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A note on the Loop Formula for the fermionic determinant

A formula expressing the fermionic determinant as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and are indexed by loops of increasing length.


Introduction
The study of the effects of virtual particles has a very long history. In particular the vacuum polarization due to electron-positron pairs was studied first by Euler and Heisenberg [1] and later by Schwinger [2].
Let us mention some of the later developments: the vacuum polarization to all orders is given by the fermion determinant. Its mathematical properties were studied later for instance by Seiler [3] in the context of constructive quantum field theory, where a stability problem arising in four dimensional quantum electrodynamics was pointed out. This latter issue was studied more extensively by [4]. The study of this issue culminated in the work of Fry [7] which also includes the effect of fermions on non-abelian gauge fields.
The quantum effects on vortex fields were analyzed by Langfeld et al [5], using Schwinger's propert time formalism. Schmidt and Stamatescu [6], also using this formalism, pointed out that the fermion (and boson) determinant on the lattice can be viewed as a gas of closed loops which can be simulated numerically via a random walk.
In this note we also consider lattice gauge theories; we derive and discuss a general loop formula for the fermion determinant. This formula provides a systematic approximation for the fermionic determinants which can be used in full QCD analyses. In particular it proved useful in problems of QCD at non-zero chemical potential where it allowed to set up the so called HD-QCD (for high density quantum chromodynamics) approximation for large mass and chemical potential [8,9,10] which was used for physically relevant simulations in leading (LO) and next-to-leading orders (NLO) [11,12,13,14]. This formula involves, however, some subtleties both in its derivation and in its interpretation and we think it is of some interest to address them here.
Let us repeat that the formula itself is not new, but in this note we provide some necessary clarifictions of its status.
This loop formula is based on the loop expansion derived long ago by one of us [15] for the logarithm of the determinant with Wilson fermions in an external gauge field. This logarithm is expressed there as a power series expansion in the hopping parameter κ, in which each term is given as a sum of closed loops. The loop formula proceeds from this expansion and expresses the full determinant as a product over determinants of a much smaller fixed dimension (the tensor product of spin and color spaces) involving closed loops on the lattice of increasing complexity [12]. It involves the resummation of nested infinite series and therefore its correctness beyond formal algebra depends on (absolute) convergence. We will discuss in more detail the derivation of the formula as well as its limitations and possible misunderstandings in using it, since one might think that the zeroes of the determinant are given by the zeroes of the factors of the product. In fact this is not the case, as we will explain in the following, and the formula should instead be understood as a systematic approximation, as illustrated below.
In particular for a finite lattice this formula expresses the determinant which is a polynomial of finite order in the hopping parameter as an infinite product. Obviously this can make sense only where the infinite product converges, which is equivalent to convergence of the expansion of its logarithm; this convergence will break down at the latest at the first zero encountered, either on the left hand side (LHS) or in one of the factors on the right hand side (RHS).
A simple example might help at this point to illustrate both derivation and problems of the loop formula. Consider We expand the logarithm in powers of κ and x, y, but for bookkeeping purposes we treat x, y as noncommuting symbols, so we consider the string xxyy as different from xyxy. This way we get Symbolically we can write where s l (x, y) stands for strings of lengh l formed from x and y. Next we introduce 'primary strings' s P l as those strings that are not repetitions of other strings and resum now first over all repetitions of primary strings ("s-resummation" in the following) and then over the primary strings. This way we get Up to cyclic permutations the primary strings of length 1 are x and y, of length 2 there is only xy, of length 3 there are xxy, yyx, xyy, yxx, xyx, yxy, of length 4 xxxy, xxyy etc.. Notice that xyxy is not a primary string but a repetition of the string xy. We now introduce equivalence classes [s P l ] for the l strings of length l differing by cyclic permutations and obtain ln(1 − κ l s P l (x, y)) .
So we obtain After exponentiation this gives Regarding now the symbols x, y again as complex numbers we turn to the question of convergence. One can see in this example that truncating the product by keeping only the factors containing up to κ 4 the LHS is only reproduced up to order κ 5 : Of course this mechanism will operate the same way at any order: truncation at order κ n will produce an approximation to the determinant up to order O(κ n+1 ).
