aa r X i v : . [ h e p - l a t ] F e b N = 2 is large Ryuichiro Kitano,

1, 2

Ryutaro Matsudo, Norikazu Yamada,

1, 2 and Masahito Yamazaki High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan Graduate University for Advanced Studies (SOKENDAI), Tsukuba 305-0801, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: February 18, 2021)We study the θ dependence of the free energy density of the four-dimensional SU (2) Yang-Millstheory at zero and high temperatures with lattice numerical simulations. The subvolume methodto create a θ = 0 domain in the lattice conﬁgurations enables us to successfully calculate theenergy density up to θ ∼ π/ . At high temperatures we obtain results predicted by the instantoncomputations. By contrast at zero temperature we ﬁnd that the theory exhibits spontaneous CPbreaking at θ = π in accordance with the large N prediction. We therefore conclude that the SU(2)Yang-Mills theory is in the large N class. The surface tension of the θ = 0 domain turns out to benegative at T = 0 . Introduction

The θ parameter of the 4d Yang-Mills theory controlsrelative weights of diﬀerent topological sectors in the pathintegral. Despite long history, it still remains as a chal-lenging problem to identify the eﬀect of the θ parameteron the non-perturbative dynamics of the theory.For the special value θ = π the Lagrangian has CPsymmetry and we can ask whether or not the CP is spon-taneously broken. In the large N limit [1] spontaneousCP violation at θ = π was demonstrated in Refs. [2–4]. For ﬁnite N a mixed anomaly between the CP sym-metry and the Z N center symmetry shows that the CPsymmetry in the conﬁning phase has to be broken [5].A similar conclusion was derived by studying restora-tion of the equivalence of local observables in SU ( N ) and SU ( N )/ Z N gauge theories in the inﬁnite volume limit [6].(See also Refs. [7–9].) While these theoretical develop-ments have narrowed down possible scenarios, an explicitnonperturbative calculation is necessary to unambigu-ously settle the fate of the CP symmetry at θ = π . Anynumerical simulation directly at θ = π , however, has beendiﬃcult due to the notorious sign problem.In Ref. [10] three of the authors of the present pa-per studied the vacuum energy density of the 4d SU (2)Yang-Mills theory around θ = 0 by lattice numerical sim-ulations. The case of SU (2) gauge group is of particularinterest since N = 2 is farthest away from the large N limit: there is a well-known parallel between 2d CP(1)and 4d SU (2) model [32], and the known vacuum at θ = π in the former [12] alludes to the appearance of gaplesstheory in the latter. By observing that the ﬁrst two nu-merical coeﬃcients in the θ expansion obey the large N scaling [33], it was inferred in Ref. [10] that the 4d SU (2)Yang-Mills theory at θ = π has spontaneous CP break-ing, contrary to the naive expectation from the 2d CP(1)model.In this work, we introduce a new method, the subvol-ume method [34], to explore the θ dependence of the freeenergy without any series expansion in θ . We demon- strate spontaneous CP violation in 4d SU (2) Yang-Millstheory at θ = π at zero temperature, in consistency withthe results of Ref. [10]. Subvolume Method and Lattice Simulation

