Monte-Carlo simulations of overlap Majorana fermions
MMonte-Carlo simulations of overlap Majorana fermions
Stefano Piemonte ∗1 , Georg Bergner †2,3 , and Camilo López ‡21 University of Regensburg, Institute for Theoretical Physics, Universitätsstr. 31,D-93040 Regensburg, Germany University of Jena, Institute for Theoretical Physics, Max-Wien-Platz 1, D-07743 Jena,Germany University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str. 9,D-48149 Münster, Germany6th May 2020
Abstract
Supersymmetric Yang-Mills (SYM) theories in four dimensions exhibit many interestingnon-perturbative phenomena that can be studied by means of Monte Carlo lattice simula-tions. However, the lattice regularization breaks supersymmetry explicitly, and in general afine tuning of a large number of parameters is required to correctly extrapolate the theoryto the continuum limit. From this perspective, it is important to preserve on the latticeas many symmetries of the original continuum action as possible. Chiral symmetry for in-stance prevents an additive renormalization of the fermion mass. A (modified) version ofchiral symmetry can be preserved exactly if the Dirac operator fulfills the Ginsparg-Wilsonrelation. In this contribution, we present an exploratory non-perturbative study of N = 1supersymmetric Yang-Mills theory using the overlap formalism to preserve chiral symmetryat non-zero lattice spacings. N = 1 SYM is an ideal benchmark toward the extension of ourstudies to more complex supersymmetric theories, as the only parameter to be tuned is thegluino mass. Overlap fermions allow therefore to simulate the theory without fine-tuning.We compare our approach to previous investigations of the same theory, and we present clearevidences for gluino condensation. Supersymmetric Yang-Mills theories (SYM) are promising extensions of the Standard Model(SM) to energies of the order and beyond the TeV scale. However, as the efforts for the experi-mental discovery of the predicted super-particles are still unsuccessful, the interest to SUSY hasbeen recently motivated by the possibility of understanding in supersymmetric theories manynon-perturbative features of strong interactions. Yang-Mills theories exhibit many emergent ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - l a t ] M a y ow-energy phenomena, like confinement and chiral symmetry breaking, that are difficult to un-derstand analytically. The conjecture that there exist supersymmetric models that share thesame physical properties if their coupling constant g is exchanged by its inverse 1 /g opens newperspectives for the analytical understanding of the behavior of Yang-Mills theories. Well-knownexamples of such connection between different models are the AdS/CFT correspondence and theSeiberg-Witten electromagnetic duality.The electromagnetic duality has been proposed originally for a certain class of theories inRef. [1, 2], such as SuperQCD and N = 2 Supersymmetric Yang-Mills theory. Lattice Monte-Carlo simulations are a useful tool to test non-perturbatively the validity of the Seiberg-Wittenduality, in particular when generalizing to N = 1 SYM or QCD the proposed mechanisms ofconfinement, such as the dual superconductor picture. The extension of the electromagneticduality to more realistic models with less or no supersymmetry requires, in fact, to control thebehavior of the theory in the limit, where the mass of the scalar and gluino fields becomes large.However, in this limit SUSY is partially or softly broken and completely different confinementscenarios could be realized. Lattice simulations would be able to investigate a-priori and ab-initiothe phase diagram of strong interactions with intact, broken, or partially-broken SUSY.The presence of many different fermion and scalar fields in the Lagrangian of N = 2 SYM andSuperQCD has the consequence to complicate the fine-tuning of lattice simulations. In general,supersymmetry is explicitly broken on the lattice and it can be restored at large distances asan accidental symmetry only after an accurate tuning of the bare couplings and masses of theaction. All couplings corresponding to relevant operators compatible with lattice symmetriesmust be considered, resulting into a number of parameters to tune of O (10), depending on thegauge group and on the number of fields in the original continuum action. N = 1 SYM is anexceptional case, since the only relevant parameter to be tuned is the gluino mass.The tuning problem can be simplified drastically if a (modified) chiral symmetry is preservedon the lattice by using Ginsparg-Wilson fermions. For instance, the fermion mass would beprotected by chiral symmetry, and Yukawa interactions would couple correctly left-handed andright-handed spinors to the corresponding scalar fields. The advantages of chiral fermions mightbe crucial for simulations of certain supersymmetric theories. Ginsparg-Wilson fermions allowto simulate in principle N = 1 SYM without any fine-tuning.Models with enlarged SUSY can be constructed from simpler supersymmetric theories livingin a space-time with extra dimensions. The pure N = 2 SYM in four dimensions can be obtainedfrom the compactification of N = 1 SYM in six dimensions [3]. Such construction is particularlyinteresting from the perspective of lattice simulations [4]. If N = 2 SYM arises as an effectivefour-dimensional theory, the scalar mass and the quartic scalar potential would be protected bygauge symmetry. This is due to the fact, that scalars emerge in the small compactification limitfrom the gauge fields living along the extra dimensions [4, 5].In this paper we explore overlap lattice simulations for the pure N = 1 SYM in four dimen-sions. The study of the overlap formulation for N = 1 SYM is a crucial step toward the extensionof Monte Carlo simulations of more complex supersymmetric theories, a step required to solvethe many numerical challenges coming from the Majorana nature of the gluino. In section 2we briefly describe the formulation of N = 1 SYM, while in section 3 we review the overlapformalism. In section 4 we discuss in details the implementation of the overlap formalism for2 = 1 SYM in four dimensions and in section 4.1 we summarize the parameters of the latticesimulations. In section 5 and 6 we present the determination of the scale and of the gluinocondensate. In section 7 and 8 we briefly discuss the expected consequences of volume reductionand of the fermion-boson cancellation of the Witten index on bulk quantities. Finally, beforethe outlook and the conclusions, in section 9 we present a first determination of the bound statespectrum using Ginsparg-Wilson chiral fermions. N = 1 Supersymmetric Yang-Mills theory
