Simulation of supersymmetric models on the lattice without a sign problem
SSimulation of supersymmetric models on the latticewithout a sign problem
David Baumgartner ∗ and Urs Wenger ∗ Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical PhysicsUniversity of BernSidlerstrasse 5CH–3012 BernSwitzerlandE-mail: [email protected] , [email protected] Simulations of supersymmetric models on the lattice with (spontaneously) broken supersymmetrysuffer from a fermion sign problem related to the vanishing of the Witten index. We propose anovel approach which solves this problem in low dimensions by formulating the path integral onthe lattice in terms of fermion loops. The formulation is based on the exact hopping expansion ofthe fermionic action and allows the explicit decomposition of the partition function into bosonicand fermionic contributions. We devise a simulation algorithm which separately samples thefermionic and bosonic sectors, as well as the relative probabilities between them. The latter thenallows a direct calculation of the Witten index and the corresponding Goldstino mode. Finally, wepresent results from simulations on the lattice for the spectrum and the Witten index for N = The XXVIII International Symposium on Lattice Field TheoryJune 14-19,2010Villasimius, Sardinia Italy ∗ 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 ] A p r imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger
1. Introduction
Supersymmetry (SUSY) is thought to be a crucial ingredient in the unification of the StandardModel interactions as well as in the solution of the hierarchy problem. On the other hand, weknow that the low energy physics is not supersymmetric, and consequently the SUSY must bebroken at low energies, either explicitly or spontaneously. Since the origin and mechanism ofspontaneous SUSY breaking is a non-perturbative effect, it can not be understood in perturbationtheory – instead, non-perturbative methods are required. One way to study non-perturbative effectsin quantum field theories is provided by the lattice regularisation. However, the lattice discretisationcomes with various problems, for example the explicit breaking of Poincaré symmetry, or theabsence of Leibniz’ rule, and it is therefore not clear at all whether and how SUSY can be realisedwithin the lattice regularisation.If the lattice discretisation, for example, enjoys some exact symmetries that allow only irrel-evant symmetry breaking operators, which become unimportant in the infrared regime, so-called’accidental’ symmetries may emerge from a non-symmetric lattice action and the full symmetrydevelops in the continuum limit. This is for example the case for (Euclidean) Poincaré symmetryin lattice QCD or SUSY in N = N ) Super-Yang-Mills theory. In the latter case, the onlyrelevant symmetry breaking operator is the gaugino mass term which violates the Z N chiral sym-metry. Therefore, a chirally invariant lattice action forbids such a term and SUSY is automaticallyrecovered in the continuum limit. For SUSY theories involving scalar fields, however, such a wayout is not available: the scalar mass term m | φ | breaks SUSY, and there is no other symmetryavailable to forbid that term. In principle, even in such cases, some symmetries can be obtainedin the continuum by fine tuning the theory with counterterms. The restoration of chiral symmetryfor Wilson fermions is one such example. For SUSY, such an approach is in general not practical,but in lower dimensions, when theories are superrenormalisable, it sometimes is [1]. Yet anotherapproach to SUSY on the lattice is to look for an exact lattice realisation of a subalgebra of thefull SUSY algebra, e.g. by combining the Poincaré and flavour symmetry group, so-called twistedSUSY (cf. [2] and references therein). This approach is applicable to systems with extended SUSYand leads to so-called Q -exact discretisations.Yet another difficulty, and maybe the most severe for supersymmetry on the lattice, is thefact that supersymmetric models with broken supersymmetry inherently suffer from a fermion signproblem that hinders Monte Carlo simulations of such models on the lattice. This can easily beseen as follows. The vanishing of the Witten index W ≡ lim β → ∞ Tr ( − ) F exp ( − β H ) , where F is the fermion number and H the Hamiltonian of the system, provides a necessary (butnot sufficient) condition for spontaneous supersymmetry breaking. On the other hand, the index isequivalent to the partition function with periodic boundary conditions, W = (cid:90) ∞ − ∞ D φ det [ / D ( φ )] e − S B [ φ ] = Z p , and the only way for the path integral to vanish is through the fermionic determinant (or Pfaffian)being indefinite, independent of the fermion discretisation. Indeed, this has been seen in many2 imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger studies of supersymmetric models on the lattice that allow spontaneous supersymmetry breaking,e.g. SUSY quantum mechanics with a supersymmetry breaking superpotential [3, 4, 5], N = N = N = ( , ) Super-Yang-Mills in 2D [11] (see, however, also [12, 13]).Here we propose a novel approach based on [14, 15] that circumvents the fermionic signproblem by formulating the path integral on the lattice in terms of fermion loops. The formulation isbased on the exact hopping expansion of the fermionic action and allows the explicit decompositionof the partition function into bosonic and fermionic contributions. Consequently, one can thendevise a simulation algorithm that separately samples the fermionic and bosonic sectors, as well asthe relative probabilities between them. This then allows a precise calculation of the Witten indexand a direct determination of the presence or absence of a Goldstino mode. Furthermore, althoughthis is less relevant in the present context, the approach eliminates critical slowing down and alsoallows simulations directly in the massless limit or at negative bare mass values [14].