Let us summarize our numerical simulation methods,highlighting in particular the subvolume method.After generating a number of gauge ﬁeld conﬁgura-tions at θ = 0 and implementing the APE smearing [21],the topological charge density q ( x ) is calculated with theﬁve-loop improved topological charge operator [16] on thelattice. The subvolume topological charge Q sub is com-puted by summing q ( x ) over a subvolume V sub as Q sub = X x ∈ V sub q ( x ) . (1)We then calculate the free energy density of the subvol-ume to be [20, 22] f sub ( θ ) = − V sub ln h cos ( θ Q sub ) i . (2)The free energy density of the whole system is obtainedby extrapolation into the inﬁnite volume: f ( θ ) = lim V sub →∞ f sub ( θ ) . (3)For numerical consistency checks it is also useful to cal-culate their θ -derivatives: d f ( θ ) d θ = lim V sub →∞ d f sub ( θ ) d θ , (4) d f sub ( θ ) d θ = 1 V sub h Q sub sin( θQ sub ) ih cos( θQ sub ) i . (5)The calculation is carried out for the lattice 4d SU (2)Yang-Mills theory at zero and ﬁnite temperatures. Forthe zero temperature simulation, we employ the conﬁgu-rations generated in our previous work [10]. In addition,two ensembles corresponding to T = 1 . T c and . T c arenewly generated at the lattice coupling β = 1 . , thesame value as the zero temperature ensemble. The simu-lation parameters of these ensembles are summarized inTab. I. N S × N T T /T c ﬁt range in n APE statistics a χ × × × × / ( aT c ) = 9 . from Ref. [24] is used to estimate T /T c , where T c is the criticaltemperature at θ = 0 . The topological charge density measured at eachsmearing step is uniformly shifted so that the values ofthe global topological charge Q = P x ∈ V q ( x ) are closestto integers. The measurement is carried out every ﬁvesmearing steps. For the APE smearing, we take α = 0 . in the notation of Ref. [25].Topological observables are distorted by topologicallumps that are taken away by smearing. The smear-ing may also alter physical topological excitations. Westudied this point in detail in Ref. [10], and developed theprocedure to undo the eﬀects of smearing. The procedureconsists of the extrapolation of the observables to the zerosmearing limit by ﬁtting the observables over a suitableinterval of the smearing steps. The ﬁt range is ﬁxed inadvance by examining the response of the global topo-logical charge to the smearing. The resulting ﬁt range isshown in Tab. I. Using the global topological charge Q determined as such, we determined the topological sus-ceptibilities χ = h Q i /V , whose values are summarizedin Tab. I for later use.Each conﬁguration is separated by ten Hybrid MonteCarlo (HMC) trajectories. Statistical errors are esti-mated by the single-elimination jack-knife method withthe binsize of 500 and 100 conﬁgurations for zero andﬁnite temperatures, respectively. Numerical Results

High Temperature

We ﬁrst explore the free energy density at T = 1 . T c ,where the instanton prediction [26], f ( θ ) ∼ χ (1 − cos θ ) ,is believed to be valid. The θ -dependence is exploredin the range of θ k = k π/ with k ∈ [1 , . Us-ing the translational invariance, a single-sized subvolume V sub /a = l × N T is taken from 64 places per a conﬁg-uration, and the size is varied by changing the values of l . The inﬁnite volume limit is obtained by ﬁtting f sub ( θ ) to f sub ( θ ) = f ( θ ) + a s ( θ ) l , (6) a f s ub ( θ ) × l θ=π /2, n APE =2545 θ=π , n

APE =2545 θ=3π /2, n

APE =2545

FIG. 1: Examples of subvolume dependence of f sub ( θ ) andthe extrapolation to the inﬁnite volume. a f ( θ ) × n APE θ=π /2 π3π /2 FIG. 2: The linear extrapolation of f ( θ ) to n APE = 0 . where s ( θ ) denotes the surface tension of the nonzero θ domain. The extrapolation of f sub ( θ ) for several val-ues of n APE and θ are shown in Fig. 1, where n APE isthe number of steps at which f sub ( θ ) is calculated. Thedata in the range of ≤ l ≤ are ﬁtted to (6) withpermissible quality. It is seen that the n APE dependenceshrinks toward the inﬁnite volume limit.The results in the inﬁnite volume limits thus obtainedare then extrapolated to the n APE = 0 limit at each valueof θ with the ﬁt range shown in Tab. I. The results areshown in Fig. 2.Finally, the θ dependence of the free energy density isshown in Fig. 3, where f ( θ ) is normalized by the topolog-ical susceptibility determined using the global topologicalcharge. The results in the large N limit, f ( θ ) /χ = θ / , -1 0 1 2 3 4 5 0 π /2 π π /2 2 π f ( θ ) / χ θ c θ / 21-cos θ∫ d θ df/d θ full volume (n APE =45)