The simplest Yang-Mills theory with a single conserved supercharge is N = 1 SYM, describingthe interactions between gluons and their fermion superpartners, the gluinos. The on-shell actionreads S N =1 SYM = Z d x (cid:26) − F µνa F aµν + i λ a γ µ ( ∇ µ ) ab λ b + θ π g F µν ˜ F µν (cid:27) , (1)being F µν and ˜ F µν the field-strength tensor and its dual. The covariant derivative ( ∇ µ ) ab actson the gluino fields λ b in the adjoint representation of the gauge group SU( N c ). The gluino is aMajorana fermion. The angle θ is set to zero in Monte-Carlo lattice simulations.Intact supersymmetry implies an exact cancellation between fermion and boson loops ina perturbative asymptotic expansion, allowing an analytic calculation of many quantities andproperties that would be out the reach of perturbation theory in non-supersymmetric theories. Aremarkable example is the exact β -function, representing the dependence of the strong couplingconstant on the scale. N = 1 supersymmetric Yang-Mills theory is asymptotically free and the β -function is known exactly, including non-perturbative contributions, from instanton calculus[6]. Asymptotic freedom implies that an ultraviolet cut-off, such as 1 /a of the lattice discretizedtheory, can be sent to infinity in the limit g → + [7]. No further tunings are required if chiralsymmetry is preserved exactly in the regularized theory, even at energy scales of the order ofthe ultraviolet cut-off. The overlap formalism allows to construct a fermion action with an exact (modified) chiralsymmetry even at non-zero lattice spacing. The overlap operator is defined as D ov ( µ ) = 1 + µ − µ γ sign( D H ) , (2)where D H = γ D W ( κ ) is the hermitian Dirac-Wilson operator. The parameter µ is proportionalto the fermion mass, up to a multiplicative renormalization constant.The mass-less overlap operator fulfills the modified chiral anticommutator { γ , D ov (0) } = 2 D ov (0) γ D ov (0) , (3)also known as Ginsparg-Wilson relation [8]. In the mass-less limit, the fermion action ¯ ψD ov ψ is3nvariant under the global chiral transformation proportional to δψ → γ (1 − D ov ) ψ (4) δ ¯ ψ → ¯ ψ (1 − D ov ) γ , (5)while the variation of the measure of the path integral does reproduce the correct continuumaxial anomaly [9]. The factor (1 − D ov ) modifies the definition of the naive continuum chiralrotations, circumventing the Nielsen-Ninomiya theorem [10] and allowing for a chiral symmetricfermion action on the lattice.The sign function of the hermitian Dirac-Wilson operator can be computed from an eigen-value decomposition, but for practical numerical calculation it is better expressed assign( D H ( κ )) = D H ( κ ) p D H ( κ ) D H ( κ ) . (6)The inverse square root function is typically approximated by means of polynomial or rationalapproximations.Chiral symmetry prevents additive renormalization of the fermion mass, therefore the renor-malized mass is proportional to µ . The Dirac-Wilson operator used in the sign function dependson the parameter κ = − m . The mass ˆ m , not to be confused with the fermion mass µ of theOverlap operator itself, can in principle take any negative value between − µ = 0 is constructed from the inverse of theoverlap operator up to a contact term S ov (0) = ( D ov (0) − − , (7)such that the propagator anticommutes with γ { γ , S ov } = 0 , (8)as it follows straight from the Ginsparg-Wilson relation. It can be shown that quark bilinearoperators and interpolators computed from S ov are automatically on-shell and off-shell O ( a )improved [11]. The fermion propagator in the mass-less limit can be also rewritten as S ov (0) = (1 − D ov (0)) D ov (0) − = (1 − D ov (0)) / D ov (0) − (1 − D ov )(0) / , (9)that is the propagator of the “rotated” fermion fields ˆ ψ ˆ ψ = (1 − D ov (0)) / ψ . (10)From the latter observation, it is natural to set the massive fermion propagator to be equal to S ov = (1 − D ov (0)) D − ( µ ) . (11)Note that the propagator defined with the prescription above is manifestly non-local, contraryto the operator D ov , in agreement with the Nielsen-Ninomiya theorem.4 Implementation of the overlap gluino algorithm
The implementation of the Hybrid Monte Carlo (HMC) algorithm for overlap gluinos has toface several challenges compared to the corresponding QCD simulations. There are two mainobstacles. The first is the general problem of topology and fermion zero modes, and the second isthe specific problem of the Majorana nature of the gluino field. Several strategies can be appliedto control these problems, like the Domain-Wall representation of the Ginsparg-Wilson relationor simulations at fixed topology, which all imply a deviation from the exact chiral symmetricaction. We apply instead a polynomial approximation of the sign function as a simple strategyto control the simulations.Supersymmetry requires that the gluino mass is exactly zero. Even in the continuum, thezero mass limit is quite peculiar and requires a careful consideration of the order of chiral andinfinite volume limit. As a consequence of chiral symmetry, related problems arise with the over-lap operator on the lattice. Each lattice configuration featuring a non-trivial topology has atleast one zero mode of the overlap operator, such that its determinant and its Boltzmann prob-abilistic weight are both exactly zero. This simple consequence of preserving chiral symmetryexactly effectively prevents topological charge fluctuations and the generation of instanton-likeconfigurations. However, zero modes and topological objects are also a fundamental feature ofthe mechanisms responsible for gluino condensation [12, 13]. The situation can be effectively