The bridge between the Dirac operator ... scaling regime in which this volume is sent to infinity at ... made for an infinite sequence of Dirac operator eigenval-.
Distribution of the k-th smallest Dirac operator eigenvalue Poul H. Damgaard The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
Shinsuke M. Nishigaki Department of Physics, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8551, Japan (June 15, 2000; revised July 14, 2003) Based on the exact relationship to Random Matrix Theory, we derive the probability distribution of the k-th smallest Dirac operator eigenvalue in the microscopic finite-volume scaling regime of QCD and related gauge theories.
arXiv:hep-th/0006111v2 14 Jul 2003
PACS number(s): 12.38.Aw, 12.38.Lg
value. We do so for the chiral unitary (β = 2), orthogonal While index theorems have for long been known to re(β = 1, corresponding to SU (2) gauge theories with funlate the number of exact zero modes of the Dirac operator damental fermions), and symplectic ensembles (β = 4, to gauge field topology, it is only during the 1990’s that corresponding to SU (Nc ≥ 2) gauge theories with adit has been realized how much further one can push such joint fermions). We will treat the most general case of exact predictions. When the gauge theory in question massive fermions, and (with one exception noted below) is in a phase with spontaneously broken chiral symmea sector of arbitrary topological charge ν. In the microtry, Goldstone’s Theorem allows for much more detailed scopic scaling limit this can be done to arbitrarily high statements, beyond the zero modes. The bridge between order k as long as the volume V is taken large enough. the Dirac operator spectrum and the Goldstone degrees This gives us an infinite sequence of distributions of Dirac of freedom is the effective partition function of the assooperator eigenvalues that all, on the macroscopic scale, ciated chiral Lagrangian [1,2]. The infinite-volume chiral build up the spectral density at the origin ρ(0). We stress condensate will be denoted by Σ. By considering the thethat there is much more information in these individual ory at finite four-volume V , one can choose a microscopic eigenvalue distributions than in the summed-over microscaling regime in which this volume is sent to infinity at scopic spectral density ρS (ζ; {µ}) itself. In particular, a rate correlated with the chiral limit m → 0 such that from the point of view of lattice gauge theory simulations µ ≡ ΣV m is kept fixed. Then exact statements can be it is far more convenient to be able to perform comparmade for an infinite sequence of Dirac operator eigenvalisons with individual eigenvalue distributions than with ues that accumulate towards the origin. This realization just the average eigenvalue density. was first made on the basis of a universal Random MaThere are by now detailed analytical predictions for trix Theory reformulation of the problem [3–6], but it has the microscopic spectral densities of the Dirac operator since then also been established directly at the level of for all three universality classes. These microscopic specthe effective Lagrangian [7,8]. tral densities show a typical oscillatory structure, which Conventionally, the exact analytical statements that clearly is closely connected to the individual eigenvalue can be made about this infrared region of the Dirac operpeaks at the microscopic scale where units are given by ator spectrum in finite-volume gauge theories have been the average eigenvalue spacing. We thus expect that the phrased in terms of the microscopic spectral n-point funcdistribution of the k-th smallest Dirac operator eigentions, and in particular the microscopic spectral density value