To determine when this expansion converges, note that for x, y ≥ 0 all terms in Eq.(2) have the same sign, so this expansion converges absolutely for κ < 1/(x + y). For general x, y we thus have absolute convergence of any reordering and resumming of the expansion for ln(1 − κ(x + y)), provided so in particular the expansion Eq.(5) converges. Exponentiating we obbtain convergence of the infinite product Eq.(7).
One can see directly that in this region neither the LHS nor any of the factors on the RHS vanishes, so in the region of convergence there is no paradox of one side vanishing with the other side nonvanishing. The infinite product on the RHS converges in this region to the LHS, which provides the analytic continuation to all of C 2 .
It should not be surprising that the location of the zeroes of the LHS is not well approximated by the zeroes of the factors on the RHS: the lowest zeroes as given by the two linear factors, (1 − κx)(1 − κy), e.g., are at κ = 1/x and κ = 1/y, whereas the true zero is at κ = 1/(x+y). Only after expanding the truncated product we obtain a first order approximation of the LHS: (1−κx)(1−κy) = 1−κ(x+y)+O(κ 2 ).
In the following we shall first present the principle leading to this formula in Sect. 2, then briefly describe its derivation for QCD with Wilson fermions, which involves some subtleties, in Sect. 3. Details and illustration of both the derivation of the loop formula and of its problems are given in the Appendices.

The idea in a nutshell
The principle behind the formula is easy to explain: let be a N × N matrix . Then which converges as long as ||M || < 1/κ or even as long as the spectral radius ρ(M ) < 1. We can interprete trM l , which involves a l-fold sum over the matrix indices as a sum over all closed paths (journeys or loops) of length l over the the index set. To be precise, we define a path of length l as a map with C l (l) = C l (0). Furthermore we define a weight for each path so that and The weight defined in Eq.(13) is clearly invariant under cyclic permutations of {0, 1, . . . , l}, so it makes sense to introduce equivalence classes [C l ] of paths C l that can be mapped onto each other by a cyclic permutation. So we finally can write As long as the series (15) converges absolutely, we can make any kind of rearrangement, summing first over (finite or infinite) subsets of paths; the result will not be affected. For instance we can define primary paths C P l of length l as paths which are not repetitions of other paths, and rewrite the formula as After exponentiating, this gives This last expression looks like a factorization of det W , but it is not, because the convergence requirement limits its validity. Absolute convergence can hold at most for |κ| < |κ c | where κ c is the zero of det W closest to the origin. But of course it may break down earlier, because convergence of the logarithmic expansion Eq.(11) does not imply absolute convergence of the path sum Eq.(16): absolute convergence of Eq.(16) requires whereas absolute convergence of Eq.(11) requires only lim sup which is a much weaker condition. In fact the latter can easily be estimated in a way independent of the 'volume' N : but it is in general not true that and therefore it is difficult to find estimates for |trM l | 1/l | independent of N . So in general there is no simple relation between the zeroes of the factors of Eq. (19) and those of det W .
In the next section we turn to the main problem of deriving a similar loop formula for the Dirac-Wilson determinant. There the index set is composed of lattice indices and internal ones (spinor, color, flavor) and the definitions of loops etc. are modified accordingly. On the other hand, the Dirac-Wilson operator connects only nearest neighbor lattice points, which leads to a tremendous simplification, since instead of the 'flight journeys' considered here, we have only paths consisting of nearest neighbour steps.

The loop formula for the Dirac-Wilson determinant
We use here the Wilson fermion formulation in d = 2 or 4 dimensions where κ is the hopping parameter, T µ are lattice translation, U µ are link matrices (assumed here in the fundamental representation of SU (3) or SL(3, C)), and are Wilson's projectors for the Dirac d.o.f.'s. In this formulation the loop formula takes a simpler form, since back-steps are forbidden. See also Appendix A.
The loop formula [15,11,12] for a finite simple square (d = 2) or hypercubic (d = 4) lattice reads: Here r is the net winding number of the path in the time direction, with periodic or antiperiodic b.c. and ǫ = +1(−1) correspondingly.