de-scribed as “zero over zero” problem [14]. A regularization is needed for a meaningful definition ofobservables like the chiral condensate. Like in lattice QCD, this requires a careful considerationof the order of thermodynamic, chiral, and continuum limit. At finite volume and finite latticespacing, a regulated version of chiral symmetry has to be considered. A related more technicalaspect is the divergent HMC fermion force whenever a zero mode is approached leading to astrongly suppressed sampling of topological sectors. The zero modes of the overlap operatorcan be avoided by a controlled approximation, as the jump of the sign function at the origin iseffectively smoothened. We explore the approximation of the sign function by a polynomial oforder N to control these problems.The integration of Majorana fermions leads to the Pfaffian of the operator D ov , that is equalto the square root of the determinant. The square root of a large sparse matrix cannot becomputed exactly, and therefore we are forced to use a second approximation besides the onerequired to compute the sign function of the overlap operator itself. Similar problems arise inQCD for the strange quark action or for isospin breaking effects in the up and down quarkaction. In these situations the overlap operator for Dirac fermions can, however, be suitablychirally decomposed so to avoid a twofold approximation [15], but such decomposition does notextend directly to Majorana fermions. In our overlap simulations we proceed in analogy withthe Wilson case, and we compute the fourth root of the positive-definite square of the hermitianoperator γ D ov Pf( D ov ) = q det ( D ov ) = q det ( γ ) det ( D ov ) = q det( γ D ov ) . (12)A rigorous treatment of Majorana fermions on the lattice leading to the above equations inthe overlap formalism has been presented in Ref. [16]. The fourth root of the square hermitianoperator is approximated by a rational fraction of two polynomials (RHMC). Pseudofermion5elds φ are introduced at the beginning of trajectory leading to the effective action S f = φ † (( γ D ov ) ) − φ , (13)that is integrated numerically. Contrary to Wilson fermion, we will demonstrate that the ap-proximated overlap operator does not lead to a sign problem. The second main reason for ourapproach based on polynomial approximations is a stable precision of the RHMC algorithm. Ifwe would consider a rational instead of the polynomial approximation of the overlap operator,an additional inner inverter inside the outer rational approximation of the RHMC would berequired and a stable algorithm is in general not guaranteed.Several possibilities to overcome the challenge of the HMC algorithm due to the non-differentiability of the sign function have been discussed in the literature [17, 18, 19, 20, 21,22, 23, 24]. Each time a zero mode of the hermitian Dirac-Wilson operator D H crosses theorigin, there is a jump in the topological charge and a discontinuity in the fermion force. Asolution is to monitor the changes of the lowest eigenvalues of D H during the HMC traject-ory and perform a refraction/reflection step [17, 18]. The Arnoldi algorithm can provide anaccurate computation of the smallest eigenvalues of D H , but the numerical cost is quite huge.Another possibility is to perform simulations at fixed topological charge, thereby forbidding dis-continuities of the force [19]. Fixed topological charge simulations might however fail to addressimportant physical features of the vacuum structure of SYM theories where topological objectsare assumed to play a relevant role, such as gluino condensation [25].Our approach is different from the preliminary studies with overlap gluinos presented inRef. [26], where an additional term has been included to the action to generate a spectral gap inthe operator D H and to fix the simulations to the topological sector zero. This is not necessarywith a polynomial approximation at finite N . Our approach is instead similar to the simulationswith Domain-Wall fermions, with the order of the polynomial approximation N replaced bythe lattice extend L in the fifth dimension. In both cases, the Ginsparg-Wilson relation isonly approximately fulfilled, and exact chiral symmetry is recovered in the limit N → ∞ or L → ∞ [27, 28, 21]. Quantities like the gluino condensate are computed after considering theinfinite volume limit first, then the extrapolation N → ∞ , and finally the continuum limit.For simplicity, we use chiral limit in the following already for the limit N → ∞ , which alwaysassumes that N is still small enough to keep the finite volume effects and lattice artefacts undercontrol.As already explained the smoothing of the sign problem by a polynomial approximationreduces the problem of near zero eigenvalues, which can be considered as a controlled way torelax the Ginsparg-Wilson relation. A polynomial of order O (100) might provide a good approx-imation of the sign function while keeping differentiability of the HMC trajectory. The errorof the polynomial approximation could be corrected afterwards by means of reweighting [23].Depending on the parameters of the simulations, in particular the fermion mass and the volumeof the lattice, there will be an optimal tuning as a compromise between the computational costand the effectiveness of the polynomial approximation. We are going to explore this possibilityin the next sections. 6 .0 0.2 0.4 0.6 0.8 1.0 x s i g n ( x ) x s i g n ( x ) Figure 1: Polynomial approximation of the sign function for n = 160 (green), n = 250 (blue) and n = 400 (orange). The inverse square root function is approximated by means of quadraticallyoptimized in the space of the orthogonal polynomials and the sign function is simply computedas sign( x ) = x/ √ x . Simulations with overlap fermions are significantly more expensive compared those with Wilsonfermions. The calculation of the force and of the inverse of the overlap Dirac operator requiresat each iteration the evaluation of the approximation of the sign function. Our main studiesto check the validity of the proposed algorithm have been done for the gauge group SU(2) ona 8 , 12 and 16 ×
32 lattice with µ = 0 . β = 1 .