2. Fermion sign problem from spontaneous SUSY breaking
Let us briefly elaborate further on the issue of spontaneous SUSY breaking (SSB), the vanish-ing of the Witten index and the connection to the fermion sign problem.It is well known that the Witten index provides a necessary, but not sufficient, condition forSSB [16]. One has W ≡ lim β → ∞ Tr ( − ) F exp ( − β H ) ⇒ (cid:40) = , (cid:54) = . From the definition it is clear that the index counts the difference between the number of n B bosonicand n F fermionic zero energy states: W ≡ lim β → ∞ (cid:2) Tr B exp ( − β H ) − Tr F exp ( − β H ) (cid:3) = n B − n F . In a field theoretic language the index is equivalent to the partition function of the system withperiodic boundary conditions imposed on both the bosonic and fermionic degrees of freedom, W = (cid:90) ∞ − ∞ D φ det [ / D ( φ )] e − S B ( φ ) = Z p . Here the determinant has been obtained by integrating out the complex valued Dirac fermionfields , while S B ( φ ) is the action for the bosonic degrees of freedom, collectively denoted by φ . Itis now clear that in order to obtain a vanishing Witten index, we need both positive and negativecontributions to the path integral, and this can only be achieved by the fermion determinant be-ing indefinite. This is the source of the fermion sign problem in the context of spontaneous SUSYbreaking, and we argue that such a sign problem must occur in any model aspiring to accommodatespontaneous SUSY breaking. Note that in the context of SUSY quantum mechanics it is misleading to speak of spontaneous or dynamical SUSYbreaking; it is rather a static breaking determined by the form of the superpotential. In case one is dealing with real-valued Majorana fermion fields, one obtains the Pfaffian Pf [ / D ( φ )] instead of thedeterminant. imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger
It is instructive to illustrate the argument in the explicit example of SUSY quantum mechanics.The continuum action of N = S = (cid:90) dt (cid:18) d φ ( t ) dt (cid:19) + P (cid:48) ( φ ( t )) + ¯ ψ ( t ) (cid:18) ddt + P (cid:48)(cid:48) ( φ ( t )) (cid:19) ψ ( t ) , (2.1)where the real field φ denotes the bosonic coordinate, while ¯ ψ and ψ denote the two fermionic co-ordinates. P ( φ ) is the superpotential and the derivatives P (cid:48) and P (cid:48)(cid:48) are taken with respect to φ . The(regulated) fermion determinant with periodic boundary conditions can be calculated analytically[17, 18], det (cid:20) ∂ t + P (cid:48)(cid:48) ( φ ) ∂ t + m (cid:21) p = sinh (cid:90) T P (cid:48)(cid:48) ( φ ) dt , and by rewriting the sinh-function in terms of two exponentials, one can separate the positive andnegative, or rather, the bosonic and fermionic contributions to the partition function,det (cid:20) ∂ t + P (cid:48)(cid:48) ( φ ) ∂ t + m (cid:21) p =
12 exp (cid:18) + (cid:90) T P (cid:48)(cid:48) ( φ ) dt (cid:19) −
12 exp (cid:18) − (cid:90) T P (cid:48)(cid:48) ( φ ) dt (cid:19) = ⇒ Z − Z . As an example, consider the superpotential P e ( φ ) = m φ + g φ , which is even under the paritytransformation φ → ˜ φ = − φ . In this case one finds P (cid:48)(cid:48) e ( φ ) ≥ Z (cid:54) = Z , i.e. no SUSYbreaking. On the other hand, for the superpotential P o ( φ ) = − µ λ φ + λ φ , which is odd underparity, one has P (cid:48)(cid:48) o ( ˜ φ ) = − P (cid:48)(cid:48) o ( φ ) and, since S B ( ˜ φ ) = S B ( φ ) , one finds Z = Z , i.e. a vanishingWitten index and the corresponding SUSY breaking. Here, the vanishing of the Witten index isguaranteed by the fact that to each configuration φ there exists another configuration ˜ φ (the paritytransformed one) that contributes to the path integral with the same