corresponds closely to the k-th peak in the miρS (ζ; {µ}) itself (here ζ ≡ ΣV λ is the microscopically croscopic eigenvalue density, a feature that indeed had rescaled Dirac operator eigenvalue λ) [3,4,9–18]. Also already been noticed in the case of just the smallest the exact distribution of just the lowest Dirac operator eigenvalue [23]. In particular, we expect that the k-th eigenvalue has been derived on the basis of the relationeigenvalue distribution as computed in the framework of ship to Random Matrix Theory [10,19–23,15,16]. In fact Random Matrix Theory is universal, which indeed turns also the distribution of the second-smallest eigenvalue in out to be the case (see below). Of course, as one further what corresponds to the massless limit of SU (Nc ≥ 3) additional check, the sequential summing-up of the ingauge theories with fermions in the fundamental repredividual eigenvalue distributions should simply build up sentation can be extracted from the work of Forrester the full microscopic eigenvalue densities, which are now and Hughes [21]. In this paper, we push these compuknown in closed analytical form. tations one step further and provide analytical expresThe chiral Random Matrix Theory ensembles are desions for both the joint probability distribution of the fined by [3–5]: first k eigenvalues and the distribution of the k-th eigen Z Nf Y † mi W det , (1) Zν(β) (m1 , . . . , mNf ) = dW e−β tr v(W W ) −W † mi i=1
1
where the integrals are over complex, real, and quaternion real (N + ν) × N matrices W for β = 2, 1, 4, respectively. These chiral random matrix ensembles provide exactly equivalent descriptions of the effective field theory partition functions in the microscopic finite-volume scaling regime [3,5]. Moreover, all microscopic spectral properties of the Dirac operator coincide exactly with the corresponding microscopically rescaled Random Matrix Theory eigenvalues [8]. This means that either formulation can be used to derive identical results, and in the case of the individual distributions of the smallest Dirac operator eigenvalues it is presently more simple to use the Random Matrix Theory formulation. Since the results turn out to be universal, i.e. independent of the detailed form of the Random Matrix Theory potential v(W † W ) [24–26], it suffices for √ us to concentrate on Gaussian ensembles with v(x) = x. This choice leads to Wigner’s semi-circle law ρ(λ) = (2/π) 2N − λ2 . The partition functions (1) of such chiral Gaussian ensembles corresponding to Nf massive flavors and topological charge ν can then be written in terms of eigenvalues {xi } of the positive-definite Wishart matrices W † W , Zν(β) (m1 , . . . , mNf )
=
Nf Y
mi
i=1
ν Z
∞
0
···
Z
0
N ∞Y
β(ν+1)/2−1 −βxi e dxi xi
Nf Y
j=1
i=1
N Y |xi − xj |β , (xi + m2j )
(2)
i>j
up to an overall irrelevant factor which is independent of the m’s. Because everything will be symmetric under ν → −ν, we for convenience take ν to be non-negative. In the following, we impose the one single technical restriction that for β = 1 the topological charge ν be odd. The unnormalized joint probability distributions for N eigenvalues {0 ≤ x1 ≤ · · · ≤ xN } in the above Random Matrix Theory ensembles take the form (β)
ρN (x1 , . . . , xN ; {m2 }) =
Nf N N Y Y Y β(ν+1)/2−1 −βxi |xi − xj |β . (xi + m2j ) e xi
(3)
i>j
j=1
i=1
The joint probability distribution of the k smallest eigenvalues {0 ≤ x1 ≤ · · · ≤ xk−1 ≤ xk } can be written as (the order of x1 , . . . , xk−1 can be relaxed): Z ∞ Z ∞ 1 1 (β) (β) 2 ΩN,k (x1 , . . . , xk−1 , xk ; {m }) ≡ (β) dxN ρN (x1 , . . . , xN ; {m2 }) dxk+1 · · · 2 (N − k)! xk xk ΞN ({m }) Nf k k Y Y Y β(ν+1)/2−1 −βxi |xi − xj |β (xi + m2j ) e xi
1
=
(β)
