We derive the fermion loop formulation for the supersymmetric nonlinear
O$(N)$ sigma model by performing a hopping expansion using Wilson fermions. In
this formulation the fermionic contribution to the partition function becomes a
sum over all possible closed non-oriented fermion loop configurations. The
interaction between the bosonic and fermionic degrees of freedom is encoded in
the constraints arising from the supersymmetry and induces flavour changing
fermion loops. For $N \ge 3$ this leads to fermion loops which are no longer
self-avoiding and hence to a potential sign problem. Since we use Wilson
fermions the bare mass needs to be tuned to the chiral point. For $N=2$ we
determine the critical point and present boson and fermion masses in the
critical regime.Comment: 7 pages, 4 figures, presented at the 31st International Symposium on
  Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, German

Steinhauer, Kyle

Wenger, Urs

English

arXiv

We derive the fermion loop formulation for the supersymmetric nonlinear O(N) sigma model by performing a hopping expansion using Wilson fermions. In this formulation the fermionic contribution to the partition function becomes a sum over all possible closed non-oriented fermion loop configurations. The interaction between the bosonic and fermionic degrees of freedom is encoded in the constraints arising from the supersymmetry and induces flavour changing fermion loops. For N ≥ 3 this leads to fermion loops which are no longer self-avoiding and hence to a potential sign problem. Since we use Wilson fermions the bare mass needs to be tuned to the chiral point. For N = 2 we determine the critical point and present boson and fermion masses in the critical regime