weight, but with opposite signstemming from the fermion determinant. Furthermore, Z = Z means that (in the limit of zerotemperature, i.e. β → ∞ ) the free energies of the bosonic and fermionic vacua are equal, and thatproves the existence of a massless, fermionic mode connecting the two vacua, i.e. the Goldstinomode.Turning now to SUSY quantum mechanics on the lattice one obtains with a Wilson type dis-cretisation (cf. next section for further details)det (cid:2) ∇ ∗ + P (cid:48)(cid:48) ( φ ) (cid:3) p = ∏ t (cid:2) + P (cid:48)(cid:48) ( φ t ) (cid:3) − , (2.2)where t now denotes a discrete lattice site index and ∇ ∗ is the backward derivative. Also in thiscase one can identify the bosonic and fermionic contributions (i.e. the first and second term of thedifference in eq.(2.2)), and we will show below that this separation is always explicit in the fermionloop formulation. As a side remark, let us note that in the limit of zero lattice spacing one findslim a → det (cid:2) ∇ ∗ + P (cid:48)(cid:48) (cid:3) −→ exp (cid:18) + (cid:90) T P (cid:48)(cid:48) ( φ ) dt (cid:19) det (cid:2) ∂ t + P (cid:48)(cid:48) ( φ ) (cid:3) , where the exponential term can be understood as coming from radiative contributions that needto be corrected by ’fine-tuning’ a corresponding counterterm [19, 18]. Reconsidering the twoexamples for the superpotential mentioned above, we find for P e with m > g ≥ (cid:2) ∇ ∗ + P (cid:48)(cid:48) e (cid:3) = ∏ t (cid:2) + m + g φ t (cid:3) − > , imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger while for P o one finds det (cid:2) ∇ ∗ + P (cid:48)(cid:48) o (cid:3) = ∏ t [ + λ φ t ] − , (2.3)which turns out to be indefinite, even when λ >
0. While this is necessary in order to enable avanishing Witten index, it imposes a serious problem on any Monte Carlo simulation, for whichpositive weights are strictly required. Moreover, the sign problem is severe in the sense that towardsthe continuum limit (i.e. when the lattice volume goes to infinity), the fluctuations of the firstsummand in eq.(2.3) around 1 tend to zero, such that W →
3. Loop formulation and separation of fermionic and bosonic sectors
In this section we illustrate the loop formulation and the separation of the partition functioninto its fermionic and bosonic sectors by means of the N = N = N = Wess-Zumino model in two dimensions
The Lagrangian of the N = L = (cid:0) ∂ µ φ (cid:1) + P (cid:48) ( φ ) +
12 ¯ ψ (cid:0) ∂ / + P (cid:48)(cid:48) ( φ ) (cid:1) ψ , (3.1)where ψ is a real, two-component Majorana field, φ a real bosonic field and P ( φ ) an arbitrarysuperpotential. Integrating out the fermionic Majorana fields yields a Pfaffian which in general, asdiscussed above, is not positive definite.On the lattice, one can use the exact reformulation of the fermionic Majorana degrees of free-dom in terms of non-intersecting, self-avoiding loops, in order to separate the contributions of thePfaffian to the various bosonic and fermionic sectors of the partition function. A similar exact re-formulation of the bosonic degrees of freedom in terms of bonds, can also be accomplished [20].While this is not necessary for the solution of the sign problem, it provides a convenient way tosimulate also those degrees of freedom without critical slowing down, and hence we will discussthis construction below.Employing the Wilson lattice discretisation for the fermionic part of the Lagrangian in eq.