Classification of sign-problem-free relativistic fermions on the basis of the Majorana positivity

We classify the sign-problem-free relativistic fermion actions on the basis of the Majorana representation. In the Majorana representation, the sign-problem-free condition is given by the semi-positivity of a Pfaffian. We show that the known sign-problem-free actions of the Dirac fermions, which are usually understood from the semi-positivity of a determinant, e.g., the action of quantum chromodynamics with nonzero chiral chemical potential or nonzero isospin chemical potential, can also be understood from the semi-positivity of a Pfaffian. We also derive new classes of the sign-problem-free relativistic fermion actions with Majorana-type source terms.


I. INTRODUCTION
The quantum Monte Carlo method is a powerful computational scheme in modern physics from particle physics to condensed matter physics. In the Monte Carlo method, the semipositivity of a weight factor is crucial. When the semi-positivity is lost, the method breaks down due to the sign fluctuation. This is called the sign problem. The sign problem is frequently induced by fermions. A famous example of the fermion sign problem is quantum chromodynamics (QCD) with nonzero baryon chemical potential [1]. Although many challenges have been done for a long time, the problem has not yet been solved. In recent years, the novel attempts to evaluate complex integral, such as the complex Langevin method and the Lefschetz thimble, are intensively discussed [2]. However, their applications are still primitive and limited because of several difficulties [3][4][5][6]. The fermion sign problem will remain as an unsolved challenge for the near future.
Under such circumstances, it is important to study sign-problem-free fermions, i.e., fermions with a semi-positive weight factor. A well-known example is the two-flavor Dirac fermion with isospin chemical potential. The conventional proof for its semi-positivity is the use of the γ 5 -hermiticity and double degeneracy of the Dirac determinant. However, this proof is specific to this case and not applicable to general cases. Although many signproblem-free fermions might be hiding, we do not know a systematic way to seek them.
It was recently proposed that the concept of the Majorana representation is useful to find sign-problem-free fermions [7][8][9]. In the Majorana representation, we can systematically prove the semi-positivity even if a determinant is not doubly degenerate or even if a weight factor is not given by a determinant. This opens up the possibility to discover new classes of sign-problem-free fermions. Several new sign-problem-free fermion models were actually found in condensed matter physics [8,9].
In this paper, we discuss sign-problem-free relativistic fermions in four dimensions on the basis of the Majorana representation. In Sec. II, we introduce the Majorana positivity condition, that is, a sufficient condition to show the semi-positivity of a Pfaffian. By using the Majorana positivity condition, we discuss sign-problem-free fermions in one-flavor case in Sec. III, and in two-flavor case in Sec. IV. We list several comments in Sec. V. Finally we summarize this paper in Sec. VI. The derivation of the Majorana positivity condition, and the definition of the Euclidean gamma matrices are summarized in Appendices A, and B.

Let the Euclidean Majorana action
where Ψ is the Majorana fermion field. Because of the Grassmannian nature of the Majorana fermion, only antisymmetric components of P contribute to the Grassmann integration, and we set P to an antisymmetric matrix without loss of generality. Then the generating functional Z is given as the real Grassmann integral, and is expressed by the Pfaffian Pf(P ) as Z = DΨ e −S = Pf(P ).
Therefore the standard numerical simulations on the basis of the Monte Carlo sampling do not suffer from the fermionic sign problem when Pf(P ) is semi-positive. Let us consider the case that the antisymmetric 2N × 2N matrix P is given by a block matrix form where P i are N × N complex matrices, and N is an even number. Because of the antisymmetry of P , P 1 = −P 1 and P 3 = −P 3 . Pf(P ) is positive semidefinite if condition 1: P 2 is semi-positive, condition 2: P 3 = −P † 1 and P 2 = P † 2 are satisfied. The proof is given in Appendix A. This is a sufficient condition for the semipositivity of a Pfaffian. We call it the Majorana positivity condition, following Ref. [8].
In general, the matrix P has the ambiguity of the basis transformation, which keeps the is given by the functional trace tr(e −βH ) [8,10]. Here, we discuss the condition in the Lagrangian formalism, where the generating functional is given by the Pfaffian Pf(P ). The conditions for P to assure the semi-positivity of Pf(P ) are slightly different from those for the Hamiltonian H to assure the semi-positivity of tr(e −βH ). We need to assume that the block matrices P i are even-dimensional matrices, which is not assumed in Refs. [8,10]. Because of the spinor structure of relativistic fermions, this assumption always holds in the following, so that it is enough to study the constraints to the Dirac operators by the conditions 1 and 2. For details, see Appendix A.
This argument can be applied to the Dirac fermion integral. The Dirac field ψ and its conjugate fieldψ are expressed by using two Majorana fields Ψ (1) and Ψ (2) [11] as and With the two-component Majorana field Ψ = (Ψ (1) , Ψ (2) ), the Dirac action becomes the The generating functional is semi-positive if P satisfies the conditions 1 and 2.

