Controlling the sign problem in finite-density quantum field theory

The European Physical Journal C, Jul 2017

Quantum field theories at finite matter densities generically possess a partition function that is exponentially suppressed with the volume compared to that of the phase quenched analog. The smallness arises from an almost uniform distribution for the phase of the fermion determinant. Large cancellations upon integration is the origin of a poor signal to noise ratio. We study three alternatives for this integration: the Gaussian approximation, the “telegraphic” approximation, and a novel expansion in terms of theory-dependent moments and universal coefficients. We have tested the methods for QCD at finite densities of heavy quarks. We find that for two of the approximations the results are extremely close—if not identical—to the full answer in the strong sign-problem regime.

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Controlling the sign problem in finite-density quantum field theory

Eur. Phys. J. C Controlling the sign problem in finite-density quantum field theory Nicolas Garron 0 Kurt Langfeld 0 0 Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool , Liverpool L69 3BX , UK Quantum field theories at finite matter densities generically possess a partition function that is exponentially suppressed with the volume compared to that of the phase quenched analog. The smallness arises from an almost uniform distribution for the phase of the fermion determinant. Large cancellations upon integration is the origin of a poor signal to noise ratio. We study three alternatives for this integration: the Gaussian approximation, the “telegraphic” approximation, and a novel expansion in terms of theory-dependent moments and universal coefficients. We have tested the methods for QCD at finite densities of heavy quarks. We find that for two of the approximations the results are extremely close-if not identical-to the full answer in the strong sign-problem regime. 1 Introduction The sign problem is known to be one of the most important challenges of modern physics. In theoretical particle physics, it prevents us from simulating finite-density QCD with standard Monte-Carlo methods. Hence most of the QCD phase diagram cannot be explored by first-principle techniques, such as lattice QCD. Many reviews can be found; see for example [ 1–9 ]. Dropping the phase factor of the quark determinant exp{i φ} from the functional integral results in a theory, say with partition function Z P Q , that is accessible by standard importance sampling Monte-Carlo simulations. Very early on, it became clear that Z P Q and the partition function of the full theory Z are only comparable for the smallest values of the chemical potential μ [ 10 ]. The deviation is quantified by the so-called phase factor expectation value, eiφ P Q = Z (μ)/Z P Q (μ) ∝ e− f V , (1) where f is the free energy difference between the full and the phase quenched theory and V is the volume (see e.g. [ 10 ]). The knowledge of this phase factor would give access to the partition function Z (μ) (we assume that Z P Q (μ) has been obtained by standard methods). In this work, we study its expectation value, eiφ P Q : it is a very small number, generically very hard to measure due to the statistical noise, which only decreases proportionally to the square root of the number of Monte-Carlo configurations. Our approach is based on the density-of-states method and in particular on the LLR formulation [ 11,12 ], which is ideally suited to calculate probability distributions of observables: it features an exponential error suppression [12] which can result in an unprecedented precision for the observable (see e.g. for an early example [ 13 ]). It is based upon a non-Markovian random walk, which immediately provides two main advantages: it bears the potential to overcome the critical slowing down for theories close to a first order phase transition [ 9,14 ], and it is not restricted to theories with a positive probabilistic weight for Monte-Carlo configurations. In fact, the method has been successfully applied to the Z3 theory at finite densities [15] and