is interesting to note,
that, presumably due to the truncation of the flow equations, the ppf-fRG and
pf-fRG converge to the same result for low $T$ even though the ground state of
the pf trimer is not in the physical Hilbert space.

Figure 2: Heisenberg dimer: Local and non-local susceptibilities (top panel:
squares and crosses, respectively) and spin projection
$C_{i}(T)=4\langle\bar{S}^{z}_{i}\bar{S}^{z}_{i}\rangle$ (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 $T\rightarrow 0$ and have been multiplied by temperature to obtain
finite values at $T=0$. The bottom panel shows the spin projection $C_{i}(T)$.
The symbols follow the convention of Fig. 2.

###  V.2 Finite temperature magnetization transition in three dimensions

One of the most remarkable properties of spin systems at finite $T$ is the
possible appearance of a magnetization transition at a critical temperature
$T_{c}$. As we consider (short-range coupled) Heisenberg systems with
continuous SO(3) spin rotation symmetry, these transitions only occur at
dimension three. Despite their classical nature, it is interesting to study
magnetic phase transitions in models of – possibly frustrated – quantum spins.
Here we investigate if ppf-fRG can detect the Néel transition in a nearest-
neighbor cubic lattice Heisenberg AFM ($J_{1}=1$) using finite-size scaling of
the correlation ratio [50, 51, 52],

$\displaystyle\xi/L=\frac{1}{2\pi}\sqrt{\chi(\mathbf{Q})/\chi(\mathbf{Q}+\bm{\delta})-1},$
(40)

where
$\chi(\mathbf{q})=\sum_{j}e^{-i\mathbf{q}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}\chi_{ij}$
is the momentum space static susceptibility, $\mathbf{Q}=(\pi,\pi,\pi)$ is the
ordering wave vector, $\bm{\delta}=(0,0,2\pi/L)$ and $L$ is a measure of the
system size.

In the ppf-fRG of the cubic lattice AFM we assume a translation invariant
infinite system, but limit the range of allowed non-trivial correlations by
restricting the vertex functions $\Gamma^{s,d}_{i_{1}i_{2}}$ to
$|\mathbf{r}_{i_{1}}-\mathbf{r}_{i_{2}}|\leq\tilde{L}$ [10]. The length scale
$\tilde{L}$ can be used for finite-size scaling [53]. To smooth out discrete-
lattice effects, we define $L=2(\frac{3}{4\pi}N)^{1/3}\simeq 2\tilde{L}$ to be
used in Eq. (40), where $N$ is the number of sites to which the reference site
is connected by a possible non-trivial vertex (including the on-site vertex).
This particular choice of $L$ corresponds to the diameter of the smeared-out
correlation-sphere including $N$ sites.

Close to the critical temperature $T_{c}$, the anticipated scaling form of the
AFM spin correlation length $\xi$ is

$\displaystyle\xi/L\sim g_{\pm}(L\absolutevalue{T-T_{c}}^{\nu}),$ (41)

so that $\xi/L$ becomes independent of $L$ at $T=T_{c}$, the sign $\pm$ refers
to the sign of $T-T_{c}$ and $\nu$ is the universal critical exponent [54].
The ppf-fRG results for the correlation length, Néel susceptibility
$\chi_{N}=\chi(\mathbf{Q})$ and the spin projection $C(T)$ are shown in Fig.
4. We indeed find a clear line-crossing in the $\xi/L$ data indicating
$T^{\text{fRG}}_{c}\simeq 0.61$ significantly below the error controlled
quantum Monte-Carlo result $T_{c}^{\text{QMC}}=0.946(1)$ [55]. This might be
related to an underestimation of the spin projection $C(T)$. The scaling
collapse in Fig. 5 shows consistency with the correct three-dimensional
Heisenberg universality class with $\nu\simeq 0.71$ [56]. In App. D, we
further investigate the truncation dependence of these quantities considering
analogous simulations using the ppf-fRG without Katanin truncation and in the
random phase approximation (RPA) where both $C(T)$ and $T_{c}$ are
overestimated and the critical exponent is consistent with the mean-field
value $\nu=0.5$. We also discuss results obtained from ppf-PA which indicates
that this method is not well suited for the assessment of magnetic ordering
transitions.

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.

