Studying and removing effects of fixed topology in a quantum mechanical model
aa r X i v : . [ h e p - l a t ] O c t Studying and removing effects of fixed topology in aquantum mechanical model
Arthur Dromard ∗ Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, GermanyE-mail: [email protected]
Marc Wagner
Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, GermanyE-mail: [email protected]
At small lattice spacing, or when using e.g. overlap fermions, lattice QCD simulations tend tobecome stuck in a single topological sector. Physical observables then differ from their full QCDcounterparts by 1 / V corrections, where V is the spacetime volume. Brower et al. and Aoki et al.have derived equations by means of a saddle point approximation, to determine and to removethese corrections. We extend these equations and apply them to a simple toy model, a quantummechanical particle on a circle in a square well potential at fixed topology. This model can besolved numerically up to arbitrary precision and allows to explore effects arising due to fixedtopology. We investigate the range of validity and accuracy of the above mentioned equations, toremove such fixed topology effects. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ tudying and removing effects of fixed topology
Arthur Dromard
1. Introduction
Topology freezing or fixing are important issues in quantum field theory, in particular in QCD.For example, when simulating chirally symmetric overlap quarks, the corresponding algorithmsdo not allow transitions between different topological sectors, i.e. they fix the topological charge(cf. e.g. [1]). Also when using other quark discretizations, e.g. Wilson fermions, topology freezingis expected at lattice spacings a < ∼ .
05 fm, which are nowadays still rather fine, but realistic [2].There are also applications, where one might fix topology on purpose. For example, when using amixed action setup with light overlap valence and Wilson sea quarks, approximate zero modes inthe valence sector are not compensated by the sea. The consequence is an ill-behaved continuumlimit [3]. Since such approximate zero modes only arise at non-vanishing topological charge, fixingtopology to zero might be a way to circumvent the problem. A possible solution to this problemis to restrict computations to a single topological sector, either by sorting the generated gauge linkconfigurations with respect to their topological charge or by directly employing so-called topologyfixing actions (cf. e.g. [4, 5, 6]).In view of these issues it is important to develop methods, which allow to obtain physicallymeaningful results (i.e. results corresponding to unfixed topology) from fixed topology simulations.The starting point for our work are the seminal papers [7, 8]. We extend the calculations from thesepapers and propose and test a corresponding method by applying it to a simple quantum mechanicaltoy model. Similar recent investigations using the Schwinger model can be found in [9, 10, 11].
2. Working at fixed topology
The partition function and the temporal correlation function of a hadron creation operator O atfixed topological charge Q and finite spacetime volume V are given by Z Q , V = ˆ DA D y D ¯ yd Q , Q [ A ] e − S E [ A , ¯ y , y ] (2.1) C Q , V ( t ) = Z Q , V ˆ DA D y D ¯ yd Q , Q [ A ] O ( t ) O ( ) e − S E [ A , ¯ y , y ] . (2.2)For sufficiently large V one can use a saddle point approximation and expand the correlation func-tion according to C Q , V ( t ) = A Q , V e − M Q , V t , M Q , V = M ( ) + M ′′ ( ) c t V (cid:18) − Q c t V (cid:19) + O ( / V ) (2.3)[7], where the expansion is in the three parameters M ′′ ( ) t / c t V , 1 / c t V and Q / c t V . M Q , V isthe mass of the hadron excited by O at fixed topological charge Q and finite spacetime volume V , M ( q ) is the hadron mass in a q -vacuum at infinite V (cf. e.g. [12]), M ( ) = M ( q = ) is thephysical hadron mass (i.e. the hadron mass at unfixed topology), M ′′ ( ) = d M ( q ) / d q | q = and c t denotes the topological susceptibility. A straightforward method to determine physical hadron masses (i.e. hadron masses at unfixedtopology) from fixed topology simulations based on the equations of the previous subsection hasbeen proposed in [7]: 2 tudying and removing effects of fixed topology
Arthur Dromard