(3.1)yields L F = ξ T C ( γ µ ˜ ∇ µ − ∇ ∗ µ ∇ µ + P (cid:48)(cid:48) ( φ )) ξ , imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger where ξ now represents the real, 2-component Grassmann field, while C = − C T is the chargeconjugation matrix and ∇ ∗ µ ∇ µ the Wilson term. Using the nilpotency of Grassmann elements onecan expand the Boltzmann factor leading to (cid:90) D ξ ∏ x (cid:18) − M ( φ x ) ξ Tx C ξ x (cid:19) ∏ x , µ (cid:0) + ξ Tx C Γ ( µ ) ξ x + ˆ µ (cid:1) , where M ( φ x ) = + P (cid:48)(cid:48) ( φ x ) , Γ ( ± µ ) = ( ∓ γ µ ) and x denotes the discrete lattice site index. Per-forming now the integration over the fermion field, at each site x the fields ξ Tx C and ξ x must beexactly paired in order to give a contribution to the path integral, so one finds (cid:90) D ξ ∏ x (cid:0) − M ( φ x ) ξ Tx C ξ x (cid:1) m ( x ) ∏ x , µ (cid:0) ξ Tx C Γ ( µ ) ξ x + ˆ µ (cid:1) b µ ( x ) , where the occupation numbers m ( x ) = , b µ ( x ) = , m ( x ) + ∑ µ (cid:0) b µ ( x ) + b µ ( x − ˆ µ ) (cid:1) = ∀ x . (3.2)This constraint is equivalent to the fact that only closed, self-avoiding paths survive the Grass-mann integration. When integrating out the fermion fields, the projectors Γ ( µ ) eventually yielda weight ω , which only depends on the geometric structure of the specific constrained path (CP)configuration (cid:96) ∈ L . In particular, one has | ω ( (cid:96) ) | = − n c / , where n c denotes the number of corners in the loop configuration, while the sign depends on thetopology of the loop configuration and will be discussed below.As mentioned above, the bosonic fields can be treated analogously [20]. On the lattice, thekinetic term ( ∂ µ φ ) yields φ x φ x − ˆ µ , and expanding this hopping term to all orders gives (cid:90) D φ ∏ x , µ ∑ n µ ( x ) n µ ( x ) ! (cid:0) φ x φ x − ˆ µ (cid:1) n µ ( x ) ∏ x exp (cid:18) − V ( φ x ) (cid:19) M ( φ x ) m ( x ) (3.3)with bosonic bond occupation numbers n µ ( x ) = , , , . . . and V ( φ x ) = + P (cid:48) ( φ x ) . In contrast tothe fermionic case, the exact reformulation requires one to include an infinite number of terms inthe hopping expansion, and hence occupation numbers up to infinity, instead of just 0 and 1 as forthe fermionic bonds. Integrating out the bosonic fields φ x yields the site weights Q ( n µ ( x ) , m ( x )) = (cid:90) d φ x exp (cid:18) − V ( φ x ) (cid:19) φ N ( x ) x M ( φ x ) m ( x ) , where N ( x ) = ∑ µ (cid:0) n µ ( x ) + n µ ( x − ˆ µ ) (cid:1) counts the number of bosonic bonds attached to a given site,while M ( φ x ) m ( x ) may contribute additional powers of φ x . So the bosonic contribution to the weightof a given configuration factorises into a product of local weights, W ( n µ ( x ) , m ( x )) = ∏ x , µ n µ ( x ) ! ∏ x Q ( n µ ( x ) , m ( x )) . imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger
In summary, the fermionic and bosonic degrees of freedom in our original partition functionare now expressed in terms of fermionic monomers and dimers and bosonic bonds, and the integra-tion over the fields has been replaced by a constrained sum over all allowed monomer-dimer-bondconfigurations, yielding Z = ∑ { (cid:96) }∈ L ∑ { n µ }∈ CP | W ( n µ ( x ) , m ( x )) · ω ( (cid:96) ) | . In particular, the partition function for the fermionic degrees of freedom is represented by asum over all non-oriented, self-avoiding fermion loops Z L = ∑ { (cid:96) }∈ L | W ( n µ ( x ) , m ( x )) · ω ( (cid:96) ) | , L ∈ L ∪ L ∪ L ∪ L , (3.4) where (cid:96) represents a fermion loop configuration in one of the four topological classes L l , l , with l , l = , ω denotes the weight of the specific loop con-figuration and depends on the geometry of the loop configuration, as discussed above. The signof the weight is solely determined by the topological class and the fermionic boundary conditions ε µ , where ε µ = , L have positive weights independent of the boundary conditions, while the sign ofthe weights for configurations in L , L and L is given by ( − ) l µ · ε µ + . So the partition func-tion Z L in eq.