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PoS(LATTICE 2013)092Loop formulation of the supersymmetric nonlinearO(N) sigma modelKyle Steinhauer∗ and Urs WengerAlbert Einstein Center for Fundamental PhysicsInstitute for Theoretical PhysicsUniversity of BernSidlerstrasse 5CH–3012 BernSwitzerlandE-mail: steinhauer@itp.unibe.ch, wenger@itp.unibe.chWe derive the fermion loop formulation for the supersymmetric nonlinear O(N) sigma modelby performing a hopping expansion using Wilson fermions. In this formulation the fermioniccontribution to the partition function becomes a sum over all possible closed non-oriented fermionloop configurations. The interaction between the bosonic and fermionic degrees of freedom isencoded in the constraints arising from the supersymmetry and induces flavour changing fermionloops. For N ≥ 3 this leads to fermion loops which are no longer self-avoiding and hence to apotential sign problem. Since we use Wilson fermions the bare mass needs to be tuned to thechiral point. For N = 2 we determine the critical point and present boson and fermion masses inthe critical regime.31st International Symposium on Lattice Field Theory - LATTICE 2013July 29 - August 3, 2013Mainz, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/source: https://doi.org/10.7892/boris.60593 | downloaded: 13.3.2017PoS(LATTICE 2013)092Loop formulation of the SUSY nonlinear O(N) sigma model Kyle Steinhauer1. MotivationDespite the fact that nature has so far not revealed any trace of supersymmetry in its ele-mentary particle spectrum, supersymmetric quantum field theories remain to be fascinating objectsto study per se. It is for example most interesting to examine various discretisation schemes forsupersymmetric field theories regularised on the lattice, e.g. using twisted supersymmetry or or-bifolding techniques, in order to understand how the supersymmetry is realised in the continuumlimit. Moreover, recent developments in simulating supersymmetric field theories in low dimen-sions efficiently and without critical slowing down [1, 2] brings the non-perturbative study of thesetheories to a new, unprecedented level of accuracy, allowing for example precise investigations ofspontaneous supersymmetry breaking phase transitions [3, 4]. In these proceedings we report onour ongoing study to apply such a programme to the supersymmetric nonlinear O(N) sigma modelregularised on the lattice. This model has already been investigated numerically using a varietyof different discretisations [5, 6, 7]. Here we concentrate on reformulating the model in terms offermion loops in order to make use of the efficient simulation algorithms and related methods tocontrol the fermion sign problem accompanying any spontaneous supersymmetry breaking [8].2. Continuum modelThe Lagrangian density of the supersymmetric nonlinear O(N) sigma model in two-dimensionalEuclidean spacetime, originally derived in [9, 10], can be written asL =12g2(∂µφ∂ µφ + iψ /∂ψ+14(ψψ)2)(2.1)where φ is a N-tuple of real scalar fields, ψ a N-tuple of real Majorana fields with ψ =ψTC and Cthe charge conjugation matrix. For the action to be O(N)-invariant and supersymmetric the fieldsmust fulfill the constraintsφ 2 = 1 and φψ = 0 . (2.2)The model described in eq.(2.1) and the constraints in eq.(2.2) are both invariant under theN = 1supersymmetry transformationsδφ = iεψ and δψ = (/∂ +i2ψψ)φεwhere ε is a constant Majorana spinor. There is an additional Z2 chiral symmetry realised byψ → iγ5ψ with γ5 = iγ0γ1. As shown in [11] the one-loop β -function coincides with the onecalculated for the model without SUSY, so it is asymptotically free for N ≥ 3. As pointed out in[12] there exists an N = 2 extension for supersymmetric nonlinear sigma models which have aKähler target manifold. This is the case for N = 3 and hence an additional SUSY can be workedout [7]. The constraints in eq.(2.2) are implemented in the partition function by inserting adequateDirac delta functions in the integration measure,Z =∫Dφ δ (φ 2−1)∫Dψ δ (φψ)e−S(φ ,ψ)=∫Dφ δ (φ 2−1)e−SB(φ)∫Dψ(N∑i≤ jφiφ jψ iψ j)e−SF (ψ) ,2PoS(LATTICE 2013)092Loop formulation of the SUSY nonlinear O(N) sigma model Kyle Steinhauer(2 +m)φ2r φ2r1√2φ2r φrφg1√2φrφgflavour diagonal constraintflavour changing constraintmass termred fermion linegreen fermion lineFigure 1: All possible vertices for N = 2 up to rotations and exchange of the two flavours denoted as r (red)and g (green) with the corresponding weights.where the fermionic measure is rewritten in the second line using the Grassmann integration rulesfor Dirac delta functions. In this form it is explicit that the fermionic constraint φψ = 0 in-duces flavour-diagonal (i = j) and flavour-changing (i 6= j) interactions between the bosonic andfermionic degrees of freedom.3. Loop formulation and sign problemWhen regularising the model on a discrete space-time lattice we follow the strategy in [7]and use the Wilson derivative for both the fermionic and the bosonic action. This strategy is wellsupported by various theoretical and numerical arguments [3, 4, 8, 13, 14, 15, 16, 17, 18]. Wecan then perform a hopping expansion to all orders