{'fraction_non_alphanumeric': 0.03222712449645118, 'fraction_numerical': 0.00943794360253213, 'mean_word_length': 4.49799620333263, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Properties of canonical determinants and a test of fugacity expansion for finite density lattice QCD with Wilson fermions

We analyze canonical determinants, i.e., grand canonical determinants projected to a fixed net quark number. The canonical determinants are the coefficients in a fugacity expansion of the grand canonical determinant and we evaluate them as the Fourier moments of the grand canonical determinant with respect to imaginary chemical potential, using a dimensional reduction technique. The analysis is done for two mass-degenerate flavors of Wilson fermions at several temperatures below and above the confinement/deconfinement crossover. We discuss various properties of the canonical determinants and analyse the convergence of the fugacity series for different temperatures.


I. INTRODUCTION
QCD with finite density is an important topic of both, experimental and theoretical studies. In particular one would like to understand the various transitions to different states of matter that are conjectured for the QCD phase diagram. Exploring these phase transitions is clearly a non-perturbative problem and non-perturbative methods must be applied for such studies. In principle the lattice QCD formulation provides a suitable nonperturbative framework, and as long as the chemical potential is zero this approach may be used to obtain reliable quantitative results.
However, if one works at finite density a severe technical problem emerges: The fermion determinant is complex at non-zero chemical potential µ and cannot be used as a probability weight in a Monte Carlo calculation. Various reweighting and expansion techniques have been explored in recent years and the reviews at the annual Lattice conference provide a glimpse at the corresponding developments [1]. An interesting option is to work not in the grand canonical approach with a chemical potential µ, but instead to use canonical partition sums with fixed net quark number q. Several studies of this alternative perspective can be found in the literature [2] - [10]. So far the focus was mainly either on reweighting techniques or full canonical simulations at a fixed quark number.
Here we explore the perspectives of a fugacity expansion of the grand canonical ensemble. The expansion coefficients are fermion determinants projected to a fixed net quark number, so-called canonical determinants. The expansion is a Laurent series in the fugacity parameter e µβ (β is the inverse temperature) which has properties different from the more conventional Taylor expansion in µβ. However, the possible application to lattice QCD at finite density is the same for both expansions: One computes the expansion coefficients in simulations at µ = 0 and uses the series to explore QCD at µ > 0.
In this article we discuss the evaluation and the properties of canonical determinants. They are computed as Fourier moments of the fermion determinant at imaginary chemical potential µβ = iϕ, which we evaluate efficiently using a recently proposed dimensional reduction formula [8]. We study the distribution properties of the canonical determinants and analyze the convergence of the fugacity expansion for various temperatures in the confined and deconfined phases. This analysis is done for a fixed numerical effort, i.e., a fixed number of 256 sampling points of ϕ in the interval (−π, π] used for the evaluation of the Fourier transformation to obtain the canonical determinants.

II. CANONICAL DETERMINANTS AND THEIR EVALUATION
The starting point for our study of canonical determinants and the fugacity expansion is Wilson's lattice Dirac operator D(µ) with chemical potential µ on lattices of size N 3 s × N t . The lattice spacing will be denoted by a, such that β = aN t . For two mass-degenerate flavors the corresponding grand canonical partition function is given by While for real chemical potential µ the grand canonical determinant is complex, for imaginary chemical potential the determinant det[D(µβ = iϕ)] is real. One may explore the generalized γ 5 -hermiticity relation for Wilson fermions, γ 5 D(µ)γ 5 = D(−µ) † , to establish the relation D (−q) = (D (q) ) * between canonical determinants with positive and negative quark numbers q. For vanishing quark number q = 0 the canonical determinant is real, i.e., D (0) ∈ R.
In principle Eq. (3) is a very elegant expression for the canonical determinants D (q) . However, the Fourier integral has to be evaluated numerically and the evaluation of the grand canonical determinant det[D(µβ = iϕ)] for the necessary ϕ-values is computationally rather costly.
