Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential
We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.
Introduction. Hermitian random matrix theory (RMT), which describes systems with real spectra, enjoys many applications in physics and beyond. Dropping the Hermiticity constraint results in matrices whose eigenvalues are, in general, complex. Examples are the Ginibre ensembles Ginibre (1965) or their chiral counterparts Halasz et al. (1997). Although these ensembles describe non-Hermitian operators, they have found many applications (see Fyodorov and Sommers (2003) for a recent review), ranging from dissipation in quantum maps Grobe et al. (1988) over quantum chromodynamics (QCD) at nonzero quark chemical potential Stephanov (1996) to the brain auditory response described by nonsymmetric correlation matrices Kwapien et al. (2000).
Observables that are typically computed in RMT are spectral correlation functions. Alternatively, one can study the distributions of individual eigenvalues, provided that the latter can be ordered. For RMT with real eigenvalues, all such distributions are known and have found a variety of important applications. For example, the largest eigenvalue follows the Tracy-Widom distribution Tracy and Widom (1994) and appears in the longest increasing sub-sequence of partitions Baik et al. (1999) or growth processes Praehofer and Spohn (2000). The smallest eigenvalue distribution in chiral RMT has become a standard tool in lattice QCD to extract the low-energy constant (LEC) that appears in chiral perturbation theory (chPT) and is related to the chiral condensate Fukaya et al. (2007). This distribution is also sensitive to the gauge field topology and can be used to distinguish different patterns of chiral symmetry breaking Edwards et al. (1999).
In this paper, we generalize some of these results to the case of non-Hermitian chiral RMT in the unitary symmetry class. We study the distributions of individual eigenvalues in the complex plane and derive analytical results for the chiral RMT introduced in Ref. Osborn (2004). Our main focus will be on QCD, but our findings are also relevant for other systems with complex eigenvalues in the same symmetry class.
In QCD, a nonzero quark chemical potential leads to a complex spectrum of the Dirac operator. In the large-volume limit, chiral RMT is equivalent Basile and Akemann (2007) to the chiral effective theory for the epsilon-regime of QCD Gasser and Leutwyler (1987), which is a particular low-energy limit of the full theory. Here, a virtue of is that couples to the second LEC in leading order of chPT, , which is related to the pion decay constant Toublan and Verbaarschot (2001). A comparison of lattice QCD data to individual complex Dirac eigenvalue distributions from RMT thus allows us to determine both and (for related methods, see Refs. Damgaard et al. (2005); Osborn and Wettig (2006)).
Unfortunately, lattice QCD with dynamical fermions at faces a serious difficulty due to the loss of reality of the action. It is very hard to obtain significant statistics in unquenched simulations, and therefore we will only compare to quenched simulations below. However, for or (where is the pion mass and is the volume) the sign problem is not severe Spl (), and our method can be used to determine from such unquenched lattice data. Therefore we also derive RMT results for unquenched QCD, thus adding to the predictions for spectral densities Akemann et al. (2005) and the average phase factor Spl ().
What is known from RMT for individual eigenvalue correlations in the complex plane? For the non-chiral, unitary Ginibre ensemble the repulsion (or spacing distribution) of complex levels was computed in Grobe et al. (1988) and successfully compared to lattice QCD data in the bulk of the spectrum Markum et al. (1999). For maximal non-Hermiticity, the distribution of the largest eigenvalue with respect to radial ordering is also known Kanzieper (2005). However, in QCD it is the eigenvalues closest to the origin that carry information about topology and LECs, and therefore we concentrate on these in the following.
The complex spectral correlation functions of the QCD Dirac operator
at were computed from different (but equivalent) chiral RMTs
in Refs. Splittorff and Verbaarschot (2004); Osborn (2004); Akemann et al. (2005) and
compared to quenched lattice QCD in
Refs. Ake (a); Osborn and Wettig (2006). Later, a Dirac operator
with exact chiral symmetry at was constructed
Bloch and Wettig (2006, 2007) and tested against chiral RMT for
topological charge . Here, we compare the data of
Ref. Bloch and Wettig (2006) to our newly derived individual complex
eigenvalue distributions, resulting in a much improved signal. For a
we refer to Ref. Akemann (2007).
