1. Matter-antimatter coexistence method for finite density QCD
as a possible solution of the sign problem
H. Suganuma (Kyoto U.)
We newly propose “matter-antimatter coexistence method ” for lattice QCD calculation at finite density, as a possible solution of the sign problem. For the matter system M with m > 0, the anti-matter system with -m < 0 is also prepared in the other spatial location. In this coexistence system, total generating functional is real and non-negative, and then one only has to deal with its real part, i.e., real part of total fermionic determinant, Re Det K. This advanced point is expected to reduce the sign problem significantly and to enable the numerical calculation in lattice QCD. Through the lattice QCD measurement only for the matter part M, one can obtain physical quantities in finite density QCD with m > 0.
arXiv: [hep-lat]
Excited QCD, Sintra, Portugal, 7-13 May 2017

2. Importance of QCD at Finite Density
Finite density QCD is fairly important to understand nuclear systems and neutron stars from the first principle of strong interaction, QCD.
From
From

3. QCD at Finite Density
Generally, finite density system is described by introducing chemical potential m and number operator N:
(Minkowski)
In Euclidean QCD, fermionic part at finite density is expressed by
(Euclidean)
This can be achieved by the simple replace of
Fermionic part at finite density is generally complex, because of

4. Sign Problem in Lattice QCD at Finite Density
Finite density QCD is fairly important to understand nuclear systems and neutron stars from the first principle of strong interaction, QCD. However, there appears a serious problem called “sign problem” in the practical lattice QCD calculation at finite density.
The origin of the sign problem is the complex value of the QCD action and fermionic determinant at finite density, even in the Euclidean metric, so that one cannot identify the action factor as a probability density in the QCD generating functional, unlike ordinary lattice QCD calculations.

5. Sign Problem in Lattice QCD at Finite Density
Finite density QCD is fairly important to understand nuclear systems and neutron stars from the first principle of strong interaction, QCD. However, there appears a serious problem called “sign problem” in the practical lattice QCD calculation at finite density. The origin of the sign problem is the complex value of the QCD action and fermionic determinant at finite density, even in the Euclidean metric, so that one cannot identify the action factor as a probability density in the QCD generating functional, unlike ordinary lattice QCD calculations. Owing to this sign problem, finite-density lattice QCD calculations cannot be generally performed for about 37 years, since the first success of lattice QCD simulation by M. Creutz in 1980.

6. General Property of QCD with chemical potential m
In the presence of chemical potential m in Euclidean metric, QCD action, generating functional, fermionic determinant satisfy
because of in continuum QCD.
cf. In the case of purely imaginary chemical potential, the theory becomes real:

7. Lattice QCD formalism (Euclidean) at finite density
[P. Hasenfratz and F. Karsch, Phys. Rept. 103 (1984) , “Finite Fermion Density on the Lattice”.]
: link variable
: plaqutte variable
m : chemical potential
k : hopping parameter
r : Wilson parameter
This can be achieved by the simple replace of

8. Lattice QCD formalism at finite density
: chemical potential
K(m): fermionic kernel
Here, hermite conjugation of K is taken for all the indices including x,y.
General Relations of QCD with chemical potential m

9. anti-matter M with m < 0
Matter-Antimatter Coexistence method
Our strategy is rather simple.
we also prepare
an anti-matter system with - m < 0 in the other spatial location.
For the matter system M with m > 0,
matter M with m > 0
anti-matter M with m < 0 -
The number of degrees of freedom in M is set to be just the same as that in .
Then, this system has a charge conjugation symmetry. In lattice QCD, however,
we need not impose charge conjugation symmetry for the fields between M and .
In this coexistence system, the total generating functional is real and non-negative. Then, we only have to deal with the real part of the total fermionic determinant.

10. Matter-Antimatter Coexistence method
The simplest example is to prepare two lattices with m and - m,
where charge conjugation symmetry exists.
matter system M with m > 0
anti-matter system with m < 0
Once the lattice-QCD gauge configuration is obtained in this system,
one can calculate physical quantities in finite density QCD with m > 0,
by the measurement only for the matter part M.

