Multi-flavor massless QED$_2$ at finite densities via Lefschetz thimbles

We consider multi-flavor massless $(1+1)$-dimensional QED with chemical potentials at finite spatial length and the zero-temperature limit. Its sign problem is solved using the mean-field calculation with complex saddle points.


Introduction
Path integral formalism has been used in a wide range to study systems of quantum field theory as well as statistical mechanics. The formalism is useful not only for the perturbative calculations but also for the numerical computations such as the lattice Monte Carlo simulations. However, it is well known that the lattice Monte Carlo simulation does not work when the action of the system becomes complex and this is called the sign problem [1,2]. It is a big obstacle when we try to compute the high-density nuclear matter based on quantum chromodynamics (QCD) [3].
In this paper, we consider (1 + 1)-dimensional quantum electrodynamics (QED 2 ) with N f massless fermions, which we call multi-flavor massless QED 2 . It is also known as Schwinger model, which is exactly solvable and mapped to the theory of free massive photons [4,5]. This model has the sign problem at finite temperature and finite number densities, but its phase structure is studied analytically by Refs. [6][7][8]. Therefore, multiflavor massless QED 2 has been used for the test of various approaches to the sign problem; the dual formulation is discussed in Ref. [9], and the study with matrix product states is used for two-flavor massless and massive QED 2 in Ref. [10]. Here, we apply the Lefshetzthimble method to obtain phase diagram of multi-flavor massless QED 2 , and study its property about the sign problem.
Using Lefschetz thimbles, we show that mean-field calculation with complex saddles becomes exact in multi-flavor massless QED 2 at finite spacial length and the zero-temperature limit. We identify all the complex saddle points of QED 2 at finite chemical potentials, and draw the phase diagram using that information. At that stage, we must know the intersection numbers between dual thimbles and the original integration region, and those quantities are identified by solving the gradient flow equation that defines Lefschetz thimbles. This is the first non-trivial example of gauge theories with the sign problem that is completely solved by the Lefschetz-thimble method.
This paper is organized as follows. In Section 2, we review the computation of massless QED 2 . In Section 3, we explain the mean-field approximation in the presence of the sign problem. In Section 4, we compute the complex saddle points and obtain the phase structures concretely for one-, two-, and three-flavor cases. To discuss the phase structure for three-flavor case, however, we need the intersection numbers. For that purpose, we numerically solve the gradient flow and visualize its structure in Section 5. We summarize our result and discuss some perspectives in Section 6.

(1 + 1)-dimensional massless QED
In this section, we review the computation of the massless QED 2 . This section does not present a new result, and the main purpose is to set the notation and make the paper self-contained.

Setup of the model
Let us first consider the path-integral expression of multi-flavor massless QED 2 on a two- Here, A = A µ dx µ is the U (1) gauge field, ψ a and ψ a are two-component spinor fields with the flavor index a = 1, . . . , N f , / D A = γ ν (∂ ν + iA ν ) with Gamma matrices γ ν , and µ a is the chemical potentials for fermion number densities n a = ψ a γ 0 ψ a . S Maxwell is the Maxwell action, The theory is now defined on the torus T 2 = [0, β] × [0, L] with the thermal boundary condition, i.e., The "∼" in the first equation means that the difference between both sides must be appropriately Dirac quantized. We review the computation of the Dirac operator on the torus T 2 according to Ref. [61]. We decompose the gauge field as The meaning of each variables is as follows: λ is a gauge parameter, so it does not affect the computation of physical quantities, such as the partition function. φ is a 2π-periodic function on torus without the zero-momentum mode, which represents the local fluctuation of the photon field. Since F 01 = −2πk/βL + ∆φ, the Maxwell action becomes The fields h 0 and h 1 are sometimes called toron fields: they do not create the electric field, but describe the nontrivial holonomy at φ = 0 and k = 0, since where n 0 and n 1 are the number of the windings of the closed curve C along x 0 -and x 1 -directions of T 2 , respectively (0 ≤ h µ < 1). The integer k refers the Chern number, which creates the constant electric field. The index theorem tells us that k represents the index of the Dirac operator, and thus we need to take into account only the k = 0 sector: otherwise, the Dirac zero modes exist and the fermion determinant becomes zero because the fermions are massless. Let us restrict ourselves to the case k = 0 in the following, and assume that the Dirac zero mode does not exist. For simplicity of the computation, we put λ = 0 by gauge fixing. By putting we get A = A + * dφ. We define the chirality matrix by γ = −iγ 0 γ 1 , then (2.10) Therefore, we obtain that In order to obtain this result, we perform the chiral rotation ψ → e γφ ψ and ψ → ψe γφ , and use the anomaly equation. It is important to notice that the global fluctuations A and the local fluctuation φ decouple completely from one another at the k = 0 sector, thanks to the absence of the Dirac zero modes. Now, the partition function reads The photon field φ becomes massive due to the chiral anomaly, and it decouples from the integration of h 0 and h 1 fields. In this paper, we are interested only in the dependence on the chemical potentials, and the h 0 and h 1 integrations are most important. In the following, we simply denote that In the next subsection, we will compute the fermion determinant under the background with nontrivial holonomies.