To alleviate the problem we use a domain decomposition technique [8] where the grand canonical fermion determinant may be rewritten exactly in a dimensionally reduced form, which holds both for real and imaginary chemical potential. Here A 0 is a real factor which depends only on the background gauge field configuration but is independent of the chemical potential µ. H 0 = H † 0 , and H are matrices that are built from propagators on sub-domains of the lattice and live on only a single time slice (see [8] for details). Thus the determinant in (4) is dimensionally reduced, i.e., the determinant is taken over a matrix with 12N 3 s rows and columns, where the factor 12 comes from the color and Dirac indices. The terms H 0 and H can be completely stored in memory and are then used many times for the evaluation of det[D(µβ = iϕ)] at the necessary values of ϕ. Due to the reduced size of the matrix where the determinant has to be evaluated, we gain a factor of O(N 3 t ), which here, at N t = 4, corresponds to a speedup by a factor of 64. The Fourier integral (3) is evaluated numerically with standard techniques [11]   at this point that also alternative dimensional reduction formulas with different properties were proposed [2,[12][13][14].
Our numerical tests are done on dynamical configurations with two flavors of Wilson fermions at µ = 0 generated with the MILC code [15]. In this exploratory study of the canonical determinants and the fugacity expansion we are here limited to a single lattice volume, 8 3 × 4. The parameters (inverse gauge coupling β gauge and hopping parameter κ) and the values for the corresponding lattice spacing a and the temperature T are taken from [5] and listed in Table I. Due to the small lattice the pion mass has to be kept rather large -it is close to 950 MeV for all our ensembles. For each set of parameters we use a statistics of 200 configurations. All errors we show are statistical errors determined with the jackknife method. In addition to the parameters of the simulation, in Table I we also list the cutoff parameter q cut used in the fugacity expansion which we discuss in more detail in Section IV.
In the range of temperatures we consider, the lowest three values of T are in the confined phase, the crossover into the deconfined phase is at T ∼ 165 MeV for our simulation parameters, and the two largest temperatures are in the deconfined phase.
As a first result we now have a look at the average log 10 det[D(µβ = iϕ)] of the logarithm of the grand canonical determinant as a function of the imaginary chemical potential parameter ϕ. This is interesting because the Fourier moments with respect to ϕ are the canonical determinants we want to study and the ϕdependence of det[D(µβ = iϕ)] already provides a first insight into their properties.
.. is the average over gauge configurations generated for two mass degenerate flavors of Wilson fermions at µ = 0 as detailed above. We study the ϕ-dependence of log 10 det[D(µβ = iϕ)] using all 256 values of ϕ in the interval (−π, π]. The integrands det[D(µβ = iϕ)] are computed with the dimensional reduction formula (4) as outlined. We show the corresponding results for all our temperature values in Fig. 1.
The figure clearly demonstrates that in the confined phase the grand canonical determinant shows only a weak dependence on ϕ, while above the crossover a pronounced variation with ϕ develops. This behavior has a simple and well known interpretation: An imaginary chemical potential µβ = iϕ is equivalent to an additional temporal boundary condition for the fermion fields, ψ( x, N t +1) = −e iϕ ψ( x, 1), where the minus sign is the usual anti-periodic temporal boundary condition for fermions. While in the confined phase the correlations are short ranged and do not give rise to a dependence on the boundary angle ϕ, in the deconfined phase correlations over distances larger than the temporal extent of the lattice generate the non-trivial response to changing ϕ. The canonical determinants D (q) are the Fourier moments of the ϕ-dependence and the first analysis in the higher modes, i.e., D (q) with larger values of |q| will play a more prominent role.

III. PROPERTIES OF THE CANONICAL DETERMINANTS
Having discussed the canonical determinants D (q) and the setting of our calculation we now come to analyzing properties of the D (q) . In Fig. 2 we show scatter plots in the complex plane for D (q) / det[D(µ = 0)], i.e., the canonical determinants normalized by the grand canonical determinant at µ = 0. This normalization is such that when one sums over q all data points D (q) / det[D(µ = 0)] in the complex plane for a single configuration the result is 1. In the plot we show results for q = 0, 1, 2, 3 and compare T = 144 MeV (top row of plots in Fig. 2), T = 164 MeV (middle) and T = 211 MeV (bottom). Note that different axis scales are used.