6. The complete summary table of allvolumes and their corresponding polynomial approximations can be found in App. A. The codeused for the simulations presented in this contribution is publicly available on GitHub [29]. It isa generic code for simulations of Yang-Mills theories for arbitrary number of colors and fermionsin adjoint or fundamental representation.The gauge action is discretized using a combination of 1 × × N = 160, N = 250 and N = 400, following the strategies and thealgorithms presented in Ref. [30]. The approximations are shown in the Fig. 1(b). Note that thediscontinuity around the origin is smoothed by the polynomial approximation itself. The linksfor the overlap operator are one-level stout-smeared with smearing parameter ρ = 0 .
15. Whilethe hopping parameter of the Dirac-Wilson approximation could in principle be chosen freely inthe interval 0 . < κ < .
25, a certain tuning is required such that zero modes of the mass-lessoverlap operator do appear. We have set κ = 0 .
2, and we will show that zero modes are presentin our simulations.The rational approximation used for the computation of the force can be chosen to be inprinciple different from the one used for the metropolis/heatbath steps. We approximate thefourth root of the Dirac operator by two rational approximations with four and eleven fractions,used in combination with multiple time-scales integrators. The rational approximations usedfor the Metropolis step require instead a higher accuracy and are the sum of 37 fractions. Thespectral intervals used to construct the rational approximations are chosen to fit all eigenvaluesof the overlap operator, including the lowest modes.The boundary conditions imposed on fermion fields are relevant when dealing with super-7 a) w /a (b) Topological charge Figure 2: a) Scale w /a as function of the order of the polynomial approximation of the signfunction for the volume 12 ( × symbols) and for N = 160 and a volume 16 ×
32 (+ symbol).The extrapolation to the chiral limit is shown by the red line with the error given by the redband. b) Monte Carlo history of the topological charge Q top on a volume 12 for N = 160(blue), N = 250 (green), N = 400 (red), excluding thermalization.symmetry simulated on a small box. As the difference in thermal statistics between bosonsand fermions breaks supersymmetry at non-zero temperature, we impose periodic boundaryconditions to all fields in all directions. The Lagrangian of N = 1 SYM in four dimensions does not depend on any dimensional coupling.The lattice spacing a is a function of the bare gauge coupling g and the scale is dynamicallygenerated by the trace anomaly through dimensional transmutation. The lattice spacing a isexpressed in terms of an infrared observable known with good accuracy, such as the Wilson flowscale w /a , defined by the flow time τ , where the condition τ ddτ ( τ h E ( τ ) i ) = 0 . h E ( τ ) i is measured from its clover antisymmetric definition. For furtherdetails, see [31]. All masses and low energy constants measured in dimensionless units of w /a are expected to scale to the continuum limit with corrections of the order O ( a ). The scale w ispresented in Fig. 2(a), and it is basically independent of the order N of the polynomial used forthe approximation of the sign function and free from finite size effects already at a volume 12 .Therefore, we extrapolate w /a to the chiral point ( N → ∞ ) using a simple constant function,obtaining w /a = 0 . w /a yields to a rather coarse lattice spacing comparedto the previous SU(2) N = 1 SYM simulations of our collaboration using unimproved Wilsonfermions. At the current stage of the project we are in particular interested in the effects of theimproved chiral symmetry at rather coarse lattice spacings. Future exploratory investigations8 a) Chiral condensate a Σ (b) Volume dependence of a Σ for N = 250 Figure 3: a) The extrapolation to the chiral limit of the gluino condensate a Σ as a function of1 /N for the volume 12 . b) Volume dependence of a Σ as a function of the volume V for theorder of polynomial of the sign function N = 250.of rather complex extended supersymmetric theories with scalar fields are difficult at fine latticespacings and we want to explore the prospects of the presented approach for these studies. Largescale simulations of complex supersymmetric models including scalars are beyond the reach ofcurrent computational resources, and it is important to keep the lattice volume as small aspossible while being able at the same time of preserving the main features of the continuumtheory being simulated.Simulations at finer lattice spacing with supersymmetry preserving periodic boundary con-ditions might result into topological freezing. In our current setup, the topological charge isfluctuating between different topological sectors, see Fig. 2(b). We measure the topologicalcharge from its field theory definition as the discretization of ˜ F µν F µν measured at the flow time t = 2 .
