Reweighting twisted boundary conditions
Andrea Bussone, Michele Della Morte, Martin Hansen, Claudio Pica
RReweighting twisted boundary conditions
A. Bussone ∗ CP3-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense M.E-mail: [email protected]
M. Della Morte
CP3-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense M.IFIC (CSIC), Calle Catedrático José Beltran, 2, 46980 Paterna, Valencia, SpainE-mail: [email protected]
M. Hansen
CP3-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense M.E-mail: [email protected]
C. Pica
CP3-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense M.E-mail: [email protected]
Preprint: CP3-Origins-2015-037 DNRF90, DIAS-2015-37, IFIC/15-65
Imposing twisted boundary conditions on the fermionic fields is a procedure extensivelyused when evaluating, for example, form factors on the lattice. Twisting is usually performedfor one flavour and only in the valence, and this causes a breaking of unitarity. In this work weexplore the possibility of restoring unitarity through the reweighting method. We first study someproperties of the approach at tree level and then we stochastically evaluate ratios of fermionicdeterminants for different boundary conditions in order to include them in the gauge averages,avoiding in this way the expensive generation of new configurations for each choice of thetwisting angle, θ . As expected the effect of reweighting is negligible in the case of large volumesbut it is important when the volumes are small and the twisting angles are large. In particular wefind a measurable effect for the plaquette and the pion correlation function in the case of θ = π / × , and we observe a systematic upward shift in the pion dispersion relation. The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] S e p eweighting twisted boundary conditions A. Bussone
1. Introduction
Non-Periodic Boundary Conditions (NPBCs) for the fermions are typically employed on thelattice in order to obtain a fine resolution of momenta in the spatial directions.
Twisting [1] amountsto imposing ψ (cid:0) x + N µ ˆ µ (cid:1) = (cid:40) e i θ µ ψ ( x ) , µ = , , ψ ( x ) , µ = , where N µ is the lattice extent in direction ˆ µ and θ j ∈ [ , π ] is an angle. Alternatively one canintroduce a constant U ( ) interaction with vanishing electric, magnetic field and electric potentialbut constant vector potential [2] (see also [3] for the equivalence of the two procedures). The aboveextra interaction is implemented by transforming the standard QCD links ( U ) in the following way U µ ( x ) = (cid:40) e i θ µ / N L U µ ( x ) , µ = , , U ( x ) , µ = . (1.1)The modification of the boundary conditions proved to be beneficial for: • Form factors: one can scan values of the exchanged momenta in the scattering process witha very fine resolution, in order to determine more accurately the form factors on the lattice.An example is the semi-leptonic decay K (cid:96) [4], used to extract the CKM element V us . • Matching between HQET and QCD: in the Schrödinger functional, one computes finite vol-ume observables for different values of θ in the valence [5]. In particular in such a setupreweighting to a unitary formulation is expected to be efficient as the volumes considered arerather small (the reweighting factors are extensive quantities). • Dispersion relation: which is used here more as a consistency check, as done also in [1].Usually one performs the twisting only in the valence, which causes breaking of unitarity. Thisis simply understood since the procedure yields different propagators for fermions in the valenceand in the sea. One way to overcome this problem is to perform direct simulations with fermionicNPBCs for each value of θ . That though, would clearly be too expensive from the computationalpoint of view. However, this kind of breaking of unitarity is expected to be a finite volume effect,in χ -PT for example that is the case [3]. This suggests that where the effect is large, reweightingmay provide a reliable way to restore unitarity.
