Lattice QCD, Random Matrix
Theory and chiral condensates
JLQCD collaboration,
Phys.Rev.Lett.98,172001(2007) [hep-lat/0702003],
Phys.Rev.D76,054503 (2007) [arXiv:0705.3322] ,
arXiv:0711.4965.
Hidenori Fukaya (Niels Bohr Institute)
for JLQCD collaboration
1
JLQCD Collaboration
KEK
S. Hashimoto, T. Kaneko, H. Matsufuru,
J. Noaki, M. Okamoto, E. Shintani, N. Yamada
RIKEN -> Niels Bohr H. Fukaya
Tsukuba
S. Aoki, T. Kanaya, Y. Kuramashi, N. Ishizuka,
Y. Taniguchi, A. Ukawa, T. Yoshie
Hiroshima
K.-I. Ishikawa, M. Okawa
YITP
H. Ohki, T. Onogi
TWQCD Collaboration
National Taiwan U. T.W.Chiu, K. Ogawa,
KEK BlueGene (10 racks, 57.3 TFlops)
2
1. Introduction
Chiral symmetry
and its spontaneous breaking are important.
– Mass gap between pion and the other hadrons
pion as (pseudo) Nambu-Goldstone boson
while the other hadrons acquire the mass ~LQCD.
– Soft pion theorem
– Chiral phase transition at finite temperature…
But QCD is highly non-perturbative.
3
1. Introduction
Lattice QCD
is the most promising approach to confirm chiral SSB
from 1-st principle calculation of QCD. But…
1. Chiral symmetry is difficult. [Nielsen & Ninomiya 1981]
Recently chiral symmetry is redefined [Luescher 1998] with
a new type of Dirac operator [Hasenfratz 1994, Neuberger
1998] satisfies the Ginsparg-Wilson [1982] relation
but numerical implementation and m->0 require a large
computational cost.
2. Large finite V effects when m-> 0.
as m->0, the pion becomes massless.
(the pseudo-Nambu-Goldstone boson.)
4
1. Introduction
This work
1.
We achieved lattice QCD simulations with exact
chiral symmetry.
•
•
•
Exact chiral symmetry with the overlap fermion.
With a new supercomputer at KEK ( 57 TFLOPS )
Speed up with new algorithms + topology fixing
On (~1.8fm)4 lattice, achieved m~3MeV !
=>
2. Finite V effects evaluated by the effective theory.
•
m, V, Q dependences of QCD Dirac spectrum are calculated
by the Chiral Random Matrix Theory (ChRMT).
-> A good agreement of Dirac spectrum with ChRMT.
–
–
Strong evidence of chiral SSB from 1st principle.
obtained
5
Contents
 1. Introduction
2. QCD Dirac spectrum & ChRMT
3. Lattice QCD with exact chiral
symmetry
4. Numerical results
5. NLO effects
6. Conclusion
6
2. QCD Dirac spectrum & ChRMT
Banks-Casher relation
[Banks &Casher 1980]
7
2. QCD Dirac spectrum & ChRMT
•
•
Banks-Casher relation
In the free theory,
r(l) is given by the surface of S3
with the radius l:
[Banks &Casher 1980]
low density
With the strong coupling
The eigenvalues feel the repulsive
force from each other→becoming
non-degenerate→ flowing to the
low-density region around zero→
results in the chiral condensate.
Σ
8
2. QCD Dirac spectrum & ChRMT
Chiral Random Matrix Theory (ChRMT)
Consider the QCD partition function at a fixed topology Q,
•
High modes ( l >> LQCD ) -> weak coupling
•
Low modes ( l << LQCD ) -> strong coupling
⇒ Let us make an assumption: For low-lying modes,
with an unknown action V(l)
⇒ ChRMT.
[Shuryak & Verbaarschot,1993, Verbaarschot & Zahed,
1993,Nishigaki et al, 1998, Damgaard & Nishigaki, 2001…]
9
2. QCD Dirac spectrum & ChRMT
Chiral Random Matrix Theory (ChRMT)
Namely, we consider the partition function (for low-modes)
•
Universality of RMT [Akemann et al. 1997] :
IF V(l) is in a certain universality class, in large n limit (n :
size of matrices) the low-mode spectrum is proven to be
equivalent, or independent of the details of V(l) (up to a scale
factor) !
•
From the symmetry, QCD should be in the same universality
class with the chiral unitary gaussian ensemble,
and share the same spectrum, up to a overall
10
2. QCD Dirac spectrum & ChRMT
Chiral Random Matrix Theory (ChRMT)
In fact, one can show that the ChRMT is equivalent to the
moduli integrals of chiral perturbation theory [Osborn et al, 1999];
The second term in the exponential is written as
where
Let us introduce Nf x Nf real matrix s1 and s２ as
11
2. QCD Dirac spectrum & ChRMT
Chiral Random Matrix Theory (ChRMT)
Then the partition function becomes
where
is a NfxNf complex matrix.
