A Numerical Study of the 2-Flavour Schwinger Model with Dynamical Overlap Hypercube Fermions
Wolfgang Bietenholz, Ivan Hip, Stanislav Shcheredin, Jan Volkholz
aa r X i v : . [ h e p - l a t ] A p r BI-TP 2011/13
A Numerical Study of the 2-FlavourSchwinger Model with DynamicalOverlap Hypercube Fermions
Wolfgang Bietenholz a , Ivan Hip b ,Stanislav Shcheredin c and Jan Volkholz da Instituto de Ciencias NuclearesUniversidad Nacional Aut´onoma de M´exicoA.P. 70-543, C.P. 04510 Distrito Federal, Mexico b Faculty of Geotechnical Engineering, University of ZagrebHallerova aleja 7, 42000 Varaˇzdin, Croatia c Fakult¨at f¨ur Physik, Universit¨at BielefeldD-33615 Bielefeld, Germany d Potsdam Institute for Climate Impact ResearchTelegrafenberg A62, D-14412 Potsdam, Germany
We present numerical results for the 2-flavour Schwinger model with dy-namical chiral lattice fermions. We insert an approximately chiral hy-percube Dirac operator into the overlap formula to construct the overlaphypercube operator. This is an exact solution to the Ginsparg-Wilson re-lation, with an excellent level of locality and scaling. Due to its similaritywith the hypercubic kernel, a low polynomial in this kernel provides a nu-merically efficient Hybrid Monte Carlo force. We measure the microscopicDirac spectrum and discuss the corresponding scale-invariant parameter,which takes a surprising form. This is an interesting case, since RandomMatrix Theory is unexplored for this setting, where the chiral condensateΣ vanishes in the chiral limit. We also measure Σ and the “pion” mass,in distinct topological sectors. In this context we discuss and probe thetopological summation of observables by various methods, as well as theevaluation of the topological susceptibility. The feasibility of this summa-tion is essential for the prospects of dynamical overlap fermions in QCD.1 ontents
The Schwinger model describes QED in two dimensions, i.e. U (1) gauge field [1]. In a Euclideanplane, the Lagrangian reads L ( ¯ ψ, ψ, A µ ) = ¯ ψ ( x ) h γ µ (i ∂ µ + gA µ ( x )) + m i ψ ( x ) + 12 F µν ( x ) F µν ( x ) . (1.1)It is a popular toy model for QCD; in particular it shares the property ofconfinement [1, 2]. On the other hand there are fundamental differences,such as the super-renormalisability of the Schwinger model, and a non-running gauge coupling g .A further qualitative difference — which is of particular interest in thiswork — is the spontaneous chiral symmetry breaking in QCD with masslessquarks. In d = 2 this effect can be mimicked to some extent for instanceby the Gross-Neveu model, where a discrete chiral symmetry breaks spon-taneously. However, in the Schwinger model with fermion mass m = 0 the2hiral symmetry is continuous, and therefore it cannot undergo sponta-neous symmetry breaking (SSB) due to the Mermin-Wagner Theorem [3].Nevertheless the chiral condensate Σ = −h ¯ ψψ i , which acts as the orderparameter for chiral symmetry breaking, takes a non-vanishing value in the1-flavour case, because of the explicit symmetry breaking due to the axialanomaly [1]. This leads to Σ( m = 0) = ( e γ / π / ) g ≃ . g (where γ isEuler’s constant).For N f ≥
2, however, the massless limit has an unbroken chiral symme-try. For N f degenerate fermion flavours of mass m the chiral condensatebehaves as [4]Σ( m ) ∝ (cid:16) m N f − β (cid:17) / ( N f +1) ⇒ δ = N f + 1 N f − , (1.2)where β = 1 /g , and δ is the critical exponent.In our study we consider N f = 2. Here there are analytical evaluationsof the proportionality constant for the case of light fermions, m ≪ g , basedon bosonisation and low energy assumptions,Σ( m ) = const . (cid:16) mβ (cid:17) / , const . = (cid:26) . . . . Ref . [5]0 . . . . Ref . [6] . (1.3)Under the same assumptions the mass of the iso-triplet (“pion”) [6] and ofthe iso-singlet (“ η particle”) [7] are predicted as M π = 2 . . . . ( m g ) / , M η = r M π + 2 g π . (1.4)The relation M π ∝ m / replaces the Gell-Mann–Oakes–Renner relation( M π ∝ √ m ) of QCD [8].For models with a finite condensate Σ( m = 0), its value can be deter-mined from the Banks-Casher plateau of the Dirac eigenvalue density atzero [9] (or near zero in a finite volume). Moreover, chiral Random MatrixTheory (RMT) has elaborated subtle techniques to predict a wiggle struc-ture on this plateau, which allows for a refined determination of Σ fromthe densities of the low-lying Dirac eigenvalues [10, 11]. This method hasbeen tested successfully in the ǫ -regime of QCD with quenched [12–14] and with dynamical [16, 17] quarks, and also in the 1-flavour Schwingermodel [18,19]. The latter studies were based on configurations, which were Strictly speaking Σ diverges logarithmically for increasing volume in the quenchedapproximation [15]. Still quenched QCD in boxes of length > ∼ . However, the established RMT techniques are not applicable in thechiral limit of the 2-flavour Schwinger model; RMT for situations withΣ( m = 0) = 0 awaits to be worked out. Nevertheless we confirm the usualRMT prediction for the unfolded level spacing distribution in a unitaryensemble. On the other hand, the microscopic spectrum does not exhibita Banks-Casher plateau. Instead we observe to high precision the scale-invariance of the product λV / , where λ is a low-lying Dirac eigenvaluein the volume V . This result remains to be understood from the RMTperspective, since it cannot be explained simply with the critical expo-nent δ = 3 given in eq. (1.2). We also discuss the densities of the Diraceigenvalues in the bulk and their scaling behaviour.Next we confront the measurement of Σ( m ), based on the full Diracspectrum, with eq. (1.3). This requires an (approximate) summation ofthe values in all topological sectors, guided by the measurements in a fewsectors. We probe several methods for this purpose and apply them toΣ and to the “pion” mass given in eq. (1.4). These approaches also in-volve a determination of the topological susceptibility. The requirement ofa topological summation is generic for simulations with dynamical chiralfermions, because the Monte Carlo histories tend to perform only very fewtopological transitions. Also other lattice fermion formulations, such as theWilson fermion and variants thereof, will run into the same problem whenthey represent light fermions on very fine lattices [22] (say a < ∼ .
05 fm inQCD). Therefore the applicability of these techniques is relevant, particu-larly in view of QCD simulations with light quarks close to the continuumlimit.Section 2 describes our lattice formulation of the Schwinger model anddiscusses some of its properties, in particular locality and scaling. Section3 presents our version of the Hybrid Monte Carlo algorithm which we usedin this study. We discuss its properties regarding conceptual conditions,and practical aspects of its performance. Section 4 deals with the Diracspectrum, the construction of a scale-invariant variable and the link toRMT. Section 5 discusses the summation of Σ and M π over the topologicalsectors, along with the evaluation of the topological susceptibility. Sec-tion 6 is devoted to our conclusions. Progress reports of this project haveappeared in several proceeding contributions [23]. That method was pioneered in the Schwinger model in Ref. [20]. It worked success-fully in some cases, but it runs into trouble when the fermion determinant fluctuatesstrongly, as it happens for very light fermions. Hence the study presented here requiresthe simulation of truly dynamical fermions, as it was first attempted in Ref. [21]. Lattice formulation with overlap Hyper-cube Fermions
We consider the lattice formulation of the Schwinger model with compactlink variables U x,µ ∈ U (1), and with the plaquette gauge action. Remark-ably, for the pure gauge theory this is indeed a perfect lattice action [24].For the fermions we employ an overlap hypercube fermion (overlap-HF)Dirac operator, which is an exact solution to the Ginsparg-Wilson Rela-tion (GWR).The GWR is a criterion for a lattice modified, exact chiral symme-try [25], which first emerged from the study of perfect actions for lat-tice fermions [24, 26–28]. Independently, chiral lattice fermions were con-structed in the Domain Wall Fermion formulation [29], which separates thezero modes of opposite chirality in an extra “dimension”. Integrating outthis extra direction leads to the overlap formula [30], which provides yetanother way to represent a chiral vector theory on the lattice [31]. Thelattice Dirac operator for Domain Wall Fermions (in the limit of an infinitewall separation) and for overlap fermions turned out to be solutions to theGWR as well [31].Its importance as a general chirality criterion was first pointed out inRefs. [28], which showed in addition that classically perfect fermion actionsobey this criterion as well. Since those formulations involve couplings overan infinite range (in d > D HF ( x, x + r ) = ρ µ ( r ) γ µ + λ ( r ) , (2.1) i.e. a vector term plus a scalar term, as in the case of the Wilson fermion,but with an extended structure ( x and x + r are lattice sites).In d = 2 these terms include only couplings to nearest neighbour latticesites and across the plaquette diagonals. We are using here the versiondenoted as CO-HF (Chirally Optimised Hypercube Fermion) in Ref. [34],which is optimal for our algorithm to be described in Section 3. For con-venience we display in Table 1 the couplings in the notation of eq. (2.1).We gauge D HF by multiplying the compact link variables along theshortest lattice paths connecting x and y = x + r ; for the diagonal the twoshortest paths are averaged [34]. Thus we arrive at the operator D HF ,xy ( U ), An extra direction is introduced, which appears as a dimension for the free fermion,but which does not carry gauge fields. ρ ( r ) λ ( r )(0 ,
0) 0 1 . ,
0) 0 . − . ,
1) 0 . − . The coupling constants of the Chirally Optimised HypercubeFermion (CO-HF) [34]. Note that ρ ( r ) is even in r and odd in r (andvice versa for ρ ( r ) ), while λ ( r ) is even in both components of r . which characterises the interacting Hypercube Fermion (HF). -1-0.5 0 0.5 1 0 0.5 1 1.5 2 2.5Re λ Im λ D ovHF (m=0.01)D ovHF (m=0)D HF -1-0.5 0 0.5 1 0 0.5 1 1.5 2 2.5Re λ Im λ D ovHF (m=0.24)D ovHF (m=0)D HF Figure 1:
The spectra of D HF and of D ovHF (with and without mass) inthe complex plane, for a typical configuration generated at β = 5 on a × lattice with m = 0 . (on the left) and at m = 0 . (on the right).The similarity of D HF to D (0)ovHF and to D ovHF ( m ) shows that the HF isapproximately chiral, and useful for an efficient computation of the HybridMonte Carlo force (see Section 3). Since the operator D HF is “ γ -Hermitian”, D † HF = γ D HF γ , the exactchirality (which got lost in the truncation) can be restored by inserting D HF into the overlap formula [31]. This yields the overlap-HF operator [33, 34] D ovHF ( m ) = (cid:16) − m (cid:17) D (0)ovHF + m ,D (0)ovHF = 1 + γ H HF p H , H HF = γ ( D HF − . (2.2) H HF is Hermitian and D (0)ovHF fulfils the GWR in its simplest form, { D (0)ovHF , γ } = D (0)ovHF γ D (0)ovHF . (2.3) The constant 1 in the formulation of H HF is fine since we deal with smooth gaugeconfigurations. For stronger gauge couplings one would prefer to increase this constant.
