Domain-wall and overlap fermions at nonzero quark chemical potential
We have recently given a construction of the overlap Dirac operator at nonzero quark chemical potential. Here, we introduce a quark chemical potential in the domain-wall fermion formalism and show that our earlier result is reproduced if the extent of the fifth dimension is taken to infinity and its lattice spacing is taken to zero. We also extend this result to include a bare quark mass, consider its continuum limit, and prove a number of properties of the overlap operator at nonzero quark chemical potential. In particular, we show that the relation between the anomaly and the index of the overlap operator remains valid.
The phase diagram of quantum chromodynamics (QCD) as a function of temperature and chemical potential has been the subject of intense studies over many years. It is physically relevant, e.g., for the study of compact stars, for ultra-relativistic heavy-ion collisions, and for the physics of the early universe. The theoretical methods that have been employed to investigate the QCD phase diagram include model calculations, effective theories, perturbative studies at high temperature and density, lattice simulations, and recently also the AdS/CFT correspondence. For an overview of the literature and a summary of the current status, we refer the reader to Ref. Stephanov:2007fk.
Lattice simulations are the predominant nonperturbative tool to study QCD from first principles. The case of nonzero temperature can be implemented on the lattice without much effort, and therefore the temperature dependence of many QCD quantities is very well understood, see Ref. Heller:2006ub for a review. This is not true for the case of nonzero baryon density or, equivalently, quark chemical potential . The reason is that at , the fermion determinant becomes complex so that standard importance sampling methods fail. This is an example of the so-called sign problem, see Ref. Splittorff:2006vj for a thorough discussion of this problem in the context of lattice QCD. A number of approaches have been invented to deal with this problem, such as reweighting along the critical line Fodor:2001au, Taylor expansion Allton:2002zi, and analytical continuation from imaginary deForcrand:2002ci; D'Elia:2002gd. Using these approaches, attempts have been made to determine the transition line and to locate the critical end-point, see Ref. Schmidt:2006us for a review. While the results are encouraging, the sign problem remains unsolved in principle.
Most of the lattice results for QCD thermodynamics have been obtained with staggered fermions, which reduce the fermion doubling problem and implement a remnant chiral symmetry on the lattice. For simulations with less than four staggered flavors, the so-called rooting problem has been the subject of some debate, see Ref. Sharpe:2006re for a review. We have no intention to enter this debate here. Rather, our aim is to investigate how a quark chemical potential can be implemented in a fermion operator that implements an exact chiral symmetry on the lattice.
In Ref. Bloch:2006cd we showed how this can be done for the overlap operator Narayanan:1994gw; Neuberger:1997fp. (For earlier work with a similar focus, see Refs. Bietenholz:1998rj; Bietenholz:1999km.) The resulting operator, , contains the sign function of a nonhermitian matrix. We also showed that quenched lattice results obtained with this operator agree with analytical predictions from nonhermitian chiral random matrix theory at Splittorff:2003cu; Osborn:2004rf; Akemann:2004dr, see also Akemann:2003wg. Furthermore, the authors of Ref. Gattringer:2007uu have shown that our construction of yields the correct energy density for free fermions.
In the present paper, we introduce a quark chemical potential in the domain-wall fermion formalism Callan:1984sa; Kaplan:1992bt; Shamir:1993zy; Furman:1994ky, which can be viewed as a particular truncation of the overlap operator. In analogy to the well-known result at , we show that in the limit in which the extent of the fifth dimension is taken to infinity and its lattice spacing is taken to zero, our earlier result for is reproduced. We also extend this result to include a bare quark mass, consider its continuum limit, and prove several properties of , including the relation between the anomaly and the index.
We should remark that at present, the topic we address here may seem to be mainly of theoretical interest since lattice simulations with such an operator, especially at , are numerically much more expensive than those with staggered fermions. However, as computers and algorithms improve, more and more lattice QCD simulations will be done with overlap and domain-wall fermions. As a first step towards such simulations at , we have already proposed and tested a new iterative method to compute the sign function of nonhermitian matrices Bloch:2007aw.
This paper is organized as follows. In Sec. II we review domain-wall fermions at , extend the domain-wall action to , and consider the limit of this action for infinite extent and zero lattice spacing of the fifth dimension. After a short side remark in Sec. III, we discuss the continuum limit of in Sec. IV. In Sec. V we prove a number of properties of that were stated, but not proven, in Ref. Bloch:2006cd. We conclude with a summary and outlook in Sec. VI.
Ii Domain-wall fermions at and relation to overlap operator
ii.1 Domain-wall fermions at
We start with the definition of the Wilson Dirac operator in four dimensions, which is given by Hasenfratz:1983ba
where is the Wilson mass, the are the lattice gauge fields with , the are the usual Euclidean Dirac matrices, the Wilson parameter has been fixed at , the 4-d lattice spacing has been set to unity, and for later convenience we have already included a quark chemical potential .