0, rounded in order to be an integer. In principle, a proper definition of the topologicalcharge is given by the chirality of the zero modes of the overlap operator. We plan to comparethe two definitions in the near future. In the present calculations, we use the flowed ˜ F µν F µν as a simple measure of the autocorrelation of our Monte Carlo chains, and we postpone to aforthcoming publication the study of the index theorem in relation to the chiral Ward identities. Simulations with overlap gluinos open new perspectives beyond the solution to problem of thetuning of supersymmetric Yang-Mills theories to the continuum limit. In particular, the vacuumstructure of N = 1 SYM might be accessible only with an exact chiral symmetric fermion action.In fact, although U A (1) axial symmetry of the form λ → exp ( iθγ ) λ (15)is broken by the anomaly, the theory has still a remnant discrete Z N c fermion symmetry forangles θ = 2 π n N c . At zero temperature, this symmetry is spontaneously broken down to Z by a non-vanishing expectation value of the gluino condensate [13, 12]. The number of vacuum9igure 4: Monte Carlo history of the gluino condensate on a lattice of volume 12 for N = 160(blue) and N = 400 (red).states at zero temperature is N c , equal to the Witten index of N = 1 SYM. In particular, if thegauge group is SU(2), the coexistence of two different phases is expected. If fermion doublers areremoved from the physical spectrum of the naive fermion action by breaking chiral symmetrywith the addition of the Wilson term, it is impossible to directly measure the gluino condensatedue to the additive renormalization of the vacuum expectation value of h ¯ λλ i . Evidences ofchiral symmetry breaking can be given only by studying the behavior of the theory at non-zerotemperatures or by using the gradient flow as an additional regularization scheme besides thelattice [32, 33].In the overlap formalism the gluino condensate Σ is free from additive renormalization. Thebare gluino condensate is defined as the expectation value of the fermion bilinearΣ = h ¯ λ (1 − D ov ) λ i = h Tr((1 − D ov ) D ov ( µ ) − ) i , (16)the trace is computed using random vectors and stochastic techniques. Note that the gluinocondensate is equivalent to the derivative of the partition function with respect to the massparameter µ . The gluino condensate Σ defined by Eq. 16 resembles the continuum h ¯ λλ i . Inthe following discussions of this section, in order to avoid confusions, we will denote the chiralcondensate by Σ instead of h ¯ λλ i .The bare gluino condensate is shown for four different polynomial approximations in Fig. 3(a),and for five different lattice volumes for N = 250 in Fig. 3(b). The gluino condensate exhibits alinear behavior as a function of 1 /N . The extrapolation to the chiral point yields to the bare con-densate a Σ = 0 . χ / dof=0.7. The non-zero value of Σ is a direct confirmationof the spontaneous breaking of chiral symmetry, being the gluino condensate proportional to itsrenormalized value only up to a multiplicative renormalization constant. The chiral condensateis almost independent on the lattice volume up to the small volume 4 .The striking feature of the chiral condensate is its smooth dependence on the order of thepolynomial used to approximate the sign function, meaning that our approach has been able tocapture the chiral properties of N = 1 SYM while regularizing the zero modes of the overlapoperator. Looking to the Monte Carlo history of Σ in Fig. 4, the mild dependence of the chiralcondensate on 1 /N is a result of two competing effects. On the one side, configurations with10igure 5: Lowest part of the spectrum of the overlap operator as obtained from an ensemble ofconfigurations on a lattice of volume 8 .small eigenvalues of the overlap operator are rarely generated going closer to the chiral limit N → ∞ , as their Boltzmann weight becomes smaller and smaller. On the other side, once aconfiguration with small eigenvalues is generated, its individual gluino condensate Σ is very large.After taking the ensemble average, the N = 160 simulation has a very similar chiral condensateto the one with N = 400, resulting from the average of a large number of configurations witha smaller Σ and few of them showing clear upward spikes from the bulk distribution. Thisphenomenon is effectively the regularization of the “zero over zero” problem given by mass-lesschiral fermions on the lattice. The non-perturbative dynamics of the theory dictates a non-vanishing expectation value of the order parameter for chiral symmetry breaking, including itsprecise value, despite the naïve reasoning that zero-modes cannot be generated in a Monte Carlosimulations and therefore gluino condensation should not occur. This also implies that there isan upper bound for N at a given volume and lattice spacing to keep the error under control.The gluino condensate is strictly positive in all our configurations. In the chiral limit, thesecond phase where the gluino condensate is negative can be reached simply after performing achiral rotation, in the same way in which all vacuum expectation values of the Polyakov loop inthe deconfined phase can be reached by a center symmetry transformation of the temporal linksin a given timeslice. The interface tension is however an interesting thermodynamic quantitythat is difficult to determine in the current setup, as tunneling during a Monte Carlo simulationof the chiral condensate from positive to negative value is impossible, even on very small volumes.It would be interesting to study the dynamics of the domain-wall interpolating between suchtwo phases, a possible solution to achieve this goal could be given by reweighting combined witha multi-canonical approach [14]. The dominant contribution to the gluino condensate comes from the eigenvalues of the overlapoperator near the origin, and it is crucial to confirm our results by studying the lowest part ofthe spectrum of D ov . The eigenvalues of the mass-less overlap operator lie on the complex plane,11n a circle of radius 1 / x = 1 /
2; in fact, the matrix 2( D ov −
1) isunitary if µ = 0, as it follows from the relation( D ov − † ( D ov −