Complex eigenvalue distributions. We start by defining the gap probability and the distribution of an individual eigenvalue in the complex plane. Suppose the partition function can be written in terms of complex eigenvalues of some operator, with a joint probability distribution function (jpdf) , symmetric in all its arguments, to be specified. (For simplicity, we consider only jpdf’s with additional symmetry , restricting ourselves to the upper half-plane .) The complex eigenvalue density correlation functions are defined as
The simplest example is just the spectral density. The gap probability is defined as the probability that there are exactly eigenvalues inside the set and eigenvalues in its complement ,
If all are known, the gap probabilities follow as in the real case Akemann and Damgaard (2004),
Parameterizing the boundary of in as , we can define the probability for eigenvalues to be inside , for the eigenvalue to be on the contour at , and for eigenvalues to be in the complement ,
(Because eigenvalues repel each other in RMT, the probability of finding two eigenvalues at is zero.) An ordering on is induced by a family of sets of increasing area with mutually nonintersecting contours. Via the Riemann mapping theorem, this can always be reduced to radial ordering. Definitions (2) and (4) are related through a variational derivative,
Employing the expansion (3), we can express the through densities. For example, for the first eigenvalue,
Results from RMT. The above considerations hold for any jpdf, including the jpdf appearing in the lattice QCD partition function in terms of complex Dirac operator eigenvalues and the jpdf of chiral RMT. We now consider the latter. The RMT for unquenched QCD with Osborn (2004) we use here is given by
The Vandermonde, , coming from the diagonalization of complex matrices of dimension (we take for convenience), leads to a repulsion of eigenvalues. (For the chiral RMTs corresponding to adjoint or two-color QCD, the Jacobians will be different, leading to different patterns of eigenvalue repulsion, see, e.g., Ref. Akemann (2007).) The weight depends on dynamical quark flavors with masses () and on the number of exactly zero eigenvalues (corresponding to the topological charge),
where is a modified Bessel function and is the chemical potential in the random matrix model. The first factor in Eq. (8) originates from the Dirac determinants. The non-Gaussian weight function results from an integration over angular and auxiliary variables Osborn (2004). For the are back on the imaginary axis. Complex RMT yields the following result for the densities Ake (b),
given in terms of the kernel of (bi-)orthogonal polynomials with respect to the weight of Eq. (8). In the quenched case (), these are given by Laguerre polynomials in the complex plane Osborn (2004). All unquenched density correlations are given explicitly in Ref. Akemann et al. (2005). A determinental expression follows for the in terms of the kernel operator times the characteristic function of . Eq. (3) is called its Fredholm determinant expansion.
As mentioned above, in the limit of large volume , RMT is
equivalent to QCD in the epsilon-regime Basile and Akemann (2007). In this
regime, the chemical potential, the quark masses, and the Dirac
eigenvalues are rescaled such that the parameters , ,
and stay finite in the large-
(large-) limit. In parentheses, we have given the scaling of these
parameters in terms of the LECs of chPT.
where is a modified Bessel function. The rescaled kernel giving all correlation functions according to Eq. (9) was derived in Refs. Osborn (2004); Akemann et al. (2005). In Fig. 1 we show as an example the density and the distribution of the first eigenvalue from Eq. (6) (in which is chosen to be semi-circular and only the first three terms are included). As in the case of real eigenvalues Akemann and Damgaard (2004), we see that the expansion converges rapidly. Higher-order terms merely assure that remains zero for large .
