The zeros of the QCD partition function
Abstract
We establish a relationship between the zeros of the partition function in the complex mass plane and the spectral properties of the Dirac operator in QCD. This relation is derived within the context of chiral Random Matrix Theory and applies to QCD when chiral symmetry is spontaneously broken. Further, we introduce and examine the concept of normal modes in chiral spectra. Using this formalism we study the consequences of a finite Thouless energy for the zeros of the partition function. This leads to the demonstration that certain features of the QCD partition function are universal.
1 Introduction
In recent years the spectral correlators of the Dirac operator in QCD have been the object of intense study using both numerical and analytic means. These correlators contain valuable information regarding both the chiral properties of the QCD vacuum and the topological structure of the gauge fields. The relation to the chiral properties of the QCD vacuum was established by Banks and Casher [1]: The eigenvalue density of the QCD Dirac operator at eigenvalue zero is proportional to the chiral condensate and is therefore an appropriate order parameter for chiral symmetry. As a complement to the BanksCasher relation, one has the YangLee picture [2] of a phase transition. In the attempt to analyze phase transitions in statistical spin models Lee and Yang [2] introduced the concept of the zeros of the finite volume partition function in the thermodynamic limit. The volume dependence of these zeros allows finite size scaling studies and subsequent identification of universality classes. In the case of chiral symmetry, one focuses on the zeros of the partition function in the complex quark mass plane. If these zeros pinch the real axis and exhibit a constant density, a discontinuity in the partition function arises at the pinch, and chiral symmetry is spontaneously broken. The mass zeros of the partition function and the low lying eigenvalues of the Dirac operator thus contain similar information about the chiral phase transition. The relation between the two is, however, involved. The partition function and its zeros are obtained by averaging over all gauge field configurations. By contrast, the eigenvalues of the Dirac operator are given for each gauge field configuration, and only a density of eigenvalues is welldefined after averaging.
The challenge of deriving relations between the zeros of the partition function and the eigenvalues of the Dirac operator was first taken up by Leutwyler and Smilga [3]. They studied QCD in a Euclidean 4dimensional box with side length subject to the constraint that
(1) 
Here is the pion mass and is the typical QCD scale. They computed the QCD partition function for equal quark masses using the effective chiral Lagrangian and found that quark masses enter only in the rescaled combination , where is the chiral condensatein the chiral limit. They further observed that the partition function zeros could be thought of as average positions of the eigenvalues. While highly suggestive, these results were not completely quantitative. The situation has changed dramatically since then. The main break through came with the introduction [4] of random matrix concepts in QCD which permits the study of the correlations of the eigenvalues of matrices drawn on a general weight constructed to ensure the chiral structure of each eigenvalue spectrum. The relation of random matrix theory to QCD in the limit (1) has been established through a number of universality studies [5], lattice QCD simulations [6],[7], and direct calculations using the effective chiral Lagrangian [8]. (For a review of random matrix theory in QCD see [9].) In terms of the spectral correlation functions, the universal limit in which QCD and chiral random matrix theory (RMT) coincide is the limit
(2) 
in which the microscopic variables
(3) 
are kept fixed and is identified as the dimensionless volume. (Here, denotes an eigenvalue of the Dirac operator and is the dimensionless quarkmass parameter.) The determination of the individual eigenvalue distributions and their most important correlators now permits direct comparison of partition function zeros and eigenvalue positions. The suggestion of Leutwyler and Smilga is remarkably accurate. The zeros and the average positions of the eigenvalues are intimately connected.
Here we shall demonstrate that this relationship can be understood as a fundamental property of the chiral ensembles. We show that the zeros are uniquely trapped by the maxima of the joint eigenvalue distribution function. This trapping appears on all scales and is thus relevant for any finite as well as in the large limit. To obtain a better understanding of the relation between the maxima of the joint eigenvalue distribution function and the zeros of the partition function, we introduce and determine the spectral “normal modes” of the chiral unitary ensemble. This provides us with a simple tool to describe the fluctuations of the eigenvalues about the maximum of the joint eigenvalue distribution.