As already discussed, for q = 0 the canonical determinants D (q=0) must be real, a fact that is obvious in the figures. The corresponding values scatter on the real axis for all temperatures. For q > 0 the points spread out also in the imaginary direction. In the confined phase (T = 144 MeV) the points for the higher Fourier modes D (q) then quickly approach the origin of the complex plane when increasing q. The situation is different in the deconfined phase (T = 211 MeV), where we observe that for increasing q the values of D (q) do not move toward the origin as quickly as for T = 144 MeV. For T = 164 MeV (middle row of plots in Fig. 2) one finds an intermediate behavior. The relative size of the canonical determinants D (q) at the different temperatures reflects the observation we have already made in the discussion of We conclude the discussion of Fig. 2 with noting, that the scatter plots of D (q) for the smaller values of q show a curious pattern, in particular the q = 1 data for T = 144 and 164 MeV: They arrange in two oblong structures that seem to have a preferred angle relative to the real axis. A similar pattern at temperatures near the crossover was observed also in canonical determinants for staggered fermions [16]. At the moment we do not understand the origin of this pattern, but it might be related to the properties of the D (q) under center transformations which rotate them according to D (q) → z q mod 3 D (q) , where z ∈ {1, e i2π/3 , e −i2π/3 } is the center element multiplied to all temporal gauge links in a fixed time slice. For pure gauge theory the spontaneous breaking of center symmetry is intimately related to the deconfinement transition, and it could be that the pattern we observe is a remnant of this mechanism. This speculation is further supported by the observation, that the phase of the Polyakov loop (the order parameter for confinement in pure gauge theory) is strongly correlated with the phases of the D (q) [16]. Also in an analysis of the canonical determinants of the quenched case [9,10] it was demonstrated that the D (q) in the broken phase very cleanly map the center orientation of the underlying center sector.
The change of the relative weight of the D (q) with temperature is manifest also in Fig. 3, where we study the size distribution |D (q) |/D (0) of the canonical determinants normalized to the trivial q = 0 sector as a function of q and again compare results for T = 144 MeV (top), T = 164 MeV (middle) and T = 211 MeV (bottom). The plots show that the distribution roughly follows a Gaussian centered around the dominant q = 0 sector. While in the confined phase the distribution is rather narrow, in the deconfined phase it widens and reflects the fact that at higher temperatures also sectors with larger quark numbers become accessible. When one turns on a chemical potential µ > 0 the canonical determinants are multiplied with powers of the fugacity factor, i.e., D (q) → e µβq D (q) , which has the effect of shifting the center of the distribution towards larger quark numbers.
Finally, in Fig. 4 we have a look at the phase θ (q) of the canonical determinants D (q) . More precisely we look at e i2θ (q) = D (q) /D (−q) (remember that D (−q) = D (q) ). In Fig. 4 we show D (q) /D (−q) as a function of q for different values of the temperature.
Below the crossover temperature the results for D (q) /D (−q) drop to 0 rather quickly with increasing q, i.e., the phases of the D (q) for q above roughly q = 5 fluctuate strongly and average to a very small number. For the largest temperatures the decrease of D (q) /D (−q) with q is slower and the phases have a sizable expectation value up to q ∼ 25 (we stress that these are statements specific to the parameters of our calculation, in particular the volume used). The decrease of the D (q) /D (−q) with increasing q also sheds light on how the fermion sign problem may be viewed in the fugacity expansion: When increasing µ the powers e µβq of the fugacity factors put a larger weight on the canonical determinants D (q) with higher quark numbers, which in turn are the ones that are suppressed due to the fluctuations of their phase. The fact that for higher temperatures the sign problem is milder is clearly visible from the slower decrease of D (q) /D (−q) for larger temperatures.

IV. CONVERGENCE PROPERTIES OF THE FUGACITY SERIES
After the study of the canonical determinants D (q) , we now can start to analyze the fugacity series (2). In a numerical study it is necessary to truncate the series and we denote the truncated fugacity expansion by where q cut denotes the highest (anti-) quark sector we take into account in the truncated series. The central question we want to study in this section is how well the truncated series S(µ) qcut approximates the full grand canonical determinant det[D(µ)] and how the quality of the approximation depends on the parameters µ and T . This is a question which we address both for individual configurations and for the gauge average. We stress at this point, that for a finite volume and non-zero temperature, the representation of det[D(µ)] by S(µ) qcut would be exact with q cut = 6N 3 s if all coefficients D (q) were known with arbitrary precision. Thus as long as the lattice is finite such that no singularities can emerge, an insufficient representation of the grand canonical determinant is only due to limited accuracy in the numerical evaluation of the Fourier integrals (3).