QCD at finite densities of heavy quarks [ 16 ]. In both cases, the probability density ρ(φ) of the phase φ has been obtained to very high precision. The phase factor expectation value is then given by eiφ P Q = dφ ρ(φ) exp{i φ} dφ ρ(φ) (2) Despite the high quality numerical result for ρ(φ), the challenge remains to extract a very small signal from the above Fourier transform. An approach, put forward in [ 15,16 ], is to first represent the numerical data for ln ρ(φ) by a fit function and then to calculate the Fourier transform of the fit function (semi-)analytically. The method produces reliable results if all the numerical data are well represented by the fit function with a small number of fit parameters [ 15,16 ]. With the advent of high precision data for ρ(φ), the main obstacle for gaining access to quantum field theories at finite densities is the above Fourier transform. The method used in [ 15,16 ] hinges on the fact that a fit function which faithfully represents the data could be found. This might not be generically the case. In this paper, we propose three alternatives to this direct method. In Sect. 3 we present the first approach, called Gaussian approximation. No fitting procedure is required; instead the phase factor is computed directly from the data. Within this framework, the integral in the numerator of (2) is known analytically. The second approximation, presented in Sect. 4 is what we call the “telegraphic” approximation. This approach can be implemented either on the fit function or directly on the data (although it might require new simulations). The integral is replaced by a simple difference. In Sect. 5, we introduce a third method, the “advanced moment expansion”, which can be seen as a variant of a cumulant expansion [ 17–20 ]. It is a systematic expansion in the deviation from the uniform distribution and as such is expected to work better in the strong sign-problem regime. We will provide evidence that the universal coefficients decrease exponentially with increasing order, providing a rapid convergence if the moments are bounded. Although the convergence is faster in the strong sign-problem regime, for the phase factor expectation value we find an excellent agreement already at the third order of the expansion, regardless of the strength of the sign problem. In this case we still rely on a fitting procedure for the density of states. However, the direct computation of the Fourier transform (2) is not needed, only the elementary moments are required. Before going through the details of these methods, we present the framework and the numerical details of our simulations in the next section. Our conclusions are presented in Sect. 6. 2 Generalities and framework 2.1 Full theory and phase quenching We consider a generic theory with a partition function Z = DUμ exp{β SYM[U ]} DetM [U ], and with a complex “matter” determinant: DetM [U ] = |DetM [U ]| exp{i φ[U ]}, φ ∈] − π, π ]. (4) With the help of the density of states ρ(s) = DUμ exp{β SYM[U ]}|DetM [U ]|δ(s − φ[U ]), (6) (7) (8) (9) (10) (11) (12) the partition function can then be recovered by a 1dimensional Fourier transform: Z = ds ρ(s) exp{i s} = ds ρ(s) cos(s). We also introduce the so-called phase quenched counter part by The expectation values of an observable A in the full and in the phase quenched theory are given as usual by DUμ A exp{β SYM[U ]}DetM [U ], DUμ A exp{β SYM[U ]}|DetM [U ]|, (3) (5) = ds ρ(s). 1 A = Z 1 A P Q = Z P Q A = Aeiφ P Q eiφ P Q implying the well-known relations Z = Z P Q eiφ P Q . In terms of the density, the phase factor expectation value is given by (2). 