Matter-antimatter coexistence method for finite density QCD toward a solution of the sign problem

TL;DR

The paper proposes a novel matter-antimatter coexistence method in lattice QCD to mitigate the sign problem at finite density by correlating matter and anti-matter systems across parallel sheets, enabling more feasible calculations.

Contribution

It introduces a new correlation approach between matter and anti-matter systems in lattice QCD to suppress the sign problem, with a pathway for physical quantity estimation via extrapolation.

Findings

Realization of a real, non-negative fermionic determinant at large correlation parameter.

Suppression of the sign problem through phase cancellation between matter and anti-matter determinants.

Potential for lattice calculations at finite density using the proposed method.

Abstract

Toward the lattice QCD calculation at finite density, we propose "matter-antimatter coexistence method", where matter and anti-matter systems are prepared on two parallel ${\bf R}^4$-sheets in five-dimensional Euclidean space-time. We put a matter system $M$ with a chemical potential $\mu \in {\bf C}$ on a ${\bf R}^4$-sheet, and also put an anti-matter system $\bar M$ with $-\mu^*$ on the other ${\bf R}^4$-sheet shifted in the fifth direction. Between the gauge variables $U_\nu \equiv e^{iagA_\nu}$ in $M$ and $\tilde U_\nu \equiv e^{iag \tilde A_\nu}$ in $\bar M$, we introduce a correlation term with a real parameter $\lambda$. In one limit of $\lambda \rightarrow \infty$, a strong constraint $\tilde U_\nu(x)=U_\nu(x)$ is realized, and therefore the total fermionic determinant becomes real and non-negative, due to the cancellation of the phase factors in $M$ and $\bar M$, although this system resembles QCD with an isospin chemical potential. In another limit of $\lambda \rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $\mu$ and $-\mu^*$. For a given finite-volume lattice, if one takes an enough large value of $\lambda$, $\tilde U_\nu(x) \simeq U_\nu(x)$ is realized and phase cancellation approximately occurs between two fermionic determinants in $M$ and $\bar M$, which suppresses the sign problem and is expected to make the lattice calculation possible. For the obtained gauge configurations of the coexistence system, matter-side quantities are evaluated through their measurement only for the matter part $M$. The physical quantities in finite density QCD are expected to be estimated by the calculations with gradually decreasing $\lambda$ and the extrapolation to $\lambda=0$. We also consider more sophisticated improvement of this method using an irrelevant-type correlation.

Equations

$$S[A,\psi ,\bar{\psi };\mu ]$$

$$\in$$

$$S[A,\psi ,\bar{\psi };\mu {]}^{\ast }=S[A,\psi ,\bar{\psi };-{\mu }^{\ast }],$$

$$[\bar{\psi }(/D+m+\mu {\gamma }_4)\psi {]}^{\ast }=\bar{\psi }(/D+m-{\mu }^{\ast }{\gamma }_4)\psi ,$$

$$\mathrm{Det}(/D+m+\mu {\gamma }_4{)}^{\ast }=\mathrm{Det}(/D+m-{\mu }^{\ast }{\gamma }_4).$$

$$[\bar{\psi }(D_F+\mu {\gamma }_4)\psi {]}^{\ast }=\bar{\psi }(D_F-{\mu }^{\ast }{\gamma }_4)\psi ,$$

$$\mathrm{Det}(D_F+\mu {\gamma }_4{)}^{\ast }=\mathrm{Det}(D_F-{\mu }^{\ast }{\gamma }_4).$$

$$S_{\lambda }\equiv \sum _{x,\nu }2\lambda \{N_c-\mathrm{Re}\mathrm{tr}[U_{\nu }(x){\tilde{U}}_{\nu }^{†}(x)]\}$$

$$S$$

$$+$$

$$Z$$

$$=$$

$$S_{\lambda }$$

$$\simeq$$

$$Z_{\mathrm{cont}}$$

$$Z$$

$$O_{\mathrm{phase}}[U,\tilde{U}]\equiv e^{i\arg [\mathrm{Det}\{(D_F[U]+\mu {\gamma }_4)(D_F[\tilde{U}]-{\mu }^{\ast }{\gamma }_4)\}]}$$

$$\mathrm{Det}\{(D_F[U]+\mu {\gamma }_4)(D_F[\tilde{U}=U]-{\mu }^{\ast }{\gamma }_4)\}\ge 0,$$

$${\bar{S}}_{\xi }$$

$$\simeq$$

Full text

I I. Introduction

Nowadays, quantum chromodynamics (QCD) has been established as the fundamental theory of strong interaction. Together with the success of perturbative QCD for high-energy processes of hadron reactions, the lattice QCD Monte Carlo simulation has been a powerful tool to analyze nonperturbative aspects of QCD, after the formulation of lattice QCD and its numerical success C7980 . Indeed, lots of studies of the QCD vacuum, hadrons and the quark-gluon plasma have been done for both zero-temperature and finite-temperature in lattice QCD.