1. Perform simulations at fixed topology for different topological charges Q and spacetimevolumes V , for which the expansion (2.3) is a good approximation, i.e. where M ′′ ( ) t / c t V ,1 / c t V and Q / c t V are sufficiently small. Determine M Q , V using (2.3) for each simulation.2. Determine the physical hadron mass M ( ) (the hadron mass at unfixed topology and infinitespacetime volume), M ′′ ( ) and c t by fitting (2.3) to the masses M Q , V obtained in step 1. In the following we improve the expansions (2.3) by explicitly calculating higher orders pro-portional to 1 / V and 1 / V . The starting point for this calculus has been a general discussion ofthese higher orders for arbitrary n -point functions at fixed topology [8]. The lengthy result, whichwe will derive in detail in an upcoming publication, is C Q , V ( t ) = A ( + x ) / exp (cid:26) − M ( ) t + c t V (cid:20) − E c t (cid:18) + x ( + x ) − (cid:19) + Q x ( + x ) (cid:21) + ( c t V ) (cid:20) + E c t (cid:18) ( + x ) ( + x ) − (cid:19) − E c t (cid:18) + x ( + x ) − (cid:19) + Q E c t (cid:18) + x ( + x ) − (cid:19)(cid:21) + ( c t V ) (cid:20) − E c t (cid:18) ( + x ) ( + x ) − (cid:19) − E c t (cid:18) + x ( + x ) − (cid:19) + E E c t (cid:18) ( + x )( + x )( + x ) − (cid:19) − Q E c t (cid:18) ( + x ) ( + x ) − (cid:19) + Q E c t (cid:18) + x ( + x ) − (cid:19) − Q E c t (cid:18) + x ( + x ) − (cid:19)(cid:21)(cid:27) + O ( / V ) , (2.4)where E n denotes the n -th derivative of the energy density of the vacuum with respect to q at q = c t = E ) and x n ≡ M ( n ) ( ) t / E n V .In principle this improved expansion can directly be used in the fitting procedure outlinedin section 2.2. Note, however, that there are six additional unknown parameters (compared to(2.3)),which have to be determined via fitting: E , E and E , M ( ) ( ) , M ( ) ( ) and M ( ) ( ) . Acompromise between improvement on the one hand and a small number of parameters on the otherhand, which seems to work well in practice (cf. section 3), is to set these six new parameters tozero. Then only 1 / V and 1 / V corrections, which are associated with the old parameters M ( ) , M ′′ ( ) and c t are taken into account.
3. Testing the method in quantum mechanics
To test the method described in section 2.2, we decided for a simple toy model, a quantum3 tudying and removing effects of fixed topology
Arthur Dromard mechanical particle on a circle in a square well potential: S E ( q ) ≡ ˆ V dt (cid:18) I j + U ( j ) (cid:19) − i q p ˆ V dt ˙ j | {z } = Q (3.1) U ( j ) ≡ ( + U if − ( p − L ) < j < +( p − L ) V denotes the finite extent of the periodic time [the analog of the spacetime volume in QCD]).This model shares some characteristic and important features of QCD: the existence of topologicalcharge (paths with topological charge Q = Q = + + q ↔ − q and the existence of both bound states and scattering states. Moreover, it can be solvednumerically up to arbitrary precision (no simulations required). For the results presented in thissection we have used I = U =
10 and L = p / (cid:1) (cid:1) (t) (cid:1) (cid:1) (t) Q=0 Q=1
Figure 1:
Classical paths with topological charge Q = Q = Since parity is a symmetry, the energy eigenstates can be classified according to P = + and P = − (in the following energy eigenvalues of P = + [ P = − ] states are denoted by E + ( q ) [ E − ( q ) ], n = , , , . . . ). The “mass” M ( q ) we are going to study in the following is defined as the energydifference between the ground state (which has P = + ) and the lowest energy eigenstate in the P = − sector, i.e. M ( q ) ≡ E − ( q ) − E + ( q ) . A suitable creation operator O for a temporal correlationfunction C ( t ) , whose exponential behavior yields M ( q ) , is O ≡ sin ( j ) . (2.3) as well as (2.4) are expansions in the small parameters M ′′ ( ) t / c t V , 1 / c t V and Q / c t V .We are interested to estimate upper bounds for these parameters such that the determination of M ( ) as outlined in section 2.2 is sufficiently precise. We proceeded as follows:1. Solve Schrödinger’s equation with H ( q ) = ( p − q / p ) / I + U ( j ) . Use the resulting en-ergy eigenvalues E + n ( q ) and E − n ( q ) to determine M ( ) , M ( ) ( ) , M ( ) ( ) , M ( ) ( ) , M ( ) ( ) , c t , E , E and E , the parameters of the C Q , V expansions (2.3) and (2.4).2. Calculate C q , V ( t ) using the energy eigenvalues from step 1 and the corresponding wave func-tions. Perform a Fourier transformation to obtain C Q , V ( t ) , the exact correlation function atfixed topology. Define and calculate the effective mass M eff Q , V ( t ) ≡ − ddt ln (cid:16) C Q , V ( t ) (cid:17) . (3.3)4 tudying and removing effects of fixed topology Arthur Dromard
3. Determine expansions for the effective mass using (2.3), (2.4) and (3.3) and compare and/orfit the resulting expressions to their exact counterpart (3.3).Note that in QCD the exact correlator C Q , V ( t ) and the corresponding exact effective mass M eff Q , V ( t ) at fixed topological charge Q and spacetime volume V will be provided by lattice simulations. In Figure 2 we show effective masses (3.3) (exact results, not expansions) as a functions of thetemporal separation for different topological charges Q . As usual at small temporal separations theeffective masses are decreasing, due to the presence of excited states. At large temporal separationsthere are also severe deviations from a constant behavior. This is in contrast ordinary quantummechanics or quantum field theory (at unfixed topology) and is caused by topology fixing. Atintermediate temporal separations there are plateau-like regions (shaded in gray in Figure 2), wherethe expansions (2.3) and (2.4) will turn out to be rather accurate approximations (cf. section 3.2.2).Note that with increasing topological charges Q the plateau-like regions of M eff Q , V ( t ) become smaller.A similar trend is observed for decreasing temporal extension V . plateau likeregion t M Q , V Q=± top. charge