Fermion determinant with nontrivial holonomies
One can compute the Dirac determinant in a way to keep the global-gauge invariance manifestly by using the zeta-function regularization (see Ref. [61] and also Appendix in Ref. [62]). To understand the formula intuitively, we derive the result for the fermion determinant heuristically by considering the fermion spectrum. After the chiral transformation, the fermion operator becomes where γ = −iγ 0 γ 1 and in the chiral notation γ = σ 3 . Therefore, the spectrum of this operator can be labeled as By performing the summation over Matsubara frequencies m, we obtain the Fermi-Dirac distribution for each fermionic mode: Contribution of the positive chiral fermions gives 17) and that of the negative chiral fermions gives Each term is invariant under h 0 → h 0 + 1, but it is not the case for h 1 → h 1 + 1. Even after taking the product over all spatial momenta n = 1, 2, . . ., the invariance under the spatial global gauge transformation is lost by To compensate this factor, one should multiply We introduce the dimensionless chemical potential µ and the dimensionless temperature τ by The µ -dependence of each determinant becomes This matches with the correct formula derived in Ref. [61] up to an uninteresting fieldindependent factor. We define the dimensionless mean-field free energy F by The explicit form of this one-loop effective potential becomes Using this one-loop effective potential, (2.13) becomes We consider about this exponential integral in the zero-temperature limit τ → 0.

Mean-field approximation with complex saddle points
In this section, we explain the mean-field approximation when the microscopic theory suffers from the sign problem [44].

Lefschetz-thimble methods and mean-field approximation
Here, we explain the general formalism to apply the mean-field approximation when the theory suffers from the sign problem. That is, we consider to apply the mean-field approximation to the path integral where the classical action S[φ] takes complex values. To consider the mean-field approximation, we introduce the order-parameter operator O[φ], and define the constrained free energy density [63,64] by is also real and one can compute phase diagram by taking the minimum of F (Φ). However, if there is the sign problem, the free energy F (Φ) is complex, and its physical meaning becomes unclear [65,66]. The partition function and the constrained free energy is related by where M is the target space of order parameters Φ. This corresponds to (2.25) in our model, where Φ = (h 0 , h 1 ) and M = (R/Z) 2 . In the mean-field approximation, we would like to evaluate this integral in the limit V → ∞ using the saddle-point approximation. Since F [Φ] is complex, this integral is an multi-dimensional oscillatory integral, and we need a technique to treat it. Here, we use the knowledge of the hyperasymptotic analysis [11][12][13], which is recently used for the study of sign problem of the lattice Monte Carlo simulation and known as the Lefschetz-thimble method . The basic idea is to deform the integration contour M into steepest descent cycles inside its complexified space M C by using the Cauchy theorem when F (Φ) is holomorphic. We denote the holomorphic coordinate of M C as Φ = (z 1 , . . . , z n ), and the set of saddle points as Using the Kähler metric on M C , ds 2 = g ij dz i ⊗ dz j , we define the gradient flow by As an important property of this differential equation, we have Therefore, along the flow line, the real part of the free energy increases while its imaginary part stays constant. This means that we can define the steepest descent and ascent cycles associated with each saddle point z σ by this gradient flow. Using solutions of the gradient flow z(t), they are defined as J σ and K σ are called Lefschetz thimble and dual thimble, respectively. They are dual quantities in terms of the intersection pairing ·, · , i.e., J σ , K τ = δ στ , which means that one can decompose M in terms of J σ as If all Re(F (z σ )) are different with each other in the limit V → ∞, we replace the integral by the saddle-point approximation, and we obtain at the leading order that We can summarize the necessary steps of the mean-field approximation with the sign problem as follows: • Complexify the target space M to M C , and find the saddle points z σ by solving the equation ∂F = 0 in M C .
• Pick up the saddle point z σ that has the minimal free energy Re(F (z σ )) with nonzero intersection number M, K σ .