Finite density QCD is also important to understand the QCD diagram, nuclear systems and neutron stars, and it is desired to perform the lattice QCD analysis as the first-principle calculation of the strong interaction. However, there appears a serious problem called the “sign problem” P83 ; FK02 in the practical lattice QCD calculation at finite density. This problem originates from the complex value including minus sign of the QCD action and the fermionic determinant at finite density even in the Euclidean metric HK84 .

At finite density with the chemical potential $\mu$, the Euclidean QCD action $S[A,\psi,\bar{\psi};\mu]$ is generally complex,

[TABLE]

with the gauge action $S_{G}[A]\in{\bf R}$ and covariant derivative $D^{\nu}\equiv\partial^{\nu}+igA^{\nu}$. (In this paper, we use hermite $\gamma$-matrices of $\gamma_{\mu}^{\dagger}=\gamma_{\mu}$ in the Euclidean metric.) Therefore, one cannot identify the action factor as a probability density in the QCD generating functional, unlike ordinary lattice QCD calculations. Also, the fermionic determinant at finite density generally takes a complex value HK84 , and its phase factor is drastically changed depending on the gauge configuration in a large-volume lattice, which makes numerical analyses difficult. This is the sign problem.

In this paper, we propose a new method of “matter-antimatter coexistence method” S17 utilizing the charge conjugation symmetry for the practical lattice QCD calculation at finite density, aiming at a possible solution of the sign problem. The organization of this paper is as follows. In Sec. II, we propose a new theoretical method of “matter-antimatter coexistence method” in Euclidean QCD at finite density, and show its actual procedure for the lattice calculation. Section III will be devoted to the summary and the conclusion.

II II. Matter-Antimatter Coexistence Method

In this section, we introduce the matter-antimatter coexistence method for general complex chemical potential $\mu\in{\bf C}$. In this method, we use phase cancellation of the fermionic determinants between a matter system with $\mu$ and an anti-matter system with $-\mu^{*}$, which generally holds in QCD at finite density.

II.1 A. General property of QCD at finite density

To begin with, we start from the general property of finite-density QCD HK84 ,

[TABLE]

for the Euclidean QCD action $S[A,\psi,\bar{\psi};\mu]$ in the presence of the chemical potential $\mu\in{\bf C}$.

For instance, in continuum QCD, one finds

[TABLE]

which leads to Eq.(3) and

[TABLE]

Also in lattice QCD, the fermionic kernel $D_{F}$ corresponding to ${\not\!\!D}+m$ generally satisfies $D_{F}^{\dagger}=\gamma_{5}D_{F}\gamma_{5}$ R12 , and therefore one finds

[TABLE]

which leads to Eq.(3) and

[TABLE]

Then, as an exceptional case, the QCD action with the pure imaginary chemical potential $\mu\in i{\bf R}$ is manifestly real, and hence its lattice calculation is free from the sign problem. However, the QCD action is generally complex at finite density.

II.2 B. Definition and setup of matter-antimatter coexistence method

Now, we show the definition and setup of our approach, the “matter-antimatter coexistence method” S17 . In this method, we consider matter and anti-matter systems on two parallel ${\bf R}^{4}$-sheets in five-dimensional Euclidean space-time. For the matter system $M$ with a chemical potential $\mu\in{\bf C}$ on a ${\bf R}^{4}$-sheet, we also prepare the anti-matter system $\bar{M}$ with $-\mu^{*}$ on the other ${\bf R}^{4}$-sheet shifted in the fifth direction, as shown in Fig. 1.

We put an ordinary fermion field $\psi(x)$ with the mass $m$ and the gauge variable $U_{\nu}(x)\equiv e^{iagA_{\nu}(x)}$ at $x\in{\bf R}^{4}$ on the matter system $M$, and put the other fermion field $\Psi(x)\equiv\psi(x+\hat{5})$ with the same mass $m$ and the gauge variable $\tilde{U}_{\nu}(x)\equiv e^{iag\tilde{A}_{\nu}(x)}\equiv U_{\nu}(x+\hat{5})$ on the anti-matter system $\bar{M}$. Here, $\hat{5}$ denotes the fifth direction vector with an arbitrary length $a_{5}$, which is independent of four-dimensional lattice spacing $a$.