1) = 14 sign( D H ) = 14 . (17)The spectrum of the smallest eigenvalues from an ensemble of 8 lattice configurations is presen-ted in Fig. 5. As a consequence of the polynomial approximation of the sign function, theeigenvalues are not lying exactly around a circle, but they are somewhat scattered towards itsinside. Real eigenvalues close to the origin, of topological nature, are generated for all threepolynomial approximations, meaning that the hopping parameter κ of the Dirac-Wilson oper-ator used in the definition of the overlap fermion action is correctly tuned. Further, we see thatthe eigenvalues are bent toward the inward of the overlap circle, and none of them appears on thenegative side of the real axis. Therefore, the sign of the Pfaffian is positive on all configurationswe have generated, and we have clear numerical evidences that the polynomial approximation ofthe sign function is able to regularize the RHMC simulations while solving the sign problem. Inagreement with the understanding of chiral symmetry breaking from the Banks-Casher relation,we observe that the spectrum of the overlap operator becomes more dense around the originand is closing the circle in the complex plane in the limit N → ∞ . As the (quenched) massparameter of the overlap operator would cross the origin going from positive to negative values,the full eigenvalue density around the origin would be responsible for the discontinuity givingrise to the non-zero gluino condensate. A full comparison between different determinations of the gluino condensate requires to set acommon scale and a common scheme to determine the multiplicative renormalization constants Z S . Previous studies with overlap fermions used the scale r /a , coming from the static interquarkpotential, instead of w /a . This difference is not an obstacle as we have been able to determinethe continuum limit extrapolation of the ratio r /w = 2 . w , we have for the bare condensate w Σ = 0 . w Σ = 0 . w Σ = 0 . w Σ = 0 . β = 2 . r /a scale extrapolated to the chiral point as a function of m res from the volume16 ×
32. Apart from the results of Ref. [26] using the overlap formalism but measuring Σfrom the eigenvalue density and Random Matrix Theory (RMT), all other values of the gluinocondensate are close to each other, confirming their equality up to an O (1) renormalizationconstant. A complete comparison to test the full agreement between the different approacheswill be possible only after the multiplicative renormalization constant Z S is computed in the samescheme at the same scale, and especially only after that w Σ is extrapolated to the continuum12imit, requirements that are currently impossible to meet.In Ref. [34], the authors argued that the renormalization group invariant (RGI) gluino mass m RGI is given by the combination m RGI = β ( α ) α m ˜ g (18)where β ( α ) is the well-known NSVZ β -function. The above relation is valid to all orders in thecoupling constant in a scheme that is able to preserve the holomorphic properties of supersym-metry, otherwise it is expected to hold only up to two loops in perturbation theory. Since thecondensate is a derivative with respect to the gluino mass m ˜ g , the RGI value of the condensatecan be expressed as w Σ RGI = αβ ( α ) w Σ . (19)This formula provides a simple perturbative estimation independent of the scheme and the scalein the limit α →
0. The strong coupling constant α = g / (4 π ) can be set to be equal to thebare lattice value in the case of the gluino condensate computed straight from the derivativeof the logarithm of the lattice partition function. The gradient-flow chiral condensate wouldrequire instead a study of the RGI scaling with respect to the gradient flow coupling. Theincompatibility of the Overlap and Domain-Wall RGI gluino condensates w Σ RGI = 0 . w Σ RGI = 0 . N should be compared to the limit of large L , the extend in the fifth dimension ofthe Domain-Wall solution. At β = 2 . L = 48 a rather large deviation by a factor of two from the L → ∞ limit, whereaseven for the low order N = 80 the polynomial approximation deviates only by a factor of 1.5from the N → ∞ limit. Therefore an overlap operator defined from a polynomial approximationincluding O (100) terms seems to be competitive and in the same regime that could be possiblyexplored today with Domain-Wall fermions. Periodic boundary conditions applied to all fields imply that the partition function being simu-lated is the Witten index, corresponding in the Hamiltonian formalism to the super-trace W ( V , L ) = Tr n ( − F exp (cid:16) − L ˆ H ( V ) (cid:17)o , (20)13eing V the three dimensional volume of the lattice box and L its temporal extent. The Wittenindex, expressed in a basis of eigenstates of the Hamiltonian ˆ H ( V ), reads W ( V , L ) = X n ∈ boson exp ( − LE n ( V )) − X n ∈ fermions exp ( − LE n ( V )) , (21)and it is equal to a constant independent of the volume V and the temporal extent L if super-symmetry is not broken explicitly, thanks to the cancellation and the matching between bosonand fermion states. The energies of each eigenstate E n ( V ) can depend on the spatial volume,however the Witten index is effectively a topological invariant quantity counting the differenceof the boson and fermion number of zero energy vacuum states [35].The idea is to use the Witten index as a probe for supersymmetry restoration on the lattice.Any deviation from a constant behavior of W ( V , L ) is a signal induced by lattice artefacts orby the polynomial approximation. However, the Witten index cannot be easily computed byMonte-Carlo simulations, as it requires knowledge of the absolute normalization of the partitionfunction. An alternative observable could be the vacuum energy density that is the orderparameter of (spontaneous) supersymmetry breaking, but the mixing with identity operator thatoccurs on the lattice due to the breaking of Lorentz symmetry prevents any straight forwardimplementation. The vacuum energy density, together with the other components of the stressenergy momentum tensor, can be computed only indirectly from lattice simulations by studyingfor instance short distance behavior of suitable correlation functions.Fortunately, the ultraviolet divergence coming from the mixing with the identity operatordoes not depend on V nor on L . We can therefore study how the Witten index reacts to asimultaneous size change for all directions of the four-dimensional box∆ ≡ L ∂ log W ( L , L ) ∂ log L (22)relative to its infinite volume value. If supersymmetry is realized on the lattice, this derivativeof the Witten index must vanish. As the size of the box L is squeezed, more and more excitedstates contribute to the sum in Eq. 21, until the point where energies close to the ultravioletlattice cut-off are reached and the approximate degeneracy between fermion and boson states isbroken. We expect the size L of the box where this breakdown