11. z m (z) Matter-Antimatter Coexistence method m0 -m0
A simple example is to consider the space-coordinate dependent chemical potential, m (z) ~ m0 sign(z), where charge conjugation symmetry exists under reflection
m (z)
matter system M with m > 0 m0 z
-m0
anti-matter system with m < 0
Once the lattice-QCD gauge configuration is obtained in this system,
one can calculate physical quantities in finite density QCD with m > 0,
by the measurement only for the matter part M.

12. z m (z) Matter-Antimatter Coexistence method m0 -m0
The other simple version is the “C-periodic” case, m (z+l) = -m (z),
where charge conjugation symmetry exists under translation
m (z)
[cf. A.S.Kronfeld, U.J.Wiese NPB357 (1991) 521;U.J.Wiese, NPB375 (1992) 45.]
matter system M
with m > 0 m0 z -l O l 2l
-m0
anti-matter system
with m < 0
L=2l is the periodicity of this system.
Once the lattice-QCD gauge configuration is obtained for large l ,
one can calculate physical quantities in finite density QCD with m > 0,
by the measurement only for the matter part M.

13. Matter-Antimatter Coexistence method
For the “matter” M with m, we prepare “anti-matter” with -m*.
General relations:
In this matter-antimatter coexistence system ,
total generating functional becomes
real and non-negative in Euclidean space-time.
We do not impose charge conjugation symmetry for the fields between M and Mbar,
but only use the same number of degrees of freedom between them.

14. Real Projection in Matter-Antimatter Coexistence method
We denote kernel of the coexistence system by
We can perform Real Projection of generating functional and fermionic determinant of matter-antimatter coexistence system:
In fact, we only have to deal with its real part , i.e.,
real part of total fermionic determinant,
This point is expected to reduce the sign problem significantly,
since the total amount of Re Det K is non-negative.

15. Matter-Antimatter Coexistence method
The only remaining problem is the sign of Re Det K,
although its total amount is non-negative.
One simple way is to deal with the sign as an operator factor in
treated as a probability density
If this sign factor does not change largely in the coexistence system,
this method would give a solution of the sign problem.
If this sign factor largely changes even in the coexistence system,
this method would fail and the sign problem remains.
Success or failure would depend on the system parameters: chemical potential m ,
bare quark mass m and space-time volume V. Particularly near m = 0 in large V,
K has quasi-zero-eigenvalues, which can give a drastic change of sign(Re Det K).

16. anti-matter M with m < 0
Charge-conjugation-symmetric Matter-Antimatter system
When charge conjugation symmetry is imposed for gauge fields between M and Mbar, total fermionic determinant is always
real and non-negative,
Then, such a charge-conjugation-symmetric system seems to be
an appropriate efficient candidate as the starting gauge configuration for the coexistence system in the practical Monte Carlo calculation.
matter M with m > 0
charge conjugation
anti-matter M with m < 0 -
This is a kind of gauge configuration to describe the coexistence system,
although this constraint on the gauge field is fairly strong.

17. - Charge-conjugation-symmetric Matter-Antimatter system
In the charge-conjugation-symmetric system,
matter and anti-matter gauge variables are related:
Suffix c denotes the charge conjugation. -
matter M
anti-matter M
charge conjugation

18. Charge-conjugation-symmetric Matter-Antimatter system
For the matter part M
: chemical potential
KM(m): fermionic kernel in the matter part M
For the anti-matter , the gauge field is set to be charge conjugate to that in the matter M. -
anti-matter M
For the charge conjugation symmetric system, total fermionic determinant is always real and non-negative, and the fermionic determinant can be regarded as a probability density in the QCD generating functional.