ΞN ({m2 }) i=1 Z ∞ Z ∞ 1 × dxN dxk+1 · · · (N − k)! xk xk
i>j
j=1
N Y
β(ν+1)/2−1 −βxi e xi
Nf Y
(xi + m2j )
j=1
j=1
i=k+1
k Y
(xi − xj )β
N Y
i>j≥k+1
|xi − xj |β ,
(4)
(β)
where ΞN stands for the normalizing integral Z ∞ Z ∞ 1 (β) (β) ΞN ({m2 }) = dx1 · · · dxN ρN (x1 , . . . , xN ; {m2 }) . N! 0 0
(5)
We now shift xi → xi + xk in the integrand of (4): (β)
Nf k k Y Y Y 1 β(ν+1)/2−1 −βxi 2 |xi − xj |β (x + m ) e x i j i (β) 2 (N − k)! ΞN ({m }) i>j j=1 i=1 Nf N k−1 N Y Y Y Y (xi + xk − xj )β |xi − xj |β . (xi + m2j + xk ) dxi e−βxi xβi (xi + xk )β(ν+1)/2−1
ΩN,k (x1 , . . . , xk−1 , xk ; {m2 }) = ×
Z
0
∞
i=k+1
e−(N −k)βxk
j=1
j=1
(6)
i>j≥k+1
To get the probability distributions of the Dirac operator eigenvalues we must change the picture back to chiral Gaussian ensembles, and take the microscopic limit √ √ ζi = πρ(0) xi = √8N xi xi → 0 , N → ∞, with the rescaled variables kept fixed. (7) m2j → 0 µj = πρ(0)mj = 8Nmj In this large-N limit the difference between partition functions based on N −k and N eigenvalues becomes insignificant. Moreover, one notices that the new terms in the integrand of (6) can be interpreted as arising from new additional 2
fermion species, with the partition function now being evaluated in a fixed topological sector of effective charge (β) ν = 1 + 2/β. Taking into account the definition (2), we then get, with Zν ({µ}) denoting the partition functions in the microscopic limit, (β)
ωk (ζ1 , . . . , ζk ; {µ}) =
k Y ζ2 µ2 |ζi | (β) ζ12 ,..., k ;{ }) ΩN,k ( N →∞ 8N 8N 8N 8N i=1
lim
2 β ν+1 2 −ν−1+ β
2
= const. e−βζk /8 ζk
Nf Y
1
1
(µ2j + ζk2 ) 2 − β
j=1
×
p (β) Z1+2/β ( µ21
+
ζk2 , . . . ,
k−1 Y
β(ν+1)−1
ζi
i=1 β
β
(ζk2 − ζi2 ) 2 −1
Nf Y
j=1
Nf k−1 Y Y µνj (ζi2 − ζj2 )β (ζi2 + µ2j ) i>j
j=1
β
}| q { zq }| q { β(ν+1)/2−1 z q q z }| { 2 2 2 2 2 2 2 2 2 2 µNf + ζk , ζk − ζ1 , . . . , ζk − ζ1 , . . . , ζk − ζk−1 , . . . , ζk − ζk−1 , ζk , . . . , ζk ) (β)
Zν (µ1 , . . . , µNf )
.
(8)
This is the main result of our paper. It shows that the joint probability distribution of the first k eigenvalues is given, apart from the relatively simple prefactor, by a ratio of finite-volume partition functions. The new partition pfunction µ2i + ζk2 , that enters in the numerator of Eq.(8) has the Nf original fermion masses shifted according to µ → i q p 2 contains β(k − 1) new fermions of masses ζk2 − ζ12 , . . . , ζk2 − ζk−1 (each mass β-fold degenerate), β(ν + 1)/2 − 1 additional degenerate fermion species of common mass ζk , and this whole partition function is evaluated in a sector of fixed topological charge ν = 1 + 2/β. While this topological index is fractional for β = 4, there is no difficulty with the evaluation of the pertinent partition function. (Indeed, it can alternatively be viewed as a partition function in a sector of topological charge ν = 0 and 3 additional massless fermions.) This expression entirely in terms of the effective field theory partition functions strongly suggests that it should be possible to derive these analytical expressions starting directly from the effective field theory, perhaps partially quenched as in Ref. [8]. The proportionality constant in Eq.(8) depends on the normalization conventions of the involved partition functions. We fix this numerical factor uniquely by the requirement that the total probability of finding any given eigenvalue is normalized to unity. (β) Correlation functions of the rescaled eigenvalues {ζk1 , . . . , ζkp−1 , ζk } are obtained from ωk (ζ1 , . . . , ζk ; {µ}) by integrating out the remaining eigenvalues in a cell 0 ≤ ζ1 ≤ · · · ≤ ζk . In particular, the distribution of the k-th smallest (β) eigenvalue ζ, pk (ζ; {µ}), is given by (β) pk (ζ; {µ})
=
Z
0
ζ
dζ1
Z
ζ
ζ1
dζ2 · · ·
Z
ζ
(β)
ζk−2
dζk−1 ωk (ζ1 , . . . , ζk−1 , ζ; {µ}) .
