Sign Problem
in Two-Color Two-Flavor QCD
with mq and mI
Kenji Fukushima
RIKEN BNL Research Center (RBRC)
in collaboration with Kei Iida (Kochi U.)
and special thanks to MOMIJI Collaboration
(K.F, K.I, A.Nakamura, C.Nonaka)
Lattice 2007 in Regensburg
Short conclusion
Monte-Carlo simulations
in finite-density two-color QCD
may be possible even with
non-degenerate quark flavors.
"Crystalline Superfluid Phase"
Lattice 2007 in Regensburg
You might think...
Two-color QCD has no sign problem???
□ It is said IT CAN HAVE.
It is easy to prove det M ( m )  det M * ( m )
□ Two degenerate flavors give
2
[det M (m )]  (non - negative)
Dirac det. is not necessarily non-negative,
but it might be non-negative in fact.
Lattice 2007 in Regensburg
Question
Two-color single-flavor QCD,
does it really have the sign problem?
(where only u-quarks are present)
Or, what about two-color
two-(non-degenerate)-flavor QCD?
(where u/d-quarks are present with
mu md diff. and/or mu md diff.)
Lattice 2007 in Regensburg
Why interesting?
Two-Color Two-Flavor System
□ Chiral Symmetry Breaking (Meson Cond.)
 ~ uu  dd
□ Superfluidity (Baryon Cond.)
 ~ ud
uu dd
Chemical Potential Difference
Total p non-zero.
Kinetic energy cost.
No dm cost.
 Crystal Structure
Lattice 2007 in Regensburg
Total p zero.
No kinetic energy cost.
dm cost.
Why "crystal" interesting??
Condensed Matter Physics
□ FFLO State
Fulde-Ferrell, Larkin-Ovchinnikov
(inside thin materials under a magnetic field)
QCD
□ Crystalline Color Superconductivity
induced as a result of the instability
(inside cores of the neutron star)
Lattice 2007 in Regensburg
Returning to the sign problem
Two-color two-(non-degenerate)flavor QCD, does it really have
the sign problem?
Determinant loses some information.
Eigenvalues tell us much more info.
M ( m ) n  (n  m) n
det M ( m )  (n  m)
Lattice 2007 in Regensburg
Not two-color; zero-density
In the continuum theory
On the lattice
Wilson term ~
M   m Dm  m
Hermite
 5 - Hermiticit y  5 M 5  M  preserved
anti-Hermite
smaller-r

Im
rDmDm
Im

larger-r
Re
Re
M  (  m)
m
M 5  (  m) 5
Lattice 2007 in Regensburg
m
m   , m  * , m   , m  *
m  , m  
Parity-violating phase
Not two-color; finite-density
Chemical potential ~ m  4 Hermite
 5 M ( m ) 5  M  ( m )

Im
Re
det M is complex in general.
m
 Sign Problem
Lattice 2007 in Regensburg
Two-color
Zero-density
Finite-density
 5 M ( m ) 5  M  ( m )
smaller-r
but

Im
 2 5C 1M ( m )C 5 2  M  ( m )
with some algebra...
larger-r
Re
m  , m   , m  , m  
m  , m  
*
m
*
Parity-violating phase
Exactly the same structure
is preserved even at finite density.
 No sign problem!
(except for the Parity-violating phase)
Lattice 2007 in Regensburg
Expected Physics
Conjecture by Splittorff-Son-Stephanov ('01)
Non - degenerate
two flavors
m I ~ mu  m d
They said that
the lattice simulation
was not possible...
but... could be.
Lattice 2007 in Regensburg
Where to search?
Prediction by Fukushima-Iida ('07)
Strong-coupling case
Weak-coupling case
Sufficient condition dm2  mI  dm1
Lattice 2007 in Regensburg
Summary
Two-color two-flavor QCD with nondegenerate u and d quarks is interesting.
□ Superfluidity with population imbalance induced
by the isospin chemical potential.
It is not QCD, but useful to clarify the crystal
structure of superfluidity at high density.
□ Crystal structure is hardly known yet in QCD.
□ MOMIJI (=MAPLE) collaboration.
Lattice 2007 in Regensburg