Topological susceptibility in lattice Yang-Mills theory with open boundary condition
Abhishek Chowdhury, A. Harindranath, Jyotirmoy Maiti, Pushan Majumdar
PP REPARED FOR SUBMISSION TO
JHEP
Topological susceptibility in lattice Yang-Mills theorywith open boundary condition
Abhishek Chowdhury a , A. Harindranath a , Jyotirmoy Maiti b and Pushan Majumdar c a Theory Division, Saha Institute of Nuclear Physics1/AF Bidhan Nagar, Kolkata 700064, India b Department of Physics, Barasat Government College,10 KNC Road, Barasat, Kolkata 700124, India c Department of Theoretical Physics,Indian Association for the Cultivation of Science, Kolkata 700032, India
E-mail: [email protected] , [email protected] , [email protected] , [email protected] A BSTRACT : We find that using open boundary condition in the temporal direction can yield theexpected value of the topological susceptibility in lattice SU(3) Yang-Mills theory. As a furthercheck, we show that the result agrees with numerical simulations employing the periodic boundarycondition. Our results support the preferability of the open boundary condition over the periodicboundary condition as the former allows for computation at smaller lattice spacings needed forcontinuum extrapolation at a lower computational cost. a r X i v : . [ h e p - l a t ] J un ontents An open problem in numerical simulation of lattice QCD is that sampling gauge configurationsover different topological sectors becomes more and more difficult as the continuum limit is ap-proached. As a consequence, autocorrelation times of physical quantities grow rapidly making thecalculation of expectation values time consuming. To partially overcome this problem, using openboundary conditions (instead of the usual periodic or anti-periodic ones) in the temporal directionof the lattice has been proposed [1]. Lattice gauge theory with such boundary conditions have nobarriers between different topological sectors. This has been shown by extensive simulations inSU(3) gauge theory [2]. Even though the open boundary conditions introduce boundary effectsand thus complicate the physics analysis, their advantage from the point of view of ergodicity andefficiency have been addressed in simulations of 2+1 flavours of O ( a ) improved Wilson quarks [3].Advantages of using open boundary conditions have also been studied in the investigation of SU(2)lattice gauge theory at weak coupling [4].In the context of topology of gauge fields, an interesting quantity to study is the topologi-cal susceptibility ( χ ) in pure Yang-Mills theory which is related to the η (cid:48) mass by the famousWitten-Veneziano formula [5–7]. For recent high precision calculations of χ with periodic bound-ary condition see, for example, Refs. [8–10]. Ref. [8] uses Ginsparg-Wilson fermion for thetopological charge density operator whereas Ref. [9] uses the algebraic definition based on fieldstrength tensor. A proposal to overcome the problem of short distance singularity in the computa-tion of topological susceptibility is given in Refs. [11, 12]. Ref. [10] employs a spectral-projectorformula which is designed to be free from singularity and compares the result with that using thealgebraic definition. The results using different approaches are in agreement with each other withinstatistical uncertainties.In this work we address the question whether an open boundary condition in the temporaldirection can yield the expected value of the topological susceptibility in SU(3) Yang-Mills theory.We employ the algebraic definition for the topological charge density used in Ref.[10] and for ameaningful comparison with Ref.[10] Wilson flow is used to smoothen the gauge field. We alsoperform simulations with periodic boundary conditions. We find that using an open boundary– 1 –attice Volume β N cnfg N τ a [ fm ] t / a O ×
48 6.21 3970 12 3 0.0667(5) 6.207(15) O ×
64 6.42 3028 20 4 0.0500(4) 11.228(31) O ×
96 6.59 2333 26 5 0.0402(3) 17.630(53) P ×
48 6.21 3500 12 3 0.0667(5) 6.197(15) P ×
64 6.42 1958 20 4 0.0500(4) 11.270(38)
Table 1 . Simulation parameters for the HMC algorithm. N is the number of integration steps, τ is thetrajectory length and t / a is the dimensionless reference Wilson flow time. condition is advantageous as it allows one to sample different topological sectors by removing thebarrier between them.Unlike the periodic lattice, any physical quantity measured on a lattice with open boundaryalso has the additional boundary term along with the bulk part (see for example Ref. [13]). Ina simulation with all other parameters kept identical, the difference between the results for somephysical quantity measured on a finite volume system with open and periodic boundary gives theboundary contribution for the system with the open boundary. As this boundary contribution di-minishes with increasing volume, result from a system with open boundary approaches the samefrom a system with periodic boundary conditions. We have generated gauge configurations in SU(3) lattice gauge theory at different lattice vol-umes and gauge couplings using the openQCD program [14]. Gauge configurations using periodicboundary conditions also have been generated for several of the same lattice parameters (necessarychanges to implement periodic boundary condition in temporal direction were made in the openQCD package for pure Yang-Mills case). Details of the simulation parameters are summarized in table1. In this table, O and P correspond to open and periodic boundary configurations respectively.Topological susceptibility is measured over N cnfg number of configurations with two succes-sive ones separated by 32 thus making the total length of simulation time to be N cnfg ×
32. Thelattice spacings quoted in table 1 are determined using the results from Refs. [15, 16]. To smoothenthe gauge configurations, Wilson flow [17–19] is used and the reference flow time t is determinedthrough the implicit equation (cid:8) t (cid:104) E ( T / ) (cid:105) (cid:9) t = t = . t is the Wilson flow time, T is the temporal extent of the lattice and E is the time slice averageof the action density given in Ref. [2]. Through this equation, the reference flow time provides areference scale to calculate the physical quantities from lattice data. An alternative to the t scaleis the w scale proposed in Ref. [20]. We don’t see any significant difference in our results usingthe two different scales. – 2 – Q simulation time Figure 1 . Trajectory history of topological charge ( Q ) versus simulation time at β = .