We assume periodic b.c. in the 'spatial' directions. Notice that the terms in Q imply unit steps on the lattice, therefore a lattice path C l in Eq. (29) is a closed chain of lattice points as produced by the lth power of Q in Eq. (27). Due to the trace condition a path must close, but it can repeatedly visit the same lattice point before closing (the Pauli principle was used in writing the determinant, after that everything is only matrix algebra). Notice that this condition also implies that on an even lattice l is even. In the following we shall speak of the C l paths in Eqs. (28-29) as primary paths. They are distinguishable, non-exactly-self-repeating lattice closed paths of length l. Since the primary paths are closed paths and the ensuing tr D,C do not depend on cyclic permutations of their points we do not identify a starting point for C l . Nonetheless, in producing such a path we can start at each of its points, therefore we get a multiplicity factor l. We define the loops L C l as the chains of links and Γ factors on the primary path C l and call them primary loops. A loop, however, may be repeated in covering the path C l before the Dirac and colour traces of tr D,C close, and the exponent s in Eq. (28) counts the repetitions in covering C l . Since stepping further after completing the primary path we only obtain identical paths, however, these repetitions do not produce a further multiplicity factor, therefore no further factor s counting the repetitions appears. These considerations have been taken into account in Eqs. (28,29).
With λ denoting the links along C l we have; The Dirac factors tr D Γ C l can be calculated for each C l geometrically [15] or numerically.
The contribution of 1-and 2-dimensional loops LC l ,C l : linear and planar loops closing on or over the lattice, in (29) further simplifies to where Eq. (33) holds for SL(3, C) (SU (3)) gauge group. In deriving Eqs. (32,33) we used the fact that the Dirac and Yang-Mills traces in Eq. (28) factorize and the former concern products of the projection operators Eqs. (25, 26): where the second equation holds for linear and planar loops [15]. In particular, e.g. for straight Polyakov loops and their inverses L C Nτ = P , L −1 C Nτ = P −1 we have For loops which explore more than two dimensions of the lattice Eq. (36) does not hold generally and therefore Eq. (29) cannot always be rewritten similar to Eq. (33). We shall speak of basic paths as primary paths without repeated visitation of a point. Notice that the number of basic paths is finite on a finite lattice. The primary paths are obtained by chaining basic paths in arbitrary order and arbitrarily often repeated (only the resulting chain as a whole should not be repeated). Hence even on a finite lattice there are infinitely many primary paths.
Notice therefore that the RHS of the loop formula on any lattice is an infinite product. Since on a finite lattice the determinant is a polynomial in κ of order N max = dN L N c with d = 2, 4 the dimension, N L the lattice volume, N c the number of colors the loop formula implies cancellations of the higher orders. This happens algebraically (i. e. without worrying about convergence), but it is justified analytically only if the nested infinite series involved converge absolutely. See the discussion in sect. 2.
On the other hand arbitrarily truncating the RHS by keeping only a finite number of factors up to some l 0 will make the loop formula into an approximation of order l 0 . The zeroes suggested by the RHS are therefore not true zeroes of the determinant but will corroborate to provide the approximation of the latter to the above order.
When we are dealing with finite temperature and µ > 0, with (anti-)periodic boundary conditions in time the coefficients for loops with positive net winding number r > 0 in the time direction will have coefficients containing powers of ζ and can therefore be of order 1 or larger: It is useful therefore in this case to reorder the loops according to powers κ lσ , the lowest order (LO) being given by straight Polyakov loops, the higher orders will involve Polyakov loops 'decorated' by spatial excursions. To LO and NLO we can use Eqs. (33,34,37) for these loops, see [11,12] for details. Notice that it is important to use here the s-resummation which brings the loops into the determinant factors, in order to avoid large numbers in the exponents. Loops without windings have no ζ factors and start at order κ 4 . For completeness the formal derivation is described in detail in Appendix B. In Appendix C we illustrate on two simple examples the evaluation of the formula and the problem of the zeroes.