6n practice we evaluate this overlap operator by means of rational Zolotarevpolynomials, as suggested in Ref. [35], after projecting out the lowest twomodes of D † HF D HF , which are treated separately.Compared to H. Neuberger’s standard overlap operator D N [31], wereplace the Wilson kernel D W by D HF [33]. Since the latter is an approxi-mate solution to the GWR already, the transition D HF → D ovHF is only amodest chiral correction, D ovHF ≈ D HF , (2.4)in contrast to the drastic transition D W → D N . This property is illustratedin Figure 1, which compares the spectra of D HF and D ovHF for typicalgauge configurations generated at m = 0 .
01 and at m = 0 .
24, both at β = 5 on a 16 ×
16 lattice. The similarity of D HF to D (0)ovHF is useful forthe computation of the overlap operator and for its favourable propertiesin addition to chirality (see below), while the similarity to D ovHF ( m ) ishelpful for our algorithm to be discussed in Section 3.Due to its perfect action background, D HF is also endowed with a goodscaling behaviour and approximate rotation symmetry, which are inheritedby D ovHF thanks to relation (2.4). That relation further provides a highlevel of locality for D (0)ovHF , since it deviates only little from the ultralo-cal operator D HF . All these properties have been tested and confirmedextensively in the quenched re-weighted study of Ref. [34].Let us reconsider here the level of locality, which is a key criterion inthe comparison of different chiral lattice fermion formulations. We test itin the usual way [36], by applying D (0)ovHF on a unit source η and measuringthe decay of the function f (r) = max x n D (0)ovHF ,xy ( U ) η y (cid:12)(cid:12)(cid:12) X µ =1 | x µ − y µ | = r o , η y = δ y (cid:18) (cid:19) . (2.5)We first consider the free fermion and demonstrate that this decay is clearlyfaster for the overlap-HF operator D ovHF than for the Neuberger operator D N , see Figure 2 on top. The plot below shows that the decay is still expo-nential for the configurations that we generated with dynamical fermionsat β = 5, which confirms the locality of our Dirac operator. This assuresthat our lattice fermion formulation is conceptually on safe grounds. Inthe range that we studied, the mass has practically no influence on thisdecay rate. We observe that D ovHF has a higher degree of locality than D N , since D ovHF at β = 5 is still clearly more local than even the free D N :the decay for the free D ovHF , f (r) ∝ exp( − . − .
45 r) by the gauge interaction, whereas D N only decays as exp( − r)even in the absence of gauge fields. 7his virtue also holds for the overlap-HF in quenched QCD [14, 37]: at β = 6 the exponent is increased by almost a factor of 2 compared to D N ,and the locality of overlap-HF is manifest down to β = 5 .
6. This enablesthe formulation of chiral fermions on coarser lattices than the use of D N ,which is of importance in view of QCD simulations at finite temperature.In that case, simulations are performed with a very small number N t oflattice sites in the Euclidean time direction. Its extension is extremelyexpensive; the computational effort grows ∝ N t [38]. The application ofthe D HF — and in future also of D ovHF — is therefore most promising infinite temperature QCD [39].The scaling behaviour was found to be excellent for both, the HF andthe overlap-HF, by considering dispersion relations in the free case and inthe 2-flavour Schwinger model with quenched re-weighted configurations,which were generated at β = 6 [34]. The HF and the overlap-HF havean even better scaling behaviour than the (truncated) classically perfectaction, which was constructed and tested for the Schwinger model in Ref.[20] (although that concept was actually designed for asymptotically freetheories [40]). Another quenched re-weighted scaling test was added inRef. [41].Throughout this work we fix β = 5 and study the effects of varyingthe lattice size and the fermion mass. So we do not investigate explicitlythe continuum extrapolation, since the scaling artifacts due to the finitelattice spacing turned out to be very small. This is illustrated by thedispersion relations of the “meson” masses shown in Figure 3: they followthe continuum behaviour up to quite large momenta, much further than theWilson fermion or the Neuberger fermion [34]. Moreover scaling artifactsare expected to be negligible also based on the large plaquette values near0 .
9, see Table 4. Hence the configurations are smooth, which correspondsto a fine lattice.On the other hand, the issue of finite size effects is relevant here, andwe will address it extensively in Sections 4 and 5. Figure 4 shows thecorrelation length ξ = 1 /M π as a function of the fermion mass, as expectedin infinite volume according to eq. (1.4). It reveals that significant finitesize effects may occur for our smallest fermion masses and volumes. Theseeffects can be very useful to investigate the distinction between topologicalsectors. In QCD they have been used to determine some of the Low EnergyConstants by means of simulations in — or close to — the ǫ -regime [12–14, 16, 17, 42–44] and the δ -regime [45]. In our study the finite size effectsare useful since they provide a suitable laboratory to probe methods ofsumming up observables measured separately in a few topological sectors.Section 5 presents pilot studies of such procedures, which might becomerelevant in lattice QCD. 8 f (r) r (taxi driver metrics)free overlap-HFfree Neuberger fermionexp(-1.5 r)exp(-r) 1e-10 1e-08 1e-06 1e-04 0.01 1 0 2 4 6 8 10 12 14 16 < f (r) > r (taxi driver metrics)m = 0.03m = 0.09m = 0.24exp(-1.45 r) Figure 2:
The locality of the overlap Dirac operators, tested by the decay ofthe function (2.5), against the taxi driver distance r = | r | + | r | . On topwe compare our overlap-HF operator D (0)ovHF to Neuberger’s standard overlapoperator D N (with a Wilson kernel) in the free case. At r = 3 ( r = 7 ), f (r) is already suppressed by more than one (two) order(s) of magnitude for D (0)ovHF . The plot below shows the exponential decay of h f (r) i based on ouroverlap-HF simulations at β = 5 . The gauge interaction reduces the decayrate just marginally, with hardly any dependence on the fermion mass m . " p i on " ene r g y p"pion" dispersion relationat m=0.01 L = 20, ν = 0 L = 20, | ν | = 1L = 32, | ν | = 1L = 32, | ν | = 2theory 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 " η pa r t i c l e " ene r g y p η dispersion relationat m=0.01 L = 20, ν = 0 L = 20, | ν | = 1L = 32, | ν | = 1L = 32, | ν | = 2theory Figure 3:
The dispersion relations for the “pion” and the “ η -particle”(isospin triplet and singlet), measured at fermion mass m = 0 . at L = 20 and L = 32 , in different topological sectors. Irrespective of the small dif-ferences, they follow in all cases very closely the theoretical curve in thecontinuum, given by eq. (1.4). Up to momentum p ≈ π/ lattice artifactsare tiny, which confirms an excellent scaling behaviour. This scaling qual-ity is similar to the overlap-HF in our previous quenched re-weighted studyat β = 6 , but in that case the Wilson fermion and the Neuberger fermionshow sizable scaling artifacts setting in at p ≈ . [34]. c o rr e l a t i on l eng t h m Figure 4:
The correlation length ξ = 1 /M π (in infinite volume) as a func-tion of the degenerate fermion mass m , according to eq. (1.4). For instancefor our lightest fermion mass, m = 0 . , it amounts to ξ ≃ . In order to simulate overlap-HFs dynamically, the standard Hybrid MonteCarlo (HMC) algorithm would use the fermionic force term¯ ψ Q − (cid:16) Q − ∂Q ovHF ∂A x,µ + ∂Q ovHF ∂A x,µ Q − (cid:17) Q − ψ , (3.1)where Q ovHF = γ D ovHF is the Hermitian overlap-HF operator, and A x,µ are the non-compact gauge link variables. However, this force term iscomputationally expensive. In particular in view of prospects for QCD weare going to explore a hopefully efficient alternative. In addition, the force(3.1) is conceptually problematic due to the discontinuous sign function H HF / p H in Q ovHF , see eq. (2.2). Under a continuous deformation ofthe configurations, the denominator vanishes at the transition betweentopological sectors. Here we refer to the proper definition of the topologicalcharge by means of the fermionic index [28] ν = n − − n + , (3.2) The original work on the HMC algorithm is Ref. [46]; pedagogical descriptions canbe found for instance in Refs. [47]. n + ( n − ) is the number of zero modes of D (0)ovHF with positive (neg-ative) chirality. Regarding the spectrum of H HF , a topological transitionmeans that an eigenvalue crosses 0, so its map by the overlap formula (2.2)flips between m and 2, i.e. it either appears as a zero mode of D (0)ovHF , orit is sent to the cutoff scale. Indeed the QCD simulations that have beenperformed with this HMC force hardly ever achieve topological transitions,see in particular Refs. [17, 48]; this issue is discussed in detail in Ref. [49].We repeat that the same problem occurs also with other lattice Dirac oper-ators for light dynamical fermions when the lattice becomes very fine [22]:the HMC history tends to get trapped in one topological sector.Here we report on HMC simulations, which are again facilitated by thepowerful property (2.4). Our algorithmic concept follows the HMC version,which was applied to improved staggered fermions of the HF-type in Ref.[50]. It used a sophisticated Dirac operator in the accept/reject step, buta simplified formulation for the force, which is quick to evaluate. In orderto simulate the dynamical overlap-HF we render the force term continuousand computationally cheap by inserting only approximate overlap operatorsin the term (3.1). To be explicit, we apply an overlap-HF to a low precision ε ′ in the external factors Q − , and we use H HF instead of Q ovHF in thederivatives (although this could be extended to a polynomial as well), ¯ ψ Q − ,ε ′ (cid:16) Q − ,ε ′ ∂H HF ∂A x,µ + ∂H HF ∂A x,µ Q − ,ε ′ (cid:17) Q − ,ε ′ ψ . (3.3)For the integration we applied the Sexton-Weingarten scheme [51] with apartial ( δτ ) correction (where δτ is the step size). The time scales for thefermionic vs. gauge force had the ratio of 1 : 5, but we did not observe muchsensitivity to this ratio.The Metropolis accept/reject step is performed with the high precisionoverlap operator D ovHF ,ε . Hence the deviations in the force are corrected,and the point to worry about is just the acceptance rate. The completesimplification, which reduces Q ovHF ,ε ′ to γ D HF , was considered in Ref. [52],which found a decreasing acceptance rate for increasing volume; that studywas based on the Scaling Optimised Hypercube Fermion (SO-HF) of Ref.[34]. However, in this respect it turns out to be highly profitable — andstill cheap — to correct the external factors to a modest precision. We We fix the sign of the index such that it matches the continuum gauge formulationof the topological charge, R d x ǫ F . However, in all considerations of this work only | ν | matters. The simplified force that we are using is not only computationally cheaper, but itis also expected to be helpful to achieve at least a few topological transitions. Hence itmight not even be favourable to extend H HF in eq. (3.3) to a polynomial, which wouldmove the force formulation closer to the dangerous sign function. number of configurations topological ν = 0 | ν | = 1 | ν | = 2 total transitions0.01 2428 307 2735 70.03 1070 508 1578 20.06 741 660 1401 70.09 919 587 1 1507 70.12 664 501 248 1413 80.18 791 563 50 1404 150.24 576 978 56 1637 17Table 2: Our statistics of configurations at L = 16 and seven fermionmasses m . The HMC trajectory lengths ℓ are given in Table 4. In allcases they consist of 20 integration steps ( δτ = ℓ/ ). The measurementsare separated by at least 200 trajectories. For m = 0 . this separationwas enlarged to 600 trajectories for a better decorrelation. As a genericproperty of dynamical overlap fermion simulations, topological transitionsare rare, as we see in particular for our lightest fermion masses. chose the algorithmic parameters for the (absolute) precisions as ε ′ = 0 .
005 (force term) , ε = 10 − (Metropolis step) , (3.4)which increases the acceptance rate by an order of magnitude compared tothe simple use of H HF throughout the force term. The force we obtain inthis way is not based on Hamiltonian dynamics, but the way it deviatesfrom it (by proceeding from γ D HF to Q ovHF ,ε ′ ) does maintain the propertyof area conservation in phase space. With this algorithm, we performed production runs at β = 5 on L × L lattices of the sizes L = 16 , , ,
28 and 32. On the 16 ×
16 lattice wecollected statistics at seven fermion masses in the range m = 0 . . . . .
24 indistinct topological sectors, as displayed in Table 2. At our lightest mass, m = 0 .
01, we performed additional simulations on larger lattices of size L = 20 . . .
32, plus runs at m = 0 . L = 32, see Table 3.Since the force term (3.3) tends to push the trajectory a bit off thehyper-surface of constant energy, we kept the trajectory length ℓ (betweenthe Metropolis steps) short. At L = 16, m ≤ .
18, we chose ℓ = 1 /
8, whichis divided into 20 integration steps ( δτ = 0 . ℓ was further reduced, see Table13 m number of configurations ν = 0 | ν | = 1 | ν | = 2 | ν | = 3 total20 0.01 435 304 73924 0.01 278 273 55128 0.01 240 180 42032 0.01 138 98 82 31832 0.06 91 293 384Table 3: Our statistics for the lattice sizes L = 20 . . . at fermion masses m = 0 . and . . The trajectory length ℓ was reduced for increasing L ,see Table 4, while the integration step was always fixed as δτ = ℓ/ . Afterthermalisation, the configurations are separated by at least 200 trajectories.We show our statistics in the topological sectors | ν | = 0 . . . . In particularat m = 0 . topological transitions were very rare, so we captured varioussectors by a multitude of “hot starts”. The configurations used for the measurements were separated by at least200 trajectories. Still the autocorrelation with respect to the observablesin Sections 4 and 5 is not always negligible, see Table 5. In particularsome problems show up for
L >
20 and higher topological charges, whichsuggests that an application of this algorithmic approach to QCD mightrequire further refinements. Here autocorrelations were taken into accountby a jackknife analysis of the measured data. That method was used forthe error calculations throughout this work; we probed a variety of binsizes and took the maximal error in each case.
Reversibility — to a good precision — is a requirement of the HMC algo-rithm. The possible danger for this crucial property could be an instabilityin the molecular dynamics trajectory due to directions with positive Lya-punov exponents, such that certain deviations from the exact trajectoryare amplified exponentially [53].To test if we are confronted with this problem, we moved forth and backwith a variable number of steps, and measured the (absolute) modificationof the gauge action, | ∆ S G | . As examples, Figure 5 (on top) shows ourresults for the reversibility precision at L = 16, δτ = 0 . At a few points in the HMC histories, where the algorithm run into convergenceproblems, we temporarily reduced ℓ below the trajectory length given in Table 4, alwaysmaintaining the dissection into δτ = ℓ/ m = 0 .
03, 0 .
12 and 0 .
24. The precision is satisfactory in all cases.It still improves significantly for increasing mass, as we also observe for δτ = 0 . m = 0 . m = 0 . m = 0 . δτ = 0 . | ∆ S G | m = 0 . m = 0 . δτ = 0 . | ∆ S G | Figure 5:
The reversibility precision with respect to the gauge action S G fora variable number of steps of length δτ = 0 . (on top) and δτ = 0 . (below), both on a × lattice at β = 5 . We show results for very differentmasses. There is no indication for any positive Lyapunov exponent. Theprecision improves as we increase m , but it is satisfactory in all cases. Being confident that our algorithm is safe, we now proceed to the ques-tion of its efficiency. The plots in Figure 6 show the acceptance rates inthe Metropolis step. In some sectors they were somewhat modest for theparameters chosen here, which is related to the aforementioned cases ofrather long autocorrelation times. We evaluated the acceptance rate by considering the acceptance probability in eachMetropolis step, regardless of the actual accept/reject decision. This is statisticallymore conclusive than just counting the acceptance ratio.