2.2 Extensive density of states For theories for which the imaginary part arises from a local action an extensive phase x ∈] − ∞, ∞[ can be defined as the sum of the local phases. This has been e.g. the case for the finite density Z3 and for heavy dense QCD [ 15,16 ]. The expectation is that the extensive phase scales with the volume. For fermionic theories with the phase φ[U ] arising from the (non-local) quark determinant, the definition of an extensive phase is not obvious. If the quark operator still admits (complex) eigenvalues, a straightforward definition would involve a sum of the phases of the eigenvalues. Another possibility was as pointed out in [21]: x [U ] = Im ln (Det M ) μ/T ∂(ln Det M ) = 0 Im ∂μ/ T = 0 μ/T Im tr M −1 ∂ M ∂μ/ T μ=μ¯ μ=μ¯ d d μ ¯ T μ ¯ . T The definition of an extensive phase factor has proven to be important to achieve the precision needed for the Fourier transform. If ρE (x ) denotes the corresponding probability distribution, the phase factor expectation value in (2) is obtained by −a1 x 2 − a2 xV4 − a3 Vx 62 + · · · . If we define a “scaling” variable by x = s/√V , the deviation from a Gaussian distribution decreases with increasing volume: 1 π eiφ P Q = Z P Q n∈Z −π 2.3 Volume dependence of the density We here consider the class of theories for which the phase of the Gibbs factor is proportional to the chemical potential μ and for which this is the only μ dependence. Scalar theories do not fall into this class since the real part of the action also acquires a μ dependence, but fermion theories in the ab initio continuum formulation might fall into this class. For these theories, let us study the dependence of ρE (s) on the physical volume V . We make explicit the μ dependence of the phase factor expectation value and point out that the partition function is positive for all μ: z(μ) = ei μ φ P Q ≥ 0. Note that we have z(0) = 1 and that from z(μ) ∈ R it follows that e−i μ φ P Q = ei μ φ P Q ⇒ z(−μ) = z(μ). Note that since z(μ) is obtained by a Fourier transform of ρ, see (2), the density of states can be recovered from z(μ) by the inverse Fourier transform (up to a normalisation constant Z P Q ≥ 0) ρE (s) = Z P Q dμ z(μ) e−isμ. 2π As argued in [ 21 ], z(μ) can be viewed as a partition function with free energy density f (μ) (a necessary condition is that z(μ) ≥ 0), leaving us with the volume dependence: z(μ) = exp{− f (μ) V }, = exp{−[c1μ2 + c2μ4 + c3μ6 + · · · ] V }, 1 eiφ P Q = Z P Q +∞ dx ρE (x ) cos(x ). −∞ The density of states ρ(s) can easily be recovered from the extended density ρE (x ). To see this, we subtract from x a multiple of 2π until s ∈ [−π, π [, s = x − 2π n, n ∈ Z, and we split the integration domain in intervals of size 2π , where the coefficients ck are volume independent. Inserting (20) into (18), we find, with an expansion in inverse powers of V , ρE (s) = const. exp s2 s4 s6 −a1 V − a2 V 3 − a3 V 5 + · · · , (13) (14) (15) (16) (17) (18) 2.4 Numerical details We use the data obtained in our previous work [ 16 ] but have also generated new simulations for reasons that we explain below. We summarise here the parameters used for the numerical simulations and the methods to obtain the density of states. The interested reader will find more details in the aforementioned reference. The lattice parameters are 84 lattice, β = 5.8, κ = 0.12. and we let the chemical potential μ vary between 1.0421 and 1.4321. We identified the “strong sign-problem region” as being 1.1 < μ < 1.4. We for each value of μ, we split the domain of the phase s ∈ [0, smax] in nint small interval of size δs and on each interval k, we compute the LLR coefficients ak . In practise we choose smax ∼ 36, δs = 0.896 and nint = 40, except for a few values of the chemical potential, for which we need a better resolution. The corresponding values are reported in Table 1. We reconstruct the probability density function for discrete values of the phase sk = kδs + δs /2, namely ρE (sk ) = exp − ai δs − ak δs /2 . k−1 i=1 (22) (23) nnint In [ 16 ] we performed a polynomial fit of ln(ρE ) and computed (13) by a semi-analytic integration (we refer to this method as “Exact”). Although the fits are of very good quality and very stable, for three values of the chemical potential, we have also ran new simulations with δs = π/5. As shown below, these new data allow us to compute ρ(s) directly from the data (without relying on any fitting procedure) and will be very useful to check the methods presented here. We have implemented this technique for three different values of the chemical potential. This is illustrated in Figs. 1, 2 and 3, where we see that the different methods give compatible results. Finally, we mention that we