As suggested in (1), chiral random matrix theory is not expected to describe all aspects of QCD. Only correlations below a certain energy length are expected to be in agreement with chiral random matrix theory [10]. In solid state physics, this energy is denoted as the Thouless energy. Recently [11] it was realized that the effects of a finite Thouless energy can be studied naturally using the language of spectral normal modes [12]. We thus perform a normal mode analyses of the chiral ensemble to formulate and establish certain universal features of the partition function zeros. This argument is independent of standard proofs of universality, and its general nature can shed some light on the way universality is realized.
In section 2 we show that for the zeros of the partition function of RMT are trapped between the maximum positions of the joint distribution function. This result holds for all scales. In section 3 we derive the normal modes of the chiral unitary ensembles and find that they are Chebyshev polynomials in the large limit. We discuss the effects of the Thouless energy in section 4. In section 5 we make the connection to the familiar microscopic spectral density. Our conclusions are contained in section 6.
2 Zeros of the partition function in Rmt
2.1 Chiral random matrix theory
We start with the partition function of chiral random matrix theory (RMT) for flavors, which is given by [4, 13]
(4) 
where denotes the Dyson index and is the Haar measure over the Gaussian distributed random matrices . is the analogue of the Dirac operator which has the chiral structure
(5) 
Here is a matrix with playing the rôle of the topological charge. Without loss of generality we assume to be positive. The chiral condensate in the chiral limit, , is related to the eigenvalue density of , , via the Banks–Casher relation [1]
(6) 
The partition function is invariant under transformations , where is a matrix and a matrix. Following the diagonalization , the partition function can be expressed in terms of the eigenvalues of (with ),
(7) 
The Vandermonde determinant, , which is the nontrivial Jacobian of the transformation from the matrices to the eigenvalues, has the form
(8) 
The partition function (7) can now be written as an integral over the joint probability density as
(9) 
with
(10) 
Unlike real QCD, RMT has the special feature that the partition function can be expressed in terms of the eigenvalues of the Dirac operator. This enables us to derive a number of statements regarding the zeros of the partition function. We now focus on the case — the universality class of QCD with 3 colours and quarks in the fundamental representation of the gauge group. (The choice corresponds to QCD with two colours in the fundamental representation; describes QCD with any number of flavours and quarks in the adjoint representation of the gauge group.)
Eq. (10) expresses an evident duality between flavor and topology: The joint probability density for massless flavours and massive flavours depends only on . This relation was proven for the QCD partition function independent of RMT in [14].
We now wish to determine the maximum of the joint probability distribution. This will allow us to put a tight bound on the zeros of the partition function. The chiral normal modes, to be discussed in section 3, describe fluctuations about the maximum of the joint probability distribution.
2.2 Extremum of the joint probability distribution
In order to determine the maximum of the joint eigenvalue probability distribution, we consider variations of with respect to the eigenvalues. (We assume the eigenvalues to be ordered with .) We introduce the coordinates and evaluate the equations
(11) 
For and topological sector this yields
(12) 
We now choose to focus on the quenched, i.e. , joint eigenvalue probability distribution. The solution to this equation reveals that the maximum of is obtained for
(13) 
where denotes the generalized Laguerre polynomials. This result follows from the observation that Laguerre’s differential equation,
(14) 
reduces to (12) at the zeros of . (The proof follows from considerations similar to those made in appendix A6 in [15]). For we can use the fact that
(15) 
to see that the density of eigenvalues in the limit is precisely that of the usual Gaussian ensembles, i.e. a semicircle with support . The partition function for is the average of a product of fermionic determinants over
(16) 
This can be readily evaluated using orthogonal polynomials, and the result for agrees with the one presented in [16]
(17) 
where
(18) 
and is the Vandermonde determinant with the negative square of the masses as arguments. For the special case of this yields the result
(19) 
which is now ready for investigation. The expression (19) coincides (up to a constant) with the series expansion for the partition function derived in [17]. From the expression found there we find that
(20) 
2.3 Trapping of the zeros
The closed forms given above allow us obtain information about the zeros of the partition from the spectral correlators. Specifically, we now show that the locations of the partition function zeros in RMT are trapped by the maxima of the joint distribution function.