Charge and complex conjugation for real-valued free energy
In Sec. 3.1, we explained the way to apply the Lefschetz thimble method to the mean-field calculations for the theory with the sign problem. However, since the free energy F (z) is complex in general, it would not be clear if the mean-field free energy F (z σ ) becomes real in the above procedure. Following Ref. [44], we explain that this is ensured under the charge conjugation of the original theory.
To explain it, we consider a fermionic system and denote the chemical potential dependence of the free energy explicitly as F = F (Φ, µ). In many examples of the fermionic system at finite densities, unbalance between the fermion and anti-fermion numbers due to the chemical potential causes the sign problem, and the free energy satisfies (3.10) In order to relate F (Φ, −µ) to the original one, we consider the charge conjugation C, which flips the sign of the chemical potential. Then, we have Indeed, this is satisfied for Polyakov-loop extended Nambu-Jona-Lasinio model [67][68][69], heavy-dense quantum chromodynamics (QCD) [65,66,[70][71][72][73][74][75][76][77][78][79], perturbative QCD with finite µ [80,81], Thirring model at finite densities [48][49][50][51][52]82] and also QED 2 in Sec. 2. Even after complexification, the anti-linear extension of the charge conjugation plays an important role, This property is called the CK-symmetry of the complexified theory in Refs. [67,68]. Under a certain condition of the Kähler metric and the charge conjugation, one can show that C · z(t) satisfies the same equation (3.5) for any solutions z(t) of the gradient flow [44]. Now, one can classify the set of saddle points z σ into three cases, (3.14) We here notice that If the mean-field theory is valid, then F (z σ ) of the selected saddle point must be real. Therefore, if one can show or assumes the validity of the mean-field approximation, it is enough to consider the gradient flow (3.5) inside the CK-invariant hyperplane, and one can pick up the CK-invariant saddle point of the minimal free energy with nonzero intersection number M, K σ . We will see that this is the case for the sign problem of multi-flavor massless QED 2 in the zero-temperature limit.
4 Lefschetz-thimble calculus for multi-flavor massless QED 2 In this section, we consider the phase diagram of multi-flavor massless QED 2 at finite L and the zero-temperature limit β → ∞. For this purpose, we compute the path integral (2.25) based on the Lefschetz-thimble method, and the mean-field calculation with the complex saddle points given in Sec. 3 turns out to be powerful for our purpose.

Complex saddle points of multi-flavor massless QED 2
We use the same symbol for the complexified toron fields, h 0 and h 1 . We define the Kähler metric on complex (h 0 , h 1 ) space (C/Z) 2 by The gradient flow to be solved in this metric is given by where the explicit form of the right hand sides is given as What we have to do for the Lefschetz-thimble method is to find the saddle points of F in the complexified (h 0 , h 1 ) space. Therefore, from (4.3) and (4.4), we need to solve (4.5) By taking the zero-temperature limit τ → 0, one can simplify this saddle-point condition as follows. In the limit τ → 0, we notice that the summation over spatial momenta n is represented by the floor function, Therefore, the saddle-point conditions (4.5) at τ 1 must be well approximated by Since the left hand sides of these equations are real and integers because they are left-and right-handed fermion numbers, so are the right hand sides. This claims that h 1 must be real and, especially 2 , For further consideration on the saddle points, we need to specify the number of fermion flavors N f . We will consider the case for N f = 1, 2, and 3 in following subsections. For the mean-field calculations, the computation of the free energy is also an important ingredient. In the limit τ → 0, one can find after a slightly technical computation that the free energy (2.24) also accepts the much simplified expression as One should notice that h 0 dependence totally disappears in the vicinity of the complex saddle points satisfying (4.7) and (4.8). This is due to the charge neutrality condition on the torus; Then, we find that the imaginary part of the constrained free energy also disappears under that condition. This fact is remarkable because use of complex saddle points weakens the sign problem of the massless multi-flavor QED 2 .