use around 1000 configurations and that the statistical errors are estimated with the bootstrap method, using 500 samples. Naturally we have checked that the errors are stable with respect to the number of samples. ! In fact, numerical results suggest that the probability distribution is Gaussian to a good extent [ 17,22–24 ], which would imply that only the cumulant φ2 c is non-vanishing. It has been argued in [21] that higher cumulants are suppressed by factors of the volume V and that, however, higher order cumulants are important for the medium and high range of chemical potentials. Throughout this paper, we define the Gaussian approximation as the approximation of the extended density of states by a normal distribution: φ4 c − · · · . (24) ρE (s) ≈ const. exp{− s2}. The phase factor expectation value (2) is then analytically obtained: (25) (26) from the standard expectation eiφ P Q = exp 1 − 4 where the subscript E indicates that the expectation values are defined with respect to the extended density ρE . We test this approach for heavy-dense QCD with partition function (7). We find the expectation value in (27) directly from the data: we take the density obtained through (23) and compute the expectation value s2 E using a trapezoidal approximation. We obtain in this way an estimate for the phase factor expectation value (26) without invoking any fitting procedure. Our numerical findings are summarised in Fig. 4. We find that the Gaussian approximation provides a surprisingly good approximation over the whole range of chemical potentials μ. Even in the strong sign-problem regime at intermediate values μ, the cancellations are well emulated and the approximate result only underestimates the true result by roughly a factor 2. 4 The “telegraphic” approximation 4.1 Methodology As can be seen in Fig. 3, ρ weakly depends on its arguments in the strong sign-problem regime and for large volumes. In this case, a Poisson re-summation of (15) should yield a rapidly converging series: dn e2πi ν n ρE (s + 2π n) ∞ dx ei ν x ρE (x ). (29) (30) (31) (32) If the sum over ν is rapidly converging, we find approximately ρ(s)/c0 ≈ 1 + 2 eiφ P Q cos(s). In the strong sign-problem regime, the amplitude of the cosine is very small, and therefore we see that ρ(s) is almost a constant. Equation (33) then offers the possibility to extract the phase factor expectation value, i.e., 1 eiφ P Q ≈ 4c0 [ρ(0) − ρ(π )]. Using (15), we therefore find eiφ P Q ≈ π2 k∈∞Z(−d1x)kρρEE(x(k) π ) . (35) −∞ We call this the telegraphic approximation. It emerges by neglecting higher contributions cν of the Poisson sum. In order to get a feeling for the resulting systematic error, we adopt, for now only, the Gaussian approximation (25) and find: c2 c1 ≈ exp 1 − 4 3 This implies that the correction to ρ(s) in (33) is of order: where we have used (26) and (32). At least in the strong signproblem regime, for which eiφ P Q is very small, we expect the telegraphic approximation to work very well. We finally point out that the telegraphic approximation can be improved in a systematic way. The order of the approximation is defined by the number of harmonics entering the density of states. E.g., in third order we have (28) ρ(s)/c0 ≈ 1 + 2 eiφ P Q cos(s) + c cos(2s) + d cos(3s), (33) (34) (36) (37) with the unknowns eiφ P Q and c, d. We generate three equations by evaluating ρ(s) at s = 0, π/3, π and solve the linear set of equations for the unknowns. We are predominantly interested in the phase factor: 1 1 1 eiφ P Q ≈ − 2 + 4 ρ(0)/c0 + 3 ρ(π/3)/c0 1 − 12 ρ(π )/c0, which can easily be converted to a discrete sum over discrete set of points of ρE (s) using (15). 