††thanks: These authors contributed equally to the work.††thanks: These
authors contributed equally to the work.

# Asymptotic sign free in interacting fermion models

Zi-Xiang Li  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China  Zhou-Quan
Wan  Institute for Advanced Study, Tsinghua University, Beijing 100084, China.
Hong Yao<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua
University, Beijing 100084, China.

###### Abstract

As an intrinsically-unbiased approach, quantum Monte Carlo (QMC) is of vital
importance in understanding correlated phases of matter. Unfortunately, it
often suffers notorious sign problem when simulating interacting fermion
models. Here, we show for the first time that there exist interacting fermion
models whose sign problem becomes less severe for larger system sizes and
eventually disappears in the thermodynamic limit, which we dub as “asymptotic
sign free”. We demonstrate asymptotically-free sign in determinant QMC for
various interacting models. Moreover, based on renormalization-group-like
ideas we propose a heuristic understanding of the feature of asymptotic sign
free. We believe that asymptotic sign free behavior could shed new lights to
deepening our understanding of sign problem. More importantly, it can provide
a promising way to decipher intriguing physics in correlated models which were
conventionally thought not accessible by QMC.

Introduction: Understanding novel correlation physics such as high-temperature
superconductivity [1, 2, 3, 4, 5] in interacting systems has been one of
central issues in modern condensed matter physics and other related fields.
However, interacting quantum many-body systems in more than one dimension,
especially those with strong correlations, are generally beyond the
solvability of analytical approaches with well theoretical control. Therefore,
developing efficient numerical approaches to study quantum correlated systems
is thus of vital importance. Quantum Monte Carlo (QMC) [6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18], which simulates quantum many-body systems by
stochastic sampling in an intrinsically-unbiased way, is among the most
important approaches. Unfortunately, QMC simulations of quantum models often
suffer from notorious sign problem, namely the weights of sampling
configurations may not be positive definite for fermionic models [19, 20] and
frustrated spin models [21, 22, 23]. Sign problem is currently the main
obstacle in applying QMC to study quantum many-body physics efficiently. It is
then highly desired to investigate general features of sign problem and to
solve sign problem of various interacting models potentially hosting
intriguing physics.

Figure 1: The schematic representation of the RG flow of interaction $U$ in
(a) and the RG-like flow of average sign $\left\langle S\right\rangle$ in (b).
(a) There is a physical quantum phase transition at critical interaction
$U=U_{c}$ between two distinct phases (one at weak coupling and the other at
strong coupling). (b) There is a sign transition point at $U=U_{*}$ between
two distinct regimes of sign scaling behaviors. When the interaction is
relatively weak ($U<U_{*}$), the average sign $\left<S\right>$ increases
asymptotically to one as the system size increases, which is qualitatively
different from the conventional exponential decaying behaviour of the average
sign for strong interactions ($U>U_{*}$). Note that the physical phase
transition point $U_{c}$ is close to the sign transition point $U_{*}$
although they are in general different.

Even though it was shown that a generic solution to the sign problem is
nondeterministic polynomial (NP) hard [24], sign problem can be cured for many
specific quantum models. In the past two decades, tremendous efforts have been
made in solving sign problems by various approaches, including designing new
algorithms [25, 26], utilizing complex-fermion symmetries [27, 28], employing
Majorana representations [29, 30], and mathematical structures [31, 32, 33]
(for a recent review, see, e.g., Ref. [6] by two of us). Indeed, various
intriguing physics has been revealed in studying strongly correlated models by
sign-free QMC simulation [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. Additionally,
new strategies such as machine learning are developed to mitigate sign problem
[63, 64, 65, 66, 67, 68] in QMC simulations. When sign problem does appear in
QMC simulation of a model, the average sign usually decays exponentially with
system size and inverse temperature [24] such that the needed computational
time increases exponentially with system size and inverse temperature,
hampering reliable QMC simulations for large system size and low temperature.