1-flavor case
Let us first consider the case N f = 1. The saddle point condition for h 1 , (4.9), says that h 1 = 1 2 . Then, Eqs. (4.7) and (4.8) reduce to the single equation, which can be trivially satisfied by choosing τ Im(h 0 ) = −µ . This saddle-point condition says that both the left-and right-handed fermion densities are always zero for any chemical potential.
One can understand this result in an easier way. The chemical potential dependence of the free energy (2.24) can be written as Sec. 5. However, all of them turns out not to be selected as correct mean fields, since they do not have the minimal free energy with non-zero intersection number, and it is reasonable because the fermion numbers at those saddle points are not quantized even at the zero-temperature limit. Therefore, we neglect such subtle possibilities for a while.
Therefore, just by shifting h 0 as h 0 − iµ /τ , the free energy becomes a real function, and the sign problem disappears. The partition function does not change under this continuous change of contours, so the chemical potential dependence does not appear in the single-flavor case.

2-flavor case
Let us next consider the case N f = 2. The saddle-point condition (4.9) requires that h 1 = 1 2 or h 1 = 0. For h 1 = 0, Eqs. (4.7) and (4.8) become 1 + µ 1 + τ Im(h 0 ) + 1 + µ 2 + τ Im(h 0 ) = 1, (4.14) Since these two conditions are equivalent, and they can be satisfied by setting τ Im(h 0 ) = − 1 2 (µ 1 + µ 2 ) because x + −x = −1 for any x ∈ R. Introducing the dimensionless isospin chemical potential by the free energy of this state is (4.17) For h 0 = 1 2 , Eqs. (4.7) and (4.8) become the single equation, Again, this is satisfied by setting τ Im(h 0 ) = − 1 2 (µ 1 + µ 2 ), and the free energy becomes Soon later, we shall show that both complex saddles have nonzero intersection number, and thus we must pick up a state with the lower free energy at a given chemical potential to determine the phase diagram. Figure 1a shows (4.17) and (4.19) with green and blue solid lines, respectively, and we can readily find that the phase transition at τ = 0 happens when µ I is a half integer. The difference of the fermion numbers, n 1 − n 2 = −2 ∂ ∂µ I F , is shown in Fig. 1b. This result is consistent with the exact computation done in Ref. [7] and also with the result by the matrix product states in Ref. [10].
To complete our argument on the 2-flavor case, let us discuss the intersection numbers at the saddle points with h 1 = 0 and h 1 = 1 2 . It is important to notice that the saddle-point conditions for both cases are solved by τ Im(h 0 ) = − 1 2 (µ 1 + µ 2 ). Indeed, the free energy F (h 0 , h 1 ) becomes a real-valued function after the shift  The above two saddle points at h 1 = 0, 1 2 correspond to local minimum of the real-valued free energy after this shift of h 0 , and thus the usual mean-field approximation becomes valid. This means that the intersection numbers of both saddle points are equal to 1: when neglecting the consistently subdominant thimbles. This argument, however, also shows that the sign problem of two-flavor QED 2 can be eliminated just by the constant shift of integration variables.

3-flavor case
Let us consider the case N f = 3 as a last example. This is the first nontrivial case, in which we cannot eliminate the sign problem by simply shifting the h 0 field. Without loss of generality, we can set µ 3 = 0 by shift of h 0 . In the following, we solve the problem under this condition. The saddle-point condition for h 1 , (4.9) says that We solve the saddle-point conditions for h 0 , (4.7) and (4.8), by separating cases.
(4.25) In Fig. 2a, we show the structure of these hexagons with corresponding fermion numbers (n 1 , n 2 ) = (2n, 2m). The free energy for this case becomes Here, we reinstate µ 3 just for keep the expression symmetric.
(4.32) The corresponding region becomes the hexagon, which is the convex hull of (2n + m + 1, n + 2m + 1), (2n + m + 5 3 , n + 2m + 1), (2n + m + 2, n + 2m + 4 3 ), (2n + m + 2, n + 2m + 2), (2n + m + 4 3 , n + 2m + 2), (2n + m + 1, n + 2m + 5 3 ) . (4.33) The result for h 1 = 1 2 ± 1 3 is summarized in Fig. 2b, where orange, pink and green regions correspond to the above cases 2a, 2b and 2c, respectively. The blank region of Fig. 2b means that there is no corresponding saddle point with h 1 = 1 2 ± 1 3 at those chemical potentials. The free energy becomes µ a 5 6 + µ a + τ Im(h 0 ) + 1 6 + µ a + τ Im(h 0 ) . (4.34) We now have solved all the possible complex saddle points, and the corresponding free energies are computed. Since all the free energies are real and independent of τ , the meanfield calculation with those complex saddle points becomes exact in the zero-temperature limit τ → 0. As we have explained in Sec. 3, we need to compute the gradient flow (4.2) to construct the dual thimbles for this method, and select the phase with minimal free energy with nonzero intersection number. To streamline the discussion, however, we postpone the computation of the gradient flow to Sec. 5, and we temporarily assume that all the above complex saddle points contribute to the partition function in the formula (3.9). This temporal assumption shall be verified by solving the gradient flow in the next section.
We can readily compute the phase boundaries of the three-flavor massless QED 2 by using the information about complex saddle points. If there are several complex solutions at a given chemical potential by comparing Figs. 2a and 2b, we select the phase with the lower free energy. As a result, we obtain the phase boundaries given in Fig. 3. One can explicitly compare this result with the exact computation done by Ref. [8] to find its correctness (see Figure 4 of Ref. [8]).