2. Reweighting
Let us suppose we want to connect simulation results for a choice of bare parameters a = { β , m , m , . . . , m n f , θ µ , . . . } to a (slightly) different set b = { β (cid:48) , m (cid:48) , m (cid:48) , . . . , m (cid:48) n f , θ (cid:48) µ , . . . } of param-eters. This can be achieved by numerically computing on the a -ensemble the reweighting factor W ab = P b / P a , which is the ratio of the two probability distributions and it is clearly an extensive2 eweighting twisted boundary conditions A. Bussone quantity, P a [ U ] = e − S G [ β , U ] ∏ n f i = det ( D [ U , θ ] + m i ) , where we have explicitly indicated the depen-dence of the Dirac operator on the twisting angle. In this way the expectation values on the b -ensemble are calculated as (cid:104) O (cid:105) b = (cid:104) (cid:101) O W ab (cid:105) a (cid:104) W ab (cid:105) a , with (cid:101) O the observable after Wick contractions, and (cid:104) . . . (cid:105) a the expectation value over the set of bareparameters a .By choosing to change only the boundary conditions from one bare set to the other we arrive at thefollowing expression of the reweighting factor W θ = det (cid:0) D W [ U , θ ] D − W [ U , ] (cid:1) = det (cid:0) D W [ U , ] D − W [ U , ] (cid:1) , where D W is the massive Wilson operator. We therfore need a stochastic method to estimate (ratioof) determinants. For this purpose we use the following integral representation of a normal matrixdeterminant with spectrum λ ( A ) that holds if and only if the real part of each eigenvalue is largerthan zero [6] 1det A = (cid:90) D [ η ] exp (cid:0) − η † A η (cid:1) < ∞ ⇐⇒ R e λ ( A ) > . (2.1)If it is so then the stochastic estimae converges. We use as probability distribution p ( η ) of the η vectors a gaussian one, then the determinant reads1det A = (cid:28) e − η † A η p ( η ) (cid:29) p ( η ) = N η N η ∑ k = e − η † k ( A − ) η k + O (cid:32) (cid:112) N η (cid:33) . It should be noted that if we require, for the case of an hermitian matrix, the existence of all gaussianmoments then all the eigenvalues must be strictly larger than one (cid:28) e − η † A η p ( η ) (cid:29) p ( η ) = (cid:90) D [ η ] exp (cid:2) − η † ( A − ) η (cid:3) < ∞ ⇐⇒ λ ( A ) > , ... (cid:28) e − N η † A η p ( η ) N (cid:29) p ( η ) = (cid:90) D [ η ] exp (cid:2) − η † [ NA − ( N − ) ] η (cid:3) < ∞ ⇐⇒ λ ( A ) > N − N −→ N → ∞ . The spectrum of the Dirac-Wilson operator at tree level is known and that allowed us to testour numerical implementation. A tree level calculation (Fig. 1(a)) shows that at fixed θ and forlarge N L the reweighting factor approaches the value 1, which is expected since it is a finite volumeeffect. Conversely, the variance grows as N L increases (Fig. 1(b)). Hence a direct estimate for large θ angles is not reliable and we need to employ a multi-step method in order to keep the error undercontrol. To this end, following [6], we factorize the relevant matrix in the following telescopic way D W ( θ ) D − W ( ) = N − ∏ l = A (cid:96) , with A (cid:96) (cid:39) + O ( δ θ (cid:96) ) , N = θδ θ (cid:96) . eweighting twisted boundary conditions A. Bussone N L W θ ( θ = . ) (a) Mean of the reweighting factor at tree levelfor θ = . as a function of N L . N L σ ( θ = . ) (b) Variance of the reweighting factor at tree levelfor θ = . as a function of N L . Figure 1:
Estimates of mean and variance of the reweighting factor employing the exact formulae for thetree level case. Each point corresponds to a cubic lattice of the form L . with A (cid:96) now near to the identity matrix as it corresponds to a small δ θ (cid:96) shift in the periodicityangle. At this point the inverse determinant and its variance are given in terms of the N analogousquantities, one for each multiple of δ θ (cid:96) in θ , as1det A = N − ∏ (cid:96) = (cid:28) exp (cid:0) − η ( (cid:96) ) , † A (cid:96) η ( (cid:96) ) (cid:1) p (cid:0) η ( (cid:96) ) (cid:1) (cid:29) p ( η ( (cid:96) ) ) , σ = N − ∑ (cid:96) = (cid:34) σ η ( (cid:96) ) ∏ k (cid:54) = (cid:96) det ( A k ) − (cid:35) .
3. Monte Carlo studies
We have calculated a number of reweighted observables in the SU ( ) gauge theory withfermions in the fundamental representation. This theory is known to exhibit confinement and chiralsymmetry breaking and it is therefore QCD-like. We have employed unimproved Wilson fermionsand the Wilson plaquette gauge action. In the table below we list the configurations used in thiswork ( m cr is estimated to be − . ( ) at β = . V β m N cnf traj. sep.16 × × × θ . To evaluate the reweightingfactor we have used the γ version of the Dirac-Wilson operator, since it is hermitian and with 2flavours it automatically satisfies the applicability condition in eq. (2.1). We found that for N η (cid:38)
200 the reweighting factor on each configuration is larger than 10 times the average value on a fewconfigurations only. This guarantees that the average values are not dominated by spikes and hence4 eweighting twisted boundary conditions
A. Bussone that there are no large statistical fluctuations of the determinant. In the figures below we show themean of the reweighting factor (all points are averaged over the total number of configurations).In Figs. 2(a) and 2(b) a good scaling for the error is clearly visible, according to N − / η and upto N η ≈ −
600 gaussian vectors is enough to saturate the statistical noise at the level of the gauge noise. θ = 0 .
3, 10 steps θ = 0 .