With large n, the integrals around the suddle point, which
satisfies
leaves the integrals over U(Nf) as
equivalent to the ChPT moduli’s integral in the e regime.
⇒
12
2. QCD Dirac spectrum & ChRMT
Eigenvalue distribution of ChRMT
Damgaard & Nishigaki [2001] analytically derived the
distribution of each eigenvalue of ChRMT.
For example, in Nf=2 and Q=0 case, it is
V
where
and
where
Nf=2, m=0 and Q=0.
-> spectral density or correlation can be calculated, too.
13
2. QCD Dirac spectrum & ChRMT
Summary of QCD Dirac spectrum
IF QCD dynamically breaks the chiral symmetry,
the Dirac spectrum in finite V should look like
r
Banks-Casher
Analytic
solution not known

Higher modes
are like free
theory ~l3
moduli
-> Let us compare with lattice ChPT
QCD
!
l
Low modes are described by ChRMT.
• the distribution of each eigenvalue is known.
• finite m and V effects controlled by the same .
14
3. Lattice QCD with
exact chiral symmetry
The overlap Dirac operator
We use Neuberger’s overlap Dirac operator
[Neuberger 1998]
(we take m0a=1.6) satisfies the Ginsparg-Wilson
[1982]
relation:
realizes ‘modified’ exact chiral symmetry on the lattice;
the action
is invariant under [Luescher 1998]
However, Hw->0 (= topology boundary ) is dangerous.
1. D is theoretically ill-defined. [Hernandez et al. 1998]
2. Numerical cost is suddenly enhanced. [Fodor et al. 2004]
15
3. Lattice QCD with
exact chiral symmetry
Topology fixing
In order to achieve |Hw| > 0 [Hernandez et al.1998, Luescher 1998,1999],
we add “topology stabilizing” term [Izubuchi et al. 2002, Vranas 2006, JLQCD 2006]
•
With Stop, topological charge , or the index of D, is fixed along
the hybrid Monte Carlo simulations -> ChRMT at fixed Q.
with• m=0.2.
Note:in aStop
-> ∞
when Hw->0
a->0.
top-> 0 when
Ergodicity
fixed
topological
sectorand
? ->S(probably)
O.K.
(Local fluctuation of topology is consistent with ChPT.)
( Note
[JLQCD, arXiv:0710.1130]
is extra Wilson fermion and twisted mass bosonic
spinor with a cut-off scale mass. )
16
3. Lattice QCD with
exact chiral symmetry
Sexton-Weingarten method
[Sexton & Weingarten 1992, Hasenbusch, 2001]
We divide the overlap fermion determinant as
with heavy m’ and performed finer (coarser) hybrid Monte Carlo step
for the former (latter) determinant -> factor 4-5 faster.
Other algorithmic efforts
1.
2.
3.
4.
5.
Zolotarev expansion of D -> 10 -(7-8) accuracy.
Relaxed conjugate gradient algorithm to invert D.
5D solver.
Multishift –conjugate gradient for the 1/Hw2.
Low-mode projections of Hw.
17
3. Lattice QCD with
exact chiral symmetry
Numerical cost
Simulation of overlap fermion was thought to be impossible;
– D_ov is a O(100) degree polynomial of D_wilson.
– The non-smooth determinant on topology boundaries
requires extra factor ~10 numerical cost.
⇒ The cost of D_ov ~ 1000 times of D_wilson’s .
However,
– Topology fixing cut the latter cost
~ 10 times faster
– New supercomputer at KEK ~60TFLOPS ~ 10 times
– Mass preconditioning
~ 5 times
– 5D solvor
~ 2 times
10*10*5*2 = 1000 !
[See recent developments: Fodor et al, 2004, DeGrand
& Schaefer, 2004, 2005, 2006 ...]
18
3. Lattice QCD with
exact chiral symmetry
Simulation summary
On a 163 32 lattice with a ~ 1.6-1.9GeV (L ~ 1.8-2fm), we
achieved 2-flavor QCD simulations with the overlap quarks with
the quark mass down to ~3MeV.
[e-regime]
Note m >50MeV with Wilson fermions in previous JLQCD works.
–
–
–
–
Iwasaki (beta=2.3,2.35) + Q fixing action
Fixed topological sector (No topology change.)
The lattice spacings a is calculated from quark potential
(Sommer scale r0).
Eigenvalues are calculated by Lanzcos algorithm.
(and projected to imaginary axis.)