We assess the quality of the approximation by considering the relative error For the study of ∆ we also evaluated the grand canonical determinant det[D(µ)] for a few values of real chemical potential, again applying the dimensional reduction formula (4). These values are then used for the assessment of the fugacity expansion using ∆. Inspecting the fugacity series (2) or (5) one observes that for increasing q there is competition of two terms: The powers of the fugacity, i.e., the factors e µβq , are terms that grow exponentially with q and their rate of growth is determined by the product µβ. The exponential growth with q is compensated by the quick decrease of the canonical determinants D (q) . This decrease is roughly Gaussian as we demonstrated in Fig. 3 and thus dominates the increasing function e µβq . However, this mechanism can be spoiled by numerical instabilities: If the higher coefficients D (q) are not known with perfect accuracy (as will always be the case when they are evaluated via numerical Fourier transformation), their fluctuations will be amplified by the factors e µβq . This effect is manifest in the behavior of S(µ) qcut when q cut is varied: First S(µ) qcut quickly saturates as a function of q cut , but for too large values, e.g., already q cut ∼ 30 for some of our parameter values, the series S(µ) qcut starts to diverge. Fortunately the D (q) decrease very rapidly: Fig. 3 shows that also for our highest temperature ensemble all D (q) with |q| > 20 essentially vanish, and the series S(µ) qcut can be truncated at small q cut . The optimal values for q cut were determined using the relative error ∆ and we list them in Table I. In Fig. 5 we show how the relative error ∆ of the fugacity expansion depends on the parameters µβ and T . For all 200 configurations in each of our ensembles we compute the canonical determinants D (q) , as well as the canonical determinant det[D(µ)] for several real values of µ. For these values of µ we can evaluate ∆. Fig. 5 shows ∆ as a function of the dimensionless combination µβ, and we compare the results for all available temperatures. For each ensemble we show ∆ for the configuration where ∆ is largest ("worst"), for the configuration where ∆ is smallest ("best"), for the median of the 200 configurations, as well as the average of ∆ over all configurations.
From the fugacity expansion has better convergence properties: At T = 144 MeV all four categories (best, worst, median and average) have a relative error ∆ smaller than 1 % for values of the chemical potential up to µβ = 0.8 and the average error over all configurations remains below 1% even up to µβ = 1.1. For T = 211 MeV an error smaller than 1 % can be maintained only up to µβ = 0.4. The reason is that here the distribution of the D (q) is much wider (compare Fig. 3), and the D (q) for larger values of |q| contribute significantly in the fugacity sum already at not too large µβ. Since these higher Fourier modes are only known with less relative accuracy, the fugacity expansion at the highest temperatures breaks down already at smaller chemical potentials. We stress that our convergence analysis is specific for the setting we use here, i.e., lattice size 8 3 ×4, parameters as listed in Table I and 256 values of ϕ in the evaluation of the Fourier integrals (3). In particular increasing the number of sampling points for ϕ in the Fourier integral will allow for a higher precision of the D (q) which leads to better convergence properties of the fugacity sum, such that higher values of µβ can be reached.
An interesting question is how the number of D (q) that need to be evaluated depends on the volume of the box. This number is roughly proportional to the width of the distribution of the D (q) as displayed in Fig. 3. In the next section we will show that this width is related to the quark number susceptibility χ q . This is an extensive quantity, i.e., it grows with the spatial volume V = (aN s ) 3 . We thus conclude that the number of D (q) that contribute significantly to the fugacity sum is proportional to the spatial volume V . In turn this means that also q cut needs to be increased linearly in V .

V. QUARK NUMBER DENSITY AND QUARK NUMBER SUSCEPTIBILITY
Let us finally discuss an exploratory calculation of the quark number density and the quark number susceptibility based on canonical determinants and the fugacity expansion. Combining the general expression for the grand canonical partition sum Z(µ) from Eq. (1) with the fugacity expansion (2) we find Both observables are intensive quantities after the normalization with the 3-volume V and are made dimensionless using suitable powers of T . For the necessary first and second derivatives one finds where we introduced the moments M n , n = 0, 1, 2 of the fugacity series as Finally we express the normalization factor 1/Z(µ) with the moment M 0 and obtain (use In the final result (11) both observables are expressed in terms of vacuum expectation values of moments of the fugacity series. These vacuum expectation values .. are computed with two flavors of Wilson fermions at µ = 0. In Fig. 6 we show our results (symbols) for n q /T 3 as function of the dimensionless combination µβ for three different temperatures. We compare the results to the outcome of the Taylor expansion presented in [17,18]. More precisely we used the terms up to fourth order from [18] and interpolated the Taylor coefficients to the temperatures used in our study. However, this comparison should be viewed with caution: The calculation in [18] is done on considerably larger lattices, has a statistics that is by a factor 20 larger than the statistics available in our exploratory study, and also is based on the staggered formulation. Nevertheless we find reasonable agreement for our results at these temperatures and range of µβ.