Conjugate Gradient iterations thatwas required per trajectory, specifically in the evaluation of D ovHF (upperplots) and in total (lower plots). Table 4 summarises the acceptance rates,as well as the number of Conjugate Gradient steps in the evaluation of theoverlap operator and in total. We add the plaquette value to characterisethe smoothness of the configurations; this serves as a point of orientationfor comparison with other models in lattice gauge theory. a cc ep t an c e r a t e mL = 16 0.2 0.3 0.4 0.5 0.6 15 20 25 30 35 a cc ep t an c e r a t e L m = 0.01
Figure 6:
The Metropolis acceptance rate on the L = 16 lattice at sevendifferent masses (on the left), and at m = 0 . on five lattice sizes (on theright). Note that the trajectory length varies, as specified in Table 4. As usual, the simulation becomes faster when we proceed from lightto moderate fermion mass. However, the number of Conjugate Gradientiterations rises only mildly as we approach the chiral limit, even down tovery light masses. The reason is that finite size effects prevent the non-zeroeigenvalues from becoming really tiny (this virtue is obviously reduced ifthe volume increases). The low-lying non-zero eigenvalues will be discussedin detail in the next section.
In this section we discuss our results for the eigenvalues of the Dirac opera-tor D (0)ovHF in eq. (2.2), after stereographic projection (a M¨obius transform)onto the half-line RI + , λ i → (cid:12)(cid:12)(cid:12)(cid:12) λ i − λ i / (cid:12)(cid:12)(cid:12)(cid:12) . (4.1) We first consider the full spectra and the resulting unfolded level spacingdistribution. This distribution is obtained as follows: one first numerates16 m ℓ acceptance number of CG iterations plaquetterate in D ovHF total value16 0.01 0.125 0.439(8) 97.2(1) 4529(4) 0.8971(2)16 0.03 0.125 0.295(15) 94.2(1) 4419(5) 0.8965(4)16 0.06 0.125 0.306(17) 93.9(1) 4405(4) 0.8974(4)16 0.09 0.125 0.355(9) 91.4(1) 4312(2) 0.8963(2)16 0.12 0.125 0.386(12) 89.6(1) 4244(3) 0.8961(3)16 0.18 0.125 0.450(9) 82.7(1) 3966(2) 0.8965(2)16 0.24 0.0625 0.706(13) 76.5(1) 3708(3) 0.8947(4)20 0.01 0.0625 0.504(13) 128.7(2) 5961(6) 0.8972(3)24 0.01 0.05 0.352(9) 149.9(6) 7129(27) 0.8969(5)28 0.01 0.04 0.242(13) 191.9(2) 9152(17) 0.8972(5)32 0.01 0.03 0.342(16) 224.9(5) 10432(18) 0.8968(5)32 0.06 0.03 0.619(10) 199.0(2) 9546(8) 0.8971(3)Table 4: The characteristic indicators for the performance of the precondi-tioned HMC algorithm: first we give the acceptance rate in the Metropolisstep at the end of each trajectory of length ℓ . The accept/reject step uses D ovHF to machine precision (see eq. (3.4)). We add the number of Con-jugate Gradient iterations needed to evaluate the operator D ovHF and intotal. Finally we quantify the smoothness of the configurations by the meanplaquette value. m | ν | plaquette Dirac eigenvalue λ chiral condensate Σ16 0.01 0 0 . . .
116 0.01 1 0 . . .
816 0.03 0 0 . . .
216 0.03 1 0 . . .
116 0.06 0 0 . . .
216 0.06 1 0 . . .
916 0.09 0 < . . .
216 0.09 1 < . . .
016 0.12 0 < . . .
316 0.12 1 0 . . .
916 0.12 2 0 . . .
916 0.18 0 < . . .
916 0.18 1 0 . . .
716 0.18 2 < . . .
616 0.24 0 < . . .
116 0.24 1 < . . .
716 0.24 2 < . . < .
520 0.01 0 < . . .
920 0.01 1 < . . .
124 0.01 2 < . . .
524 0.01 3 0 . . .
428 0.01 1 0 . . .
728 0.01 3 0 . . .
032 0.01 0 < . . .
232 0.01 1 < . . .
632 0.01 2 1 . . .
932 0.06 0 < . . .
832 0.06 1 < . . . The integrated autocorrelation times τ int = + P τ ≥ C ( τ ) (where C ( τ ) is the autocorrelation function) over a total trajectory length 25, forthe mean plaquette value, the leading non-zero Dirac eigenvalue λ (relevantin Section 4), and the chiral condensate Σ (relevant in Section 5).
75 80 85 90 95 100 0 0.05 0.1 0.15 0.2 0.25 C G i t e r a t i on s i n D o v H F / t r a j . m L = 16 80 100 120 140 160 180 200 220 240 15 20 25 30 35 C G i t e r a t i on s i n D o v H F / t r a j . Lm = 0.01 C G i t e r a t i on s / t r a j . m L = 16 4000 5000 6000 7000 8000 9000 10000 11000 15 20 25 30 35 C G i t e r a t i on s / t r a j . LL = 16 m = 0.01
Figure 7:
The number of Conjugate Gradient iterations per trajectory in D ovHF (upper plots), and including all operations (lower plots). We showresults at L = 16 and various masses (plots on the left), and at m = 0 . on various lattice sizes (plots on the right). the eigenvalues of single configurations in ascending order, λ i (here we omitthe eigenvalues with negative imaginary parts before the mapping (4.1)).Next we put all eigenvalues in a set of configurations together and numeratethem again. Now we consider pairs of eigenvalues λ i , λ i +1 (of the sameconfiguration), and count by how many ordinal numbers they differ in theoverall order. This difference — divided by the number of configurationsinvolved — is the unfolded level spacing. Random Matrix Theory (RMT) predicts three shapes for the statis-tical distribution of these level spacings. They refer to the three possiblepatterns of spontaneous chiral flavour symmetry breaking (for N f flavours), SU (2 N f ) → SO (2 N f ) orthogonal SU ( N f ) ⊗ SU ( N f ) → SU ( N f ) unitary SU (2 N f ) → Sp (2 N f ) symplectic . (4.2)At least in 4d Yang-Mills theory with chiral fermions, this set of patterns19 c u m u l a t i v e den s i t y unfolded level spacingorthogonalunitarysymplecticdata for L=16, m=0.01data for L=32, m=0.01 Figure 8:
The cumulative density of the unfolded level spacing distribution.We show the curves corresponding to the RMT prediction for the orthogo-nal, the unitary and the symplectic ensemble, and confront them with oursimulation data (for our lightest fermion mass, m = 0 . ). We clearlyobserve agreement with the RMT formula for the unitary ensemble. For L = 16 there is still a slight deviation for level spacings > ∼ . . As we in-crease the size to L = 32 , even this deviation practically disappears, i.e.the agreement becomes very precise. is complete [54]. They correspond to the real, complex and pseudo-realfermion representation of the gauge group. (In the real and pseudo-realcase, fermion and anti-fermion representations are equivalent, hence theunbroken symmetry is enlarged to SU (2 N f ).) Ref. [56] elaborated thecorresponding formulae for the unfolded level spacing distributions. Forlattice QCD (with chiral quarks) the prediction of the unitary ensemblehas been confirmed [12, 57], but the case of the 2-flavour Schwinger modelis theoretically less clear, because there is no spontaneous breaking of thechiral flavour symmetry.Our results (for m = 0 .
01, as an example) are shown in Figure 8. We An overview of the conceivable types of chiral symmetry breaking with an isomor-phic representation by non-unitary groups is given in Ref. [55]. m h λ , ν =0 i h λ , | ν | =1 i h λ , | ν | =2 i h λ , | ν | =3 i
16 0.01 0.1328(6) 0.175(2)16 0.03 0.130(2) 0.173(2)16 0.06 0.125(2) 0.173(1)16 0.09 0.115(2) 0.172(2)16 0.12 0.108(2) 0.166(2) 0.216(3)16 0.18 0.108(2) 0.166(2) 0.221(4)16 0.24 0.109(3) 0.163(2) 0.215(4)20 0.01 0.102(2) 0.127(2)24 0.01 0.125(4) 0.148(6)28 0.01 0.082(3) 0.120(5)32 0.01 0.057(3) 0.076(3) 0.084(3)32 0.06 0.030(3) 0.054(3)Table 6:
The lowest non-zero eigenvalue of D (0)ovHF (after the stereographicprojection (4.1)) for different masses and lattice sizes, in distinct topologicalsectors. see very clear agreement with the RMT formula for the unitary ensemble.The tiny deviation that we observe for L = 16 is a finite size effect, whichis manifestly suppressed as we enlarge the lattice size to L = 32. We now focus on the leading non-zero eigenvalue λ , based on the statisticspresented in Tables 2 and 3. The results for h λ i are given in Table 6. Wementioned before that chiral RMT has been worked out for the case ofa finite condensate Σ( m →
0) [10, 11], with successful applications in the ǫ -regime of QCD. This is not the situation we are dealing with; here Σ van-ishes in the chiral limit, as eq. (1.3) shows. The situation is qualitativelysimilar for fermions interacting through Yang-Mills gauge fields above thecritical temperature of the chiral phase transition. Also there the under-standing of the Dirac spectra is controversial; for numerical studies andconjectures we refer to Refs. [58, 59].In infinite volume, V → ∞ , the chiral condensate is given by the Diracspectrum as Σ = Z dλ ρ ( λ ) λ + m ( ρ : eigenvalue density) . (4.3)Along with the prediction quoted in Section 1, Σ ∝ m / , this suggests [19] ρ ( λ > ∼ ∝ λ / , (4.4)21n contrast to the Banks-Casher plateau [9] that one obtains in the standardsetting (with Σ( m → = 0). In that case, the density for the re-scaledsmall eigenvalues λ i Σ V is scale-invariant (at fixed m Σ V ) [60]. In our case,the generic relation h λ i i ∝ [ V ρ ( λ > ∼ − suggests that the parameter ζ i = λ i V / W ζ (for small λ i ; V = L ) (4.5)should adopt this rˆole, at fixed µ ζ = mV / W ζ — or simply at small m . W ζ is a constant of dimension [mass] / , which is (in this context) analogousto Σ in the standard setting. λ V , λ V , λ V , λ V ν = 0L=16L=20L=32 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 λ V , λ V , λ V , λ V | ν | = 1L=16L=20L=28L=32 Figure 9:
Cumulative densities of λ i V / ∝ ζ i , for i = 1 . . . , at mass m = 0 . and topological charge ν = 0 (on the left) resp. | ν | = 1 (on theright). We see that ζ i deviates from scale-invariance. Hence we probed the corresponding finite-size scaling, but it is not confirmed. This is illustrated in Figure 9 for our lightest fermion mass, m = 0 .