Q=± Q=± Q=± eff plateau-likeregion Figure 2:
Effective masses M eff Q , V as functions of the temporal separation t for different topological charges Q and fixed V = / c t . Figure 3 shows again the gray region of Figure 2, where the three panels correspond to Q = , ± , ±
2. This time not only the exact effective mass is plotted (blue curve), but also variousexpansions: green curve , the expansion (2.3) from [7] (three parameters); cyan curve , our improved expansion (2.4) with nine parameters (cf. section 2.3); red curve , our improved expansion (2.4) with three parameters (cf. section 2.3).Clearly the two improved expansions (cyan, red) are much closer to the exact result (blue), than theunimproved expansion (green). Since there does not seem to be a qualitative difference betweenthe two improved expansions, the version with only three parameters (red) seems to be the bestcandidate for our model to determine the mass M ( ) at unfixed topology via fitting.5 tudying and removing effects of fixed topology Arthur Dromard
Figure 3 as well as similar plots for many different topological charges Q , temporal extensions V and parameters of the model allow to crudely estimate a region, where the deviations betweenour improved expansions (2.4) and the exact result (3.3) is < ∼ | M ′′ ( ) | t / c t V < ∼ .
5, 1 / c t V < ∼ . Q / c t V < ∼ Q=0
M’’ 0 t (cid:1) t V t M’’ 0 t (cid:0) t V t Q=±1 Q=±2
M’’ 0 t (cid:2) t V t M Q , V eff M Q , V eff hep-lat/0302005 |M''(0)|t<0.5V χ t |M''(0)|t<0.5V χ t |M''(0)|t<0.5V χ t M Q , V eff Figure 3:
Comparison of the exact effective mass M eff Q , V and various expansions as functions of the temporalseparation t for different topological charges Q and fixed V = / c t . In Figure 4 we mimic the method to determine a physical mass (i.e. at unfixed topology)outlined in section 2.2. We use the exact result for the effective mass to generate masses at fixedtopological charge and finite temporal extent, M Q , V ≡ M eff Q , V ( t = ) (the dots in Figure 4, step 1 insection 2.2). Then we perform a single fit of either the expansion (2.3) from [7] or our improvedversion (2.4) with three parameters at t =
10 inserted in (3.3) to these masses M Q , V (only valuesfulfilling 1 / c t V < ∼ . Q / c t V < ∼ M ( ) (the physical mass at unfixed topology), M ′′ ( ) and c t (the curves in Figure 4, step 2 in section 2.2).Both expansions give rather accurate results for M ( ) (the error is of the order of 0 . c t (an error of a few percent). Note that the error for both M ( ) and c t issignificantly smaller, when using the improved expansion (2.4), as shown in Figure 4.
4. Conclusion
We have tested a method to extract physical masses (i.e. masses at unfixed topology) fromcalculations or simulations at fixed topology and finite temporal extent or spacetime volume. Themethod provides accurate results with errors significantly below 1%, when applied to a quantummechanical toy model. Therefore, it might be a promising candidate to eliminate unwanted fixedtopology effects also in QCD.The method is based on an expansion in 1 / V (inverse powers of the spacetime volume). Wehave improved this expansion from [7] by including higher orders proportional to 1 / V and 1 / V and we demonstrated that these higher orders significantly reduce the associated error.We also explored the range of validity of the method, when applied to our quantum mechanicaltoy model: | M ′′ ( ) | t / c t V < ∼ .
5, 1 / c t V < ∼ . Q / c t V < ∼ . c t V = O ( ) .6 tudying and removing effects of fixed topology Arthur Dromard exact results error M M'' χ T Q=0
Q=1
Q=2
Q=3 h(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) o(cid:10)(cid:11)(cid:12)(cid:13) m(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)t(cid:20)(cid:21)(cid:22) (cid:23)(cid:24) tanon (cid:25)(cid:26)(cid:27)(cid:28)(cid:29) (cid:30)(cid:31) !
Figure 4:
Determining the physical mass (i.e. the mass at unfixed topology) from fixed topology results.
Acknowledgments
We thank Wolfgang Bietenholz, Krzysztof Cichy, Christopher Czaban, Dennis Dietrich, Gre-gorio Herdoiza and Karl Jansen for discussions. M.W. acknowledges support by the Emmy NoetherProgramme of the DFG (German Research Foundation), grant WA 3000/1-1. This work was sup-ported in part by the Helmholtz International Center for FAIR within the framework of the LOEWEprogram launched by the State of Hesse.
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