Gradient flow inside CK-invariant space
In order to check the intersection numbers of the dual thimbles, we solve the gradient flow numerically for three-flavor massless QED 2 in this section. We first emphasize the technical importance of the CK-transformation in order to visualize the gradient flow, and apply this technique to numerically solve the flow for N f = 3. All the following procedures are valid also for larger N f 's. The big technical issue of the mean-field approximation with complex saddles is that we have to compute the intersection numbers between (R/Z) 2 and dual thimbles K σ inside the complexified space (C/Z) 2 . Since it is impossible to draw figures in the four-dimensional manifold, we do not know a manifestly clear way to compute such quantities. The CKtransformation plays an important role to manage this issue for the massless QED 2 .
Since the original theory has the symmetry under charge conjugation, one can construct the CK symmetry discussed in Sec. 3.2 [44,67,68], Especially if Re(h 0 ) = 0 and Im(h 1 ) = 0, every physical quantity becomes real. One can also notice that the saddle points obtained in the previous section satisfy this restriction. Therefore, in the following, we solve the gradient flow under the condition with real x and y. Indeed, if one starts the gradient flow from a generic point inside the CK-invariant plane, then the flow line lies inside the CK-invariant plane [44]. The gradient We can now visualize the gradient flow inside the CK-invariant plane to compute the nontrivial quantities, intersection numbers. We numerically solve this equation to find out the structure of the gradient flow for the case N f = 3, and judge the contribution to the free energy from complex saddle points. We show the gradient flow and the schematic illustration of Lefschetz and dual thimbles in Fig. 4 at (µ 1 , µ 2 ) = (0.75, 0.2). The horizontal axis represents x = Re(h 1 ), and the vertical axis does y = τ Im(h 0 ). We can find black blobs at h 1 = 1 2 , 1 2 ± 1 3 , and they are saddle points computed in Sec. 4.4. There are blue dotted line segments through those saddles, which consist of approximate saddles at τ = 0.1. This is because the saddle-point condition at τ = 0, (4.7) and (4.8), does not designate the specific value of τ Im(h 0 ). At sufficiently low temperatures, the set of approximate saddle points form a line segment. The line segment of approximate saddles separate gradient flows, which moves almost horizontally towards right and left, and the flow lines emanating from it form a diamond-like shape.
Black squares show subdominant saddles, at which h 1 is not appropriately quantized as (4.22), and they are irrelevant in the mean-field calculation.
We can now comment on when the sign problem of the reweighting is severe. Only inside the diamond-shaped regions, overall number density vanishes, 3 a=1 n a = 0, and the sign problem along the h 0 -integration becomes absent. Therefore, if the real axis is away from the diamond-shaped region of the dominant saddle in Fig. 4, the conventional reweighting suffers from the severe sign problem. We can solve this problem by deforming the integration contours to Lefschetz thimbles.
The existence of the line segment of approximate saddles, however, makes it difficult to solve the gradient flow with good accuracy especially around the saddle points. This brings us the numerical problem to compute the Lefschetz thimbles J σ and dual thimbles K σ . However, what we need for the mean-field calculation with complex saddle points is the intersection number between the original integration cycle and dual thimbles K σ . Since the intersection number is a topological quantity, we do not need to compute K σ exactly, and we can approximate K σ as the line segment that contains z σ and the flow sucked into the edges of the line segment. In Fig. 4, we show this approximate Lefschetz thimbles J σ by red solid lines, and the approximate dual thimbles K σ by green dashed lines combined with blue dotted lines. That is, if the line segment or the flow line sucked into the line segment intersects with the real axis, we must take the complex saddle into the Lefschetz thimble decomposition of the partition function.
This confirms that all the saddle points discussed in Sec. 4 contribute, and it justifies the mean-field calculation done in the previous section. Therefore, the phase boundaries for N f = 3 given in Fig. 3 are exact at the zero-temperature τ = 0.