4.2 Numerical implementation Again, we use Heavy-Dense QCD to test this approximation. Having in hands the density of state—either ρE obtained from the fit or ρ from the date through (15)—it is straightforward to implement numerically (35). If we take the results from the fit, we find that this approximation provide results extremely close to the “exact” ones: except for a few values of μ in the weak sign-problem regime, the results (central value and variance) are actually indistinguishable. For example, for μ = 1.0821, we find ln eiφ ePxQact = −1.992175 ± 2.910279 × 10−3, ln eiφ aPpQprox = −1.992174 ± 2.910306 × 10−3. (38) (39) We show our results for the various μ in Table 2 and Fig. 5. It is essential that the numerical data are accurate enough to produce significant results upon the cancellations in (35). Clearly, the LLR approach in combination with the fit of the density of states succeeds. In the following, we will test the “naive” implementation for which we compute ρ(s) directly from the data (without relying on any fitting procedure). For this purpose, we use our new simulation data with δs = π/5. In this case we have ρ(s) for s = π/10, 3π/10, . . ., but do not have ρ(0) nor ρ(π ). Therefore we use a variant of (34): Although in the strong sign-problem regime μ = 1.2921, we could not extract a signal directly from the data (without a representation of the density of states by a fit), for the two other values of μ we find reasonable agreement. This shows that the telegraphic approximation cannot master the cancellations without further technical advances. We stress, however, that the telegraphic approximation greatly assists the LLR approach by reducing the Fouier integration of the density of states to an alternating sum (see (35)). 5 The advanced moments approach 5.1 General formulation The starting point is the expansion of the density of states: The coefficients d j depend on the underlying theory, and N0 ≥ 2 will define the order of the expansion. Our conjecture is that the coefficients d j are suppressed by powers of the volume with increasing j . For QCD, this conjecture is supported by the strong coupling expansion and the hadron resonance gas model [ 21 ]. There is also some numerical evidence by the WHOT-QCD collaboration [ 22–24 ]. Last but not least, this conjecture becomes true for the limited class of theories considered in Sect. 2.2. Using (47) in (14), we can express the phase factor expectation in terms of the theorydependent coefficients d j : 1 eiφ P Q = Z P Q j=1 d j I2 j , where d0 has dropped out upon integration, and where I2 j = ds s2 j cos(s) = (−1) j−l+1 2(2 j )! π 2l−1. (2l − 1))! The values I2k can be efficiently calculated by the recursion I2k = −2(2k)π 2k−1 − (2k)(2k − 1)I2k−2, with the initial condition I0 = 0. Our strategy to access the coefficients d j in an actual numerical simulation is to calculate combinations as the simple moments s2n . Using the truncation (47) for a given N0, we find s2n+2 with 2π 2i+2 j+1 Ai j = 2i + 2 j + 1 . = 1 N0−1 Z P Q j=0 Anj d j Keeping in mind that we have s2n+2 available from a numerical simulation, the idea is to choose a set of n-values and to consider (51) as a linear set of equations to obtain the unknowns d j . Note that for n = −1, s2n+2 = 1 = 0 follows from the symmetry ρ(−s) = ρ(s) and does not contain theory specific information. We hence choose n = 0, . . . , N0 − 1 and obtain ( A−1) jn s2n+2 . Inserting this into (48), we obtain d j Z P Q = We now have at our fingertips the moment expansion of the phase factor for a given order N0. We have not yet achieved γN +1 n = k(N ) n−1 − α2k γkn / k(NN ) M2N +2 N 1 = k(N ) N n=0 = s2N +2 N 1 + n=1 k(NN ) kn(N−)1 − kn(N ) s2n+2 − α2k γkn s2n k n=1 α2k γkn s2n , where we have changed the order of the double sum. We therefore find the recursion a systematic expansion, featuring increments of decreasing size (when we increase the order N0). To this aim, we define the first advanced moment M4 for N0 = 2 by We then define recursively for N = 2, . . . , (N0 − 1): N0−1 n=1 α2N +2 = k(NN ), and finally we achieve the systematic expansion We stress that the coefficients α2n+2 are universal, i.e., the only dependence on the theory under investigations enters via the moments Mk . Last but not least, we would like to have an explicit representation of the advanced moments M in terms of the simple expectations values sn . We define M2k = γki s2i . k i=1 By construction of the advance moments, we have the normalisation γkk = 1. Although for high order N 1 the intermediate coefficients γNi can become very large (we will show this below), the field theories of interest, i.e., finite-density