Between the gauge variables $U_{\nu}(x)$ in $M$ and $\tilde{U}_{\nu}(x)$ in $\bar{M}$, we introduce a correlation term such as

[TABLE]

with a real parameter $\lambda$ ($\geq 0$) in the lattice formalism. This correlation acts on $U_{\nu}(x)$ and $\tilde{U}_{\nu}(x)$ at the same four-dimensional coordinate $x\in{\bf R}^{4}$, and connects two different situations: $\tilde{U}_{\nu}(x)=U_{\nu}(x)$ in $\lambda\rightarrow\infty$ and two separated QCD systems in $\lambda\rightarrow 0$.

In fact, the total lattice action in this method is expressed as

[TABLE]

with the gauge action $S_{G}[U]\in{\bf R}$ and the fermionic kernel $D_{F}[U]$ in lattice QCD. After integrating out the fermion fields $\psi$ and $\Psi$, the generating functional of this theory reads

[TABLE]

II.3 C. Four-dimensional continuum limit

Near the four-dimensional continuum limit of $a\simeq 0$, this additional term becomes

[TABLE]

with $\lambda_{\rm phys}\equiv\lambda a^{-2}$, and the generating functional goes to

[TABLE]

with the continuum gauge action $S_{G}[A]\in{\bf R}$ and $\tilde{D}^{\nu}\equiv\partial^{\nu}+ig\tilde{A}^{\nu}$.

II.4 D. Lattice calculation procedure

Next, we show the actual procedure of this method for the lattice QCD calculation at finite density. In the practical lattice calculation with the Monte Carlo method, the fermionic determinant in $Z$ is factorized into its amplitude and phase factor as

[TABLE]

and the phase factor of the total fermionic determinant

[TABLE]

is treated as an “operator” instead of a probability factor, while all other real non-negative factors in $Z$ can be treated as the probability density.

The additional term $S_{\lambda}$ connects the following two different situations as the two limits of the parameter $\lambda$:

1. In one limit of $\lambda\rightarrow\infty$, a strong constraint $\tilde{U}_{\nu}(x)=U_{\nu}(x)$ is realized, and the phase factors of two fermionic determinants ${\rm Det}(D_{F}[U]+\mu\gamma_{4})$ and ${\rm Det}(D_{F}[\tilde{U}]-\mu^{*}\gamma_{4})$ are completely cancelled, owing to Eq. (7). Therefore, the total fermionic determinant is real and non-negative,

[TABLE]

and the numerical calculation becomes possible without the sign problem. Note however that this system resembles QCD with an isospin chemical potential SS01 , which is different from finite density QCD.

2. In another limit of $\lambda\rightarrow 0$, this system goes to “two separated ordinary QCD systems” with the chemical potential of $\mu$ and $-\mu^{*}$, although the cancellation of the phase factors cannot be expected between the two fermionic determinants ${\rm Det}(D_{F}[U]+\mu\gamma_{4})$ and ${\rm Det}(D_{F}[\tilde{U}]-\mu^{*}\gamma_{4})$ for significantly different $U_{\nu}(x)$ and $\tilde{U}_{\nu}(x)$, which are independently generated in the Monte Carlo simulation.

In fact, in other words, this approach links QCD with a chemical potential and QCD with an isospin-chemical potential.

For a given four-dimensional finite-volume lattice, if one takes an enough large value of $\lambda$, $\tilde{U}_{\nu}(x)\simeq U_{\nu}(x)$ is realized, and approximate phase cancellation occurs between the two fermionic determinants ${\rm Det}(D_{F}[U]+\mu\gamma_{4})$ and ${\rm Det}(D_{F}[\tilde{U}]-\mu^{*}\gamma_{4})$ in $M$ and $\bar{M}$. Then, we expect a modest behavior of the phase factor $O_{\rm phase}[U,\tilde{U}]$ in Eq.(27), which leads to feasibility of the numerical lattice calculation with suppression of the sign problem.

Once the lattice gauge configurations of the coexistence system are obtained with the most importance sampling in the Monte Carlo simulation, matter-side quantities can be evaluated through their measurement only for the matter part $M$ with $\mu$.