Conclusions
The loop formula Eqs. (27-29) expresses the determinant of the N ×N Wilson fermionic matrix, with N = N Lattice × N Dirac × N Colour , as an infinite product of determinants of a fixed (independent of N Lattice ), small dimension (N Dirac × N Colour ) 2 . The loop formula is based on the loop expansion given in [15], which itself represents a hopping parameter expansion, and its derivation requires careful reordering of the terms and resummation of infinite nested series. This way one obtains an algebraically correct representation, whose use as an approximation requires, however, a convergence analysis.
The infinite product of small determinants on the RHS can be ordered according to the length of the loops or to the order of the hopping parameter. The individual zeroes of the RHS are not zeroes of the LHS but the RHS converges to the big, exact determinant of the LHS in the region below the lowest zeroes. Truncation of the RHS after a given loop length produces approximations of the LHS after expanding the product and retaining the terms up to the corresponding order.
The formula provides a valid series of approximations in its range of absolute convergence; this domain does, however, not include the zeroes of the full determinant nor those of any approximants. It can thus not be used for an approximate determination of the zeroes of the full determinant, but the approximants can still be useful, e.g. in problems of QCD at finite chemical potential (or density). The formula may be interpreted as providing an ensemble of loops; this suggests that higher orders, which are hard to evaluate algebraically, might be instead be produced by stochastic generation of loops as in [6].
A Grand Canonical Partition function for QCD with Wilson fermions at µ > 0 (one flavour). Here M is the "bare mass", M 0 the bare mass at µ = 0, U denote the link variables and T lattice translations. For completeness we introduced anisotropy factors γ G , γ F which have to be tuned, by requiring isotropy of physical quantities at T = µ = 0 (hadron masses, string tension etc), defining so a unique physical anysotropy of the lattice spacings η = a σ /a τ which will enter the physical temperature, chemical potential, masses, etc. In the following γ G = γ F = 1. The exponential prescription for µ ensures cancelling of divergences in the small a limit.

B Formal derivation
In the following we shall formally derive the loop formula. For illustration we shall refer to Fig. 1. With we have at first step Notice: (1) Due to the traces we obtain only connected, closed loops on the lattice (without backtracking for Wilson fermions).
(2) We consider now primary loops L C l . They are of all possible lengths l and may close on or over the toroidal lattice.
Example: In 1 4 trQ 4 the loop tr D,C S 1 V 1 S −1 2 U 2 appears with multiplicity 4 (since it can be started at each node) while leading each time to the same contribution by cyclicity of tr D,C -cf Fig.(1) right. (3) Each primary loop can appear repeated any number of times s. As already pointed out, because of the indistiguishability in the repetition there is no further multiplicity.
Example: On a 1-dim periodic lattice of length 2 the term 1 2 trQ 2 has 2 loops closing over the lattice, U 1 U 2 and U 2 U 1 , contributing equally tr D,C U 1 U 2 = tr D,C U 2 U 1 , see Fig. 1 left. Since the corresponding paths are identical we speak of them as one primary loop with multiplicity 2, contributing thus 2 × 1 2 tr D,C U 1 U 2 in the loop formula.
The term 1 4 trQ 4 of order n = 4 contains (U 1 + U 2 ) 4 and has just 2 different terms, U 1 U 2 U 1 U 2 and U 2 U 1 U 2 U 1 , allowed by the lattice trace (notice that a term like U 2 1 U 2 2 cannot appear since each link is associated with a lattice step). One can see U 2 U 1 U 2 U 1 as the first permutation of U 1 U 2 U 1 U 2 , further permutations do not bring new loops. The contributions of these two loops to the ensuing tr D,C are equal by cyclicity of the trace. Therefore we speak here of just one loop, say U 1 U 2 U 1 U 2 . Since it is a genuine repetition it does not represent a primary loop, but the s = 2 repetition (U 1 U 2 ) 2 of the primary loop U 1 U 2 of multiplicity 2 and contributes therefore 2 × 1 4 tr D,C (U 1 U 2 ) 2 in the loop sum.