59 and lattice volume48 ×
96 for open boundary condition (top) and periodic boundary condition (bottom). The data shown is atWilson flow time t / a = The open boundary condition has been proposed to help the tunneling of the system between dif-ferent topological sectors characterized by the corresponding topological charge ( Q ) as one ap-proaches the continuum limit. To that end we first compare the trajectory history of Q for openversus periodic boundary conditions for a reasonably small lattice spacing. In figure 1 we plot thefluctuation of Q versus simulation time at β = . ( a = . ) and lattice volume 48 ×
96 foropen boundary condition (top) and periodic boundary condition (bottom) both starting from ran-dom configurations. The data shown is at Wilson flow time t / a =
2. Unless otherwise stated, allthe data presented in the following are at the reference Wilson flow time ( t ). It is evident that withopen boundary condition, thermalization is reached very fast whereas with periodic boundary con-dition it takes a long time just to reach thermalization. It is also evident that after thermalization,autocorrelation length is much larger for the periodic boundary condition compared to the openboundary condition. We have checked that the variation is not so marked for periodic boundaryconditions at larger lattice spacings.Next we look at the distribution of Q . In figure 2 along with time histories, we plot thehistogram obtained for Q . Top one (blue) is open boundary condition and bottom (red) is periodic– 3 –
700 1400-10-50510 10 20 30-10-505100 700 1400-10-50510 100 200 300-10-50510 Q N cnfg Figure 2 . Distribution of Q versus N cn f g . Top one (blue) is open boundary condition and bottom (red) isperiodic boundary condition at β = .
42 and lattice volume is 32 × boundary condition at β = .
42 and lattice volume is 32 ×
64. We note that (1) as expected fromthe boundary conditions, top (blue) Q is not an integer whereas for bottom (red), it is an integerand (2) even for this coupling ( β = .
42) which is lower compared to figure 1, taking the samenumber of configurations, the top one gives much better spanning than the bottom. In the plot ofhistograms in this figure, we have used bin sizes of 0 . q ( x ) ). We denote q ( x ) integrated over the spatial volume at fixed Euclidean time x by Q ( x ) . Thechange in the behaviour of Q ( x ) as a function of time slice x reveals the effect of open boundaryin the temporal direction. The distribution of Q ( x ) versus N cn f g is presented in figure 3 for theensemble O where x = , , , , β = .
42 andlattice volume 32 ×
64. The distribution of Q ( x ) is calculated with bin size of 0 .
01. As we movefrom close to the boundary to deeper in the bulk, the spanning of Q ( x ) steadily increases andfinally settles down in the bulk region. The same behaviour is also observed at the other end of thetemporal lattice.The topological susceptibility is defined as χ = (cid:104) Q (cid:105) V – 4 – Q ( x ) N cnfg Figure 3 . Distribution of Q ( x ) versus N cn f g for the ensemble O where x = , , , , β = .
42 and lattice volume 32 × where V is the space-time volume. To investigate the effect of open boundary on susceptibility wedefine a subvolume susceptibility [21] as follows: χ ( ∆ x ) = (cid:104) ˜ Q (cid:105) ˜ V where ˜ Q is the q ( x ) integrated over the spatial volume and temporal length ( ∆ x ) which is takensymmetrically over the mid point of the temporal direction. The subvolume ˜ V is the product ofspatial volume and ∆ x . In figure 4 we plot χ versus ∆ x for the ensembles O , O and O . Due toopen boundary in the temporal direction, there is slight dip close to the temporal boundary whichis consistent with the behaviour of Q ( x ) as shown in figure 3. We find that, overall, the effect ofthe open boundary on the subvolume susceptibility is within the statistical uncertainties.It is interesting to study the stability of χ with respect to Wilson flow time. In figure 5, we showthe behaviour of χ for both open and periodic boundary condition under Wilson flow plotted versusthe flow time for different lattice spacings and lattice volumes. For very early flow times, χ showsnon-monotonous behaviour for both open and periodic boundary condition. For later flow times, χ converges from above to a plateau for open boundary condition whereas it converges from belowfor the periodic boundary condition. The values of susceptibility extracted at the reference flowtime t are given in table 2 and plotted in figure 6. In the figure 6, we show χ / in dimensionful– 5 – χ a ∆ x O O O Figure 4 . Subvolume susceptibility ( χ ) versus temporal length ( ∆ x ) for the ensembles O , O and O . Lattice a χ / − χ / [ MeV ] O O O P P Table 2 . Topological susceptibility. unit plotted against a for both open and periodic boundary condition for different lattice spacingsand volumes. We find that the results for open and periodic lattices are very close to each other ata given physical volume.For comparison, data from Ref. [10] for periodic boundary condition is also plotted. Alsoshown are the linear fits to the data Ref. [10] (green lines) and the data for open boundary condition(blue lines). The extracted value of χ / for the open boundary condition data is 184.7 (1.7) MeVwhich compares well with the result 187.4 (3.9) MeV of Ref. [10].– 6 – t/r χ r β = 6.21 open β = 6.21 periodic β = 6.42 open β = 6.42 periodic β = 6.59 open Figure 5 . Behaviour of topological susceptibility for both open and periodic boundary condition underWilson flow plotted versus the flow time for different lattice spacings and lattice volumes.