for the fermionic and the bosonic variables inorder to obtain an exact reformulation of the partition function in terms of non-oriented, closedfermion loops and constrained bosonic bond configurations (boson loops) [8]. These loops can inprinciple be simulated effectively by enlarging the configuration space by an open fermionic string[1, 2] and bosonic worms [19], respectively. The fermion loop formulation in particular solvesthe fermion sign problem since it naturally decomposes the contributions to the partition functioninto bosonic and fermionic ones, each with a definite sign depending on the employed boundaryconditions [8]. The control of these signs is particularly important in the context of spontaneoussupersymmetry breaking when the partition function for periodic boundary conditions, i.e. theWitten index, vanishes due to the exact cancellation of the bosonic and fermionic contributions.Unfortunately, the model as discretised above nevertheless suffers from a sign problem. Firstly,the fermionic constraint φψ = 0 requires on each lattice site exactly one term of the form φiφ jψ iψ jinducing the interactions between fermions and bosons. Since the flavour-nondiagonal constraintwith i 6= j changes the flavour content within a fermion loop, a loop can be self-intersecting forN ≥ 3 and hence looses the crucial property of having a definite sign depending only on its windingtopology. Secondly, the Wilson term introduced in the bosonic sector generates a next-to-nearestneighbour diagonal hopping term with the wrong sign leading to an overall sign which fluctuatesunder local changes of the bosonic bond configuration. In order to avoid these complications werestrict ourselves in the following to N = 2 for which the fermion loops remain self-avoiding andin addition treat the bosonic degrees of freedom in the standard way, i.e., without employing ahopping expansion. The partition function for a system with V lattice sites can then be written asZ =(1g2)V ∫D φ δ (φ 2−1)e−SB(φ)∑{l}w(l,φ)3PoS(LATTICE 2013)092Loop formulation of the SUSY nonlinear O(N) sigma model Kyle Steinhauer-1-0.500.511.6 1.8 2 2.2 2.4 2.6 2.8 3Zpp/Zap2 +m64× 6432× 3216× 168× 84× 4-0.600.62.1 2.2 2.3linear fit0.160.170.180.190.20.210.220.230.240.250 0.05 0.1 0.15mc1/Llinear fitquadratic fitmcFigure 2: Left plot: Zpp/Zap as a function of the bare mass m at g = 0.5 for various lattice extents. The smallinset shows a linear fit used to obtain the critical mass defined by Zpp(mc) = 0. Right plot: Critical massesmc(L) at g = 0.5 for various lattice extents together with an extrapolation to the thermodynamic limit.where the weight w(l,φ) for a given loop configuration l is determined solely by the geometry ofthe loops and, through the constraints in eq.(2.2), depends on the bosonic field φ . As an interestingside remark we note that the coupling g has no influence on the fermion loops except indirectlythrough the bosonic fields. In Figure 1 we show all allowed vertices up to rotations and exchangeof the two flavours denoted as r (red) and g (green). The additional factors are 1/√2 from the Diracalgebra structure for each corner in the loop [20] and (2+m) from the fermionic monomer term(with m being the bare fermion mass). Note that the weights corresponding to the flavour changingconstraints are still not guaranteed to be positive definite since the bosonic field components appearlinearly in those. For certain values of the parameters m and g this leads to a sign fluctuating underlocal changes of the bosonic field and this is why we consider the phase quenched model withw(l,φ)→ |w(l,φ)| in the following. The correct model can then be recovered by reweighting withthe sign σ(w) and this seems to work well in some of the interesting parameter regimes.4. Results for N = 2The supersymmetric nonlinear O(N) sigma model is defined in the chiral limit where both thefermions and bosons are massless. However, since the Wilson term explicitly breaks chiral symme-try even at zero bare fermion mass, the chiral limit of the regularised theory is not defined simplyby the vanishing of that mass. Instead, it needs to be tuned to the critical value mc where the corre-lation length of the fermion diverges and the fermion develops a zero mode yielding Zpp(mc) = 0for the partition function with periodic boundary conditions in both directions [20, 21].The behaviour of Zpp/Zap when varying the bare mass is illustrated in the left panel of fig. 2for various lattice extents L at g = 0.5. The inset illustrates for the 16× 16 lattice how the valuesof mc(L) are extracted by performing linear fits to Zpp/Zap in the region close to zero. It is evidentthat this definition of the critical mass allows rather accurate determinations of mc and hence leadsto precise extrapolations to the thermodynamic limit. In the right panel of fig. 2 we show such an4PoS(LATTICE 2013)092Loop formulation of the SUSY nonlinear O(N) sigma model Kyle Steinhauer00.050.10.150.20.250.30.351.6 1.8 2 2.2 2.4 2.6 2.82 +mmcLs = 32, mψLs = 32, mφLs = 8, mψLs = 8, mφ00.010.020.030.040.050.060.070.080 0.02 0.04 0.06 0.081/LmψmφFigure 3: Left plot: Boson and fermion