(3.4), where all sectors contribute positively, represents a system with unspecifiedfermionic boundary conditions [21], while a partition function with fermionic b.c. periodic in thespatial and anti-periodic in the temporal direction, respectively, is described by the combination Z pa = Z L − Z L + Z L + Z L . This combination represents the system at finite temperature. Analogously, the partition functionwith fermionic b.c. periodic in all directions – the Witten index – is given by Z pp = Z L − Z L − Z L − Z L . The interpretation of the Witten index in terms of the partition functions Z L ij is straightforward.Any fermion loop winding non-trivially around the lattice carries fermion number F =
1, henceconfigurations with an odd number of windings, i.e. configurations in Z L , Z L and Z L , alsocarry fermion number F =
1, while configurations with no, or an even number of windings, i.e. in Z L , have F =
0. The partition function Z L may therefore be interpreted as representing thebosonic vacuum, while the combination Z L + Z L + Z L corresponds to the fermionic vacuum.Consequently, the latter contributes to the Witten index W ≡ Z pp with opposite sign relative to thebosonic vacuum. Since each of the four partition functions is positive, vanishing of the Wittenindex implies Z L = Z L + Z L + Z L . N = supersymmetric quantum mechanics The loop formulation for the N = imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger fermionic part, the continuum action in eq.(2.1) reads S L = ∑ x ( P (cid:48) ( φ x ) + φ x ) − φ x φ x − + ( + P (cid:48)(cid:48) ( φ x )) ¯ ψ x ψ x − ¯ ψ x ψ x − , where x now denotes the one-dimensional, discrete lattice site index. Note that in one dimension thefermionic lattice derivative, including the contribution from the Wilson term with Wilson parameter r =
1, becomes a simple, directed hop ¯ ψ x ψ x − which, in the loop formulation, can be describedby the (directed) bond occupation number b ( x ) = ,
1. Integrating out the fermionic degrees offreedom yields a constraint for the fermion monomer and bond occupation numbers, analogous toeq.(3.2), namely m ( x ) + ( b ( x ) + b ( x − )) = ∀ x . So for N = m ( x ) = , b ( x ) = x , and a fermionic one where m ( x ) = , b ( x ) = x . Since the latter corresponds to aclosed fermion loop, it will pick up a minus sign from the Grassmann integration, relative to thebosonic contribution.On top of the two fermion loop configurations one may treat the bosonic fields in the sameway as before and employ a hopping expansion to all orders. After rearranging the bosonic fieldsthe integration can eventually be performed separately at each site and one ends up with the weight W ( n ( x ) , m ( x )) = ∏ x n ( x ) ! (cid:90) d φ x φ n ( x )+ n ( x − ) x e − V ( φ x ) ( + P (cid:48)(cid:48) ( φ x )) m ( x ) (3.5)for a given bosonic and fermionic bond configuration, with V ( φ x ) = + P (cid:48) ( φ ) . In terms of theseweights the partition function can now be written as Z L = (cid:90) D φ D ¯ ψ D ψ e − S L = ∑ { (cid:96) }∈ L ∑ { n }∈ CP | W ( n ( x ) , m ( x )) | , L ∈ L ∪ L , (3.6) where the second sum is over all allowed bosonic bond configurations { n } ∈ CP, and (cid:96) representsone of the two fermion loop configurations in the topological classes L or L , respectively. Asbefore, the sign of the weight is solely determined by the topological class and the fermionicboundary condition. If l = , ε = , ( − ) l · ( ε + ) .Choosing anti-periodic fermionic boundary conditions ε = Z a = Z L + Z L , which is simply the partition function for the system at finite temperature, while choosing periodicfermionic boundary