Here, we demonstrate a novel behavior of sign problem, which is qualitatively
distinct from conventional sign problem mentioned above. For the first time,
we numerically observed that the sign problem of various interacting models
are increasingly mitigated when their system size is increased, although the
sign-problem of those models cannot be fully solved by any known methods so
far. Specifically, the average value of sign can approach asymptotically to
one, namely sign problem in the simulation asymptotically vanishes, in the
limit of infinite system size. We dub this intriguing phenomenon as
“asymptotic sign-free” (ASF). Our study of ASF mainly focuses on Dirac fermion
systems, namely models whose non-interacting limit features Dirac cones,
although we will discuss other types of models as well.

To understand this novel observation, we propose a renormalization group (RG)
like explanation of ASF, which might provide a possible general theoretical
framework to fathom the asymptotic behavior of sign problem in interacting
systems. The spirit of RG-like scenario to understand the scaling behaviour of
sign problem in interacting models is illustrated in Fig. 1. We believe that
the asymptotic sign free behavior unveiled in our study will shed new insight
to understanding the nature of sign problem, and potentially paves a promising
avenue to studying intriguing physics in correlated fermion models with large
size which were previously thought not accessible by QMC.

Sign problem in QMC: In this paper we employ determinant QMC (DQMC), which is
an intrinsically-unbiased QMC algorithm, to simulate interacting fermion
models. Both finite-temperature and ground-state properties of a model can be
investigated by DQMC. For finite-temperature simulations, the expectation
value of a physical observable can be computed as:
$\langle\hat{O}\rangle=\frac{\mathrm{Tr}{e^{-\beta\hat{H}}\hat{O}}}{\mathrm{Tr}{e^{-\beta\hat{H}}}}$,
where $\beta$ is the inverse temperature and $\hat{H}$ is the Hamiltonian.
Employing Hubbard-Stratonovich (HS) transformation of the interacting terms in
the Hamiltonian which introduces auxiliary fields, the expectation value can
be expressed: $\langle\hat{O}\rangle=\frac{\sum_{z}O(z)w(z)}{\sum_{z}w(z)}$,
where $O(z)$ is the expectation value of observable $\hat{O}$ for auxiliary
field configuration $z$ and $w(z)$ is the Boltzmann weight determining the
sampling probability in simulations.

The sign problem appears once the weight $w(z)$ is not positive definite. When
the sign $S(z)\\!=\\!\frac{w(z)}{|w(z)|}$ can be positive or negative, its
average value $\langle S\rangle=\frac{\sum_{z}w(z)}{\sum_{z}|w(z)|}$ for a
sign-problematic model usually decays exponentally with system size $N$ and
inversion temperature $\beta$, $\langle S\rangle\sim e^{-\kappa\beta N}$
($\kappa>0$) [24], which renders QMC simulations not feasible for large system
size and low temperature. In the following, we perform large-scale QMC
simulations on several quantum many-body models featuring Dirac fermions to
demonstrate that asymptotic sign free behavior can emerge, namely the average
sign $\langle S\rangle$ increases with system size, rendering reliable QMC
studies of those models with large system size feasible although those models
are sign-problematic.

Figure 2: The average sign $\left<S\right>$ in QMC simulations of the
honeycomb Hubbard model at half-filling using a special type of Hubbard-
Stratonovich transformation. (a) $\left<S\right>$ as a function of interaction
strength $U$. Here $U=U_{*}$ represents a transition between sign scaling
behaviour. For $U<U_{*}$, the sign problem becomes less severe for larger
system while for $U>U_{*}$ opposite behaviours occurs. Inset is the result of
data collapse using $\left<S\right>$ and $(U-U_{*})L^{1/\nu}$. (b) The scaling
of $1-\left<S\right>$ as a function of linear system size $L$. The calculation
is performed for linear system size $L=9,12,15,18$ with inverse temperature
$\beta=2L$. For $U<U_{*}$, the average sign $\left<S\right>\approx
1-e^{-\alpha L}$ ($\alpha>0$) and it approaches to 1 in the thermodynamic
limit, namely asymptotic sign free.