occurs to be small if our O ( a )improved lattice action is able to keep the magnitude of lattice artefacts under control at ourcurrent lattice spacings.The partial derivative with respect to the box size L = N a can be computed straightfor-wardly with the chain rule as the derivative of the partition function with respect to the baregauge coupling ∆ = 4 N c g ∂g∂ ln( a ) ( h s g i − h s g i ) , (23)since the lattice spacing a is a function of g , a ≡ a ( g ). The derivative ∆ is proportional tothe expectation value of the gauge energy density h s g i minus its infinite-volume vacuum value,up to a multiplicative constant proportional to the β -function. The dependence of the lattice Note that ∆ does not include terms proportional to the chiral condensate, as the bare mass is proportionalto the renormalized mass with overlap fermions and it is equal to zero in our simulations. W defined as ∆ ≡ L ∂ log W ( L ,L ) ∂ log L forperiodic boundary conditions applied to the gluino field for the simulations with the polynomialapproximation N = 250 for various lattice of volumes V . Significant deviations from zero of ∆appear from the smallest volume V = 3 .spacing on the gauge coupling can be estimated using the NSZV expression as ∂g∂ ln( a ) = − g π N c − N c g π , (24)that is exact up to one-loop order in perturbation theory in the lattice regularization scheme.As the Witten index is a constant, we expect the quantity ∆ to be zero. Deviations from zerowill appear for anti-periodic fermion boundary conditions in time direction as a result of thedifferent statistics of fermion and bosons. In this case ∆ is related to the thermal average ofthe trace of the energy-momentum tensor, that raises sharply near the deconfinement phasetransition. In Fig. 6 the Witten index ∆ is compatible with zero up to a small lattice box of size4 , while the deviation from zero is observed starting from a lattice box 3 . We can thereforeconclude that overlap fermions are able to keep an effective degeneracy of fermion and bosonstates on boxes as coarse as 4 . Along the same ideas as the Seiberg-Witten electromagnetic duality, several recent theoreticalstudies opened the possibility for an understanding of confinement in four-dimensional N =1 SYM on R × S based on a semiclassical analysis at small compactification radii. Thecancellations occurring between boson and fermion states guaranteed by supersymmetry for theWitten index, allows to analytically compute the structure of the phase diagram. N = 1 SYMis expected to Abelianize for small compactification radii, with confinement preserved by non-perturbative effects [36, 37, 38, 39]. If there are no phase transitions disconnecting the smallfrom the large compactification regimes, semiclassical calculations might be able to provide ananalytical understanding of confinement of a four dimensional interacting gauge theory. Thefirst lattice simulations of the compactified N = 1 SYM have been able to show evidences forcontinuity, but the small compactification regime has been difficult to explore given the explicitbreaking of chiral symmetry induced by Wilson fermions [40, 41]. Overlap fermions would allow15 a) Periodic vs antiperiodic (b) Volume comparison 4 vs 8 Figure 7: a) Distribution of the Polyakov loop for periodic (red) and antiperiodic (blue) on avolume 4 for N = 250. b) Distribution of the Polyakov loop for periodic boundary conditionson a volume 4 (red) and 8 (blue) for N = 250.to explore better the small compactification regime, in particular when considering whetherchiral symmetry is remains spontaneously broken.The compactification of more than one space-time dimension has been studied in the contextof the Eguchi-Kawai volume reduction [37, 38]. In this exploratory study we can provide someevidence for volume independence of bulk quantities such as the gluino condensate, as shownin Fig. 3(b). The chiral condensate shows a complete volume independence up to lattices ofsize 6 , while it drops for 4 . Given the small number of lattice points of a volume equal to 4 ,it is difficult to tell if such deviation is a genuine volume dependence of the chiral condensateor rather a lattice artefact. The gauge action, proportional to a combination of square andrectangular Wilson loops, appears to be instead constant even at 4 . The flat behavior followsfrom the Witten index shown in Fig. 6, which is proportional to the change of the gauge action.The Polyakov loop P L has a symmetric distribution also for the volume 4 for periodic boundaryconditions, while a double peak asymmetric distribution is present for antiperiodic boundaryconditions applied to the gluino field, see Fig. 7(a). The distribution of P L becomes broader asthe volume gets smaller and smaller, see Fig. 7(b), in agreement with the predicted broadeningof the effective potential of the Polyakov loop for small volumes. The bound states of N = 1 SYM are expected to be organized in supermultiplets of particleswith equal mass [42, 43, 44, 45]. The predicted lowest chiral supermultiplet consists of a scalar(0 + ) and pseudoscalar (0 − ) bosonic states, that can be identified with a mixture of glueballsand flavor-singlet mesons, and by an exotic spin-1 / F µν ( x ) σ µν λ ( x ). Twelve levels of APE-smearing with smearing parameter α = 0 .
15 are applied to the link fields used to construct the16 a)
12 + correlator (b) Gluino-glue mass as a function of N Figure 8: a) Correlator of the gluino-glue on the lattice volume 12 and the order of the ap-proximation of the sign function equal to N = 250. b) Mass of the gluino-glue as a function ofthe order of the polynomial approximation for the volume 12 . The red line and the shaded arerepresent the central value and the error of the fit to a constant.Figure 9: Correlator of the 0 + glueball on the volume 16 ×
32 and the order of the approximationof the sign function equal to N = 160.clover version of F µν ( x ). The gluino-glue correlator is presented in Fig. 8(a). The fitted gluinomass is independent of the polynomial approximation used in the sign function, as shown inFig. 8(b). The fitted chiral value of the gluino mass in units of w /a is w m gg = 1 . w m gg = 0 . ++ sector suffers from large statistical errors. The relevant 0 + operatorsare the fermion bilinear ¯ λλ and the glueball-like operator F µν F µν . From our previous investig-ations with Wilson fermions, we have observed that the largest overlap with the ground stateis given by glueball-like operators, and therefore we consider a large variational basis of 1 × × α = 0 .