19. Lattice QCD formalism (Euclidean) at finite density
[P. Hasenfratz and F. Karsch, Phys. Rept. 103 (1984) , “Finite Fermion Density on the Lattice”.]
: link variable
: plaqutte variable
m : chemical potential
k : hopping parameter
r : Wilson parameter
This can be achieved by the simple replace of

20. Charge-conjugation-symmetric Matter-Antimatter system
For the matter part M
: chemical potential
KM(m): fermionic kernel in the matter part M
For the anti-matter , the link-variable is set to be charge conjugate to that in the matter M. -
anti-matter M
h.c. of K is taken for all the indices including x,y.
For the charge conjugation symmetric system, total fermionic determinant is always real and non-negative, and it can be regarded as a probability density in lattice QCD.

21. - Charge-conjugation-symmetric Matter-Antimatter system
To generate the “charge-conjugation-symmetric system” in lattice QCD, we adopt “charge-conjugation-symmetric update” for the link variable in the lattice-QCD Monte Carlo simulation. -
matter M
anti-matter M
charge conjugation
cf. periodic update on periodic lattice
Through this charge-conjugation-symmetric update for the link-variable,
charge conjugation symmetry is kept, and one can generate lattice-QCD
gauge configurations corresponding to charge-conjugation-symmetric
matter-antimatter system in the Monte Carlo calculation.

22. anti-matter M with m < 0
Matter-Antimatter Coexistence method after C-symmetric generation
In lattice QCD, it is better to start from this charge-conjugation-symmetric system,
because total fermionic determinant is always real and non-negative.
matter M with m > 0
charge conjugation
anti-matter M with m < 0 -
Starting from C-symmetric gauge configuration, we use Real Projection in
Matter-Antimatter Coexistence method, and update the gauge configuration:
Through the update, we expect some persistency of the sign of Re Det K in the
mater-antimatter coexistence system, although it depends on system parameters.

23. Fermionic Kernel of lattice QCD at Finite Density
Positive Real
Real
density effect, complex
Real
Wilson parameter: r = 1
: chemical potential
hopping parameter:
in usual lattice QCD calculation
In particular, k is fairly small for large bare mass quark.
WHOT-QCD, PRD 84, (2011).
Positive Real
density effect, complex
plaquette average
Polyakov loop
For odd n, Tr Q n = 0
For n = 2, Tr Q 2 = 0 because of
At least, Re det K is positive for large quark mass, small chemical potential, confinement phase.

24. anti-matter M with m < 0
Matter-Antimatter Coexistence method after C-symmetric generation
In lattice QCD, it is better to start from this charge-conjugation-symmetric system,
because total fermionic determinant is always real and non-negative.
matter M with m > 0
charge conjugation
anti-matter M with m < 0 -
Starting from C-symmetric gauge configuration, we use Real Projection in
Matter-Antimatter Coexistence method, and update the gauge configuration:
Through the update, we expect some persistency of the sign of Re Det K in the
mater-antimatter coexistence system, although it depends on system parameters.

25. anti-matter M with m < 0
Matter-Antimatter Coexistence method in Lattice QCD calculation
Once one can generate lattice-QCD gauge configuration of
the matter-antimatter coexistence system,
matter M with m > 0
anti-matter M with m < 0 -
Through the measurement only for the matter part M,
one can calculate physical quantities in finite density QCD with m > 0,
like the ordinary lattice-QCD calculation.

26. - Summary and Conclusion
We have proposed “matter-antimatter coexistence method ” for lattice QCD calculation at finite density, as a possible solution of the sign problem. - For the matter system M with m > 0, we also prepare anti-matter system with -m < 0 in other spatial location. - In this coexistence system, total generating functional Z is real and non-negative, and then we only have to deal with its real part, i.e., real part of the total fermionic determinant, Re Det K. This advanced point is expected to reduce the sign problem significantly and to enable the numerical calculation in lattice QCD. - Once such gauge configurations are obtained, one can evaluate physical quantities in finite density QCD with m > 0, through lattice QCD calculation only for the matter part M. - The next important step is to perform the numerical lattice QCD calculation. -
matter M with m > 0
anti-matter M with m < 0
arXiv: [hep-lat]