The honeycomb Hubbard model: The interacting effect on the Dirac fermions has
stimulated enormous interests in the past many years, including interaction-
driven quantum phase transitions [34, 35, 36, 69, 70, 71, 72, 73, 74, 75, 76,
77, 78] and topological phases [79, 80, 48, 49, 81, 82, 83, 84, 85, 86]. One
simple but important model featuring interacting Dirac fermions is the
spin-1/2 Hubbard model on the honeycomb lattice. We now consider this model:

$\displaystyle H=-t\sum_{\left\langle
ij\right\rangle,\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+h.c.)+U\sum_{i}n_{i\uparrow}n_{i\downarrow},$
(1)

$c^{\dagger}_{i\sigma}$ creates an electron on site $i$ with spin polarization
$\sigma=\uparrow$/$\downarrow$, $t$ is the NN hopping, and $U$ is the onsite
repulsion. Hereafter we set $t=1$ as unit of energy. The honeycomb Hubbard
model at half filling is sign free when choosing appropriate HS
transformations and has been shown to exhibit a quantum phase transition at
$U\\!=\\!U_{c}\\!\approx\\!3.85$ between Dirac semimetal and antiferromagnetic
Mott insulator [87]. Here we employ a special HS transformation to purposely
make sign problem occur in this model by decoupling the Hubbard interaction
differently between terms on A sublattices and B sublattices. Specifically, we
decouple the Hubbard interaction in the spin channel as follows:
$e^{-\Delta\tau
U(n_{i\uparrow}-\frac{1}{2})(n_{i\downarrow}-\frac{1}{2})}=\frac{1}{2}e^{-U\Delta\tau/4}\sum_{s_{i}=\pm
1}e^{\lambda s_{i}c^{\dagger}_{i}\vec{\sigma}\cdot\vec{n}_{i}c_{i}}$, where
$c^{\dagger}_{i}=(c^{\dagger}_{i\uparrow},c^{\dagger}_{i\downarrow})$,
$\Delta\tau$ is the time-slice in Trotter decomposition, $\lambda$ is a
constant defined as $\cosh\lambda=\exp(U\Delta\tau/2)$, $s_{i}$ is axillary
field, $\vec{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ are Pauli matrices,
and $\vec{n}_{i}$ is a unit vector. Specifically, we set $\vec{n}_{i}=\hat{z}$
for $i\in$ A sublattice and $\vec{n}_{i}=\vec{n}=(\sin\theta,0,\cos\theta)$
for $i\in$ B sublattice. When $\theta=0$, this HS transformation becomes the
conventional one. When $\theta\neq 0$, QMC simulations will suffer a sign
problem and the severity of sign problem can be continuously tuned by varying
$\theta$. Using this special HS transformation with fixed $\theta$, we
investigate how the sign problem of this model depend on the system size under
study.

For $\theta=\pi/2$, curves of the average sign $\langle S\rangle$ for QMC
simulations of the model on the lattice with $L\\!\times\\!L$ unit cells show
crossing behavior around $U=U_{*}\approx 2.79$, as shown in Fig. 2(a). The
crossing suggests that $U=U_{*}$ represents a transition between different
sign scaling behaviours. For the sign transition at $U=U_{*}$, we obtain the
nominal critical exponent $\nu_{*}\approx 1.2$, which appears to be close to
the physical correlation-length exponent $\nu\approx 1.02$ of the physical
quantum phase transition at $U=U_{c}$ [87]. Remarkably, the average sign
$\langle S\rangle$ in the relatively weak interaction regime $U<U_{*}$
increases with the system size $L$ with an approximate scaling $\left\langle
S\right\rangle\approx 1-e^{-\alpha L}$ ($\alpha>0$) as shown in Fig. 2(b).
Consequently, the average sign flows to one in the thermodynamic limit, namely
asymptotic sign free. Conversely, when Hubbard interaction is relatively
strong, namely $U>U_{*}$, the average sign decays with system size $L$ and the
model exhibits conventional sign problematic behaviour. We further calculate
the average sign in QMC simulations of this model using different values of
$\theta$ in the HS transformation, as shown in Fig. S1; similar crossing
occurs at a slightly different value of $U$ but the asymptotic sign free
behavior is robust in the weak interaction regime.