01, in the sectors of topological charge ν = 0 and | ν | = 1. As aquantitative measure, the integrated variance is given — and comparedto better approaches — in Table 7. Note, however, that the derivation ofrelation (4.4) may be invalidated by inserting an explicitly mass-dependentspectral density, ρ ( λ, m ), in eq. (4.3).Next we consider another scenario, with a reduced exponent of V inthe re-scaling factor. Now we assume the scale-invariant variable to be Z i = λ i V / W Z ( W Z of dimension [mass] / ). This scenario is motivatedby the fact that it belongs to a theoretically well explored universality class:it corresponds to ρ ( λ > ∼ ∝ λ / , which is the spectral density obtainedby RMT in the Gaussian approximation . There is a precise prediction forthe corresponding spectral density in terms of Airy functions [58], ρ Airy ( Z ) ∝ Z [Ai( − Z )] + [Ai ′ ( − Z )] ( ∼ √ Z/π at Z ≫ . (4.6)22 ρ Airy
L = 16L = 20L = 32
Figure 10:
Eigenvalue histograms for Z ∝ λ V / , at m = 0 . , ν = 0 compared to the spectral density ρ Airy in eq. (4.6), which RMT predictsin the Gaussian approximation. There is no convincing support for thisscenario.
Figure 10 compares the Airy function density (4.6) to the histogramsthat we obtained in various volumes at m = 0 .
01 and ν = 0. Our dataexhibit a far more marked wiggle structure, hence the agreement is notreally convincing.The finite size scaling quality of Z is also captured in Table 7. It issignificantly better than the one of ζ , but still not fully satisfactory.Let us finally proceed to the most successful approach, which was iden-tified empirically. It turns out that our data are in excellent agreementwith a scale-invariant parameter z i = λ i V / W z ( W z : constant of dimension [mass] / ) , (4.7)which implies a microscopic spectral density ρ ( λ ) ∝ λ / . The scale invari-ance of z . . . z is illustrated in Figure 11, again for our lightest fermionmass, m = 0 .
01, in the sectors of topological charge ν = 0 and | ν | = 1,which can be compared directly to Figures 9. A quantitative confrontationwith the previous two ans¨atze is included in Table 7.We also tested the behaviour if the re-scaled mass is kept approximatelyconstant, as an alternative to just keeping m small. In Figure 12 we com-pare h ζ i i for different lattice sizes, L = 16 and 32, again in the sectors | ν | = 0 and 1, for µ ζ ≈ const. We add the corresponding test with h z i i and µ z = mV / W z ≈ const., which reveals again a clearly superior finite sizescaling.Our data favour this last scenario unambiguously. A hint for a possibleexplanation can be found in Ref. [5], which introduced the dimensionless23ndex | ν | p / / / / / / int (n)int A measure for the agreement between the cumulative densities ofthe re-scaled eigenvalues s i = λ i V p , at m = 0 . and | ν | = 0 , . Weshow Var int = P i =1 R ds i [ P max k =1 ( ρ L k ( s i ) − ρ m ( s i )) ] / (max − , the inte-grated variance, where the sum over k includes L k = 16 , , at ν = 0 ( max = 3 ), and L k = 16 , , , at | ν | = 1 ( max = 4 ). ρ m ( s i ) is themean value over the volumes involved. The quantity Var (n)int is normalisedby dividing through the relevant interval in s , where . < ρ m ( s ) < . .We confirm that the power p = 5 / yields by far the least variance, i.e. thebest agreement. λ V , λ V , λ V , λ V ν = 0L=16L=20L=32 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 λ V , λ V , λ V , λ V | ν | = 1L=16L=20L=28L=32 Figure 11:
Cumulative densities of λ i V / ∝ z i , for i = 1 . . . , at mass m = 0 . and topological charge ν = 0 (on the left) resp. | ν | = 1 (on theright). In contrast to ζ i and Z i , the variable z i is scale-invariant to animpressive precision. parameter l := √ mL / / ( βπ ) / (4.8)to distinguish different regimes. The aforementioned behaviour Σ ∝ m / is expected for l ≫
1, whereas l ≪ ≪ L/ √ πβ implies Σ ∝ mL . For m = 0 .
01 we are in an intermediate regime, l = 0 . . . . . L/ √ πβ =8 . . . . . ∝ m / plausible.However, a precise explanation for this behaviour remains to be workedout. In particular in the framework of RMT — extended to this extraor-dinary setting — this might be feasible, but it is far from trivial [61].24 < λ > V / V µ /W ζ = 1.92 vs. 1.81, ν = 0| ν | = 1 3 4 5 6 7 200 400 600 800 1000 < λ > V / V µ /W z = 0.96 vs. 0.76, ν = 0| ν | = 1 Figure 12:
Finite size scaling for h ζ i at µ ζ ∝ mV / ≈ const. (left) and h z i at µ z ∝ mV / ≈ const. (right). These plots confirm again that z performs much better as a scale-invariant variable. At last we take a look at λ as one of the bulk eigenvalues, and we findan optimal finite size scaling for λ L . , see Figure 13 (left). The ploton the right shows that this factor works well also for the re-scaled fullcumulative density (including all eigenvalues up to the considered value).Based on the fact that the spectral cutoff λ max = 2 is fixed in any volume, c u m u l a t i v e den s i t y λ ν =0) L λ ν | =1) L L=16L=20L=32 0 2 4 6 8 10 12 14 0 5 10 15 20 25 ρ c u m u l a t i v e λ L L=16L=20L=321.2 + 0.029 ( λ L ) Figure 13:
For λ , a low-lying bulk eigenvalue, the scaling factor is shiftedto L . . The re-scaled full spectral cumulative densities in different volumes(for m = 0 . , ν = 0 ) agree well, and turn into the bulk behaviour ρ ( λ ) ∝ λ (resp. ρ cumulative ( λ ) ∝ λ ), which is expected in d = 2 . it is now tempting to speculate that the volume factor for a good finitesize scaling gradually decreases from V / . . . V . However, consideringeigenvalues above the regime shown in Figure 13, but below the cutoffregime, we could not find any consistent scaling factor. Indeed there is nocompelling reason for such a factor to exist.25 Topological summation and susceptibility