The Laguerre polynomials are polynomials orthogonal on the interval with weight function . They have three properties which are useful for our purpose:

For orthogonal polynomials in general, the zeros of the th order polynomial and the endpoints of the weight function define intervals. Exactly one zero of the orthogonal polynomial of order lies in each of these intervals.

For fixed , the th zero of the Laguerre polynomial, , is a monotonically increasing function of , thus
(21) 
The generalized Laguerre polynomials are related to one another via
(22)
In the last section we saw that the massless joint distribution function has its maxima when the eigenvalues are located at the zeros of and that the partition function for and is proportional to . We are interested in relating the zeros of the partition function to the position of the eigenvalues at the maximum of the joint distribution function. Since the zeros of follow from those of by a rotation from the real to the imaginary axes in , we restrict our attention in the following considerations to Laguerre polynomials of positive argument.
It follows from (21) that the zeros of are trapped between the corresponding zeros of and . According to (22), the zeros of lie at the extrema of . The extrema of are evidently trapped by the zeros of (and the end points, if necessary). The result is that the th zero of is trapped between the th and st zero of and, given the nature of the argument, is expected to be closer to the lower value. In other words, the zeros of are trapped by the most probable values for the eigenvalues of . This result is an exact property of the chiral unitary ensembles and is consequently valid on every scale including the microscopic scale.
The trapping just derived relates the zeros of the partition function and the location of the maximum in the joint eigenvalue distribution of the quenched ensemble. We can also relate the zeros of the partition function to the maximum of its own integration kernel, . For we have by flavortopology duality. The other limit, , decouples one flavor and leaves . So, with increasing , the extrema of move smoothly from those of to those of , i.e. from the zeros of to the zeros of . By (21), they must pass the zeros of the partition function for some intermediates values of the mass . The relation between the collective maximum of and the average eigenvalue positions will be reconsidered in section 5.
3 Normal modes in Rmt
We have seen that the maximum of the massless joint distribution function is obtained when the eigenvalues are located at the zeros of the Laguerre polynomials. In order to study the properties of fluctuations about this maximum, it is useful to make a Gaussian approximation to which leads to the form
(23) 
where is the position of the th eigenvalue relative to , its value at the collective maximum of . The matrix is defined as
(24) 
evaluated at the maximum. Concentrating again on the case , we find that the diagonal elements of are
(25) 
and that the offdiagonal elements are
(26) 
We now consider the eigenvalue equation for the real symmetric matrix :
(27) 
The eigenvectors, , are the (normalized) normal modes of the RMT spectrum. They describe the statistically independent correlated fluctuations of the eigenvalues of the random matrix about their most probable values. The normal modes provide an alternate description of the eigenvalues of any given random matrix since
(28) 
We can locate the eigenvalues by specifying either the or the amplitudes as convenient. The eigenvalues, , provide a measure of the magnitude of these fluctuations.
The derivation of these eigenvalues and eigenvectors can be performed as in [12]. The resulting eigenvalues are
(29) 
As in [12], we find a linear dispersion relation valid for all and . The linearity of (29) is a reflection of the wellknown rigidity of random matrix spectra. Furthermore, this result is independent of . Just as in the Gaussian case, the eigenvectors are found to be Chebyshev polynomials in the large limit (i.e., with corrections of order ):
(30) 
The normalization of the eigenvectors is
(31) 
where is again the semicircle
(32) 
Note that only odd normal Chebyshev polynomials appear. This is a consequence of chiral symmetry, which ensures that all nonzero eigenvalues come in pairs . Equation (23) can now be written as
(33) 
Following (28), the coefficients are constructed as
(34) 
and are statistically independent. It is obvious from (33) that that the mean square amplitude for the th normal mode is
(35) 
We can use the normal modes to construct a Gaussian approximation to the partition function as
(36) 
with the given by (13) and given by (28). The Gaussian approximation (33) also permits a simple approximate calculation of the number variance [12]. This calculation reveals that the familiar logarithmic behaviour of the number variance (i.e., the “spectral rigidity” of the random matrix ensembles) is a direct consequence of the linearity of the dispersion relation (29) for all .