FIG. 3: The θ dependence of f ( θ ) at T = 1 . T c . and the instanton prediction, f ( θ ) /χ = 1 − cos θ , areshown by the solid and dotted curves, respectively. Theresult is consistent with the instanton prediction as itshould be. Note that the observed periodicity θ ∼ θ + 2 π is far from obvious in the subvolume method, as the sur-face tension term in Eq. (6) is not π periodic. Furtherconsistency check is made by comparing the result withthat obtained from the numerical integration of df ( θ ) /dθ ,which is shown as the dashed curve. We observe no sig-niﬁcant deviations in the comparison.The free energy obtained on the full volume showsthe similar behavior even without any extrapolation, asshown in the plot. One may think that this is the simplestmethod to estimate f ( θ ) . However, this method alwaysyields, if available, df ( θ ) /dθ | θ = π = 0 independently ofwhether the spontaneous CP violation occurs or not. Inorder to obtain correct inﬁnite-volume results one wouldhave to repeat the full volume calculation on lattices withvarying full volumes and take the inﬁnite volume limit.The subvolume method skips such a tedious procedure.We performed the same analysis at T = 1 . T c , whichleads to the results almost identical to those in Fig. 3. Zero Temperature

We repeat the same procedure described above at T =0 with subvolumes V sub /a = l from 512 places per aconﬁguration. This time, the calculation sometimes failswhen θ Q sub becomes large because the statistical ﬂuctu-ation makes the jack-knife samples of h cos ( θ Q sub ) i neg-ative, which can be regarded as a reﬂection of the signproblem in this approach. We nevertheless have foundenough data points for our analysis.Figure 4 shows the inﬁnite volume limit of f sub ( θ ) . Inthis case, the ﬁt range in l is set to be ≤ l ≤ keep-ing chi-squared values acceptable. Figure 5 shows thelinear extrapolation to the n APE = 0 limit. The linear ﬁtworks well for the chosen ﬁt range in Tab. I. Finally, theresulting f ( θ ) and df ( θ ) /dθ are shown in Fig. 6 together a f s ub ( θ ) × l θ=π /2, n APE =2040 θ=π , n

APE =2040 θ=3π /2, n

APE =2040

FIG. 4: The linear extrapolation of f sub ( θ ) to the inﬁnitevolume limit. a f ( θ ) × n APE θ=π /2 π3π /2 FIG. 5: The linear extrapolation of the energy density a f ( θ ) to the n APE = 0 limit. with the predictions from the large N and the instantoncalculus. The numerical consistency between f ( θ ) and df ( θ ) /dθ is checked and shown in the dashed curves. Inthe T = 0 case, the calculation around θ = 2 π fails andno result is available. It is important to note that clearly d f ( θ ) /dθ | θ = π = 0 . This means that the CP symmetry at θ = π is spontaneously broken at the zero temperature as d f ( θ ) /dθ = − i h q ( x ) i is CP odd. One can conclude thatthe 4d SU(2) Yang-Mills theory is in the large- N classunlike the 2d CP model.In Fig. 7, the resulting surface tension s ( θ ) is com-pared with that for T = 1 . T c . It turns out that thetension strongly depends on both θ and temperature T ,and is negative at T = 0 . π /2 π π /2 2 π f ( θ ) / χ θ T = 0 θ / 21-cos θ∫ d θ df/d θ full volume (n APE =40)-1 0 1 2 3 4 5 0 π /2 π π /2 2 π d f ( θ ) / d θ / χ θ T = 0 θ sin θ∆ ffull volume (n APE =40)

FIG. 6: The θ dependence of energy density f ( θ ) (top) andits derivative df ( θ ) /dθ (bottom). Discussion and Conclusion