5. The measure of the glueball mass has been possible only on the16 ×
32 lattice with the order of the approximation of the sign function equal to N = 160,where the fitted mass is aM + = 1 . /N for the volume 16 ×
32. The red-dashed line is aquadratic interpolation of the two available data with a constant term equal to zero.partially quenched chiral perturbation theory. Its mass can be extracted from the connectedpart of the correlator of the η particle defined by the operator ¯ λγ λ . In the chiral limit, thepion mass is expected to vanish and the η mass is dominated by the disconnected parts of the0 − correlators. The square of the pion mass is, according to the approach of partially quenchedchiral perturbation theory, proportional to the gluino mass. In our overlap simulations, thegluino mass is set to zero, but the finite order of the approximation leads (similar to the finite L for Domain-Wall fermions) to a finite effective mass. It can be measured by the partiallyconserved axial current relation or the square of the adjoint pion mass. The zero mass limitis obtained in the limit of infinite N without any need for fine-tuning. The pion mass wasmeasured only on the 16 ×
32 ensembles, where we have a sufficiently large temporal extentfor the exponential fit, see Fig. 10. As expected, the pion mass is decreasing from N = 160 to N = 250.
10 Outlook
Based on this first exploratory studies, Ginsparg-Wilson fermions provide a starting point forseveral interesting further investigations of supersymmetric theories. In the following, we list afew directions which we are considering for our next studies.The index theorem relates the difference of the zero eigenvalues with positive and negativechirality of the mass-less overlap operator to the topological charge [9]. Since gluinos are Major-ana fermions transforming in the adjoint representation of SU(2), the number of real eigenvaluesof the Dirac-Wilson operator is a multiple of four. There is the possibility for a configuration,depending on the boundary conditions, to have for instance just a couple of isolated negativeeigenvalues so that Index( D adjoint ) = 2 N c Q top . Such a configuration would correspond to a“fractional topological charge” instanton, semiclassical objects that might be responsible for thegluino condensation [25, 47, 48]. The investigation of fractional topological charges has notbeen addressed in unquenched simulations, and it would be a relevant application for futuresimulations with overlap gluinos.If two dimensions are compactfied, and if their length is reduced to zero, the lower dimen-sional effective theory has an enlarged supersymmetry. The two-dimensional model would be18n ideal benchmark to study the renormalization properties of theories with extended super-symmetry constructed from N = 1 SYM with overlap fermions, to connect our simulations toprevious results [5, 49, 50, 51].The improvements guaranteed by overlap fermions in supersymmetric Yang-Mills theorywith more than one conserved supercharge or coupled to matter chiral superfields have not beenfully addressed so far. Several difficulties appeared even in the simpler case of the Wess-Zuminomodel, in particular for the consistency between the Majorana condition and chiral symmetryin the Yukawa couplings [52]. Further difficulties arise from the large number of couplings tobe tuned. Even if the fermion mass is protected by chiral symmetry, the bare scalar mass mustbe tuned such that its renormalized value is zero and there are the Yukawa and the quarticcouplings that must be tuned to be equal to the gauge coupling. A large number of simulationswould be required only to extrapolate to the supersymmetric limit. Some preliminary studyabout this possibility in the context of SuperQCD have been presented in Ref. [53, 54].
11 Conclusions
The main advantage of preserving exact chiral symmetry is that N = 1 Super Yang-Mills canbe simulated on the lattice without the need of fine-tuning any parameter. Further, the scalingof physical observables with overlap fermions to the continuum limit is automatically O ( a )improved. The exploratory study of overlap gluino simulations presented in this contribution ispromising, as the polynomial approximation of the sign function is able to regularize mass-lesschiral fermions, while avoiding at the same time topological freezing and the sign problem. Fromthis perspective, our setup is close to the Domain-Wall implementation of the Ginsparg-Wilsonrelation, with the role of the size of the fifth dimension replaced by the order N of the polynomialapproximation.The infrared observables we have studied in the current paper are either independent onthe order N of the polynomial approximation, such as the gluino-glue mass or the scale w /a ,or they show a mild linear dependence, as in the case of the gluino condensate. Our resultsare in agreement with the spontaneous breaking of chiral symmetry, as the gluino condensateis clearly non-vanishing in the chiral limit N → ∞ . The bare value of the gluino condensatein dimensionless units is close to the value determined by previous investigations of the sametheory using Domain-Wall fermions or using Wilson fermions with the gradient flow, confirmingtheir agreement up to an O (1) renormalization constant.The Witten index, and in particular its derivative, allows to check and confirm the expectedboson-degeneracy required by unbroken supersymmetry. Our numerical results are compatiblewith a constant Witten index up to lattice boxes as small as 4 at the current lattice spacing.As glueball states are affected by a bad signal-to-noise ratio, further studies and larger statisticswill be required to confirm the expected fermion-boson degeneracy directly from the bound statespectrum. Acknowledgments
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