Conclusion and perspective
We show that the mean-field approximation with complex saddle points gives the rigorous result on the phase structure of multi-flavor massless QED 2 at zero temperature and finite densities. We derive the saddle-point condition and concretely solve it for N f = 1, 2, and 3 cases. By computing the gradient flow, we check the intersection number to list up the candidates of complex mean fields, and draw the phase diagram by comparing the meanfield free energies with nonzero intersection numbers.
Multi-flavor massless QED 2 suffers from the sign problem, and the conventional reweighting technique does not work for general large chemical potentials. We have shown that the Lefschetz-thimble method completely solves this problem at T = 0 for all the chemical potentials: After deforming the integration path into the complex space using the gradient flow, we can use the saddle-point method and the reweighting factor becomes consistent with 1. This means that the lattice Monte Carlo simulation on Lefschetz thimbles solves the exponential complexity of multi-flavor massless QED 2 at finite densities.
We would like to emphasize that this is highly nontrivial result. Indeed, there is an example of the fermion sign problem, the one-site Hubbard model, where the exponential complexity does remain even after the deformation of the integration path with the gradient flow [48]. In that example, complex zeros of the fermion determinant separate the original integration cycle into multiple Lefschetz thimbles, and the summing up them gives an exponentially small reweighting factor in terms of the inverse temperature β. Both one-site Hubbard model and multi-flavor massless QED 2 are strongly-coupled fermionic systems, and they have the first-order phase transition induced by jumps of the fermion number. Why do these examples have the big difference in the Lefschetz-thimble approach?
In the one-site Hubbard model [48], there are infinitely many Lefschetz thimbles that contribute in the zero-temperature limit. Integration on each Lefschetz thimble at the saddle point z σ is represented by exp(−βF (z σ )) in our notation, and the difference of the free energy β(F (z σ ) − F (z τ )) remains finite in the limit β → ∞. This means that each Lefschetz-thimble integral gives a comparable contribution, and the interference among them affects significantly in the one-site Hubbard model. On the other hand, F (z σ )'s in the multi-flavor massless QED 2 are independent of β, and thus β(F (z σ ) − F (z τ )) → ±∞ in the limit β → ∞. One of the Lefschetz thimbles dominate the others, and the destructive interference does not occur in massless QED 2 .
Another speculative remark is that the charge neutrality condition (4.11) at the saddle point plays an important role to eliminate the sign problem in multi-flavor massless QED 2 . In this model, the unbalance of oppositely charged particles creates the sign problem when integrating over the phase of the temporal Wilson loop, h 0 , because of the violation of charge neutrality at each real configuration. Use of the complexified fields makes the charge neutrality manifest, and the h 0 dependence of the free energy totally disappears in the vicinity of the complex saddle points. It is an important future study to understand what is a consequence of the color neutrality condition of other QCD-like gauge theories.
As a final remark, let us comment on the limitation of this study when we discuss the QCD sign problem. One of the biggest difference between massless QED 2 and the chiral limit of QCD is the property of the chiral symmetry. In QED 2 , the chiral symmetry is broken by the axial anomaly, and the massless Nambu-Goldstone boson does not appear. Indeed, the spontaneous symmetry breaking of the continuous symmetry is forbidden in (1 + 1)dimensional field theories [83,84]. On the other hand, QCD has the spontaneous chiral symmetry breaking, and massless pions exist. The existence of charged massless pions gives the big obstacle when applying the conventional reweighting technique to the sign problem of QCD at the zero temperature and finite density, which causes the early-onset problem of baryon number densities at half of the pion mass [85][86][87][88][89][90][91][92][93][94][95]. It is not yet known whether the massless pions bring any difficulties to the Lefschetz-thimble approach, and it cannot be discussed with massless QED 2 because of the different origin of chiral symmetry breaking. To identify the consequence of the chiral symmetry breaking to the Lefschetz thimbles must become a benchmark in the future study on applications of Lefschetz thimbles to the sign problem.