##  VI Conclusion

In summary, we have considered simple small spin clusters as examples to show
that working with the pf representation (1) generally requires the PF
projection to the physical $S=1/2$ subspace at all temperatures. This
challenges the existing approach in the pf-fRG literature, which omits the
projection and focuses on zero temperature. We have leveraged the pf-fRG to
include the PF projection in potential form which requires only minor
technical modifications and therefore can be readily adopted to existing
variants of the method. The discrete nature of finite-temperature Matsubara
frequencies simplifies the numerical implementation considerably.

In conclusion, for small benchmark models the proposed ppf-fRG yields
quantitatively reliable end-of-flow results at large and intermediate
temperatures but fails at small temperatures $T\lesssim 0.4J$ due to the
omission of higher-order in $J/T$ diagrams, a problem that also could not be
alleviated by adopting a parquet scheme. We note that a detailed understanding
for the surprisingly poor performance of the latter is lacking at this point.
For forthcoming applications, we suggest that the deviation of the spin
projection $C_{i}(T)$ from the exact value of unity by $10\%$ or more signals
the quantitative failure of the ppf-fRG.

This failure of the (p)pf-fRG at low and zero temperature also questions an
earlier proposal to remove unphysical contributions of the pf representation
in the context of the pf-fRG at $T=0$. The idea rests on the addition of an
on-site term $-J_{0}\sum_{\mu=x,y,z}\bar{S}^{\mu}_{i}\bar{S}^{\mu}_{i}$ [23]
with $J_{0}>0$ penalizing the $S=0$ sector energetically versus the $S=1/2$
sector. If the treatment of the pf Hamiltonian was exact, the above idea would
be a valid $T=0$ alternative to the PF trick (the latter works at any $T$).
However, due to the unavoidable approximations in the diagrammatic methods at
hand, results obtained from the on-site term [23, 24] should be considered
with appropriate care.

For infinite systems in three dimensions, we have demonstrated that it is
possible to robustly detect magnetic ordering by finite-size scaling of the
ppf-fRG data. This establishes an alternative to the less physical and
implementation dependent concepts of flow-divergence or flow-breakdown at
cutoff scale $\Lambda_{\star}$ which, by mean-field arguments, can be related
to $T_{c}$ [19, 22, 24]. However, although the correct type of Néel order has
been predicted by the ppf-fRG for the nearest-neighbor AFM Heisenberg model on
the cubic lattice, it remains unclear why the transition appears at a critical
temperature more than $30\%$ below benchmark results. A possible hint might be
the spin projection $C_{i}(T=T_{c}^{\mathrm{fRG}})\simeq 0.88$ which is out of
the range identified as reliable above.

Finally, there exist two alternatives to the PF projection scheme that we
would like to mention. First, in a field theory framework, a functional delta-
function representation can be used to constrain the pf operators to the
physical subspace. This then leads to the notion of a bosonic gauge field
which plays an important role in the (mean-field) theory of spin liquids [5].
On the computational side, however, the introduction of such a bosonic field
would require a multitude of additional vertex functions [12], a formidable
challenge yet to be faced. Second, it is well known that a faithful quantum
spin $S=1/2$ representation exists in terms of Majorana fermions [57, 58]. The
associated pseudo-Majorana fRG (pm-fRG) has been developed only recently [41].
The data from pm-fRG applied to the benchmark clusters treated in Sec. V.1 are
shown by the gray lines in Fig. 2 (dimer) and Fig. 3 (trimer). The pm-fRG
results are very similar to those of the ppf-fRG, with the same difficulties
at small $T$. However, for the detection of the magnetic phase transition in
the cubic lattice case, the pm-fRG determines $T_{c}$ much more accurately
only $5\%$ below the exact result [53].

## Acknowledgements

We thank Johannes Reuther, Nils Niggemann, Jan von Delft, Ronny Thomale and
Marc Ritter for fruitful discussions. The authors gratefully acknowledge the
Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this
project by providing computing time through the John von Neumann Institute for
Computing (NIC) on the GCS Supercomputer JUWELS at Jülich Supercomputing
Centre (JSC). The computations in this work were, in part, run at facilities
supported by the Scientific Computing Core at the Flatiron Institute, a
division of the Simons Foundation. We acknowledge the usage of the Noctua2
cluster at the Paderborn Center for Parallel Computing (PC2). The authors
acknowledge financial support by a MCQST-START fellowship and by the Munich
Quantum Valley, which is supported by the Bavarian state government with funds
from the Hightech Agenda Bayern Plus.