19
Runs
• Run 1 (epsilon-regime) Nf=2: 163x32, a=0.11fm
e-regime (msea ~ 3MeV)
– 1,100 trajectories with length 0.5
– 20-60 min/traj on BG/L 1024 nodes
– Q=0
• Run 2 (p-regime) Nf=2: 163x32, a=0.12fm
6 quark masses covering (1/6~1) ms
– 10,000 trajectories with length 0.5
– 20-60 min/traj on BG/L 1024 nodes
– Q=0, Q=−2,−4 (msea ~ ms/2)

Run 3 (p-regime)
(in progress)
-
-
Nf=2+1 : 163x48, a=0.11fm
2 strange quark masses around physical ms
5 ud quark masses covering (1/6~1)ms
Trajectory length = 1
About 2 hours/traj on BG/L 1024 nodes
4. Numerical results
In the following, we mainly focus on the data with m=3MeV.
Bulk spectrum
Almost consistent with the Banks-Casher’s scenario !
–
–
–
Low-modes’
accumulation.
The height
suggests
 ~ (240MeV)3.
gap from 0.
⇒ need ChRMT analysis
for the precise
measurement of  !
21
4. Numerical results
Low-mode spectrum
Lowest eigenvalues qualitatively agree with ChRMT.
RMT
Lattice
1
4.30
[4.30]
2
7.62
7.25(13)
3
10.83
9.88(21)
4
14.01
12.58(28)
[] is used as an input.
~5-10% lower -> Probably NLO 1/V effects.
k=1 data -> 
= [240(6)(11) MeV]3
statistical
NLO effect
22
4. Numerical results
Low-mode spectrum
Cumulative histogram
is useful to compare the shape of the distribution.
RMT
lattice
1
1.234
1.215(48)
2
1.316
1.453(83)
3
1.373
1.587(97)
4
1.414
1.54(10)
The width agrees with RMT within ~2s.
[Related works: DeGrand et al.2006, Lang et al, 2006,
Hasenfratz et al, 2007…]
23
4. Numerical results
Heavier quark masses
For heavier quark masses, [30~160MeV], the good agreement
with RMT is not expected, due to finite m effects
of non-zero modes.
But our data of the ratio of the eigenvalues still show a qualitative
agreement.
NOTE
• massless Nf=2 Q=0 gives
the same spectrum with
Nf=0, Q=2. (flavor-topology
duality)
• m -> large limit is
consistent with QChRMT.
24
4. Numerical results
Heavier quark masses
However, the value of , determined by the lowest-eigenvalue,
significantly depends on the quark mass.
But, the chiral limit is still consistent with the data with 3MeV.
25
4. Numerical results
Renormalization
Since =[240(2)(6)]3 is the lattice bare value, it should be
renormalized.
We calculated
1.
the renormalization factor in a non-perturbative RI/MOM
scheme on the lattice,
(non-perturbative)
2.
3.
(tree)
match with MS bar scheme, with the perturbation theory,
and obtained
26
4. Numerical results
Systematic errors
•
finite m ->
small.
As seen in the chiral extrapolation
of  , m~3MeV is very close to
the chiral limit.
•
finite lattice spacing a -> O(a2) -> (probably) small.
the observables with overlap Dirac operator are automatically
free from O(a) error,
•
NLO finite V effects -> ~ 10%.
1.
2.
Higher eigenvalue feel pressure from bulk modes.
higher k data are smaller than RMT. (5-10%)
1-loop ChPT calculation also suggests ~ 10% .
statistical
systematic
27
5. NLO V effects
Meson correlators compared with ChPT
With a comparison of meson correlators with (partially
quenched) ChPT, we obtain
[P.H.Damgaard & HF, Nucl.Phys.B793(2008)160]
where NLO V correction is taken into account.
[JLQCD, arXiv:0711.4965]
28
5. NLO V effects
Meson correlators compared with ChPT
But how about NNLO ? O(a2) ? -> need larger lattices.
29
6. Conclusion
•
•
•
•
•
We achieved lattice QCD simulations with
exactly chiral symmetric Dirac operator,
On (~2fm)4 lattice, simulated Nf=2 dynamical
quarks with m~3MeV,
found a good consistency with Banks-Casher’s
scenario,
compared with ChRMT where finite V and m effects
are taken into account,
found a good agreement with ChRMT,
– Strong evidence of chiral SSB from 1st principle.
– obtained
30
6. Conclusion
The other works
–
–
–
–
–
–
–
–
Hadron spectrum [arXiv:0710.0929]
Test of ChPT (chiral log)
Pion form factor [arXiv:0710.2390]
difference [arXiv:0710.0691]
BK [arXiv:0710.0462]
Topological susceptibility [arXiv:0710.1130]
2+1 flavor simulations [arXiv:0710.2730]
…
31
6. Conclusion
The future works
– Large volume (L~3fm)
– Finer lattice (a ~ 0.08fm)
We need 24348 lattice (or larger).
We plan to start it with a~0.11fm, ma=0.015
(ms/6) [not enough to e-regime] in March
2008.
32