quantum field theory in the strong sign-problem regime, should give advanced moments within bounds. In this case, the convergence is then left to the coefficients αn. Inserting (62) into (59), we find after renaming of indices (48) (49) (50) (51) (52) (53) (54) (55) N N k=2 k=max(n,2) N k=max(n,2) γN +1 N +1 = 1 , where 1 ≤ n ≤ N and 2 ≤ N ≤ N0 − 1. The recursion can be solved in closed form for i ∈ {2, . . . , N0} and j ∈ {1, . . . , N0}: γii = 1, γ21 = γi j = k(1) k0(1) , γi j = 0 for j > i, 1 k(ji−−11) − k(ji−−12) , i > j and i > 2. k(i−1) i−1 5.2 The first advanced moments For illustration purposes, we will explicitly calculate the first few advanced moments. The main task is to obtain the coefficients ki(N ), which emerge from the solution of a linear set of equations; see (55). For the leading order N0 = 2, we find ⎛ π 3 ( Ai j ) = 2 ⎜ 3 ⎜⎝ π 5 5 π 5 ⎞ 5 π 7 ⎠⎟⎟ , 7 (I2 j ) = 0 −4π . k(1) 1 175 = α4 = − 2π 6 , k0(1)/ k1(1) Hence, the first advanced moment, see (56), is given by M4 = s4 − 53 π 2 s2 . At next to leading order, i.e., N0 = 3, we have ⎛ π 3 ⎜ 3 ( Ai j ) = 2 ⎜⎜⎜ π 5 ⎜⎝⎜⎜ π57 7 π 5 5 π 7 7 π 9 9 π 7 ⎞ 7 ⎟ 11 ⎛ π 9 ⎟⎟⎟ , (I2 j ) = ⎝ 9 ⎟⎟ π 11 ⎠⎟ The solution of the corresponding linear system is given by k(2) 0 k(2) 1 k(2) 2 = − 9845 2π 2π−6 33 , 315 16π 2 − 231 = 4 π 8 , 4851 2π 2 − 27 = − 8 π 10 = α6. From (66), we then find for the coefficients γ γ31 = k(2) 0 − α4γ21 k(2) 2 = 251 π 4, (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) We have computed the moment coefficients up to order N0 = 5. We find for the coefficient matrix (k ≥ 2, i ≥ 1): (76) (77) 1 0 ⎟⎟⎟⎟⎟⎟⎟ 0 ⎟⎟⎟⎟ 1 ⎠ We finally perform a consistency check. For a truncation of the density of states at order N0, all the moments up to M2N0 contribute to the phase factor expectation value at this order, see (61). If we consider (47) as exact for the moment in the sense that all simple moments s2n are calculated with this density, then the phase factor expectation value is obtained exactly by summing all contributions including the term containing M2N0 . Since this result is already exact, all moments M2k with k > N0 must vanish. For example, assume that the density is given by ρ(s) = d0 + d1s2, (N0 = 2), then e.g. M6 (and all higher moments need to vanish for all choices for d0 and d1. This devises a consistency check. We find for the present example s6 10 n 20 30 s4 s2 = 15 Inserting these simple moments into M6, (76), we find that all terms cancel and that M6 indeed vanishes for all choices of d0 and d1. If we consider, in a quantum field theory setting, the expansion (47) as an expansion with respect to some inverse power of the volume, the moments M2n are then suppressed by these powers. 5.3 Convergence For high orders N0, the coefficients γ in the definition (62) of the advanced moments M2k can become very large. In this section, we will assume that for functions ρ(s) arising in a quantum field theory setting the moments remain within bounds. This occurs due to cancellations between simple moments s2i , as we will show below. In this case, the expansion (61) of the phase factor expectation value in terms of the advanced moments is dictated by behaviour of the coefficients α2k for large k. These coefficients are universal: they do not depend on the underlying theory, i.e., ρ(s). They arise from the solution of the linear system (55), which reads in a shorthand notation k = A−1 I, and it is this linear system that we are going to study in greater detail. Since the matrix A in (52) is symmetric and positive, we perform a Cholesky decomposition and solve for k: A = L L T , L y = I, L T k = y, where L is a lower triangular matrix. Note that if the system L y = b is solved at order N0 and if subsequently the order N0 is increased, the first N0 components of the solution y are unaffected by the increase due to the triangular form of L. The same is true for the matrix L: increasing the order from N0 to N0 + 1 does not affect the first N0 rows and columns. We are interested in the N0 dependence of the last component of k: k(NN00) = yN0 /L N0,N0 . The Cholesky decomposition gives Lii = Aii − 1 Li j = Li j ( Ai j − i−1 k=1 j−1 Li2k , k=1 Lik L jk ), i > j (78) (79) (80) (81) We have solved this iteration analytically for values N0 up to 30. We find that for large n the data is well described. We find In essence, the advanced moments approach from Sect. 5 is an efficient numerical method to evaluate the Fourier transFig. 