3, 4 steps N η h W θ i (a) Mean of the reweighting factor for θ = . as afunction of N η . θ = π/
2, 20 steps θ = π/
2, 15 steps N η h W θ i (b) Mean of the reweighting factor for θ = π / as afunction of N η . Figure 2:
Monte Carlo history of the reweighting factor, averaged over the entire number of configurations.The figures correspond to a volume V = × . Reweighting may give a significant effect in small volumes. We have first looked at the pla-quette because it does not explicitly depend on the BCs since, in general, in Wilson loops for eachlink entering the loop there is another one in the opposite direction and the exponential factorsin eq. (1.1)) cancel among the two. We are neglecting autocorrelations, because measurementsare separated by 10 to 20 molecular dynamics units. A binning procedure, including bins fromlength one up to 10, shows no significant autocorrelation effects. In Fig. 3(a) we show results afterreweighting only one flavor, by taking the square root of the reweighting factor. No effects arevisible in this case within errors. In Fig. 3(b) instead we have included the reweighting factor forboth flavors and one can see a permil effect, recognizable because the plaquette is determined veryaccurately.The second quantity that we have studied is the pion dispersion relation. In the pion correlatorwe twisted only one flavor in the valence, hence the lowest energy state becomes a pion withmomentum (cid:126) p = ± (cid:126) θ / L . We have reweighted the correlators in order to take into account the twistin the sea and we have extracted the effective mass after symmetrizing the correlators in time.In Fig. 4 we show results for the dispersion relation and we compare them to un-reweighteddata (i.e., with twisting only in the valence), to the continuum prediction ( aE ) = ( am π ) + θ / N L and to the lattice free (boson) theory prediction cosh ( aE ) = + cosh ( am π )+ ( θ / N L ) . Over the5 eweighting twisted boundary conditions A. Bussone unreweighted plaquette θ = π/
2, 20 steps θ = π/
2, 15 steps N η h L i (a) Plaquette with only one flavor reweighted. unreweighted plaquette θ = π/
2, 20 steps θ = π/
2, 15 steps N η h L i (b) Plaquette with both flavors reweighted.
Figure 3:
Monte Carlo history of the reweighted plaquette. The figures correspond to a volume V = × . entire range of θ values explored there is no clear effect within errors. However there is a systematiceffect upward. The discrepancy between the reweighted data and the lattice free prediction isconceivably due to cutoff effects and non-perturbative dynamics. lattice free curvecontinuum curvereweighted datavalence twist data θ/N L E ( θ ) Figure 4:
Pion dispersion relation for V = × . Each point corresponds to the best determination of thereweighting factor. Obviously, in large volume we do not expect the situation to improve with respect to smallvolumes, concerning both statistical accuracy and significance of the reweighting. We have lookedat the pion dispersion relation for two rather different values of m π . We expect to find a larger effectfor the lighter pion but already at these volumes it appears as one can neglect the effects coming6 eweighting twisted boundary conditions A. Bussone from the breaking of unitarity. In Fig. 5 we show the results for the two pion masses. No sizeableeffect is detectable in either case. lattice free curvecontinuum curvereweighted datavalence twist data θ/N L E ( θ ) (a) Case of m (cid:39) − . which corresponds to a heavypion. lattice free curvecontinuum curvereweighted datavalence twist data θ/N L E ( θ ) (b) Case of m (cid:39) − . which corresponds to a lighterpion. Figure 5:
Pion dispersion relation for V = × . The reweighting factor at a given θ is obtained by atelescopic product involving all the previous ones.
4. Conclusions
We have presented an application of the reweighting method to the case of the spatial peri-odicity of fermionic fields. We have studied the approach at tree level and by mean of numericalsimulations. For the latter we have looked at the plaquette and at the pion dispersion relation insmall and large volumes. In both cases we have found the effects to be at the sub-percent level forvalues of θ up to π / References [1] G. M. de Divitiis, R. Petronzio and N. Tantalo, “On the discretization of physical momenta in latticeQCD,” Phys. Lett. B (2004) 408 [hep-lat/0405002].[2] P. F. Bedaque, Phys. Lett. B (2004) 82 [nucl-th/0402051].[3] C. T. Sachrajda and G. Villadoro, Phys. Lett. B (2005) 73 [hep-lat/0411033].[4] P. A. Boyle, A. Juttner, R. D. Kenway, C. T. Sachrajda, S. Sasaki, A. Soni, R. J. Tweedie andJ. M. Zanotti, Phys. Rev. Lett. (2008) 141601 [arXiv:0710.5136 [hep-lat]].[5] M. Della Morte et al. [ALPHA Collaboration], JHEP (2014) 060 [arXiv:1312.1566 [hep-lat]].[6] J. Finkenrath, F. Knechtli and B. Leder, Nucl. Phys. B (2013) 441 [arXiv:1306.3962 [hep-lat]].[7] J. M. Flynn et al. [UKQCD Collaboration], Phys. Lett. B (2006) 313 [hep-lat/0506016].(2006) 313 [hep-lat/0506016].