A Reduction FormulaforFermion Determinant
Keitaro Nagata, Atsushi Nakamura(Hiroshima U. RIISE)
in XQCD-J collaboration
S. Motoki(KEK), Y. Nakagawa(Niigata), T. Saito(Kochi)
New Type of Fermions on Lattice,
2012/02/16 at YITPDet (D) is a key in fd LQCD
• Fermion determinant det D(mu)– chemical potential– sign problem• Techinques in fdLQCD need the evaluation of det D(mu)• Determinant calculation is heavy– Time O(N^3) for NxN matrix– Memory O(N^2)

An idea• There is a formula– to perform the temporal part of det D(mu) by hand
– and reduce the rank of the determinant• The rank of the determinant is reduced, and difficulty ofthe determinant evaluation is relaxed– CPU time and memory is suppressed ~ 1/Nt^2

References• Derivation for Staggered type
– Gibbs, PLB 172, 53 ('86). Hasenfratz &Toussaint, NPB371, 539('92).

• Derivation for Wilson type
–Borici, PTP. Suppl. 153, 335 ('04). Alexandru &Wenger, PRD83, 034502 ('11).
KN&AN, PRD82,094027 ('10).

• Derivation for continuum case– Adams, PRL92, 162002 ('04), PRD70, 045002 ('04).

• Application– reweighting : Glasgow group e.g., Barbour&Bell, Nucl.Phys. B372 (1992)
385-402. Fodor & Katz, JHEP 0203 (2002) 014.
– canonical formalism : Hasenfratz &Toussaint('92). de Forcrand &Kratochivila, hep-lat/0509143.
–Hadron spectrum. Fodor, Szabo & Toth, JHEP 0708, 092('07).
– see also refs in KN&AN, PRD82,094027 ('10).

Basic concept of the reduction formula• Fermion matrix as t-t matrix– spatial (diag), temporal (n.n + b.c)– temporal hop accompanies chemical potential• Performing temporal determinant by hand– rank reduces to N/Nt (memory & CPU time ~1/Nt^2. )– det D is an analytic function of mu.

• Reduction formula– suppress CPU time and memory 1/Nt^2– provides an analytic form of mu-dependence• Limitation– direct method is heavy– limited to small lattice size

Focus on temporal part of quark action∆( µ ) = I − κ (spatial hop + clover)− κ [e µa (1 − γ 4 )U 4 ( x)δ x ', x + 4̂ + e − µa (1 + γ 4 )U 4+ ( x' )δ x ', x − 4̂ ]• Quark action as t-t matrix– temporal hop accompanies chemical potential
ψ γ 0ψ → ψ γ 4ψ– spatial (diag), temporal (n.n + b.c)spatial hop on
i-th time slice

hop to nexttime sliceD(mu) in t-t matrix

Reduced matrix• After calculating temporal part ofdeterminantdet ∆( µ ) = C0ξ − Nr det(Q + ξ ),Q ~ A1 A2  ANt• reduced matrix Q― propagation from a point at initialtime to a point at final time
― rank Nr= 4 Nc x Ns^3, (= full size/Nt)― independent of mu• overall factor C0― spatial loops― independent of mu

A14NcNs^3 planetwo expressions of det D(mu)• Calculating the eigenvalues of Q

det(Q − λI ) = 0det ∆( µ ) = C0ξ − Nr / 2 det(Q + ξ ),Nr

det ∆( µ ) = C0ξ − Nr / 2 ∏ (λn + ξ )n =1• Fugacity expansion

det ∆( µ ) ~Q : (Nr, Nr ) matrix
Nr=4Nc Ns^3Nr / 2ncξ
∑nn = − Nr / 2ev's or cn determine the mu-dependence ofdet D(mu)Temperature(beta) dependence 1T<Tc

T=Tc• At low T : Z3-like distribution• At high T : breaking of Z3 and broadening• Blank observed at high T is related to the Polyakov loop
(Alexandu&Wenger , PRD83, 034502('11).

T>Tc

Symmetry of eigenvalues
λ↔

1

λ*
clover-Wilson + RG-gauge(Nf=2)
Volume : 8^3x4quark mass : mps/mV ~ 0.8
Configurations : HMC at mu=0

Temperature(beta) dependence 2
Histogram : Absolute (left) and phase (right)
high T

T<Tc

T>Tc
T=Tc

• At low T : Z3-like distribution• At high T : breaking of Z3 and broadening

Gap of ev's near unit circle
Left : eigenvalues close to unity
Right : max. ev among the smaller halfgap

ln max | λn ||λn |<1

The gap decreases as T increases.