Figure 3: (a) The results of the average sign $\left\langle S\right\rangle$ in
zero-temperature QMC for the spinless fermionic model with repulsive
interaction $V>0$. The calculation is performed for linear system size
$L=9,12,15,18$. The projector parameter is fixed $\Theta=60$. (b) The results
of the average sign $\left\langle S\right\rangle$ in finite-temperature QMC
for the same model as in (a). The calculation is performed for $L=9,12,15,18$.
The inverse temperature $\beta$ scales with $L$ as $\beta=3L$. The insets
explicitly show the results of average sign $\left\langle S\right\rangle$
versus $L$ for several values of $V$ in the weak interaction regime.

Repulsive spinless honeycomb model: In addition to the spinful Hubbard model,
we investigated the spinless fermion model on honeycomb lattice with the
following Hamiltonian:

$\displaystyle H=-t\sum_{\left\langle
ij\right\rangle}(c^{\dagger}_{i}c_{j}+h.c.)+V\sum_{\left\langle
ij\right\rangle}(n_{i}-\frac{1}{2})(n_{j}-\frac{1}{2}),$ (2)

where $c_{i}$ annihilates a spinless fermion on site $i$,
$n_{i}=c^{\dagger}_{i}c_{i}$ is the fermion number operator, $t$ is the
hopping amplitude of fermions on the NN bond, and $V>0$ represents density
repulsion. When $V=0$, the model in Eq. (2) exhibits massless Dirac fermions.
We focus on half filling, namely Fermi level lies at the Dirac point.

To perform QMC study of the model above, one needs to do HS transformation to
decouple the density-density interaction and there are various ways to do it
(details of the HS transformation in different channels are shown in the
Supplementary Materials). If the interaction term is decoupled in the density
channel, the QMC simulations displays sign problem; namely the weight $w(z)$
is not positive definite. It was known that there is a quantum phase
transition at $V=V_{c}$ between the Dirac semimetal at weak interactions and
CDW ordering at strong repulsions [35, 34, 88, 89, 90]. To investigate the
nature of sign problem of this model, we calculate the average sign in the
sign-problematic decoupling channel, and explore how the behavior of sign
problem depends on interaction strength $V$ and on system size $L$.

Figure 4: (a) The results of average sign $\left\langle S\right\rangle$ in
zero-temperature QMC of the spinless fermionic model with attractive
interaction $V<0$. The calculation is performed for linear system size
$L=9,12,15,18$. The projector parameter is fixed $\Theta=60$. (b) The results
of average sign $\left\langle S\right\rangle$ in finite-temperature QMC of the
same model as in (a). The calculation is performed for $L=9,12,15,18$. The
inverse temperature $\beta$ scales with $L$ as $\beta=3L$. The insets
explicitly show the results of average sign $\left\langle S\right\rangle$
versus $L$ for several values of $V$ in the weak interaction regime.

Results of $\left\langle S\right\rangle$ in both finite-temperature and
ground-state QMC simulations are shown in Fig. 3. Intriguingly, with
increasing system size $L$, the average sign $\left\langle S\right\rangle$
displays two distinct behaviors in the regimes of small and large $V$. For
zero-temperature QMC simulations, as shown in Fig. 3(a), the value of
$\left\langle S\right\rangle$ increases and approaches to $1$ with increasing
system size $L$ in the weak interacting regime ($V<V_{*}$), indicating that
the sign problem asymptotically vanishes in the thermodynamic limit. The
results of average sign $\left\langle S\right\rangle$ as function of system
size $L$ are shown in the insets of Fig. 3, clearly showing the increasing
behaviour of $\left\langle S\right\rangle$ with $L$ in the weak interaction
regime. However, when the interaction is relatively strong ($V>V_{*}$),
$\left\langle S\right\rangle$ decays exponentially as a function of $L$,
consistent with the conventional scaling behaviour of sign problem. For
finite-temperature QMC simulations, as presented in Fig. 3(b) where inverse
temperature $\beta$ scales linearly with system size $L$, the results also
provide compelling evidence of asymptotically free behaviour of sign problem
in the weak interaction regime. Thus, our QMC simulations of the model with
repulsive interaction clearly shows the ASF behaviour when the interaction is
relatively weak. Note that the sign transition point at $V=V_{*}$ between two
distinct scaling behaviours in the sign problem is close to physical quantum
phase transition point $V=V_{c}\approx 1.355$ [34, 35].