masses at coupling g = 0.5 as a function of the bare mass for twodifferent lattice volumes. Right plot: Boson and fermion masses for g = 0.5 at the critical mass mc versusthe inverse lattice extent.extrapolation at fixed coupling g = 0.5. Eventually, this procedure can be repeated for a range ofcouplings in order to obtain mc(g) in the thermodynamic limit. We note however that the behaviourof the system changes significantly for g & 0.8 where we expect the occurrence of a Kosterlitz-Thouless phase transition.Next we investigate the boson and fermion mass spectrum of the theory. The fermion correlatorcan be measured with high accuracy using the open fermionic string update introduced in [1]. In theleft panel of fig. 3 we show the observed masses at fixed coupling g = 0.5 for the two volumes withspatial extent Ls = 8 and 32 and temporal extent Lt = 32. The boson mass mφ depends noticeablyon the volume, but is essentially independent of the bare fermion mass and remains constant acrossthe phase transition. The fermion mass mψ on the other hand drops when the bare mass is decreasedtowards its critical value and becomes degenerate with the boson mass in the regime close to thecritical point mc (denoted by the blue bar). The mass degeneracy is further examined in the rightpanel of fig. 3 where the two masses mφ and mψ at the critical mass mc for g = 0.5 are plottedversus the inverse lattice extent. The data indicate that the mass degeneracy between the boson andfermion mass at the critical point survives the continuum limit where the model becomes chirallyinvariant.We emphasise again that the above calculations are made in the phase quenched model. Theonly sources of negative contributions are the flavour changing constraints φrφgψrψg. We areinvestigating their occurrence in the left panel of fig. 4 where the flavour changing constraint densityis plotted versus the bare mass for various couplings g on a lattice of size 8× 8. For large g thenumber of flavour changing interactions grows for decreasing m < mc while for smaller couplingsthe number remains rather small. On the other hand, for large values of m > mc the density offlavour changing interactions vanishes for any coupling. In this region there are predominantlyflavour diagonal constraints contributing positively with φ 2r or φ 2g . As a consequence the averagesign 〈σ〉 plotted in the right panel of fig. 4 appears to behave reasonably well and this suggests thatapproaching the chiral limit from m > mc is under good control.5PoS(LATTICE 2013)092Loop formulation of the SUSY nonlinear O(N) sigma model Kyle Steinhauer00.10.20.30.40.50.5 1 1.5 2 2.5 3 3.5 4flavourchangingconstraints/V2 +mg = 1.0g = 0.8g = 0.6g = 0.4g = 0.200.20.40.60.810.5 1 1.5 2 2.5 3 3.5 4〈σ〉2 +mFigure 4: Left plot: Density of flavour changing constraints as a function of the bare mass for a range ofcouplings. Right plot: Same but for the average sign 〈σ〉.5. Summary and outlookWe constructed a fermion loop formulation of the supersymmetric nonlinear O(N) sigmamodel on an Euclidean space-time lattice using the Wilson derivative for both the bosonic andfermionic fields. This formulation maintains both the O(N) symmetry and the constraints arisingfrom the supersymmetry and in principle allows a straightforward simulation of the model. ForN = 2 we determined the critical mass, defining the massless model in the continuum limit, fora range of couplings and showed how the fermion and boson masses become degenerate at thispoint. A careful analysis of Ward identities is necessary in order to clarify whether or not the su-persymmetry is fully restored in the continuum limit, but in principle it is now possible to determinethe full phase diagram of the model, i.e., the range of parameters for which the supersymmetry isspontaneously broken, if at all.One obstacle towards this goal is the fluctuating sign stemming from the bosonic degreesof freedom entering in the constraints φψ = 0. However, it is not clear whether this sign problemsurvives the continuum limit since it is a pure artefact of the particular regularisation chosen. In fact,the sign problem can be completely avoided by introducing the Wilson term only in the fermionicsector and performing the hopping expansion also for the bosonic degrees of freedom as outlined insection 3, but of course it remains to be seen whether supersymmetry is restored in the continuumalso for such a regularisation.References[1] U. Wenger, Efficient simulation of relativistic fermions via vertex models, Phys.Rev. D80 (2009)071503, [arXiv:0812.3565].[2] U. Wenger, Simulating Wilson fermions without critical slowing down, PoS LAT2009 (2009) 022,[arXiv:0911.4099].[3] D. Baumgartner, K. Steinhauer, and U. Wenger, Supersymmetry breaking on the lattice: the N=1Wess-Zumino model, PoS LATTICE2011 (2011) 253, [arXiv:1111.6042].6PoS(LATTICE 2013)092Loop formulation of the SUSY nonlinear O(N) sigma model Kyle Steinhauer[4] U. Wenger, K. Steinhauer, and D. 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Loop formulation of the supersymmetric nonlinear O(N) sigma model

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