In this work we have shown that the open boundary condition in the temporal direction can yieldthe expected value of the topological susceptibility in lattice SU(3) Yang-Mills theory. The resultsagree with numerical simulations employing periodic boundary condition. The advantage of openboundary conditions over periodic boundary conditions (see, however, Ref. [22]) are illustrated infigure 1.As further avenues of investigation, detailed comparison between Wilson flow and conven-tional smearing techniques used for smoothening gauge fields and the same between differentalgebraic as well as chirally improved fermionic definitions of topological charge density are inprogress. It is also interesting to compute the topological charge density correlator (see Ref.[23]and the references therein) using open boundary.
Acknowledgements
Numerical calculations are carried out on the Cray XT5 and Cray XE6 systems supportedby the 11th-12th Five Year Plan Projects of the Theory Division, SINP under the DAE, Govt. ofIndia. We thank Richard Chang for the prompt maintenance of the systems and the help in data– 7 – a (fm ) χ / ( M e V ) CERN periodicopenperiodic
Figure 6 . χ / in dimensionful unit plotted versus a for both open and periodic boundary condition fordifferent lattice spacings and lattice volumes. For comparison, data from Ref.[10] for periodic boundarycondition is also plotted. Also shown are the linear fits to the data Ref.[10] (green lines) and the data foropen boundary condition (blue lines). management. This work was in part based on the publicly available lattice gauge theory code openQCD [14]. References [1] M. Luscher, PoS LATTICE , 015 (2010) [arXiv:1009.5877 [hep-lat]].[2] M. Luscher and S. Schaefer, JHEP , 036 (2011) [arXiv:1105.4749 [hep-lat]].[3] M. Luscher and S. Schaefer, Comput. Phys. Commun. , 519 (2013) [arXiv:1206.2809 [hep-lat]].[4] M. Grady, arXiv:1104.3331 [hep-lat].[5] E. Witten, Nucl. Phys. B , 269 (1979).[6] G. Veneziano, Nucl. Phys. B , 213 (1979).[7] E. Seiler, Phys. Lett. B , 355 (2002) [hep-th/0111125].[8] L. Del Debbio, L. Giusti and C. Pica, Phys. Rev. Lett. , 032003 (2005) [hep-th/0407052].[9] S. Durr, Z. Fodor, C. Hoelbling and T. Kurth, JHEP , 055 (2007) [hep-lat/0612021]. – 8 –
10] M. Luscher and F. Palombi, JHEP , 110 (2010) [arXiv:1008.0732 [hep-lat]].[11] M. Luscher, Phys. Lett. B , 296 (2004) [hep-th/0404034].[12] L. Giusti and M. Luscher, JHEP , 013 (2009) [arXiv:0812.3638 [hep-lat]].[13] H. Asakawa and M. Suzuki, J. Phys. A: Math. Gen. , 7811 (1996).[14] http://luscher.web.cern.ch/luscher/openQCD/ [15] M. Guagnelli et al. [ALPHA Collaboration], Nucl. Phys. B , 389 (1998) [hep-lat/9806005].[16] S. Necco and R. Sommer, Nucl. Phys. B , 328 (2002) [hep-lat/0108008].[17] M. Luscher, Commun. Math. Phys. , 899 (2010) [arXiv:0907.5491 [hep-lat]].[18] M. Luscher, JHEP , 071 (2010) [arXiv:1006.4518 [hep-lat]].[19] M. Luscher and P. Weisz, JHEP , 051 (2011) [arXiv:1101.0963 [hep-th]].[20] S. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth and L. Lellouch et al. ,JHEP , 010 (2012) [arXiv:1203.4469 [hep-lat]].[21] P. de Forcrand, M. Garcia Perez, J. E. Hetrick, E. Laermann, J. F. Lagae and I. O. Stamatescu, Nucl.Phys. Proc. Suppl. , 578 (1999) [hep-lat/9810033].[22] G. McGlynn and R. D. Mawhinney, arXiv:1311.3695 [hep-lat].[23] A. Chowdhury, A. K. De, A. Harindranath, J. Maiti and S. Mondal, JHEP , 029 (2012)[arXiv:1208.4235 [hep-lat]]., 029 (2012)[arXiv:1208.4235 [hep-lat]].