Gap is related to pion mass
1amπ = − ln max | λn |2
Nt |λn |<1
Gibbs('86). Eigenvalues and mpi
See also, Fodor, Szabo, Toth ('06). Eigenvalues and hadron spectrum

Left : mpi/T. Right : mpi/Tpc• At low T, mpi/T is well fitted with a/T, a = 4 Tpc (mq heavy)• At high T, mpi approaches to a constantCheck 1. Volume dependence
Histograms : |ev| (Left), arg(ev) (Right)
blue : 10^3 x4, red : 8^3 x 4• The small V-dependence suggests
long distance propagations are small• Further investigation for small quark mass is importantCheck 2. quark mass dependence
mps / mv = 0.6 (red), 0.8 (blue)
Histograms : |ev| (Left), arg(ev) (Right)
confinement (top), deconfinement(bottom)

Gap shrinksTemperature(Nt) dependence
Nt=8(T/Tc=0.5)

Nt=4 (T/Tc=1)

10^5• As Nt increases (T decrease)― larger half of ev's become larger― smaller half of ev's become smaller

λ↔

1

λ*

Temperature(Nt) dependence 2Power-law of ev's with an exponent Nt| λ |= lNt• Q is a product of Nt-matrices

Q ~ A1 A2  ANt• In equilibrium, it is expected

Ai = A + δAi• Power law

Q ~ A Nt

Scale separationdet ∆( µ ) = C0ξ

− Nr / 2

Nr / 2

Nr / 2∏ (1 / λ + ξ ) ∏ (λ + ξ )*nnn =1Nr / 2det ∆( µ ) = C0ξ − Nr / 2 ∏ (lnn =1n =1*− Nt

Nr / 2+ ξ ) ∏ (ln + ξ )Ntn =1

Nt larger

λ =| l | Ntlow T limit
Eigenvalues at low T

T → 0, ( Nt → ∞)λL ~ l Nt → ∞, λS ~ l − Nt → 0
Fugacity depends on Tdet ∆( µ ) = C0ξ

− Nr / 2

Nr / 2∏ (lnn =1*− Nt

Nr / 2+ ξ ) ∏ (ln + ξ )Ntn =1

Deriavtion of low-T limit of det D(mu) needs a careful
calculation

Symmetry of Reduced matrix
det ∆( µ ) = C0ξ

− Nr / 2

Nr / 2∏ (lnn =1

1. mu/T=fixed (xi=const.)*− Nt

Nr / 2+ ξ ) ∏ (ln + ξ )Ntn =1

Nr / 2det ∆( µ ) = C0 ∏ λnL ∈ Rn =1

2. mu=fixed (xi->infty, T->0)
a. small mu e µa < lb. large mue µa > lNr / 2

det ∆( µ ) = C0 ∏ λnL ∈ R
n =1det ∆( µ ) = C0ξ − Nr / 2 det Q ∈ RInterpretation
ξ − Nr / 2 = exp(2 N c N f N s3 )Nr / 2det ∆( µ ) = C0 ∏ λnL ∈ R<n>=0n =1det ∆( µ ) = C0ξ − Nr / 2 det Q ∈ R<n> = 2 Nc Nf [lattice unit](all states are occupied)"Silver Blaze problem"
Independence of det D on mu at low T : Cohen,
PRL91,222001 ('03)Adams, PRD70, 045002 ('04).

Lee-Yang zero• Lee-Yang theorem : origin of thermodynamical
singularities, i.e., phase transition∂F1 ∂ZF = −T ln Z (ξ ),∞ −T, ξ = exp(− µ / T )∂µZ ∂µZ (ξ ) = 0 ⇒ F ′ → ∞Canonical formalism of QCD• Grand partition function is expanded in powers offugacity• Nq (= Max. # of the quark) is given by (d.o.f x V )• Negative power terms come from the anti-quark

Canonical formalism of QCD• We employ the Glasgow method– Reduction formula + reweighting

Nr(det ∆( µ )) 2 = ∑ cn ξ nn = − Nr

• Approaches to improve overlap for canonical and LY••••

Fodor & Katz, Phys.Lett. B534 (2002) 87-92
de Forcrand & Kratochivila, hep-lat/0509143
Ejiri & Yoneyama, arXiv:0911.2257 [hep-lat]
Li, Alexandru & Liu, Phys.Rev. D84 (2011) 071503

Canonical partition function

large phase fluctuation
at low T• Zn decreases exponentially– broadening at high T : increase of effective d.o.f• Zn (mod(n,3) !=3) does not vanish even in confinement phase– because of importance sampling at mu=0– including all Z3 sectors by hand leads to cancelation

Lee-Yang Zero 1• In the vinicity of mu =0, Z(mu) is dominated by terms near n=0• The behavior of the root distribution near xi=1 can be studied by
the first terms.
• (Density depends on N' ( f(x) = 1 + x + x^2 + ... + x^N)
• This approximation should not be used for the study of the orderof the phase. )Lee-Yang ZeroT<Tc

T=Tc

T>Tc

• Root distribution shows two different behavior• at low T : two circle (Zn = exp( - a |n|))• at high T : unit circle + 3 branches (Zn = exp ( -a n^2))• branch point is located at near RW endpoint

Examples for Polynomial's Root

Lee-Yang Zero 2T<Tc

T=Tc

T>Tc

CEP

• LYzeros seems to approach to real axis near mu/T = 1• Further investigations are needed for– overlap problem & sign problem– thermodyanamic limitSummary• Reduction formula– relax the difficulty to calculate fermion determinant– provides the analytic expression of det D(mu)– although it is still heavy, the formula is sometimes useful tostudy finite density systems.
• Properties of reduced matrix– angular distribution is related to Z3– gap is related to chiral symmetry breaking– the whole eigenvalues are important in overlap &signproblem• In this talk, we have studied– numerical properties of transfer matrix– low-T limit of det D(mu)– canonical partition function & Lee-Yang zero