conditions ε = Z p = Z L − Z L representing the Witten index. Here, the interpretation is particularly intuitive: the two fermionloop configurations simply represent the bosonic and fermionic vacuum, while Z L and Z L repre-sent the bosonic and fermionic partition function in the corresponding sectors. The Witten index8 imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger vanishes whenever Z L = Z L , i.e. when the contributions from the bosonic and fermionic sectorscancel. In this case, the free energy of the bosonic and fermionic vacuum is equal, and this isequivalent to saying that there exists a gapless, fermionic excitation which oscillates between thetwo vacua, i.e. the Goldstino mode.
4. Solution of the fermion sign problem
In this section we briefly describe the simulation algorithm and explain how it eventuallysolves the fermion sign problem. The loop system can most efficiently be simulated by enlargingthe configuration space by open strings. Following [20] the bosonic bonds are updated by insertingtwo bosonic sources, which sample directly the bosonic 2-point correlation function. Similarly,the fermion bonds are most efficiently updated by simulating a fermionic string [14] that sam-ples the configuration space of the fermionic 2-point correlation function, instead of the standardconfiguration space of loops.For the N = { ξ Tx C , ξ y } at position x and y , while for N = { ¯ ψ x , ψ y } .The algorithm proceeds by locally updating the endpoints of the open fermionic string using asimple Metropolis or heat bath step according to the weights of the corresponding 2-point function(cf. [14] for details). When one end is shifted from, say, x to one of its neighbouring sites y , afermionic dimer on the corresponding bond is destroyed or created depending on whether the bondis occupied or not. Contact with the partition functions Z L ij and Z L i , respectively, is made eachtime the open string closes. This then provides the proper normalisation for the expectation valueof the 2-point function.The solution of the fermion sign problem discussed above relies on the correct determinationof the relative weights between the bosonic and fermionic sectors, and the simulation algorithmdescribed in [14] achieves this in a most efficient way. The open fermionic string tunnels betweenloop configurations in the various topological homotopy classes L , L , L , L in two, and L , L in one dimension, thereby determining the relative weights between the partition functions Z L , Z L , Z L , Z L , or Z L , Z L , respectively. From the relative weights, the Witten index (orany other partition function of interest) can be reconstructed a posteriori.Let us emphasise that the open string algorithm and the corresponding absence of critical slow-ing down at the critical point as reported in [14] is crucial for the solution of the sign problem. Sincethe algorithm updates the configurations according to the fermionic 2-point correlation function,they are updated equally well on all length scales up to a scale of the order of the largest fermioniccorrelation length. This is in fact the reason why critical slowing down is essentially absent evenat a critical point when the correlation length becomes infinite. Now, in order to have W = L → ∞ ) the Goldstino mode has to become massless. Since the algorithm ensures anefficient update of that mode, the tunneling between the bosonic and fermionic vacua is guaranteedand the Witten index indeed vanishes in practice.9 imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger m B , F L mL=10mL=19mL=31 m B , F / m mL=10mL=19mL=31 Figure 1:
Spectrum for the perturbatively improved standard discretisation at g / m = .