By performing the lattice calculations with gradually decreasing $\lambda$ and their extrapolation to $\lambda=0$, we expect to estimate the physical quantities in finite density QCD with the chemical potential $\mu$. (This procedure may resemble the chiral extrapolation, where the current quark mass $m$ is gradually reduced and the lattice data is extrapolated to $m=0$.)

II.5 E. More sophisticated correlation between matter and antimatter systems

As a problem in this method, there could appear an obscure effect from the additional correlation at the finite value of $\lambda$. On this point, we here consider a possible remedy in this framework.

So far, we have demonstrated this method by taking the simplest correlation of $S_{\lambda}$ in Eq.(8). In this method, however, there is some variety on the choice of the correlation between $U_{\nu}(x)$ in $M$ and $\tilde{U}_{\nu}(x)$ in $\bar{M}$.

In fact, the validity of the data extrapolation can be checked by various extrapolations with different type of the additional correlation.

In particular, it is interesting to consider more sophisticated correlation such as

[TABLE]

with a dimensionless non-negative real parameter $\xi$. At the classical level, this correlation is an irrelevant interaction and it gives vanishing contributions in the continuum limit $a\rightarrow 0$, like the Wilson term $-\frac{1}{2}ar\bar{\psi}D^{2}\psi$ R12 .

By the use of this irrelevant-type correlation, the effect from the additional term is expected to be reduced in the actual lattice calculation.

III III. Summary and conclusion

We have proposed the “matter-antimatter coexistence method” toward the lattice calculation of finite density QCD. In this method, we have prepared matter $M$ with $\mu$ and anti-matter $\bar{M}$ with $-\mu^{*}$ on two parallel ${\bf R}^{4}$-sheets in five-dimensional Euclidean space-time, and have introduced a correlation term $S_{\lambda}\equiv\sum_{x,\nu}2\lambda\{N_{c}-{\rm Re~{}tr}[U_{\nu}(x)\tilde{U}_{\nu}^{\dagger}(x)]\}\simeq\sum_{x}\frac{1}{2}\lambda a^{2}\{A_{\nu}^{a}(x)-\tilde{A}_{\nu}^{a}(x)\}^{2}$ between the gauge variables $U_{\nu}=e^{iagA_{\nu}}$ in $M$ and $\tilde{U}_{\nu}=e^{iag\tilde{A}_{\nu}}$ in $\bar{M}$. In one limit of $\lambda\rightarrow\infty$, owing to $\tilde{U}_{\nu}(x)=U_{\nu}(x)$, the total fermionic determinant is real and non-negative, and the sign problem is absent. In another limit of $\lambda\rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $\mu$ and $-\mu^{*}$.

For a given finite-volume lattice, if one takes an enough large value of $\lambda$, $\tilde{U}_{\nu}(x)\simeq U_{\nu}(x)$ is realized and phase cancellation approximately occurs between two fermionic determinants in $M$ and $\bar{M}$, which is expected to suppress the sign problem and to make the lattice calculation possible. For the obtained gauge configurations of the coexistence system, matter-side quantities can be evaluated by their measurement only for the matter part $M$. By gradually reducing $\lambda$ and the extrapolation to $\lambda=0$, it is expected to obtain estimation of the physical quantities in finite density QCD with $\mu$.

The next step is to perform the actual lattice QCD calculation at finite density using this method. It would be useful to combine this method with the other known ways such as the hopping parameter expansion ASSS14 , the complex Langevin method P83 and the reweighting technique FK02 . For example, if the hopping parameter expansion is utilized, huge calculations of the fermionic determinant can be avoided, and a low-cost analysis with the quenched gauge configuration becomes possible, since the additional term $S_{\lambda}$ only includes gauge variables. In addition to the actual lattice calculation, the effect from the additional term is to be investigated carefully.

Efficiency of this method would strongly depend on the system parameters, such as the space-time volume $V$, the quark mass $m$, the temperature $T$ and the chemical potential $\mu$. For instance, near the chiral limit $m=0$ in large $V$, the fermionic determinant tends to possess quasi-zero-eigenvalues, which permits a drastic change of the phase in the fermionic determinant, although the zero fermionic-determinant case would give no significant contribution in the QCD generating functional. In any case, this method is expected to enlarge calculable area of the QCD phase diagram on $(T,\mu,m,V)$.

Acknowledgements

The author is supported in part by the Grant for Scientific Research [(C)15K05076] from the Ministry of Education, Science and Technology of Japan.