The chiral condensate Σ can be measured using the complete Dirac spec-trum, Σ = 1 V X i λ i + m , (5.1)where we still refer to the eigenvalues λ i after the projection (4.1). Herewe do not need any assumptions from RMT or the ǫ -regime. We want toinvestigate the link to the analytical formula for Σ( m ) in eq. (1.3).However, since there are only few topological transitions in the HMChistory (cf. Table 2), we can only measureΣ | ν | = −h ¯ ψ ψ i| | ν | , (5.2) i.e. the chiral condensate in separate topological sectors. Hence an appro-priate summation has to be performed. This challenge is generic for HMCsimulations with dynamical light fermions on fine lattices, so it is relevant— in particular in view of QCD — to explore such topological summations.Here we encounter an interesting test bed to probe various methods for thispurpose.For very light fermions, the dominant contribution to Σ ν =0 is due tothe zero modes. Hence a suitable notation isΣ ν = | ν | mV + ε | ν | , ε > ε > ε > · · · > . (5.3)The inequalities at the end correspond to a general property of stochasticHermitian matrices (such as γ D ovHF ): the presence of zero modes pushesthe small non-zero eigenvalues to higher (absolute) values.Our results for the direct measurement of Σ ν are given in Table 8. Thehierarchy anticipated in relation (5.3) is consistently confirmed. In additionwe observe the inequality ε | ν | ( V ) > ε | ν | ( V ) if V > V (5.4)to hold. If the volume is enlarged, the non-zero eigenvalues reach out tosmaller values. We see that this effect supersedes the pre-factor 1 /V ineq. (5.1), so that ε | ν | increases. The validity of inequalities (5.3), (5.4) isillustrated in Figure 14. Moreover, we recognise a smooth mass and volumedependence of the terms ε | ν | . 26 m h Σ ν =0 i h Σ | ν | =1 i h Σ | ν | =2 i h Σ | ν | =3 i h Σ | ν | =4 ih ε ν =0 i h ε | ν | =1 i h ε | ν | =2 i h ε | ν | =3 i h ε | ν | =4 i
16 0.01 0.01273(8) 0.39968(5)0.01273(8) 0.00905(5)16 0.03 0.0374(4) 0.1573(2)0.0374(4) 0.0271(2)16 0.06 0.0713(8) 0.1174(2)0.0713(8) 0.0523(2)16 0.09 0.0985(8) 0.1182(3)0.0985(8) 0.0748(3)16 0.12 0.1174(6) 0.1275(4) 0.1468(4)0.1174(6) 0.0949(4) 0.0817(4)16 0.18 0.1434(3) 0.1469(3) 0.1548(4)0.1434(3) 0.1252(3) 0.1114(4)16 0.24 0.1633(4) 0.1652(2) 0.1697(4) 0.1758(9) 0.1841(7)0.1633(4) 0.1489(2) 0.1371(4) 0.1270(9) 0.1190(7)20 0.01 0.0141(3) 0.2608(3)0.0141(3) 0.0108(3)24 0.01 0.3572(2) 0.5296(2)0.0100(2) 0.0088(2)28 0.01 0.1408(2) 0.3923(2)0.0132(2) 0.0096(2)32 0.01 0.0181(5) 0.1112(2) 0.2073(4)0.0181(5) 0.0135(2) 0.0120(4)32 0.06 0.0883(7) 0.093(1)0.0883(7) 0.077(1)Table 8:
Results for the directly measured chiral condensate at differentmasses and lattice sizes, in distinct topological sectors. We observe fullagreement with inequalities (5.3) and (5.4). As m increases at fixed L , thedominant rˆole of the zero mode contributions to Σ ν =0 is diminished. As L increases at fixed m , however, ε | ν | is enhanced. In this method we assume a Gaussian distribution of the topological charges,so that Σ can be written asΣ = ∞ X ν = −∞ p ( | ν | ) Σ | ν | , p ( | ν | ) = exp (cid:16) − ν V χ t (cid:17)P ∞ ν = −∞ exp (cid:16) − ν V χ t (cid:17) , (5.5)27 ε ε ε ε ε ε ε ε ε Figure 14:
The terms ε | ν | introduced in eq. (5.3), which represent the contri-bution of the non-zero modes to the chiral condensate in a fixed topologicalsector. These plots reveal a smooth and monotonous dependence on thefermion mass (on the left, at L = 16 ) and on the volume (on the right, at m = 0 . ). where χ t is the topological susceptibility. At least in QCD the chargedistribution is indeed Gaussian to a good approximation (see for instancethe index histograms in Refs. [14]). A high statistics study only found atiny deviation, which tends to vanish in the large volume limit [62].If we have data in the sectors up to charge Q , i.e. Q is the maximumof the simulated sectors | ν | , we insert the measured values Σ . . . Σ Q , withthe maximum and minimum of the statistical error bar. For higher chargeswe make use of inequality (5.3) to fix the minimal and maximal values asΣ | ν | , min = | ν | mV , Σ | ν | , max = Σ | ν | , min + ε Q . (5.6)In the cases L = 24 and 28 we do not have data for all the sectors with | ν | < Q . Here we employ in addition inequality (5.4) and the results in thenext smaller (larger) volume to fix Σ | ν | , min (Σ | ν | , max ).If we insert some susceptibility χ t into eq. (5.5), we obtain a value forΣ. Its uncertainty is modest, because most sectors, which have not beenmeasured, have exponentially suppressed probabilities p ( | ν | ). By requiringagreement with the theoretical prediction of Ref. [6] (given in eq. (1.3)) wenow determine χ t ; the results are given in Table 9 and Figure 15.Table 8 shows that only in the case of light fermions the predictionof Ref. [6] can be reproduced. For L = 16 and m ≥ . all the Σ ν arelarger than the prediction, hence there is no way to reproduce it with aweighted sum. In the case L = 32, m = 0 .
06 the value of h Σ ν =0 i is tooclose to the prediction for Σ to extract a sensible result for χ t . At L = 16, m = 0 .
09 a value for χ t which achieves this can still be found, but the result28 m Σ of Ref. [6] χ t h ν i
16 0.01 0.04888 0.0006586(5) 0.1686(1)16 0.03 0.07050 0.00117(1) 0.299(3)16 0.06 0.08883 0.00159(8) 0.407(20)20 0.01 0.04888 0.000500(2) 0.200(1)24 0.01 0.04888 0.000408(16) 0.235(9)28 0.01 0.04888 0.000367(15) 0.288(12)32 0.01 0.04888 0.000341(4) 0.349(4)Table 9:
Results for the topological susceptibility χ t based on the methoddescribed in Subsection 5.1, which assumes Gaussian charge distributionand the chiral condensate according to Ref. [6]. On the L = 16 lattice thismethod does not work for m ≥ . , due to the latter assumption. χ t mL = 16 0.0067 m conjecture 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0 0.001 0.002 0.003 0.004 0.005 χ t Figure 15:
The topological susceptibility χ t determined by the method de-scribed in Subsection 5.1. On the L = 16 lattice we see that Smilga’s for-mula cannot be valid at m ≥ . , hence also this method is not applicableanymore. For the smaller masses our results for the susceptibility suggest abehaviour χ t ∝ √ m . It is compared to the conjecture in eq. (5.8), based onRefs. [18, 63]. The plot on the right refers to m = 0 . . An infinite volumeextrapolation, which assumes χ t to be consistent with a linear dependenceon /V , leads to χ t = 0 . . does not make sense either, since it is below the χ t values determined at m = 0 .
03 and 0 .
06 (see Figure 15). We conclude that Smilga’s formulais not applicable at m ≥ .
09, which assigns an explicit meaning to hisassumption m ≪ / √ β ≃ . χ t ( m ) ∝ √ m . (5.7)Alternative results (with quenched configurations and re-weighting) were29iven in Ref. [18]. On the theoretical side, Ref. [63] conjectured for N f degenerate flavours in the large volume limit1 χ t = N f Σ (1) m + 1 χ t,q , Σ (1) ≃ . g , (5.8)where Σ (1) := Σ N f =1 ( m = 0) (cf. Section 1), and χ t,q = χ t ( m → ∞ ) is thequenched value. Actually this mass dependence of χ t was conjectured inthe framework of QCD. If we apply the same form in the Schwinger model,and insert the value χ t,q ≈ . g = 0 . χ t ≈ . L = 16, thus suppressing thefinite size effects. In particular at m = 0 .
06 eq. (5.8) implies χ t ≃ . m = 0 .