(9)
We note that this gives a very simple representation of the probability distribution of the k-th smallest Dirac operator eigenvalue. In particular, already for the most elementary case of just the lowest Dirac operator eigenvalue in the β = 2 universality class we obtain a very simple expression for an arbitrary topological sector ν. In the normalization convention of, e.g., [7], ν (2)
p1 (ζ, {µ}) =
Nf
ζ −ζ 2 /4 Y µj e 2 j=1
q z }| { p ν Z2 ( µ21 + ζ 2 , . . . , µ2Nf + ζ 2 , ζ, . . . , ζ) Zν (µ1 , . . . , µNf )
a more compact and convenient expression than the one provided in Ref. [23]. For example, in the quenched case of Nf = 0 it yields the simple relation
,
(10)
where the indices i and j run between −ν/2 + 1 and ν/2 − 1, and the Pfaffian is thus taken over a matrix of size (ν − 1) × (ν − 1) [2]. For example, for ν = 1 this formula gives
ζ −ζ 2 /4 e det[I2+i−j (ζ)] , (11) 2 where the determinant is over a matrix of size ν × ν. It is worth pointing out that also the corresponding quenched distributions for β = 1 become exceedingly simple: (2)
p1 (ζ) =
(1)
p1 (ζ) = const. ζ (3−ν)/2 e−ζ
2
/8
(1)
ζ −ζ 2 /8 e , 4
(13)
1 −ζ 2 /8 e I3 (ζ) , 2
(14)
p1 (ζ) = for ν = 3 we obtain
Pf [(i − j)Ii+j+3 (ζ)] ,
(1)
p1 (ζ) =
(12) 3
while for ν = 5 it gives (1)
p1 (ζ) =
(this is a technical restriction that we expect to be lifted eventually). We shall here for convenience provide the precise detailed expressions for all these known cases. In (β) exhibiting explicit forms of ωk (ζ1 , . . . , ζk ; {µ}) by substituting these expressions, we choose to set ν = 2/β − 1 in Eq.(2) and introduce additional β(ν +1)/2−1 massless flavors instead. The number of flavors in this alternative picture, Nf + β(ν + 1)/2 − 1, will be denoted by n. For the β = 4 case, we thus consider here only the case with an odd number of massless flavors, so that the ‘effective’ number of massless flavors is even, n ≡ 2a. We then get
1 −ζ 2 /8 3I3 (ζ)2 − 4I2 (ζ)I4 (ζ) + I1 (ζ)I5 (ζ) , e ζ (15)
all of these being correctly normalized. By now, the effective finite-volume partition functions are known in analytically closed forms for β = 2 [28] and β = 1 [16]. They are also known for a general even number of pairwise degenerate fermions for β = 4 [16]
(2) ωk (ζ1 , . . . , ζk ; µ1 , . . . , µn )
= const. e
−ζk2 /4
2
(1)
ωk (ζ1 , . . . , ζk ; µ1 , . . . , µn ) = const. e−ζk /8
Qn 2 2 2 2 2 j=1 (ζk − ζj ) l=1 (ζk + µl ) |ζi | Qk−1 Qk−1 Qn 2 2 2 2 2 i=1 j=1 i>j (ζi − ζj ) l=1 (ζj + µl ) |ζi |
k−1 Y
(ζk2 − ζj2 )
2
(4)
k Y
|ζi |
k−1 Y
(ζk2 − ζj2 )4
i=1
j=1
j=1
A(1)
B (1)
A(4)
j=1,...,n+2k−2
a Y Pf [B (4) ] (ζk2 + µ2l )2 , Pf [A(4) ] l=1
(16b)
(16c)
(17a) (17b)
j=1,...,n+2k−2