Attractive spinless honeycomb model: For the spinless fermion honeycomb model
in Eq. (2) with attractive interaction ($V<0$), its QMC simulation is sign-
problematic for any known HS transformations. This model was shown to exhibit
pair density wave (PDW) [91, 92] ordering in certain range of attractive
interaction and also exotic quantum critical point with emergent supersymmetry
[91, 93, 94, 39, 40, 95]. Here, we investigate the scaling behavior of sign
problem in this sign-problematic model for both weak and strong attractive
interaction. Again, we do the HS transformation in the density channel and
perform simulations of the model both at zero temperature and at finite
temperature. Surprisingly, in a large regime of weak interactions, the sign
problem is asymptotically free, implying the accurate ground-state properties
can be achieved reliably by large-scale QMC simulations although the model is
not strictly sign free.

For zero-temperature simulations of the model, the results explicitly reveal
that in the regime $|V|\leq 1.2$ the average sign increases and approaches to
one with increasing $L$, implying asymptotic sign free behavior in this model,
as shown in Fig. 4(a). As $|V|>1.2$, the average sign vanishes asymptotically
with increasing system size. This behaviour qualitatively holds in the results
of finite-temperature QMC simulation, as shown in Fig. 4(b), further
corroborating the existence of asymptotic sign free behavior in attractive
spinless honeycomb model. Hence, although sign problem is not strictly solved
in spinless model Eq. (2) with $V<0$, it is still feasible to access the
ground-state properties of this model with large system size by QMC simulation
in a large regime of interactions and to possibly fathom intriguing physics
such as pair-density-wave superconductivity and emergent SUSY, which is left
to future study.

Renormalization-group-like flow of sign: The numerical results above provide
compelling evidences that various interacting Dirac-fermion models can exhibit
a new type of sign behaviour, dubbed as asymptotic sign free. Consequently,
the sign problem can feature two qualitatively distinct scaling behaviours,
depending on the models under study as well as the HS transformations employed
in QMC simulations. A natural question is why there exist two types of
distinct behaviours in the scaling of average sign with system size.

Here we propose a RG-like explanation. For massless Dirac fermion systems in
2D, weak short-range four-fermion interactions are irrelevant in RG, thus
rendering the non-interacting limit ($U=0$) a stable fixed point model. For
$U=0$, it is obvious that it is sign free with $\left\langle
S\right\rangle=1$. The schematic RG flow of interactions for the Dirac-fermion
system is given in Fig. 1(a). There exists a critical point $U=U_{c}$
separating two distinct phases: the Dirac semimetal phase in which
interactions are irrelevant and the broken-symmetry phase in which
interactions are relevant. Analogous to the RG flow of interactions as the
length scale is increased, we schematically plot the flow of average sign as
the system size is increased, as shown in Fig. 1(b). There are also two
distinct behaviours of sign problem separated by $U=U_{*}$: asymptotic sign
free regime and conventional sign-problematic regime. For asymptotic sign free
regime ($U<U_{*}$), as unveiled in this paper, the interaction is irrelevant
and flows to the non-interacting limit; similarly the average sign also flows
to $\left\langle S\right\rangle=1$, implying asymptotic sign free. Conversely,
for sufficiently strong interactions, the interaction is relevant and the
average sign $\left\langle S\right\rangle$ flows to the point $\left\langle
S\right\rangle=0$ as the system size is increased, namely conventional sign-
problematic regime.

Figure 5: The schematic representation of the flow of sign around a stable
fixed-point-model which is sign free. We conjecture that around a sign free
stable-fixed-point model there exists a finite regime which is asymptotic sign
free.