01, our results attain thevicinity of this prediction: at L = 32 we obtained χ t ≈ . χ t ≈ . χ t on 1 /V ; that assumptionleads to a smaller value of χ t ( V = ∞ ) = 0 . . The procedure of the previous subsection is robust for light fermions (bar-ing finite size effects on Σ). However, since it uses the analytic result forΣ as an input to determine χ t , it does not evaluate Σ itself from the nu-merical results in distinct topological sectors. In Subsections 5.3 and 5.4we will test a method to do so, following Ref. [64]. For convenience were-derive here in a concise form the formula for an approximate topologicalsummation that was given in Ref. [64] for the pion mass; we generalise it toarbitrary observables. This consideration follows the lines of Refs. [64, 65],pointing out in particular which assumptions are involved, so the subse-quent subsections can refer to them.First we assume the fermion field to be integrated out. Thus we refer toan effective gauge action S eff [ U ], which keeps track of the fermion determi-nant. So the partition function takes the form Z = R D U exp {− S eff [ U ] } .Next we introduce a Kronecker δ as a filter of gauge configurations [ U ] witha specific topological charge ν , δ ν,ν [ U ] = 12 π Z π − π dθ exp { i θ ( ν − ν [ U ]) } . (5.9)30his formulation involves the vacuum angle θ , and it allows us to write a“partition function” restricted to one topological sector as Z ν = Z D U e − S eff [ U ] δ ν,ν [ U ] = 12 π Z π − π dθ e i θν Z ( θ ) , (5.10)where Z ( θ ) = R D U exp {− S eff [ U ] − i θν [ U ] } is the (complete) partitionfunction for a general vacuum angle (and Z = Z (0)).If some observable O is measured only in one topological sector, thecorresponding expectation value is given by ¯ O ν = 1 Z ν Z D U e − S eff [ U ] δ ν,ν [ U ] O [ U ] = 12 πZ ν Z π − π dθ e i θν Z ( θ ) ¯ O ( θ ) . (5.11)The relation − Z ′′ ( θ ) | θ =0 = h ν i = V χ t (5.12)suggests Z ( θ ) = Z exp( − V χ t θ / . (5.13)Inserting this ansatz into eq. (5.10) leads to Z ν Z = 12 π Z π − π dθ exp (cid:16) − V χ t θ + i θν (cid:17) , (5.14)where we recognise the stationary phase θ s,ν = i νV χ t = i ν h ν i . (5.15)As an approximation, we extend the bounds in the integral (5.14) to ±∞ .For this step, it is favourable if V χ t is large. Then we obtain Z ν Z ≃ √ πV χ t exp (cid:16) − ν V χ t (cid:17) . (5.16)This formula is obviously consistent with an integral approximation for Z = P ν Z ν , which is again best justified for large V χ t .If we insert ansatz (5.13) into eq. (5.11), we obtain another integral withthe stationary phase θ s,ν . By repeating its approximation as a Gaussianintegral, and employing relation (5.16), we arrive at¯ O ν = Z πZ ν Z π − π dθ ¯ O ( θ ) exp (cid:16) − V χ t θ + i θν (cid:17) ≃ r V χ t π Z ∞−∞ dθ ¯ O ( θ ) exp (cid:16) − V χ t θ − θ s,ν ) (cid:17) . (5.17) We use the notation ¯ O , rather than hOi , because in the following this will be morepractical for indicating the dependence on θ . | θ − θ s,ν | to contribute significantly to thisintegral, we may also approximate¯ O ( θ ) ≃ ¯ O ( θ s,ν ) + 12 ¯ O ′′ ( θ ) | θ s,ν ( θ − θ s,ν ) . (5.18)Let us further assume | θ s | to be small, so we can replace the first term inthis formula for ¯ O ( θ ) by¯ O ( θ s,ν ) ≃ ¯ O (0) + 12 ¯ O ′′ ( θ ) | θ s,ν . (5.19)Hence a further property, which is favourable for our approximation, is asmall topological charge | ν | , in addition to a large value of V χ t .The approximation (5.19) also implies ¯ O ′′ ( θ ) | θ s ≃ ¯ O ′′ ( θ ) | . So we canexpress the (numerically measurable!) restricted expectation value as¯ O ν ≃ r V χ t π Z π − π dθ h ¯ O + 12 ¯ O ′′ [( θ − θ s,ν ) + θ s,ν ] i exp (cid:16) − V χ t θ − θ s,ν ) (cid:17) , (5.20)where ¯ O and ¯ O ′′ are taken at θ = 0. Extending once more the boundariesto ±∞ leads to the final form¯ O ν ≈ ¯ O + 12 ¯ O ′′ V χ t (cid:16) − ν V χ t (cid:17) . (5.21)This is the same structure as Ref. [64] obtained for the pion mass. It is con-sistent that the limit V χ t → ∞ renders all topological sectors equivalent,so that all ¯ O ν coincide with ¯ O .The numerical measurement with few topological transitions providesresults for the left-hand-side of eq. (5.21). On the right-hand-side ¯ O , ¯ O ′′ and χ t are unknown. We are interested in ¯ O and χ t , and measurements of¯ O ν in various topological sectors and volumes allow in principle for theirevaluation (as far as the above approximations make sense). Up to now,this intriguing and possibly powerful technique has not been tested withsimulation data. This will be pioneered in the next two subsections. Now we insert the chiral condensate Σ as our observable ¯ O . As our inputwe have data for some Σ ν , i.e. for the left-hand-side of eq. (5.21), but onthe right-hand side Σ, Σ ′′ and χ t are unknown. In view of their evaluation,it is convenient to re-write eq. (5.21) asΣ ν ≈ Σ − AV + BV ν A := − Σ ′′ χ t , B := − Σ ′′ χ t , χ t = AB . (5.22)32ased on data from different topological sectors in a fixed volume V wecan only evaluate B ; it is not possible to obtain Σ and χ without includingdifferent volumes.By considering two sectors with charges | ν | = k and ℓ (and k = ℓ ) atfixed V and m , a result for B is obtained as1 V B k,ℓ = V Σ k − Σ ℓ k − ℓ = 1 m ( k + ℓ ) + V ε k − ε ℓ k − ℓ . (5.23)If more than two Σ | ν | values have been measured, the approximate agree-ment between the emerging B k,ℓ represents a consistency condition. InTable 10 we give corresponding results for B k,ℓ /V , derived from the datain Table 8. m L B , /V B , /V B , /V B , /V B , /V . . . . . . . . . . . . . . Results for the term B k,ℓ /V obtained in each case from two valuesof Σ | ν | in a fixed volume V and at a fixed mass m , according to eq. (5.23). For small fermion masses, the results for B k,ℓ are dominated by the semi-classical term 1 / [ m ( k + ℓ )], and thus strongly dependent on the topologicalsectors involved. This feature is suppressed, however, when m increases; inparticular for L = 16 , m = 0 .
24 we do observe approximate agreement fordifferent choices of k and ℓ , in striking contrast to the semi-classical result.This similarity, which is illustrated in Figure 16, is a remarkable quantumeffect, due to the fluctuation terms ε | ν | . Still we observe systematically thatthe B k,ℓ values decrease if higher topological charges are involved. Thatalso reduces the reliability of our assumption of a small | θ s,ν | — whichfavours the approximation (5.19) — so we consider B , most reliable.In order to proceed, i.e. to get access also to A and thus to χ and Σ,we have to confront data from different volumes. Considering two volumes V and V , but keeping m and | ν | fixed, eq. (5.22) impliesΣ | ν | ( V ) − Σ | ν | ( V ) = A (cid:16) V − V (cid:17) + Bν (cid:16) V − V (cid:17) . (5.24)Most convenient is the sector ν = 0, where we can read off A withoutinvolving the variable B . At | ν | = 1 it is most obvious to insert B asΣ ( V ) − Σ ( V ) = A (cid:16) V − V (cid:17) + B , ( V ) V − B , ( V ) V , (5.25)33 B k , / V knumerical resultssemi-classical Figure 16:
The terms B k, /V for k = 1 . . . , at m = 0 . , L = 16 . For thenumerical result we observe a remarkable stability in k , in contrast to thesemi-classical values. which is identical to the result obtained from Σ ( V ), Σ ( V ). This isexpected to be the most reliable value for A . Our results obtained in thisway are listed in Table 11. m m = 0 . m = 0 . L , L ) (20 ,
16) (32 ,
16) (32 ,
20) (32 , A Results for the term A in eq. (5.22), obtained from Σ or Σ intwo volumes, at a fixed mass m , according to eqs. (5.24) and (5.25). The idea is to use these results in both volumes involved. This is sensibleif A is approximately constant in the volume, but that is not confirmed inour data set for m = 0 .
01. Since the assumptions tend to hold better forlarger volumes, we use the value of A obtained in ( L , L ) = (32 ,
16) todetermine Σ in L = 16, and A from (32 ,
20) for Σ in L = 20 and 32. Thisleads toΣ ≃ . L = 16) , Σ ≃ . L = 20 or 32) . (5.26)These results for Σ are nicely consistent, but more than a factor of 2 belowthe value expected in infinite volume. Hence for m = 0 .
01 this methodworks in an intrinsically consistent way, but the results for Σ are reducedby strong finite size effects.Similarly, if we now evaluate χ t by referring to eq. (5.22) we obtainvalues between 10 − and 10 − , i.e. well below the results in Subsection5.1. Note that the applicability of the method used here — based on the34pproximations in Subsection 5.2 — is indeed questionable for m = 0 . V χ t .So the mass m = 0 .
06 is more promising. We cannot test the volumeindependence of A from our data, but based on L = 16 and 32 we obtain m = 0 .
06 : A = 5 . , Σ = 0 . . (5.27)Let us focus on the size L = 32 and involve B , , which leads to χ t = 0 . . (5.28)The method that we used in Subsection 5.1 to evaluate χ t does not workin this case as we mentioned before (due to the sizable uncertainty of Σ itbasically just constrains χ t < . χ t ≃ . . . (5.29)This result is in excellent agreement with the analytical predictions basedon Ref. [6], Σ = 0 . m = 0 .
01 the ultimate results for χ t and the Σ (summed over all topologicalsectors) seem to be affected by strong finite size effects.The situation improves as we proceed to m = 0 .
06. In this case thecorrelation length (in infinite volume) is only 4 . χ t and Σ. For practical reasons, it is favourable to consider the decay of a currentcorrelation function for measurements of the “pion” mass, rather than thepseudoscalar density [7]. In this way we obtained the results in Table 12.Again we can test the topological summation for the fermion masses m = 0 .
01 and 0 .
06. Here we proceed in a manner different from Subsection5.3: if we insert the data of Table 12 into the formula M π, | ν | = M π − AV + BV ν , (5.30)the unknown terms M π , A and B are over-determined. We choose theoptimal values for them by a least square fit.Let us start again with m = 0 .