##  Appendix A ppf-fRG flow equations for the Heisenberg case

The one-loop flow equations for the pf-fRG as found in Refs. [16, 59] do not
rely on the vertex symmetries that are broken by the PF potential term (2),
see the discussion in Sec. IV. Here, we write these flow equations for the
Heisenberg case where the vertices are parameterized in a density and spin
part, $\Gamma^{s}$ and $\Gamma^{d}$, see Sec. IV.3.
The cutoff dependent propagator is defined as
$G^{\Lambda}_{j}(\omega)=-ig^{\Lambda}_{j}(\omega)=\frac{\theta^{\Lambda}(\omega)}{i\omega+i\theta^{\Lambda}(\omega)\gamma_{j}(\omega)}$
where $\Sigma_{j}(\omega)=-i\gamma_{j}(\omega)$ and we chose the regulator to
be Lorentzian
$\theta^{\Lambda}(\omega)=\frac{\omega^{2}}{\omega^{2}+\Lambda^{2}}$. The flow
equation for the spin part of the vertex reads

$\displaystyle\frac{d}{d\Lambda}\Gamma^{s,\Lambda}_{i_{1}i_{2}}(s,t,u)$
$\displaystyle=T\sum_{\omega}$ (42) $\displaystyle
P^{\Lambda}_{i_{1}i_{2}}(\omega,s-\omega)\times$
$\displaystyle[-2\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}+\Gamma^{d,\Lambda}\Gamma^{s,\Lambda}+\Gamma^{s,\Lambda}\Gamma^{d,\Lambda}](s,\omega_{1^{\prime}}-\omega,\omega-\omega_{2^{\prime}})_{i_{1}i_{2}}(s,\omega-\omega_{1},\omega-\omega_{2})_{i_{1}i_{2}}$
$\displaystyle-\sum_{j}P^{\Lambda}_{jj}(\omega,\omega-t)\times$
$\displaystyle[2\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}](\omega_{1}+\omega,t,\omega_{1^{\prime}}-\omega)_{i_{1}j}(\omega+\omega_{2^{\prime}},t,\omega-\omega_{2})_{ji_{2}}$
$\displaystyle+P^{\Lambda}_{i_{2}i_{2}}(\omega,\omega-t)\times$
$\displaystyle[\Gamma^{s,\Lambda}\Gamma^{d,\Lambda}-\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}](\omega_{1}+\omega,t,\omega_{1^{\prime}}-\omega)_{i_{1}i_{2}}(\omega+\omega_{2^{\prime}},\omega-\omega_{2},t)_{i_{2}i_{2}}$
$\displaystyle+P^{\Lambda}_{i_{1}i_{1}}(\omega,\omega-t)\times$
$\displaystyle[\Gamma^{d,\Lambda}\Gamma^{s,\Lambda}-\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}](\omega_{1}+\omega,\omega_{1^{\prime}}-\omega,t)_{i_{1}i_{1}}(\omega+\omega_{2^{\prime}},t,\omega-\omega_{2})_{i_{1}i_{2}}$
$\displaystyle+P^{\Lambda}_{i_{1}i_{2}}(\omega,\omega+u)\times$
$\displaystyle[2\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}+\Gamma^{d,\Lambda}\Gamma^{s,\Lambda}+\Gamma^{s,\Lambda}\Gamma^{d,\Lambda}](\omega_{1^{\prime}}+\omega,\omega_{2}-\omega,u)_{i_{1}i_{2}}(\omega+\omega_{1},\omega-\omega_{2^{\prime}},u)_{i_{1}i_{2}}$