0. In the left plotthe excitation energies m B (circles) and m F (squares) and the lattice spacing are expressed in units of thelattice extent L , while in the right plot they are expressed in units of the bare mass parameter m .
5. Results for N = supersymmetric quantum mechanics We are now ready to present some results for the spectrum and the Witten index for the caseof N = Here we present our results for the spectrum using the standard discretisation including acounterterm. As discussed in [19], for the standard discretisation described above, the correctcontinuum limit is spoiled by radiative corrections . This can be corrected by adding a counter-term of the form ∑ x P (cid:48) ( φ x ) [19]. By doing so, one ensures that all observables reach the correctcontinuum limit and that the full supersymmetry is eventually restored.As an example we consider a superpotential with unbroken supersymmetry, i.e. P e ( φ ) = m φ + g φ , and choose the coupling g / m = .
0. The results are presented in figure 1 wherewe show the lowest lying excitation energies for the boson (circles) and the fermion (squares) asa function of the lattice spacing a for various values of fixed mL . In the left plot, the quantitiesare expressed in units of the lattice extent L , while in the right plot, they are expressed in units ofthe bare mass parameter m in order to illustrate the common scaling behaviour. The leading latticeartifacts turn out to be O ( a ) for both the fermion and boson masses. At finite lattice spacing thesupersymmetry is explicitly broken by the discretisation, and hence the boson and fermion massesare not degenerate. In the continuum limit, however, the supersymmetry is restored and the massesbecome degenerate. Q -exact discretisation As briefly discussed in the introduction, for models with extended supersymmetry it is some-times possible to preserve some of the supersymmetries exactly at finite lattice spacing [22]. Theso-called Q -exact discretisations preserve a suitable sub-algebra of the full supersymmetry algebra, Note, however, that in one dimension these corrections are finite. imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger m B , F L mL=10mL=19mL=31 m B , F / m m F , mL=17.0m B , mL=17.0m F , mL=31.0m B , mL=31.0m F , mL=10.0m B , mL=10.0 Figure 2:
Spectrum for the Q -exact discretisation at g / m = .
0. In the left plot the excitation energies m B (circles) and m F (squares) and the lattice spacing are expressed in units of the lattice extent L , while in theright plot they are expressed in units of the bare mass parameter m . The dashed line denotes the exact valueobtained with Numerov’s method. i.e. a linear combination of the available supersymmetries. In the context of N = Q -exact action is obtained from the standard action by adding, e.g., theterm ∑ x P (cid:48) ( φ x ) ∇ ∗ φ x , but other forms are also possible [23, 18]. Since the term contains a derivative,there are additional hopping terms that need to be considered in the hopping expansion [24].In the following we concentrate again on the superpotential P e with unbroken supersymmetry.Using the Q -exact discretisation one expects degenerate fermion and boson masses even at finitelattice spacing [22] and this is beautifully confirmed by our results at g / m = . O ( a ) for both the fermion and boson masses.Finally, in the right plot we show the results of a high precision simulation that serves thepurpose of checking the correctness of the new simulation algorithm, as well as our procedures forthe extraction of the fermion and boson masses. Indeed, we can confirm the mass degeneracy toa precision better than a few per mill at all lattice spacings, and the continuum value of the massgap agrees with the exact result in the continuum obtained with Numerov’s method (dashed line)also within a few per mill. Due to the loop formulation and the efficiency of the new simulationalgorithm, these results can be obtained with a very modest computational effort. Let us turn to the Witten index W ∝ Z p / Z a , i.e. the partition function with periodic boundaryconditions Z p , normalised to the finite temperature partition function Z a . We start with the super-potential P e ( φ ) = m φ + g φ for which supersymmetry is unbroken and W (cid:54) =
0. The results arepresented in figure 3, where we show Z p / Z a in the left plot as a function of the bare mass parameter am for the coupling g / m = . Q -exact discretisation. The continuum limit is reached as am →
0, and we indeed find that Z p / Z a →