6 The exponential rate K , see (82), as a function of the order n of the expansion that very quickly Lnn reaches an asymptotic regime which is well described by Ln+1,n+1 = K π 2 Lnn, 1 K = 4 , see Fig. 6. Asymptotically, we therefore find the exponential increase: L N0,N0 ∝ Unfortunately, we could not prove any of these asymptotic behaviours analytically, but we have verified (85) by also solving the linear system L T k = y for k. Our analytical result for N0 = 2 to N0 = 32 is shown in Fig. 8. We find the remarkable result that the expansion coefficients α2N0 are exponentially decreasing with N0 suggesting a rapid convergence of the advanced moment expansion as long as the moments M2n are bounded. 5.4 Application to HDQCD (82) (83) (85) 1.004 of the LLR method, they are extracted with a very good statistical precision. We also observe that going from s2 to s8 , the relative error increases very slowly. We turn now to the advanced moments: since all the elementary moments are positive, the relative signs in (88)–(94) imply that important cancellations occur. At leading order (LO), we have Fig. 8 The behaviour of the coefficient α2N0 = kNN00 of the expansion in terms of advanced moments form (14) for sufficiently smooth integrands ρ(s). In this section, we test the method in the quantum field theory context of QCD at finite densities of heavy quarks (HDQCD). Our preliminary results have been reported in [ 25 ]. Here we are interested in the strong sign-problem region (in which μ ∼ 1.3): in Fig. 3, we show that the density is almost constant whereas for μ ∼ 1 and μ ∼ 1.4, the density has variation of order 1 (see Figs. 1, 2). Hence, we expect that the advanced moment expansion will have a better convergence in the strong sign-problem regime. From now on, we focus on the severe sign problem region, μ = 1.2921. Once the density is known, we can compute the elementary moments (again using our fit results and semianalytic integration). They are reported in Table 3. By virtue (86) (87) (88) (89) (90) (91) (92) (93) (94) As expected, strong cancellations between the simple moments occur making it mandatory to determine the simple moments with high precision. The analysis has been carried out using the bootstrap resampling method, and we point out that strong correlations are at work to obtain the advanced moments at the level of precision reported here. The numerical values are also reported in Table 4. One should note that the overall signs of the advanced moments oscillate; however, αi Mi is a positive quantity, as can be seen in (61) or in the numerical values. The phase factor expectation value (61) is then given by eiφ When the order of the expansion increases, the statistical error decreases and that the results converges quickly to the “exact” answer eiφ = 2.189(324) × 10−6, obtained by fitting the extensive density ρE and by carrying out the Fourier transform using the fit, as in [ 16 ]. (In the latter we quote 2.37(21) × 10−6, the small difference in the central value comes from the fact that we use a different δs .) We observe a rapid convergence here. Since the phase factor is a small number, it is useful to look at the logarithm of this quantity. We find log eiφ P Q = −13.032 ± 0.152 (Full). The advanced moment method yields log eiφ P Q = −13.445 ± 0.152 + O(α6 M6) = −13.065 ± 0.152 + O(α8 M8) = −13.033 ± 0.152 + O(α10 M10) (LO), (NLO), (NNLO). It is remarkable that not only the central value but also the variance is very well approximated by our expansions. Indeed for this value of μ, the full (relative) variance is already given by the first order. Of course the quality of the approximation depends on the variation of ρ (and therefore on the strength of the sign problem). We now vary the value of μ in the range 1 < μ < 1.4 and compare the results of the phase factor expectation value