The results obtained in the above Dirac fermion models strongly imply that the
existence of a sign-free stable fixed point model is essential for the
emergence of asymptotic sign free behaviour in QMC simulations. From this
perspective, we propose a conjecture based on our numerical observations and
arguments from RG: for an interacting model, if there exists a stable fixed
point that is sign-free, a finite asymptotic sign free regime should emerge
around this sign free stable fixed point model, as schematically illustrated
in Fig. 5. We emphasize that despite the emergence of asymptotic sign free
regime, the physical phase transition point is in general not identical to the
sign transition point although they can be close to each other. A related
conjecture is the following: for lattice models featuring generic Fermi
surfaces in two or higher dimensions, the non-interacting limit are unstable
against infinitesimal attractive interactions; consequently asymptotic sign
free regime should not appear around the non-interacting model with Fermi
surfaces. This conjecture is numerically confirmed by our calculation of
average sign for doped spinless fermion model with relatively weak
interaction, as discussed in the Supplementary Materials, which clearly shows
that sign average decays exponentially with system size, consistent with the
conventional behaviour of sign problem.

Discussions and concluding remarks: Recently Ref. [96] showed numerical
evidences that sign problem in QMC can be most severe close to physical
quantum critical points in certain interacting fermion models. The present
study shows that the sign can be asymptotic free, even close to the quantum
critical point. Moreover, the sign transition point $U=U_{*}$ is in general
different from the physical quantum critical point $U=U_{c}$ although they can
be close to each other. Intrinsic relation between nominal critical exponent
$\nu_{*}$ of sign transition and $\nu$ of physical phase transition remains
elusive and is left for future study.

In conclusion, we showed convincing evidences for the first time that sign
problem can disappear asymptotically with increasing system size, totally
beyond the conventional understanding that the average sign decays
exponentially with system size. Moreover, we proposed a RG-like picture to
heuristically understand sign problem in interacting models and build the
possible correspondence between physical RG flow and the scaling behaviour of
sign problem. Even when sign problem appears in QMC simulations of a given
model, it remains feasible to explore quantum correlation physics at low
temperature and large system sizes if the model is asymptotic sign free.
Consequently, we believe that our finding in this paper might open a new and
promising way to investigate intriguing correlation physics such as high-
temperature superconductivity and emergent supersymmetry through asymptotic
sign free simulations.

Acknowledgement: We would like to thank Shi-Xin Zhang, Yuan Wan and Yuan-Yao
He for helpful discussions. This work is supported in part by the NSFC under
Grant No. 11825404 (ZQW and HY), the MOSTC Grants No. 2018YFA0305604 and No.
2021YFA1400100 (HY), the CAS Strategic Priority Research Program under Grant
No. XDB28000000 (HY), and the start-up grant of IOP-CAS (ZXL).

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##  I Supplementary Materials

###  I.1 A. The details of quantum Monte-Carlo simulation

Determinant quantum Monte-Carlo is commonly used to study the finite-
temperature or zero-temperature properties of interacting fermionic systems.
For finite-temperature algorithm, people deal with quantum partition function
of the model: $Z=\mathrm{Tr}{e^{-\beta\hat{H}}}$, where $\hat{H}$ is the
Hamiltonian under consideration and $\beta$ is inverse temperature. The
ensemble average of the observable is obtained by the QMC simulation:
$\left\langle
O\right\rangle=\frac{\mathrm{Tr}{[e^{-\beta\hat{H}}\hat{O}}]}{\mathrm{Tr}{e^{-\beta\hat{H}}}}$.
For zero-temperature algorithm, the ground-state expectation value of given
observable is evaluated by:

$\displaystyle\langle\hat{O}\rangle=\frac{\langle\psi_{G}|\hat{O}|\psi_{G}\rangle}{\langle\psi_{G}|\psi_{G}\rangle}=\lim_{\Theta\rightarrow\infty}\frac{\langle\psi_{T}|e^{-\frac{\Theta}{2}H}\hat{O}e^{-\frac{\Theta}{2}H}|\psi_{T}\rangle}{\left\langle\psi_{T}\left|e^{-\Theta
H}\right|\psi_{T}\right\rangle}$ (S1)