We have chosen to consider the normal modes for the case and . This choice is some what arbitrary; it would be equally sensible to start with the and normal modes. We will employ this Gaussian approximation below to consider the sensitivity of the partition function zeros to the effects of a Thouless energy. Since the resulting shifts are small, this arbitrariness will be of no consequence.
4 Effects of a Thouless energy
Normal modes describe the correlated fluctuations of eigenvalues about their most probable values. As we have seen (29), the normal modes for RMT obey a linear dispersion relation. By contrast, uncorrelated eigenvalues obey a quadratic dispersion relation, and the mean square amplitude of the lowest mode with is larger by a factor of [12]. In QCD, it is expected that spectral correlations in a sufficiently small energy domain will follow the predictions of RMT. On larger energy scales, spectral correlations die out. The characteristic energy which divides these regions is the Thouless energy, usually denoted by . In 4dimensional QCD the Thouless energy is estimated to be [10]
(37) 
where is the mean level spacing. This behaviour has been verified in lattice studies [18]. The connection between the Thouless energy and the normal modes of the eigenvalue spectrum has been investigated in [11] for the case of sparse matrices. There it was found that “almost all” normal modes obey the linear dispersion relation discussed above with remarkable accuracy. Reflecting the presence of a Thouless energy, a small fraction (i.e., approximately ) of long wave length modes are strongly enhanced. The mean square amplitude of the longest wave length mode with approaches the value appropriate for uncorrelated eigenvalues. Since such enhancement can be readily incorporated in our Gaussian approximation to the partition function (36), normal modes provide us with a natural and convenient tool to study the effects of a Thouless energy on the zeros of the partition function. So far we have seen that the most probable eigenvalues are located at the zeros of the Laguerre polynomial and that the zeros of the partition function are given by the zeros of . The Gaussian approximation (36) allows us to see how this result is modified by the presence of a Thouless energy. To mimic the effects of the Thouless energy, we shall enhance the long wavelength modes in the partition function.
Our aim is to demonstrate that the zeros of the partition function in the microscopic region remain virtually unaffected even if the enhancement of the soft modes is substantial. Since we are concerned only with the microscopic zeros, every long wavelength mode contributes to a “breathing” of the spectrum. In order to investigate the influence of the soft modes, it is sufficient to evaluate the strength (i.e., mean square amplitude) of this effective breathing.
For concreteness we start with longest wavelengths modes with fluctuations as given by the Gaussian approximation. Let denote the values at the maximum of . The fluctuations introduced in the Gaussian approximation by the longest wavelengths modes are
(38) 
From (30) we know that the normal modes are . This gives for the linear coefficient
(39) 
Additionally we know from (26) that the normal modes are linearly independent. With this information the variance of becomes in leading order of
(40) 
Here we used the facts that the linear terms vanish and that with (29) for the quadratic terms. The result is simply a correction.
In order to study the effects of a Thouless energy, we now enhance the mean square amplitudes of these soft modes by a factor of . This factor provides a smooth interpolation from the behaviour of uncorrelated soft modes (for ) to that of RMT (for ). This interpolation is completely consistent with the results of [11]. We now find that
(41) 
The result still shows strong suppression in . The decision to single out soft modes for enhancement is not essential. We could declare any fraction of the long wave length modes which vanishes in the limit as “soft”. A similar interpolation between the limits of uncorrelated and RMT modes will always lead to a value of which vanishes as . In short, the effects of a Thouless energy are expected to have a negligible effect on over the entire microscopic spectrum.