[D0 (µi , µj )]i,j=1,...,n (n : even) , [D0 (µi , µj )]i,j=1,...,n [R0 (µj )]j=1,...,n = (n : odd) , [−R0 (µi )]i=1,...,n 0 h i h i ˜i , µ ζ ˜ ) D (˜ µ , µ ˜ ) D ( 1 i j 1 j i=1,..,k−1 i i,j=1,...,n h i j=1,...,n (n + k : odd) , h ˜ ˜ ˜ D (˜ µ , ζ ) D ( ζ , ζ ) 1 i j 1 i j i=1,...,n ij=1,...,k−1 h ii,j=1,...,k−1 h i h ˜ D1 (˜ µi , µ ˜j ) D1 (ζi , µ ˜j ) i=1,..,k−1 R1 (˜ µj ) = h i i,j=1,...,n h i j=1,...,n h i j=1,...,n D1 (˜ µi , ζ˜j ) i=1,...,n D1 (ζ˜i , ζ˜j ) R1 (ζ˜j ) (n + k : even) , i,j=1,...,k−1 j=1,...,k−1 j=1,...,k−1 h i h i −R1 (˜ µi ) −R1 (ζ˜i ) 0 i=1,...,n i=1,...,k−1 Z 1 I2α+1 (ζ) I2α (tζ) I2α+1 (tζ ′ ) I2α+1 (tζ) I2α (tζ ′ ) ′ 2 , Rα (ζ) = − , Dα (ζ, ζ ) = dt t 2α ′2α+1 2α+1 ′2α ζ ζ ζ ζ ζ 2α+1 0 (µ , µ )] (a : even) , [I 0 i j i,j=1,...,a [I0 (µi , µj )]i,j=1,...,a [Q0 (µj )]j=1,...,a = (a : odd) , [−Q0 (µi )]i=1,...,a 0
4
(16a)
l=1
p p ˜i ≡ ζk2 + µ2i ) where the matrices A(β) and B (β) are given by (ζ˜i ≡ ζk2 − ζi2 ), µ h i A(2) = µj−1 I (µ ) , j−1 i i i,j=1,...,n h i h i h i j−3 j−3 j−4 (2) ˜ ˜ ˜ ˜ B = µ ˜i Ij−3 (˜ µi ) ζi Ij−3 (ζi ) i=1,...,k−1 ζi Ij−4 (ζi ) i=1,...,k−1 , i=1,...,n j=1,...,n+2k−2
det[B (2) ] , det[A(2) ]
n Y Pf [B (1) ] (ζk2 + µ2l ) , Pf [A(1) ]
k Y
i=1
ωk (ζ1 , . . . , ζk ; µ1 , µ1 , . . . , µa , µa ) = const. e−ζk /2
Qk−1
k Y
(17c)
(17d)
(17e)
B (4)
i h I4 (˜ µi , µ ˜j ) h i i,j=1,...,a ˜ µi , ζj ) i=1,...,a I4 (˜ h j=1,...,k−1 i ˜ −S (˜ µ , ζ ) 4 i j i=1,...,a i j=1,...,k−1 h I4 (˜ µi , µ ˜j ) = h i i,j=1,...,a Q4 (˜ µ ) j j=1,...,a i h ˜ I (˜ µ , ζ ) 4 i j i=1,...,a j=1,...,k−1 i h −S4 (˜ µi , ζ˜j ) i=1,...,a j=1,...,k−1
Z
1
Z
1
h i h i I4 (ζ˜i , µ ˜j ) i=1,..,k−1 S4 (ζ˜i , µ ˜j ) i=1,..,k−1 h i j=1,...,a h i j=1,...,a ˜ ˜ ˜ ˜ I4 (ζi , ζj ) S4 (ζi , ζj ) i,j=1,...,k−1 i,j=1,..,k−1 h i h i D4 (ζ˜i , ζ˜j ) −S4 (ζ˜i , ζ˜j ) i,j=1,...,k−1 i,j=1,..,k−1 h i h i h i −Q4 (˜ µi ) I4 (ζ˜i , µ ˜j ) i=1,..,k−1 S4 (ζ˜i , µ ˜j ) i=1,..,k−1 i=1,...,a j=1,...,a h i j=1,...,a h i 0 Q4 (ζ˜j ) O4 (ζ˜j ) j=1,...,k−1 j=1,...,k−1 h i h i h i −Q4 (ζ˜i ) I4 (ζ˜i , ζ˜j ) S4 (ζ˜i , ζ˜j ) i,j=1,...,k−1 ii,j=1,..,k−1 h ii=1,...,k−1 h i h ˜ ˜ ˜ ˜ ˜ D4 (ζi , ζj ) −O4 (ζi ) −S4 (ζi , ζj ) i=1,...,k−1
i,j=1,...,k−1
′
′
i,j=1,..,k−1
(a : even),
(a : odd),
(17f)
Z
1
Iα−1 (2tζ) Iα−1 (2tζ) Iα−1 (2tuζ ) Iα−1 (2tuζ) Iα−1 (2tζ ) , Qα (ζ) = dt − , α−1 ′α−1 α−1 ′α−1 ζ ζ ζ ζ ζ α−1 0 0 0 Z 1 Z 1 Z 1 Iα−1 (2tζ) Iα (2tuζ ′ ) Iα−1 (2tuζ) Iα (2tζ ′ ) Iα (2tζ) Sα (ζ, ζ ′ ) = dt t2 du , O (ζ) = dt t u − , α α−1 ′α α−1 ′α ζ ζ ζ ζ ζα 0 0 0 Z 1 Z 1 Iα (2tζ) Iα (2tuζ ′ ) Iα (2tuζ) Iα (2tζ ′ ) dt t3 Dα (ζ, ζ ′ ) = du u . − ζα ζ ′α ζα ζ ′α 0 0
Iα (ζ, ζ ′ ) =
dt t
du
(β)