and the density part flows according to

$\displaystyle\frac{d}{d\Lambda}\Gamma^{d,\Lambda}_{i_{1}i_{2}}(s,t,u)$
$\displaystyle=T\sum_{\omega}$ (43) $\displaystyle
P^{\Lambda}_{i_{1}i_{2}}(\omega,s-\omega)\times$
$\displaystyle[\Gamma^{d,\Lambda}\Gamma^{d,\Lambda}+3\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}](s,\omega_{1^{\prime}}-\omega,\omega-\omega_{2^{\prime}})_{i_{1}i_{2}}(s,\omega-\omega_{1},\omega-\omega_{2})_{i_{1}i_{2}}$
$\displaystyle-\sum_{j}P^{\Lambda}_{jj}(\omega,\omega-t)\times$
$\displaystyle[2\Gamma^{d,\Lambda}\Gamma^{d,\Lambda}](\omega_{1}+\omega,t,\omega_{1^{\prime}}-\omega)_{i_{1}j}(\omega+\omega_{2^{\prime}},t,\omega-\omega_{2})_{ji_{2}}$
$\displaystyle+P^{\Lambda}_{i_{2}i_{2}}(\omega,\omega-t)\times$
$\displaystyle[3\Gamma^{d,\Lambda}\Gamma^{s,\Lambda}+\Gamma^{d,\Lambda}\Gamma^{d,\Lambda}](\omega_{1}+\omega,t,\omega_{1^{\prime}}-\omega)_{i_{1}i_{2}}(\omega+\omega_{2^{\prime}},\omega-\omega_{2},t)_{i_{2}i_{2}}$
$\displaystyle+P^{\Lambda}_{i_{1}i_{1}}(\omega,\omega-t)\times$
$\displaystyle[3\Gamma^{s,\Lambda}\Gamma^{d,\Lambda}+\Gamma^{d,\Lambda}\Gamma^{d,\Lambda}](\omega_{1}+\omega,\omega_{1^{\prime}}-\omega,t)_{i_{1}i_{1}}(\omega+\omega_{2^{\prime}},t,\omega-\omega_{2})_{i_{1}i_{2}}$
$\displaystyle+P^{\Lambda}_{i_{1}i_{2}}(\omega,\omega+u)\times$
$\displaystyle[\Gamma^{d,\Lambda}\Gamma^{d,\Lambda}+3\Gamma^{s,\Lambda}\Gamma^{s,\Lambda}](\omega_{1^{\prime}}+\omega,\omega_{2}-\omega,u)_{i_{1}i_{2}}(\omega+\omega_{1},\omega-\omega_{2^{\prime}},u)_{i_{1}i_{2}}$

where

$P^{\Lambda}_{ij}(\omega,\omega^{\prime})=\left(-ig^{\Lambda}_{i}(\omega)\right)S^{\Lambda}_{j}(\omega^{\prime})+\left(-ig_{j}^{\Lambda}(\omega^{\prime})\right)S^{\Lambda}_{i}(\omega),$
(44)

and $S^{\Lambda}_{i}(\omega)$ is the single scale propagator

$\displaystyle
S^{\Lambda}_{j}(\omega)=-\frac{\partial}{\partial\Lambda}G^{\Lambda}_{j}(\omega)=\left(-ig_{j}^{\Lambda}(\omega)\right)\frac{\partial}{\partial\Lambda}\frac{i\omega}{\theta^{\lambda}(\omega)}\left(-ig_{j}^{\Lambda}(\omega)\right)=-i\left(g_{j}^{\Lambda}(\omega)\right)^{2}\frac{\partial}{\partial\Lambda}\frac{\omega}{\theta^{\lambda}(\omega)}.$
(45)

Finally, the flow equation for the self energy is

$\displaystyle\frac{d}{d\Lambda}\gamma^{\Lambda}_{i}(\omega_{1})=T\sum_{\omega_{2}}\sum_{j}[2\Gamma_{d}(\omega_{1}+\omega_{2},0,\omega_{1}-\omega_{2})_{ij}-\delta_{ij}(3\Gamma_{s}+\Gamma_{d})(\omega_{1}+\omega_{2},\omega_{1}-\omega_{2},0)_{ii}]S_{j}(\omega_{2}).$
(46)

In the Katanin truncation scheme, the partial derivative in the single scale
propagator becomes a full derivative only in the flow equations for the vertex
[16],

$\displaystyle
S^{\Lambda}_{j}(\omega)=-\frac{\differential}{\differential\Lambda}G^{\Lambda}_{j}(\omega)=-i(g_{j}^{\Lambda}(\omega))^{2}\left(\frac{\partial}{\partial\Lambda}\frac{\omega}{\theta^{\lambda}(\omega)}+\frac{\differential}{\differential\Lambda}\gamma_{j}^{\Lambda}(\omega)\right).$
(47)