where $\Theta$ is projector parameter, $\left|\psi_{G}\right\rangle$ is the
ground-state wave function of the Hamiltonian $\hat{H}$, and
$\left|\psi_{T}\right\rangle$ is a slater-determinant trial wave function that
has non-zero overlap with $\left|\psi_{G}\right\rangle$. The Determinant QMC
involves two main steps as following: (1) Trotter-Suzuki decomposition, which
discretizes inverse temperature $\beta$ (projector parameter $\Theta$) into
small imaginary-time slices:
$\Delta_{\tau}=\frac{\beta}{N_{\tau}}$($\Delta_{\tau}=\frac{\Theta}{N_{\tau}}$)
in finite-temperature (zero-temperature) algorithm. (2) Hubbard-Stratonovic
(HS) transformation, under which four-fermions terms in Hamiltonian are
decoupled to the non-interacting bilinear-fermion operators coupled to
classical auxiliary fields. As we discuss below, the appearance of sign
problem depends on the decoupling channel of H-S transformation. In our study,
we fix $\Delta_{\tau}=0.1$ in the study of spinful Hubbard and
$\Delta_{\tau}=0.05$ in the study of spinless t-V model, which is sufficient
to guarantee the convergence of results. In finite-temperature simulation, we
scale $\beta$ linearly with system size $L$. In zero-temperature simulation,
we have checked that $\Theta=60$ is sufficiently large to access the accurate
ground-state physical observables in spinless t-V for the system sizes under
consideration in our simulation. Hence, we fix $\Theta=60$ and calculate
average sign for different $L$ in the simulation on spinless t-V models.

For spinful Hubbard model, we implement an unconventional H-S transformation
by decoupling the Hubbard interaction differently between terms on A
sublattices and B sublattices. We decouple the Hubbard interaction in the spin
channel as following:

$\displaystyle e^{-\Delta\tau
U(n_{i\uparrow}-\frac{1}{2})(n_{i\downarrow}-\frac{1}{2})}=\frac{1}{2}e^{-U\Delta\tau/4}\sum_{s_{i}=\pm
1}e^{\lambda s_{i}c^{\dagger}_{i}\vec{\sigma}\cdot\vec{n}_{i}c_{i}}$ (S2)

where $c^{\dagger}_{i}=(c^{\dagger}_{i\uparrow},c^{\dagger}_{i\downarrow})$,
$\cosh\lambda=e^{\frac{U\Delta_{\tau}}{2}}$ and $s_{i}$ is auxiliary field
defined on site $i$. Specifically, we set $\vec{n}_{i}=\hat{z}$ for $i$ in A
sublattice and $\vec{n}_{i}=\vec{n}=(\sin\theta,0,\cos\theta)$ on B
sublattice. As $\theta=0$, this HS transformation becomes the conventional one
and the model is sign free at half filling. When $\theta\neq 0$, the sign
problem appears and the severity of sign problem depends on the value of
$\theta$. In the maintext, we present the results of the simulation by
choosing $\theta=\frac{\pi}{2}$.

For repulsive spinless t-V model, the model is sign-free at half filling if
the interaction is decoupled in the hopping channel, as introduced in previous
works. The interaction can be decoupled in hopping channel as:

$\displaystyle
e^{-\Delta_{\tau}V(n_{i}-\frac{1}{2})(n_{j}-\frac{1}{2})}=\frac{1}{2}e^{-V\Delta_{\tau}/4}\sum_{s_{ij}=\pm
1}e^{\lambda s_{ij}(c^{\dagger}_{i}c_{j}+h.c.)}$ (S3)

where $\cosh\lambda=e^{\frac{V\Delta_{\tau}}{2}}$. The absence of sign problem
is identified from various perspectives, including Majorana time-reversal
symmetry [29, 30], split orthogonal group [33] and Majorana reflection
positivity [32]. For attractive interaction, the sign problem of the model
cannot be eliminated in any known algorithms of QMC simulation. Hence the
ground-state properties of the spinless t-V model with attractive remains
elusive. In our study, to investigate the properties of sign problem, we
perform H-S transformation of spinless t-V model in a sign-problematic
channel, namely density channel. In this decoupling channel, the t-V model is
sign-problematic for both repulsive and attractive interaction, even at half
filling. For repulsive interaction $V>0$:

$\displaystyle
e^{-\Delta_{\tau}V(n_{i}-\frac{1}{2})(n_{j}-\frac{1}{2})}=\frac{1}{2}e^{-V\Delta_{\tau}/4}\sum_{s_{ij}=\pm
1}e^{\lambda s_{ij}(n_{i}-n_{j})}$ (S4)

where $\cosh\lambda=e^{\frac{V\Delta_{\tau}}{2}}$ and $s_{ij}$ is auxiliary
field defined on bond $ij$. For attractive interaction $V<0$:

$\displaystyle
e^{-\Delta_{\tau}V(n_{i}-\frac{1}{2})(n_{j}-\frac{1}{2})}=\frac{1}{2}e^{-V\Delta_{\tau}/4}\sum_{s_{ij}=\pm
1}e^{\lambda s_{ij}(n_{i}+n_{j}-\frac{1}{2})}$ (S5)

where $\cosh\lambda=e^{\frac{-V\Delta_{\tau}}{2}}$ and $s_{ij}$ is auxiliary
field defined on bond $ij$. Sign problems exist for both repulsive and
attractive interactions in this decoupling channel. We perform simulation on
spinless t-V model in this decoupling channel, and demonstrate the existence
of asymptotic sign-free regime at half filling for both repulsive and
attractive interactions.

Figure S1: The results of finite temperature QMC for the average sign
$\left<S\right>$ as a function of interaction strength $U$ in honeycomb
Hubbard model at half filling using partially rotated HS transformations with
(a) $\theta=\pi/4$; (b) $\theta=\pi/8$; (c) $\theta=\pi/16$. The crossing
region of $\left<S\right>-U$ curves has a slight shift when $\theta$ is
changing. The calculation is performed for linear system size $L=9,12,15,18$.
The inverse temperature $\beta$ scales with linear system size as $\beta=2L$.
The simulation parameter $\Delta\tau=0.1$.

###  I.2 B. Additional results of sign problem in honeycomb Hubbard model

In this section, we demonstrate that the ASF region with weak interaction
exists using different HS transformations in honeycomb Hubbard model at half
filling. We show the results of average sign $\left<S\right>$ as a function of
interaction strength U using partially rotated HS transformations with
rotation parameter $\theta=\pi/4,\pi/8,\pi/16$ in Fig. S1. In all these three
cases, the average sign show different scaling behaviors between strong and
weak interaction region separated by a critical $U_{*}$, indicting asymptotic
sign free region exist for different HS transformations. The critical
interaction $U_{*}$ has a shift when changing parameter $\theta$ ($U_{*}\sim
2.85,2.9,3.0$ for $\theta=\pi/4,\pi/8,\pi/16$, respectively). Since its value
can change for different HS transformation, we do not imply any direct
relationship between $U_{*}$ and the physical critical point $U_{c}$.
Nevertheless, the asymptotic sign free behaviors still exist in the weak
interaction region using different HS transformation in honeycomb Hubbard
model at half filling.

Figure S2: The finite-temperature QMC results of average sign for doped
repulsive (a) and attractive (b) spinless t-V model. The interaction strength
is fixed $V=0.8$. The doping level is $p=0.04$. The inverse temperature
$\beta$ scales with linear system size $L$ as $\beta=3L$.

###  I.3 C. The sign problem in doped spinless t-V model with Fermi surfaces

In the main text, we have shown that for both repulsive and attractive
spinless t-V model at half filling, the models are asymptotic sign-free in a
large parameter region. In this section, we discuss the behaviour of sign
problem occurring in spinless t-V model away from half filling. We perform
finite-temperature QMC simulation and scale inverse temperature $\beta$ with
linear system size $L$ as $\beta=3L$. In finite-temperature QMC simulation,
doping is achieved by tuning chemical potential. In Fig. S2, we present the
results of sign averages at a typical interaction strength $|V|=0.8$ deep in
the asymptotic sign-free regime at half filling. The results of average sign
$\left\langle S\right\rangle$ as a function of linear system size $L$ for
repulsive and attractive interactions are shown in Fig. S2 (a) and (b),
respectively, clearly indicating the values of $\left\langle S\right\rangle$
decay with system size in both cases. As system size increases, the
$\left\langle S\right\rangle$ displays exponentially decaying behaviour
consistent with the conventional scaling behaviour of sign problem in QMC.
Taking other interaction strengths and doping levels, the average sign
exhibits qualitatively same behaviours, decaying with increasing system size
and lowering temperature.