ˆ Here K stand for the integral operators with conζ voluting kernels K (β) (ς, ς ′ ; {µ}) over an interval 0 ≤ ς, ς ′ ≤ ζ. The universality of the probability distribu(β) tion pk (ζ; {µ}) is hence guaranteed. It is nevertheless instructive to see how this manifests itself in our explicit computation. For a generic Random Matrix Theory potential the exponential factor e−β(N −k)xk which is produced by the shift xi → xi + xk is simply replaced by 2 exp −(β/8) πρ(0) xk 1+O(1/N ) , and thus yields an 2 identical factor e−βζk /8 in the microscopic limit (7) [23]. Based on the universality theorems in Refs. [24,26] one readily establishes that also the remaining ratio of partition functions, and in particular the full expression (8), is universal.
In all cases the normalization constants are fixed uniquely by the requirement that the probabilities sum up to unity. By construction, the individual distributions (β) pk (ζ; {µ}) sum up to the microscopic spectral density (β) ρS (ζ; {µ}): (β)
ρS (ζ; {µ}) =
∞ X
k=1
(β)
pk (ζ; {µ}) .
(18)
(2)
To illustrate this, we plot in Fig.1 pk (ζ) for k= 1, . . . , 4, (2)
their sum, and ρS (ζ) = |ζ| J02 (ζ) + J12 (ζ) /2 for the
quenched (Nf = 0) chiral unitary ensemble with ν = 0. One clearly sees how the microscopical spectral density gradually builds up from the individual eigenvalue distributions. We finally turn to the issue of universality. It was proven in Refs. [26,27] that the diagonal elements of the quaternion kernels K (1,4) (ζ, ζ ′ ; {µ}) for the orthogonal and symplectic ensembles can be constructed from the scalar kernel K (2) (ζ, ζ ′ ; {µ}) of a unitary ensemble with a related weight function. As the scalar kernel in the microscopic limit (7) is insensitive to the details of the potential v(x) either in the absence [24,25] or in the presence of finite and non-zero masses µ [14], so are the corre(β) sponding quaternion kernels. Furthermore, pk (ζ; {µ}) can be expressed in terms of the Fredholm determinant of K (β) (ζ, ζ ′ ; {µ}) [29,30]: (β)
pk (ζ; {µ}) =
PHD would like to thank the Institute for Nuclear Theory at the University of Washington for its hospitality and DOE for partial support during the completion of this work. We thank R. Niclasen for help with the figure preparation. The work of PHD was supported in part by EU TMR Grant no. ERBFMRXCT97-0122 and the work of SMN was supported in part by JSPS, and by Grant-inAid no. 411044 from the Ministry of Education, Science, and Culture, Japan.
(−1)k ∂ ∂ k−1 ˆ (β) ) det(1 − t K . ζ (k − 1)! ∂ζ ∂t t=1
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(19) 5
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0.5
0.4
ρS(ζ) 0.3
p1(ζ)
0.2
p3(ζ)
p2(ζ)
p4(ζ)
0.1
0
0
5
ζ
10
15
FIG. 1. Microscopic spectral density ρS (ζ) (thin line), k-th smallest eigenvalue distribution pk (ζ), k = 1, 2, 3, 4 (solid lines), P4 and their sum p (ζ) (broken line) for the quenched chiral unitary ensemble (β = 2, Nf = ν = 0). k=1 k
6