##  Appendix B Observables from vertex functions

Following Ref. [25] the susceptibilities
$\chi_{ij}(\Omega)\equiv\chi_{ij}^{zz}(\Omega)$ for the Heisenberg case can be
computed from the self energy and vertices

$\displaystyle\chi_{ij}^{zz}(\Omega)=$
$\displaystyle\int_{0}^{\beta}\mathrm{d}\tau
e^{i\Omega\tau}\sum_{\alpha_{1},\alpha_{1^{\prime}},\alpha_{2},\alpha_{2^{\prime}}}\frac{1}{4}\sigma_{\alpha_{1}\alpha_{1^{\prime}}}^{z}\sigma_{\alpha_{2}\alpha_{2^{\prime}}}^{z}\langle\mathcal{T}_{\tau}f^{\dagger}_{i\alpha_{1}}(\tau)f_{i\alpha_{1^{\prime}}}(\tau)f^{\dagger}_{j\alpha_{2}}(0)f_{j\alpha_{2^{\prime}}}(0)\rangle$
(48) $\displaystyle=$
$\displaystyle-\frac{\delta_{ij}}{2\beta}\sum_{\omega}G_{i}(\omega)G_{i}(\omega+\Omega)$
(49)
$\displaystyle-\frac{\delta_{ij}}{2\beta^{2}}\sum_{\omega,\omega^{\prime}}G_{i}(\omega)G_{i}(\omega+\Omega)G_{i}(\omega^{\prime})G_{i}(\omega^{\prime}+\Omega)\times[\Gamma^{s,\Lambda}-\Gamma^{d,\Lambda}](\omega+\omega^{\prime}+\Omega,\omega-\omega^{\prime},\Omega)_{ii}$
$\displaystyle-\frac{1}{2\beta^{2}}\sum_{\omega,\omega^{\prime}}G_{i}(\omega)G_{i}(\omega+\Omega)G_{j}(\omega^{\prime})G_{j}(\omega^{\prime}+\Omega)\times
2\Gamma^{s,\Lambda}(\omega+\omega^{\prime}+\Omega,\Omega,\omega-\omega^{\prime})_{ij}.$

To compute the equal time susceptibility we have to sum over all bosonic
Matsubara frequencies $\Omega$ and use the infinitesimal positive imaginary
time $\delta\tau$,

$\displaystyle\langle S^{z}_{i}S^{z}_{j}\rangle=$
$\displaystyle\sum_{\alpha_{1},\alpha_{1^{\prime}},\alpha_{2},\alpha_{2^{\prime}}}\frac{1}{4}\sigma_{\alpha_{1}\alpha_{1^{\prime}}}^{z}\sigma_{\alpha_{2}\alpha_{2^{\prime}}}^{z}\langle\mathcal{T}_{\tau}f^{\dagger}_{i\alpha_{1}}(+\delta\tau)f_{i\alpha_{1^{\prime}}}(+\delta\tau)f^{\dagger}_{j\alpha_{2}}(0)f_{j\alpha_{2^{\prime}}}(0)\rangle$
(50) $\displaystyle=$
$\displaystyle-\frac{\delta_{ij}}{2\beta^{2}}\lim_{\delta\tau\xrightarrow[]{}0}[\sum_{\omega}e^{i\delta\tau\omega}G_{i}(\omega)][\sum_{\omega}e^{-i\delta\tau\omega}G_{i}(\omega)]$
(51)
$\displaystyle-\frac{\delta_{ij}}{2\beta^{3}}\sum_{\omega,\omega^{\prime},\Omega}G_{i}(\omega)G_{i}(\omega+\Omega)G_{i}(\omega^{\prime})G_{i}(\omega^{\prime}+\Omega)\times[\Gamma^{s}-\Gamma^{d}](\omega+\omega^{\prime}+\Omega,\omega-\omega^{\prime},\Omega)_{ii}$
$\displaystyle-\frac{1}{2\beta^{3}}\sum_{\omega,\omega^{\prime},\Omega}G_{i}(\omega)G_{i}(\omega+\Omega)G_{j}(\omega^{\prime})G_{j}(\omega^{\prime}+\Omega)\times
2\Gamma^{s}(\omega+\omega^{\prime}+\Omega,\Omega,\omega-\omega^{\prime})_{ij}.$

In the bubble term, the limit $\delta\tau\xrightarrow[]{}0$ has to be taken
with care and can not be straightforwardly computed numerically. We compute it
by adding and subtracting the sum over the bare part of the propagator and
calculate the second sum analytically. The remaining sum can be calculated
numerically.