III. ONE FLAVOR
We consider the one-flavor Dirac fermion action with the Dirac operators with (The matrix elements are explicitly shown in Appendix B.) Because of the Grassmannian nature of ψ and ψ c , the symmetric parts of h and h do not contribute to the integral. Thus, Introducing the Majorana field Ψ = (Ψ (1) , Ψ (2) ) = (R (1) , L (1) , R (2) , L (2) ), and changing the basis as ( with where H S RL = (H RL + H LR )/2 and so on. The block matrix P 2 must be semi-positive to satisfy the condition 1. Since the matrix elements of P 2 are given by H S RL , H AS RL , h AS RL , and h AS RL , they give constraints on M , M 5 , Σ µν , d µ , d µ , d 5µ , and d 5µ . As for the condition 2, and P 2 = P † 2 is satisfied when Equations (23)- (26), and the semi-positivity of P 2 guarantee the semi-positivity of the Pfaffian. The result is summarized in Table I. term imaginary chemical potential  imaginary orbit-rotation coupling [34] imaginary axial gauge field chiral chemical potential [35][36][37][38] imaginary spin-rotation coupling [34]  For example, the standard QCD Dirac operator is with D µ = ∂ µ + iA a µ T a = −D † µ , and real positive M ∈ R + . It satisfies Eqs.(23)-(26), and is positive definite. Thus it is sign-problem free. On the other hand, the lattice Wilson-Dirac operator does not satisfy the condition 1, because the Wilson term M is not semi-positive.
This is consistent with the fact that the one-flavor Wilson fermion has the sign problem.
Other known sign-problem-free terms, such as chiral chemical potential, are also explained by this Majorana positivity argument, as shown in

IV. TWO FLAVORS
We consider the two-flavor Dirac fermion action, where ψ is the two-flavor Dirac field. The Dirac operators are given by Eqs. By using the two-flavor Majorana fermions, Ψ = (u (1) , u (2) , d (1) , d (2) ), the action (30) reads where We change variables of the Grassmann integration as and then the generating functional becomes The block matrix P 2 C γ 5 must be semi-positive to satisfy the condition 1, which gives the constraints on the off-diagonal components of all the parameters in flavor space. The condition 2 is written as From Eq. (37), we have From Eq. (38), we also have Equations (39)- (43) and the semi-positivity of P 2 C γ 5 guarantee the semi-positivity of the Pfaffian. The result is shown in Table II.
For two flavors, the sign-problem-free classes are enlarged. A known example is the two-flavor QCD Dirac operator with the degenerate mass M and nonzero isospin chemical potential µ π , with D µ = ∂ µ + iA a µ T a = −D † µ , and real M , µ π ∈ R. The Dirac operator (44) satisfies Eqs. (39)- (43) and the semi-positivity condition P 2 C γ 5 = 0. Thus the isospin chemical potential is sign-problem free unlike the baryon chemical potential. Another example is the Wilson-Dirac operator. The Wilson term M satisfies Eq. (39). Thus the Wilson-Dirac operator is semi-positive for two flavors, while it is not for one flavor.

V. MISCELLANEOUS
Several comments are listed here: • The derived conditions are sufficient conditions, not necessary conditions, for signproblem-free classes. Other sign-problem-free classes will be possible. For example, the fermions obtained by the basis transformation from sign-problem-free fermions, such as spatially twisted chemical potential [44], are also sign-problem free.
• The classification is independent of whether the parameter is a dynamical field or an gauge field isospin chemical potential [23,26,[40][41][42] isospin electric field [43] isospin axial gauge field chiral chemical potential [35][36][37][38] II. Summary table of sign-problem-free terms of the two-flavor Dirac fermion. In the column of condition 1, the checkmark stands for the constraint by the semi-positivity of P 2 C γ 5 to the off diagonal components. external source. When the parameter is dynamical, the integral is taken after the fermion integral.
• While all the components are sign-problem free for dynamical gauge fields, the corresponding physical situations depend on the components for external gauge fields.
An external magnetic field is sign-problem free but an external electric field has the sign problem [45]. Similarly, an external axial electric field is sign-problem free but an external axial magnetic field has the sign problem.
• The classification is applicable to the theory with four-fermion interactions. A fourfermion action is converted to a bilinear-fermion action with an auxiliary field by the Hubbard-Stratonovich transformation. When the resultant fermion action satisfies the Majorana positivity condition, the theory is sign-problem free.

VI. SUMMARY
We discussed sign-problem-free relativistic fermions on the basis of the Majorana positivity. The results are summarized in Tables I and II. All known sign-problem-free terms were classified and some new sign-problem-free terms were found. The results will be imme- We here prove the Majorana positivity condition. From the condition 1, P 2 is written as where U is an unitary matrix, and Λ is a diagonal matrix, whose components are nonnegative.
From the condition 2, P is equivalently written by the form Then P is rewritten as where and the determinant is semi-definite, so that the our goal is to show the semi-positivity of Pf(P ). We expand Pf(P ) by the number r to count how many times the off-diagonal component i1 N contributes. For a certain value of r, the contribution reads The charge conjugation matrix is satsfying C * = C, C = −C, C † C = 1, C 2 = −1.
We also define The matrix elements of Eq. (14) are those of Eq. (15) are and those of Eq. (16) are