1, i.e. W (cid:54) = L the partition function ratio goes to zero with am → T is inverselyproportional to the extent of the lattice L . So if we plot the data as a function of m / T = mL , mL = mL → ∞ corresponds to T →
0. This is11 imulating supersymmetric models without a sign problem
David Baumgartner and Urs Wenger -1 -0.5 0 0.5 1a m00.20.40.60.81 Z p / Z a L = 8L = 16L = 32L = 64L = 128 -10 -5 0 5 10m L00.20.40.60.81 Z p / Z a L = 8L = 16L = 32L = 64L = 128
Figure 3:
Simulation results for the Witten index W ∝ Z p / Z a as a function of the bare mass parameter am (left plot), and as a function of the inverse temperature mL (right plot), for a system with unbroken supersymmetry using the Q -exact discretisation at g / m = . Z p / Z a L = 8L = 16L = 32L = 64L = 128 Z p / Z a m L = 1.0m L = 5.0m L = 10.0m L = 30.0 Figure 4:
Simulation results for the Witten index W ∝ Z p / Z a as a function of the inverse temperature mL (left plot), and as a function of the lattice spacing a / L at fixed temperature (right plot), for a system with broken supersymmetry using the standard, perturbatively improved discretisation at λ / m = . illustrated in the right plot of figure 3, from where we find Z p / Z a → mL (cid:38)
5. The data shows a rather good scaling behaviour towards the continuum limit( L → ∞ at fixed mL ). In the continuum, the value Z p / Z a = mL , i.e. towards zero temperature mL → ∞ .Note that with the formulation and algorithm presented here, it is also possible to simulate atnegative bare mass m <
0. In this case, we still find Z p / Z a → P o ( φ ) = − m λ φ + λ φ , for which su-persymmetry is broken, we should expect a vanishing Witten index. Our results for this case arepresented in figure 4, where we show the Witten index W ∝ Z p / Z a as a function of mL in the leftplot, and as a function of the lattice spacing a / L at fixed values of mL in the right plot using thestandard, perturbatively improved discretisation at λ / m = .
0. In this case we find that while theWitten index W ∝ Z p / Z a approaches zero at ’infinite temperature’ mL → mL in the continuum limit, even though for large values of mL the12 imulating supersymmetric models without a sign problem David Baumgartner and Urs Wenger scaling towards the continuum limit is reached only at very fine lattice spacings. In fact, for large mL the approach W →
6. Summary and outlook
We have discussed the occurrence of a fermion sign problem in the context of spontaneoussupersymmetry breaking on the lattice and its relevance for the vanishing of the Witten index,regulated as a path integral on the lattice. We then argued that with the help of the fermion loopexpansion one can achieve an explicit separation of the bosonic and fermionic contributions to thepath integral in such a way that the source of the sign problem, namely the cancellation between thebosonic and fermionic contributions to the partition function with periodic boundary conditions, isisolated. The solution of the fermion sign problem is then achieved by devising an algorithmwhich separately samples the bosonic and fermionic contributions to the partition functions and, inaddition, also samples the relative weights between them, essentially without any critical slowingdown. In such a way one is able to calculate the Witten index on the lattice without suffering fromthe fermion sign problem even when the index vanishes. The absence of critical slowing down isessentially due to the fact that the algorithm directly samples the massless Goldstino mode whichmediates the tunnelings between the bosonic and fermionic vacua.As examples we described in some detail the exact reformulation of the lattice path integralin terms of fermionic bonds and monomers, and bosonic bonds for the N = N = Q -exact discretisationwhich preserves a linear combination of the two supersymmetries. In the latter case the boson andfermion spectra are degenerate even at finite lattice spacing, while in the former case they becomedegenerate only in the continuum limit. Finally, we also present lattice calculations of the Wittenindex for broken and unbroken supersymmetry using the standard discretisation with a counter-term. For both cases we are able to reproduce the correct Witten index in the continuum limit.For broken supersymmetry the approach to the continuum limit is exponentially slow in the latticespacing. Using the loop formulation and the fermion worm algorithm [14] the exponentially slowapproach as well as the sign problem is no obstacle in practice.Obviously, the approach presented here is particularly interesting for the N = References [1] M. F. L. Golterman and D. N. Petcher,
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