obtained in [ 16 ] with the method proposed here. It is interesting to note that even in the weak sign-problem region, in which the density ρ fluctuates between 0 and 1, the NLO and NNLO approximations already yield reasonable approximations. This is illustrated in Figs. 9 and 10. Our numerical results can be found in Table 5. We quote the “full answer” as (95) (96) (97) (98) (99) (100) (101) (102) obtained in [ 16 ] and the relative difference with the method presented here, for the first three orders. (Here we implement the advanced moments method with the same δs as in [ 16 ].) The NLO approximation works at the percent level over the full available range, even in the weak sign-problem region. 6 Conclusions There are two main possibilities in addressing finite-density quantum field theory: (i) facing the large cancellations that give rise to the smallness of the partition function or (ii) to reformulate to an equivalent theory say by dualisation [ 6 ] or by a complexification of the fields [ 5 ]. Method (ii) would be preferred if the approach exists and if exactness can be 1.042 1.062 1.082 1.102 1.122 1.142 1.162 1.182 1.202 1.232 1.252 1.272 1.292 1.312 1.332 1.352 1.372 1.392 ln exp(i φ) guaranteed. The appeal of method (i) is that it is universally applicable if a way is found to control the cancellations. A first success for direction (i) emerged with the advent of Wang–Landau type techniques and, most notably, the LLR method [ 11 ]: due to the feature of exponential error suppression of the LLR approach [ 12 ], high precision data for the density of states ρ (s) of finding a particular phase s over many orders of magnitude has become available. The partition function now emerges as Fourier transform of ρ (s). Due to large cancellations, this Fourier transform is a challenge in its own right. The recent success reported in [ 15 ] and in [ 16 ] hinge on the ability to find a fit function for ln ρ (s) that well represents hundreds of numerical data points with relatively few fit parameters. This situation is unsatisfactory since the quest for this fit function might not be always successful. The present paper explores three methods to perform the Fourier transform: • The Gaussian approximation of the extensive density of states ρE is most easily implemented, but hard to improve in a systematic way. For the example of HDQCD, we found this approximation yields the right order of magnitude throughout and only misses the exact phase factor by a factor of 2 when the sign problem is strongest. • The telegraphic approximation yields the phase factor through an alternating (discrete) sum of the extensive density of states ρE . The relative systematic error is of the order of the phase factor itself, which makes the approximation excellent in the strong sign-problem regime. • The advanced moment approach is a systematic expansion of this Fourier transform with respect to the deviations of ρ (s) from uniformity. The expansion therefore works best in the strong sign-problem regime. The expansion is independent of the quantum field theory setting and can be applied to the Fourier transform of any sufficiently smooth function ρ (s), s ∈ [−π, π ]. At the heart of expansion are the so-called advanced moments. We have thoroughly derived these moments and the theoryindependent expansion coefficients α. We found evidence that the expansion coefficients decrease exponentially with increasing order, thus guaranteeing rapid convergence if ρ (s) admits moments M that are bounded. We have tested and validated the advanced moment expansion in the context of HDQCD: we have confirmed that the expansion converges very quickly. It works best in the strong sign-problem region as expected, although at third order the results agree with the “full” answer at the sub-percent level even in the weak sign-problem regime. Acknowledgements We are grateful to B. Lucini and A. Rago for helpful discussions. 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Nicolas Garron, Kurt Langfeld. Controlling the sign problem in finite-density quantum field theory, The European Physical Journal C, 2017, 470, DOI: 10.1140/epjc/s10052-017-5039-7