$\displaystyle\lim_{\delta\tau\xrightarrow[]{}0}\sum_{\omega}e^{i\delta\tau\omega}G(\omega)$
$\displaystyle=\underbrace{\sum_{\omega}(G(\omega)+\frac{i}{\omega+T\frac{\pi}{2}})}_{\text{finite}}-\lim_{\delta\tau\xrightarrow[]{}0}\sum_{\omega}e^{i\delta\tau\omega}\frac{i}{\omega+\frac{\pi}{2\beta}}$
(52)
$\displaystyle=\sum_{\omega}(G(\omega)+\frac{i}{\omega+\frac{\pi}{2\beta}})+i\beta(\frac{1}{2}-\frac{i}{2}).$

##  Appendix C Perturbative check for ppf-fRG and ppf-PA at large temperatures

When considering large or intermediate temperatures, the one-loop ppf-fRG and
the ppf-PA are error controlled with respect to the exact solution. In the
one-loop truncation, diagrams of order $\frac{J^{3}}{T^{2}}$ and higher are
neglected when calculating self energy or vertex. Therefore, the difference of
the ED solution with the fRG solution should scale with $\frac{J^{3}}{T^{2}}$.
This behavior can be seen in Fig. 6(a) for the self-energy of the Heisenberg
dimer. For small temperatures, when $\frac{J}{T}$ becomes large, the scaling
breaks down. In the ppf-PA, diagrams of order $\frac{J^{5}}{T^{4}}$ are
neglected for the self energy. Compared to one-loop, this error scaling in
ppf-PA is challenging to observe and can only be seen for very large frequency
grids and at high temperatures, see Fig. 6(b).

(a)  (b)

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.

##  Appendix D Finite temperature phase transitions for other truncation
schemes

In Sec. V.2 the ppf-fRG was applied to asess the magnetic phase transition in
the cubic lattice AFM Heisenberg model. Here, we additionally investigate the
ppf-fRG without Katanin truncation, the RPA [10, 16], and the ppf-PA as
further benchmark. The numerical RPA calculation is implemented using the ppf-
fRG with Katanin truncation but a restriction to the terms including a site-
sum $\sum_{j}$ on the right-hand side of Eqns. (42), (43) and (46). The RPA
confirms that the implementation of the PF term can reproduce the analytic
spin mean-field result $T_{c}=1.5$ and $\nu=0.5$ for Heisenberg spins, see
Fig. 7 (a), (c). The one-loop scheme without Katanin truncation yields
$T_{c}=1.29$ as well as a critical exponent consistent with the mean-field
result $\nu=0.5$, see Fig 7 (b), (d). In both the RPA and the one-loop
truncation without Katanin, the spin projection exceeds unity.

(a)  (b)

(c)  (d)

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 $C(T)=1$.

We finally consider the ppf-PA formalism. As can be seen in the upper panel of
Fig. 8, we indeed find a smeared-out line crossing for $\xi/L$, with the three
largest system sizes crossing at $T_{c}\simeq 0.855$. However, the
corresponding Néel susceptibilities (see the middle panel in Fig. 8) are
substantially smaller than those obtained in ppf-fRG and a proper scaling
collapse can be found neither for $\nu=0.5$ nor $\nu=0.71$. On the numerical
side, we find that the rate of convergence drops considerably for $T/J\lesssim
1.0$ and the results, including the location of the intersection point of the
$\xi$-scaling, become highly sensitive to numerical details such as the
specific choice of mixing factors and solution algorithm. This could indicate
that the ppf-PA fixed points in vicinity of the critical regime are strongly
repulsive and hard to access in numerical calculations. In conclusion, for the
assessment of magnetic phase transition the ppf-PA seems less reliable and
consistent than the one-loop ppf-fRG scheme.

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 $T_{c}^{\mathrm{ppf-PA}}\simeq
0.855$. A corresponding scaling collapse consistent with $\nu=0.71$ or
$\nu=0.5$ could not be found. For $C(T)$ all five lines lie on top of each
other.

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