01: if we include all 11 sector thatwe simulated, we obtain M π = 0 . m M π, M π, M π, M π, M theory π ( L = ∞ )16 0.01 0.041(1) 0.271(4) 0.07116 0.03 0.123(5) 0.275(4) 0.14816 0.06 0.214(6) 0.310(3) 0.23516 0.09 0.302(5) 0.358(2) 0.30816 0.12 0.359(5) 0.413(2) 0.494(5) 0.37416 0.18 0.498(4) 0.525(2) 0.589(5) 0.49016 0.24 0.631(6) 0.648(2) 0.700(3) 0.59320 0.01 0.038(2) 0.209(4) 0.07124 0.01 0.257(8) 0.32(1) 0.07128 0.01 0.146(4) 0.25(1) 0.07132 0.01 0.05(1) 0.160(8) 0.192(6) 0.07132 0.06 0.23(1) 0.232(7) 0.235Table 12: The “pion” masses measured in various volumes, at differentfermion masses, in the topological sectors | ν | = 0 . . . . The last columndisplays the (infinite volume) prediction of Ref. [6] (cf. eq. (1.4)). | ν | > M π = 0 . ξ ≃
14, cf. Figure 4, hence strong finitesize effects are expected, and they generically enhance the mass gap.Therefore, we now skip the smallest volumes. We need at least twovolumes to determine the term A , so we restrict the consideration to L = 28and 32, and we include | ν | ≤
2; thus we are left with four sectors. Thissmall number of input data causes a large error, but the optimal value ofthe least square fit moves very close to the theoretical prediction (see Table12), m = 0 .
01 : M π = 0 . . For m = 0 .
06 we only have four sectors to deal with, but the finite sizeeffects are much less severe, and we arrive again at a sensible result, m = 0 .
06 : M π = 0 . , χ t = 0 . . (5.31)As we already saw in Subsection 5.3, this method is not very useful for thedetermination of χ t ; it is always plagued by a large uncertainty, as in theexample given here (for m = 0 .
01 this is even worse). The three resultsthat we obtained for χ t at m = 0 .
06 (in Subsection 5.1, 5.3 and 5.4) differwithin the same magnitude. The result in Table 9 is larger, but only based on L = 16. h ν i should not be too small. Members of the JLQCD Collaboration proposed an alternative method toevaluate χ t even in one single topological sector based on the topologicalcharge density , ρ t ( x ). Ref. [65] derived the “model independent formula” lim | x |→∞ h ρ t ( x ) ρ t (0) i| | ν | ≃ − χ t V + 1 V (cid:16) ν + c χ t (cid:17) + O ( V − ) , (5.32)which captures even a possible deviation from the Gaussian charge distri-bution by a non-vanishing kurtosis c = ( h ν i − h ν i ) /V . However, thisterm is known to be tiny, so we neglect it in the following.The issue is to search for a plateau value of the charge density cor-relation at large distances, which differs from the constant ν /V (in thesector with topological charge | ν | ). This shift ∆ = − χ t /V is negativebecause fluctuations in ρ t ( x ) have to compensate. However, ∆ tends tobe small and hard to resolve numerically. For the absolute value | ∆ | itis favourable if m increases (although we are actually interested in quasi-chiral fermions), but it is unfavourable if V increases (although we needaccess to the asymptotic value in eq. (5.32), and O ( V − ) should be negli-gible). Moreover, in analogy to the approximations that lead to eq. (5.21),the derivation of eq. (5.32) involves the assumptions h ν i = V χ t is large , | ν |h ν i is small . (5.33)Our estimates for χ t suggest that we do not have any setting which isreally adequate for the first of these two conditions. Sometimes, however,such approximations apply reasonably well even if the assumptions do notstrictly hold (this is the experience in QCD with simulation data matchedto formulae of the ǫ - or δ -regime, and in the preceding two subsections tosome extent).Since our configurations are smooth, it is not problematic to use thenaive lattice formulation of the topological charge density, ρ t = ǫ F .(There is a cleaner formulation for Ginsparg-Wilson fermions [25], but itis tedious in practice.) As examples, the results for L = 16 in the topolog-ically neutral sector for three masses are shown in Figure 17. In particular A variant of this method, which uses the η ′ -correlation of the pseudo-scalar density,has been applied to two-flavour QCD in Ref. [66]. < ρ t ( x , x ) ρ t ( x , ) > | ν = x m=0.01m=0.03m=0.06 Figure 17:
The topological charge correlation function for L = 16 , ν = 0 and m = 0 . , . and . . we see that the correlation over short (but non-vanishing) distances is neg-ative, since a given link variable contributes with opposite signs to ρ t inits adjacent plaquettes. The jackknife errors for the quantity h ρ t ( x ) ρ t (0) i| are typically in the range (1 . . . · − .In the most promising case, m = 0 .
06, we evaluated 741 topologicallyneutral configurations and we have identified the magnitude for χ t ≈ . ≈ − · − . In order to safely resolve this shift we wouldtherefore need about 50 000 to 100 000 configurations. We conclude thatthis method requires a very large statistics, so we cannot apply it. We presented a study of the 2-flavour Schwinger model with dynamicalchiral fermions. We applied the overlap Hypercube Fermion (HF), andconfirmed its features as a promising formulation of a chiral lattice fermion.This is manifest by its excellent locality and scaling behaviour. Moreover,our study allowed us to explore a number of conceptually interesting andrelevant issues, and to test new methods to handle them.This is certainly of interest, even in a toy model, since simulations withdynamical overlap fermions have been explored only poorly so far.For our simulation we designed a specifically suitable variant of theHybrid Monte Carlo (HMC) algorithm, demonstrated its correctness andtested its efficiency. It is sufficient to insert a low polynomial approxima-tion for the overlap operator to compute the force term, while the highprecision overlap operator is employed in the accept/reject step. It is anopen question if this strategy — perhaps with further refinements — can38e carried over to the simulation of QCD with dynamical chiral quarks.Next we discussed the spectrum of the Dirac operator. Random MatrixTheory is not yet worked out for this type of model, with a vanishing chiralcondensate at zero fermion mass. Nevertheless the unfolded level spacingdensity follows the standard RMT formula for the unitary ensemble.The prediction Σ( m ) ∝ m / [4–6] suggests a microscopic spectral den-sity ρ ( λ > ∼ ∝ λ / , and the scale-invariant variable ζ ∝ λV / (if weassume no explicit mass dependence of ρ ). This conjecture does, however,not agree with our data. An alternative scenario with ρ ( λ > ∼ ∝ λ / hasa theoretical background as well (Gaussian distribution), but the data donot really support it either. Instead they favour z ∝ λV / as the scale-invariant variable, and therefore ρ ( λ > ∼ ∝ λ / .A hint for an interpretation of this surprising result can be found inRef. [5], which derived the behaviours Σ ∝ m / and Σ ∝ mL in two limit-ing cases. Regarding the parameter that characterises these extreme cases,our settings are in an intermediate regime, which appears compatible withthe relation Σ ∝ m / that we observed. For a precise theoretical clarifica-tion, we hope for the corresponding RMT formulae to be elaborated.Direct measurements of the chiral condensate and of the mass of the iso-triplet (“pion”) could only be performed in fixed topological sectors, sincethe HMC histories contain only few topological transitions. This limitationis generic for the simulation of light fermions close to the continuum limit.It is therefore a major challenge to explore techniques for the topologicalsummation of such measurements. Assuming a Gaussian distribution ofthe topological charges, the data can be used to evaluate the topologicalsusceptibility. In order to determine the actual observable, we exploredapproximate summation techniques, which led to sensible results in someparameter window, where the term h ν i = V χ t is not too small.Topological summations could become relevant for future QCD simula-tions. Nowadays they are carried out with very light dynamical quarks —such that the pion mass is close to its physical value — on finer and finerlattices. In particular in applications of dynamical overlap fermions [17]the HMC history tends to be trapped in the topologically trivial sector forthe entire simulation, which endangers ergodicity and does not provide thephysical result (unless the volume is very large). For Wilson-type latticefermions (which break the chiral symmetry explicitly), the problem is lessobvious on the currently used lattice spacings, but on still finer lattices(with a < ∼ .
05 fm) it is expected to show up as well [22]. Hence topologi-cal summation methods are of interest for non-perturbative studies of lowenergy nuclear physics based on first principles of QCD. Our results tell us39o be cautious with such summations, but at the same time they providehope for their feasibility.
Acknowledgements
We are indebted to Martin Hasenbusch for helpful adviceon algorithmic aspects, to Poul Damgaard and Jacques Verbaarschot for shar-ing their deep insight into Random Matrix Theory, and to Stephan D¨urr, Hide-nori Fukaya and Jim Hetrick for very useful comments. Most production runswere performed on the clusters of the “Norddeutscher Verbund f¨ur Hoch- undH¨ochstleistungsrechnen” (HLRN). We thank Hinnerk St¨uben for technical assis-tance, as well as Edwin Laermann and Michael M¨uller-Preußker for their supportat Bielefeld and Humboldt University, respectively. This work was supportedin part by the Croatian Ministry of Science, Education and Sports (project No.0160013), and by the Deutsche Forschungsgemeinschaft (DFG) through Son-derforschungsbereich Transregio 55 (SFB/TR55) “Hadron Physics from LatticeQCD”, which is coordinated by the University of Regensburg.
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