Domain-wall fermions Callan:1984sa; Kaplan:1992bt; Shamir:1993zy; Furman:1994ky are constructed by introducing an additional fifth dimension with lattice spacing and extent . The fermion fields now have an additional index , while the gauge fields remain four-dimensional and do not depend on . At , the domain-wall fermion action is given by Shamir:1993zy; Furman:1994ky
where , the chiral projection operators and are defined by , and the fermion fields satisfy the following boundary conditions in the fifth dimension,
The quantity is a bare quark mass parameter. In Eq. (II.1), the Wilson mass has to be in the range to obtain renormalizable solutions in the fifth dimension and to avoid the existence of doublers Kaplan:1992bt; Shamir:1993zy.
To take the limit, the domain-wall fermion action of Eq. (II.1) is supplemented by a pseudo-fermion action to cancel divergences due to the heavy fermions in the large- limit Narayanan:1992wx; Frolov:1992ck; Frolov:1993zr; Narayanan:1993sk; Furman:1994ky; Vranas:1997da; Neuberger:1997bg. We use the pseudo-fermion action of Ref. Vranas:1997da, which is given by Eq. (II.1) with anti-periodic boundary conditions, i.e., in Eq. (4), but in which bosonic fields are used instead of the fermionic ones. As the fermion action is only defined up to a constant normalization factor, we choose to multiply the pseudo-fermion action by .
ii.2 Domain-wall fermions at
We define the domain-wall fermion action at to be the same action as in Eq. (II.1), except that is replaced by . With very minor modifications, the arguments of Ref. Kaplan:1992bt leading to the bounds on the Wilson mass apply to the case of as well, and we again obtain the requirement .
ii.3 limit of domain-wall fermions at
For , the connection between domain-wall fermions and the overlap operator in the limit has been exhibited in a number of earlier works, e.g., Refs. Neuberger:1997bg; Kikukawa:1999sy; Borici:1999zw; Edwards:2000qv. To make the presentation self-contained, we now retrace some of the steps taken in these papers, in particular Refs. Borici:1999zw; Edwards:2000qv, with small modifications suitable for our purposes.
The idea is to introduce successive spinor transformations to diagonalize the Dirac operator in the fifth dimension and integrate out the fermion fields. To this end, we start with the transformation
To simplify the first term in Eq. (II.3), we introduce the transformation
which is diagonal in the fifth dimension. The nontrivial Jacobian of this transformation can be ignored since it is cancelled by the corresponding Jacobian for the pseudo-fermions. We also define an operator by
where we have used
(See below for comments on the invertibility of and .) The structure of Eq. (II.3) suggests to transform the for according to
|with inverse transformation|
|The for are transformed according to|
The and can now be integrated out trivially. Finally, we integrate out and and obtain, together with the corresponding contribution of the pseudo-fermions,
with an effective 4-d operator given by
We now take a closer look at the transfer matrix of Eq. (8), which can be rewritten as
with . For , is -hermitian, i.e., it satisfies , and thus is hermitian. From this it follows that is also hermitian. The transfer matrix can be related to a 4-d Hamiltonian by writing it in the form
For , is hermitian.
Up to this point, everything went through as in Refs. Borici:1999zw; Edwards:2000qv. We now move on to the case of , in which ceases to be -hermitian. Therefore, neither nor nor are hermitian. To obtain the limit of Eq. (17), we consider the matrix function
where is given by Eq. (19) with a nonhermitian matrix . A function of an arbitrary complex matrix can be defined by Golub1989
where the integral is defined component-wise and is a collection of closed contours in such that is analytic inside and on and such that encloses the spectrum of . We are therefore interested in
and is an eigenvalue of . If (), () as , i.e., we can write , where denotes the sign function. For , () if (), and hence . Let us now consider with , for which
From this expression we obtain immediately
where the last equality defines the sign function of a complex number. This can also be written as
with the branch cut of the square root along the negative real axis. We thus obtain
where the sign function of a nonhermitian matrix is defined formally by Eq. (22) in combination with Eq. (26) or (27). A simpler form for can be obtained if can be diagonalized, i.e., if it can be written in the form with and , where is the dimension of . It then follows from Eq. (22) that
so that the matrix sign function can be defined by Roberts1980
Note that if cannot be diagonalized, one can use the Jordan canonical form instead, see Ref. Bloch:2007aw for details.
In the course of the derivation, we have assumed (i) that the operators and are invertible and (ii) that the elements of the diagonal matrix are not zero or purely imaginary so that the sign function is well-defined. For any of these assumptions to be violated, the gauge-field would have to be fine-tuned. This happens on a gauge field set of measure zero and can therefore be ignored in practice. The same remark applies to the possibility that might not be diagonalizable.
Iii Side remark
Note that for , there is an expression for the overlap operator that is equivalent to Eq. (32), i.e.,
One could be tempted to use this expression for as well, with replaced by . However, the resulting operator is not equivalent to Eq. (32) for due to the lack of -hermiticity of . In particular, it does not satisfy the Ginsparg-Wilson condition and has no exact zero modes at finite lattice spacing. Thus, the expression (33) is not suitable for an extension to .