The question is now how we can evaluate the effect on the zeros from the enhanced long wave length modes. To this end we introduce the distribution function of the fluctuations
(42) 
and fix and by the value of found above. is the normalization. In the RMT case we have and , while in the case where the long wavelength modes are enhanced we have and .
The effect of the first normal modes is different for large and small eigenvalues. Whereas for the smallest eigenvalues it amounts to a breathing it means incoherent fluctuations for the larger ones. We now evaluate the effect of this breathing on the microscopic zeros. Recall that the partition function for and is the average of the fermionic determinant with respect to the joint probability distribution, and that . In the microscopic limit where the quantity is fixed we have
(43) 
To investigate the correction to the microscopic partition function zeros under the influence of the longest wavelength normal modes we thus have to consider the following integral
(44) 
where the normalization factor is
(45) 
The factor comes from the rescaling in the fermionic determinant. For the evaluation of (44) we make use of the series expansion of the cosine
(46) 
This yields
(47) 
Since , we can use that [19]
(48) 
and finally find that in the case where we consider RMT long wavelength modes the correction term to is of order , and in the case of enhanced wavelength modes it is of order . In both cases the zeros of the partition function are unaffected in the microscopic limit.
These results suggest a new kind of “universality” of the microscopic partition function. As a specific example this universality shows that the microscopic zeros are unaffected even if the longest wavelength normal modes are enhanced in such a way that they interpolate between Poissonian and RMT statistics.
5 The microscopic limit
So far we have been discussing the joint distribution function, the zeros of the partition function, and the positions that specify the collective maximum. Here we link this to the more familiar microscopiceigenvalue density. For a finite the eigenvalue density is found from the joint distribution function as
(49) 
The doublemicroscopic spectral density is then defined as [4, 20, 21]
(50) 
and similarly for all other spectral correlations. The functional form of the microscopic eigenvalue density has been derived in [21] and for and topological charge it reads
(51) 
The partition function in the microscopic limit for one flavor is proportional to . The extrema of (51) are given by
(52) 
and are obviously functions of . If the derivative in (52) is evaluated at then the solutions of (52) corresponding to the maxima of coincide with the zeros of the microscopic partition function, , for an imaginary argument. To conclude: The peaks of the microscopic eigenvalue density are in one to one correspondence with the zeros of the partition function and by the trapping proven in section 2.3 also to the positions which maximize the joint probability distribution.
6 Conclusions and outlook
In the present paper we have established an intimate relationship between zeros of the partition function and the spectral properties of the Dirac operator. The relation is derived within chiral random matrix theory and applies to QCD Dirac spectra and partition function zeros near the origin (in the microscopic regime). Through the introduction of spectral normal modes we have tested the validity of the relationship when a finite Thouless energy is introduced. The observed independence of the microscopic zeros compliments the existing universality studies. The present study treats one flavour. For more flavours with degenerate masses, it is known that the zeros of the microscopic partition function are not confined to the imaginary axis [22]. While this makes the relation between the eigenvalues and the zeros somewhat less direct, there is no reason to expect that the normal mode analyis should not apply for any number of flavours.
We remark that the normal mode analysis is generically applicable and not a special feature of random matrix theory. In particular, the normal mode analysis lends itself to a study of the spectral properties of the Dirac operator in lattice QCD. Such a study would be truly interesting in that it would shed new light on the role of random matrix like correlations in lattice gauge theories. Almost all of the normal modes are expected to be given by random matrix theory while only the very long wavelength modes are determined by the detailed dynamics.
In a broader perspective this study may also be seen as the first step towards the establishment of a onetoone correspondence between the zeros and the most likely eigenvalue positions whenever the short range spectral correlations are random matrix like. If such a general relation were established then it could be used to argue that the critical exponents for the YangLee edge in QCD must coincide with the ones for the gap in the spectral density of the Dirac operator. Such relationship, if true, would allow for substantial simplifications when trying to determine critical exponents by lattice techniques.
Acknowledgments:
The authors wants to thank Poul Henrik Damgaard and
Thomas Wilke for useful discussions.
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