(4) Identification of primary loops. Example: The loops S 1 V 1 S −1 2 U −1 1 and S 1 V 1 S −1 2 U 2 are basic loops, the first one is a plaquette, the second a decorated Polyakov loop, both appearing with multiplicity 4.
is a primary loop, consisting of the iterated basic loop Plaquette and the basic Polyakov loop and has multiplicity 12. In contrast to this, the loop is not a primary loop but the s = 3 repetion of the basic loop plaquette S 1 V 1 S −1 2 U −1 1 and appears therefore with multiplicity 4.
We now make the 2-nd step to obtain: Notice: (5) The summands in Eq. (47) are Dirac and colour traces, that is just complex numbers. Each summand corresponding to a primary loop is of the form since this is just the expansion of a logarithm (s-resummation). It is the repeated covering of primary loops which resumes to the logarithms. (6) Using Eq. (50) we can rewrite Eq. (47) as Since the summands commute we can exponentiate and rewrite Eq.(47) as by inverting the general formula Eq.(50). This is Eq. (29).
As remarked before, the zeroes suggested by the RHS factors of the loop formula generally are not zeroes of the LHS, but keeping the lowest order factors up to some order may provide reasonable approximations for the LHS, depending on the parameters and configuration.
A particular situation occurs for the case of large mass and chemical potential. Here the temporal loops disentangle in the limit κ → 0, µ → ∞, ζ = κe µ = f ixed and the determinant reduces to the "HD-QCD" determinant in LO, here for antiperiodic boundary conditions (apbc) (ǫ = −1) Notice that since in this case the expansion parameter C can be of order 1, the s-resummation to Eq. (29) is important for avoiding large exponents in Eq. (28).

C Simple examples
In the following we treat two examples which both illustrate the loop formula and its problems. We use mimicking a 2-dim lattice theory at finite µ. We use U(1) links to reduce the dimensionality of W . Notice that we have see also Eq. (34). Since we are interested in chemical potential problems we keep ζ fixed in this illustrations and consider various κ orders.

C.2 Case (2)
We consider 2 short, connected Polyakov loops (a 2 × 2 lattice with pbc in 1 direction and free b.c in the other). See Fig. 1, right plot and Table 1. Using again gauge transformations to a maximal gauge fixing we obtain Fig. 2, right plot and Table 2. We have ǫ = 1(−1) for pbc(apbc).  Basic and primary loops of length up to l = 6, with corresponding Dirac coefficients C i and multiplicities m i : 1 − 2 − −1 : The loop formula gives: where L i are primary loops, obtained by chaining together basic loops.
For calculations we can use the maximal gauge, setting U 2 = V 2 = S 2 = 1, see Table 2. Then L 1 = U, L 2 = V are the two straight Polyakov loops, XV U −1 and X the two plaquettes L 3 , L 4 and L 5 = V X, L 6 = U X −1 are basic decorated Polyakov loops and L 7 = X V U, L 8 = X −1 U 2 , L 9 = X V 2 , L 1 0 = X −1 U V, L 11 = L 12 = U V are primary decorated Polyakov loops obtained by chaining a basic straight Polyakov loop and a basic decorated Polyakov loop. Notice that we can attach further straight Polyakov loops and obtain primary loops of arbitrary length in order κ 2 but we stopped at length l = 6 for this illustration.
Since the lattice is even we take only even power of Q in evaluating Eq. (72). We shall only consider loops up to length 6. To order 0 in κ we find: 1 2n Q 2n = 1 n (4ζ ǫU ) n + 1 n 4ζ ǫU −1 n + 1 n (4ζ ǫV ) n + 1 n 4ζ ǫV −1 n Notice that these basic loops are of lengths 2 , U = U 1 U 2 , V = V 1 V 2 . The calculation was performed up to order 16 , which is 2n max and one can check that up to this order there is no deviation from the logarithm series. Assuming that the series continues as expected we find to this order A similar result follows from the decorated Polyakov loops which appear with a factor κ 2 (for simplicity we no longer write the nondominant inverse loops which have factorsζ ): Finaly the non-basic primary loops L 7 − L 12 give We obtain thus the HD determinant to order κ 2 including all loops up to length 6, in complete agreement with the exact determinant to this order