Iv Continuum limit of
In this (and only this) section we reintroduce the 4-d lattice spacing that was set to unity earlier and write the Wilson Dirac operator, with the argument suppressed, in the form Hernandez:1998et
where with and is the massless Wilson operator, . We therefore have for
and we find
Equation (32) thus becomes
Since for the Wilson Dirac operator becomes the continuum Dirac operator, this is also true for , up to a normalization factor. There was no need in the above derivation to use the -hermiticity of , which is lacking for . The only input required was the fact that anticommutes with in the continuum limit, which holds for as well.
V Properties of
In Ref. Bloch:2006cd we stated, but did not prove, a number of properties of at . We supply the missing proofs here, assuming throughout this section.
Property 1: For , is no longer -hermitian but satisfies
instead. From Eq. (32) with we obtain
from which Eq. (39) follows because of
In the second step, we used the well-known fact that satisfies Eq. (39), which follows from Eq. (II.1) and from . In the third step, we used the fact that the sign function satisfies for any matrix , which follows from Eq. (22) with , or from Eq. (31). Note that the proof of Property 1 also goes through for .
Property 2: satisfies a Ginsparg-Wilson relation Ginsparg:1981bj of the form
Setting in Eq. (32) and using the shorthand for , we have
Property 3: All eigenvalues of that are not equal to 0 or 2 come in pairs (with eigenvector ) and (with eigenvector ). We first note, with all arguments suppressed, that
from which we conclude that
where we have again used . Note that for , all eigenvalues lie on the circle with radius 1 and center at 1, in which case and are complex conjugates of each other.
Property 4: The mapping projects the pair and to a complex conjugate pair . This follows from elementary algebra.
Property 5 makes statements about the eigenvectors of corresponding to eigenvalue 0 or 2. In these cases, Property 3 implies that and are degenerate eigenvectors of . This means that commutes with in the corresponding degenerate subspace and can thus be diagonalized in this subspace. Because of the eigenvalues of are , i.e., the eigenvectors of corresponding to or 2 can be arranged to have definite chirality. In the following we denote by the number of eigenvectors corresponding to or 2 with . Consider now the operator , where the argument has been suppressed. It is easily shown that if is an eigenvector of with eigenvalue , then and are degenerate eigenvectors of with eigenvalue . For , we now construct the vectors . According to Property 3, and are linearly independent in this case, and therefore the two vectors are nonzero and linearly independent. Moreover, they are also eigenvectors of with eigenvalue , respectively. We now consider in a basis consisting of the and of the eigenvectors of corresponding to and 2 with definite chirality. Since in this basis the operators and are simultaneously diagonal with eigenvalues and , respectively, we have
where in the second line is the (accidental) degeneracy of the eigenvalue of and the factor of in front of the sum removes a double counting of eigenvalues. An analogous argument holds for , which also satisfies the Ginsparg-Wilson relation and for which the roles of and are interchanged, leading to
where . From we conclude that
as stated in Ref. Bloch:2006cd. From we also conclude that
The relation (50) between the anomaly and the index of was already proven for in Refs. Narayanan:1993sk; Hasenfratz:1998ri; Luscher:1998pqa. Our simple derivation shows that it remains valid at . (The method introduced in Ref. Luscher:1998pqa also works at without modifications.) Eq. (50) was used in Ref. Bloch:2006cd to explain an observed shift in the number of zero modes of as a function of .
This means that for , is a normal operator, whereas for , we cannot conclude anything from Eq. (51) about the normality of . This suggests that for , is not a normal operator (at least generically). This expectation is confirmed numerically. It is interesting to note that the operator (51) is equal to the operator we defined in the proof of Property 5.
We have extended the domain-wall formalism to nonzero quark chemical potential and have shown that in the limit in which and we obtain an expression for the overlap Dirac operator that is identical to our earlier result Bloch:2006cd. We have also included a bare quark mass, considered the continuum limit, and proven a number of analytical properties of this operator.
In actual lattice simulations, the use of Eq. (32) will be hindered by two problems. The first is the infamous sign problem that plagues lattice QCD at . We have nothing new to say about this problem here. The second problem is that not much is known about efficient numerical computations of the sign function of a nonhermitian matrix. As remarked earlier, we have started to address the second problem in Ref. Bloch:2007aw.
Work is in progress in several directions. First, we will continue our algorithmic developments to compute the sign function of nonhermitian matrices, with particular emphasis on novel deflation schemes. Second, we are currently testing the predictions of nonhermitian random matrix theory also for the unquenched theory, which can be done by reweighting on the small lattices that we have studied so far. We are also studying the average phase factor of the fermion determinant, for which some analytical predictions exist in the epsilon-regime of QCD Splittorff:2006fu; Splittorff:2007ck.