For increasing , the quenched density Eq. (10) rapidly becomes rotationally invariant close to the origin. In terms of the new variable , it becomes
In this limit, we can derive a closed expression for the gap probability ASI (). Because of the rotational symmetry we choose to be a semi-circle of radius and obtain
where we have introduced the incomplete Bessel function for , and zero otherwise. Our quenched expression Eq. (12) generalizes the corresponding result of Ref. Grobe et al. (1988) for the non-chiral Ginibre ensemble, which is given in terms of incomplete exponentials .
Comparison with lattice data. We now come to the comparison of our analytical results to quenched lattice QCD data. For details of the simulation we refer to Ref. Bloch and Wettig (2006). The gauge fields were generated in the quenched approximation on a lattice at (see Bloch and Wettig (2006) for an explanation of these choices). The Dirac operator introduced in Ref. Bloch and Wettig (2006) is a generalization of the overlap Dirac operator ove () to . It satisfies a Ginsparg-Wilson relation Ginsparg and Wilson (1982) and has exact zero modes at finite lattice spacing. We can therefore test our predictions in different sectors of topological charge . In Ref. Bloch and Wettig (2006) complete spectra of the generalized overlap operator were computed for several values of and large numbers of configurations, and these data are used in the comparisons to the RMT results below. We also used the fit parameters and from Ref. Bloch and Wettig (2006) to determine and , i.e., no additional fits were performed.
For the contours we again choose semi-circles, for all values of . Since we prefer to show 2D plots we have integrated over the phase of the complex number and display only the radial dependence. Results for are shown in Fig. 3 for and , corresponding to and , and in Fig. 4 for and , corresponding to and . (The lattice spacing has been set to unity.)
For all values of we compare the data to the expansion Eq. (6), in which only the first three terms were used.
For the data were found to be approximately rotationally invariant, and we also compare them to the exact result in the large- limit from Eqs. (12) and (13). (Because of the rotational invariance, only the ratio could be determined for in Ref. Bloch and Wettig (2006), see Eq. (11). In this case the value of used in Eq. (6) is an extrapolation, assuming that is independent of .) The agreement between data and analytical curves is excellent except for at (see Fig. 4). In these two cases we have left the range of validity of RMT.
We emphasize that while the rise of the distributions from zero was in principle already tested in Ref. Bloch and Wettig (2006) through the density (see Fig. 1 or 2), their decrease represents a new, parameter-free test. Note also that because of the integration over the phase, the signal is much better than in Ref. Bloch and Wettig (2006). This allows us, for the first time, to successfully test the RMT predictions for .
Figures 3 and 4 also show the effect
of truncating the Fredholm expansion (6): The analytical
curves bend up for large after (almost) touching zero.
Higher-order terms in the expansion (6) only affect the tail
of the distributions. They will “repair” the bending-up and ensure
that the tails remain zero, just as the data. The same effect was
observed earlier for real eigenvalue distributions
Akemann and Damgaard (2004). This feature of our approximation can be seen
most clearly when comparing to the exact result in the large-
limit, see Fig. 4 (right), in which we can observe how
the expansion converges in the case of large .
Conclusions. We have shown that the distributions of
individual complex eigenvalues from non-Hermitian RMT agree very well
with the corresponding distributions of the complex eigenvalues of the
quenched QCD Dirac operator closest to the origin in three different
topological sectors. As in the Hermitian case, these distributions
are much easier to compare with than the density, in which a plateau
may not be observable due to appreciable finite-volume corrections.
Our analytical results are also relevant for other non-Hermitian
systems in the chiral unitary symmetry class. In the future, it would
be interesting to compute (and apply) similar results for the
symplectic symmetry classes.
Acknowledgments. We thank K. Splittorff and J.J.M. Verbaarschot for useful discussions. This work was supported by EPSRC grant EP/D031613/1 (GA & LS), by EU network ENRAGE MRTN-CT-2004-005616 (GA), and by DFG grant FOR 465 (JB & TW).
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