We also attempt a comparison of our results for the quark number susceptibility χ q at µ = 0 to the results from the Taylor expansion. The corresponding numbers are listed in Table II (again interpolated to our values of T ). We find that up to temperatures near the crossover the agreement of the two approaches is reasonable. Given the fact that the calculation [18] and our exploratory study differ considerable in statistics and volume, this agreement is satisfactory. Above the crossover we see, however, a rather strong discrepancy. We suspect two main reasons for this discrepancy: 1) The rather small volumes that are available for our calculation, and 2) the fact that for the larger values of T higher Fourier modes contribute significantly (the factor q 2 in the second moment M 2 of Eq. (10) enhances them further) which in the current setting of 256 sampling points for the numerical evaluation of the Fourier integral are not sufficiently accurate.

VI. SUMMARY AND DISCUSSION
In this exploratory study we analyzed the canonical determinants of 2-flavor lattice QCD with Wilson fermions. The canonical determinants are the coefficients in the fugacity expansion of the fermion determinant and may be calculated as the Fourier modes with respect to imaginary chemical potential. We speed up this evaluation by using a dimensional reduction formula for the grand canonical determinant.  We illustrate that a sizable dependence of the fermion determinant on the imaginary chemical potential sets in only at temperatures near the crossover temperature, which already shows that at the crossover an enhancement of the higher quark numbers can be expected. Studying the size distribution of the canonical determinants we find that their distribution is roughly Gaussian in the net quark number q with a width that starts to increase at the crossover. The average phases of the canonical determinants drop quickly with q, but the drop is slowed above the crossover temperature. This analysis sheds light on the complex phase problem from the point of view of the fugacity series.
We continue with a systematical analysis of the convergence properties of the fugacity expansion by comparing the truncated fugacity series to an exact evaluation of the grand canonical determinant at several values of the chemical potential. It is shown that for lower temperatures we can obtain very good accuracy up to µβ ∼ 0.8, while above the crossover we only reach µβ ∼ 0.4. We stress that these results are specific for the numerical effort we invest here, in particular 256 sampling points for the evaluation of the Fourier moments. On a finite lattice the fugacity expansion is a finite Laurent series, and in principle it is possible to compute all coefficients such that the series representation of the grand canonical determinant becomes exact.
Finally we use the fugacity series to explore the evaluation of the quark number density and the quark number susceptibility through first and second moments of the truncated fugacity series. This is an interesting alternative to a standard calculation where the quark number q is a binomial in the quark fields, q ∝ d 4 xψ(x)γ 4 ψ(x) which after integrating out the fermions is related to the trace of the quark propagator, q ∝ Tr γ 4 D −1 . In this form q is difficult to evaluate numerically using, e.g., stochastic estimators. Once the canonical determinants D (q) are available the expressions in terms of the moments of the fugacity expansion are considerably simpler and we demonstrate that the approach is compatible with the conventional method up to temperatures where we have sufficiently control over the fugacity series.
Conceptually the fugacity expansion falls in the same category as the Taylor expansion: A series is used to extrapolate the information from a Monte Carlo simulation at µ = 0 to finite values of the chemical potential. Thus both, Taylor-and fugacity expansion face the same overlap problem, i.e., the µ = 0 configurations used for evaluating the expansion coefficients may have rather little overlap with the configurations that dominate physics at finite µ. This limits both expansion approaches to small values of µ.
Concerning other properties, Taylor-and fugacity expansion are different: At finite volume the fugacity series is a finite Laurent series in the fugacity parameter e µβ , while the Taylor series is an infinite series even at finite volume. Thus the two series will have different properties and one should explore which of the two approaches provides a better expansion around µ = 0. To study this question we currently compare the two series in an effective theory for QCD, where a flux representation free of the complex phase problem allows one to obtain high precision reference data from a Monte Carlo simulation [19]. These will be used to study the quality of the approximation from Taylor-and fugacity series.
We conclude with stressing that this exploratory work should be considered only as a first step towards a full development of fugacity expansion as a reliable tool in finite density lattice QCD. Certainly further improvement of the numerical techniques and more tests will be necessary to assess the full potential of the approach.