Exotic Phases of a Higgs-Yukawa Model with Reduced Staggered Fermions
EExotic Phases of a Higgs-Yukawa Model withReduced Staggered Fermions
Simon Catterall ∗ a , Nouman Butt a and David Schaich b a Department of Physics, Syracuse University, Syracuse, New York 13244, United States b Department of Mathematical Sciences, University of Liverpool,Liverpool L69 7ZL, United KingdomE-mail: [email protected] , [email protected] , [email protected] We investigate the phase structure of a four dimensional SO(4) invariant lattice Higgs-Yukawamodel comprising four reduced staggered fermions interacting with a real scalar field. Thefermions belong to the fundamental representation of the symmetry group while the three scalarfield components transform in the self-dual representation of SO(4). We explore the phase dia-gram and find evidence of a continuous transition between a phase where the fermions are mass-less to one where the fermions acquire mass. This transition is not associated with symmetrybreaking and there is no obvious local order parameter. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n iggs-Yukawa Model Simon Catterall
1. Lattice Action and Symmetries
Conventionally fermions acquire a mass gap through either explicit or spontaneous symmetrybreaking. It is well known that certain Higgs-Yukawa lattice models can be constructed that gapfermions without breaking symmetries using strong four fermion interactions. However until re-cently these symmetric massive phases have been regarded as lattice artifacts separated from thecontinuum by broken symmetry phases and/or first order phase transitions. The model describedin this proceedings seems to circumvent these constraints and appears to offer the possibility of al-lowing for a continuum limit in which fermions acquire mass without breaking symmetries. Apartfrom the intrinsic interest in such a dynamics it opens up the possibility of formulating chiral latticefermions using the Eichten-Preskill approach described in [1].The action for the model is S = (cid:88) x ψ a [ η. ∆ ab + Gσ + ab ] ψ b + 14 (cid:88) x ( σ + ab ) − κ (cid:88) x,µ (cid:2) σ + ab ( x ) σ + ab ( x + µ ) + σ + ab ( x ) σ + ab ( x − µ ) (cid:3) (1.1)where repeated indices are to be contracted and η µ ( x ) = ( − (cid:80) µ − i =1 x i are the usual staggeredfermion phases. The discrete derivative is ∆ abµ ψ b = 12 δ ab [ ψ b ( x + µ ) − ψ b ( x − µ )] . (1.2)The self-dual scalar field σ + ab is defined as σ + ab = P + abcd σ cd = 12 (cid:20) σ ab + 12 (cid:15) abcd σ cd (cid:21) (1.3)with P + projecting the antisymmetric matrix field σ ( x ) to its self-dual component which trans-forms under only one of the SU(2) factors comprising SO(4). It is the presence of the additionalSU(2) symmetry of the fermion operator that allows one to prove the associated Pfaffian operatoris real, positive semi-definite and hence the model can be simulated without encountering a signproblem.When κ = 0 we can integrate out the scalar field and obtain the pure four fermi model studiedin Refs. [2, 3]. While a single peak is observed in a certain fermion susceptibility the eventualconsensus of this earlier work was that this peak straddled a very narrow symmetry broken phasebordered by two phase transitions [4]. In Ref. [5] it was argued that it should be possible toeliminate this broken phase by tuning an additional scalar kinetic term. In concrete terms it shouldbe clear from the form of the action that κ > favors ferromagnetic ordering of the scalar field andassociated fermion bilinear. This is to be contrasted with the preferred antiferromagnetic orderingobserved in the earlier pure four fermi work. Hence by tuning κ it should be logically possibleto reach a phase with no spontaneous ordering of the fermion bilinear. Furthermore, general RGarguments suggest that the renormalizability of the theory should necessitate the addition of such aterm. Results for this model appeared in Ref. [6]. In addition to its manifest SO(4) symmetry, the action is also invariant under a shift symmetry ψ ( x ) → ξ ρ ψ ( x + ρ ) (1.4) In 3d RG arguments do not require such a term and indeed no intermediate phase is seen [7]. iggs-Yukawa Model Simon Catterall with ξ µ ( x ) = ( − (cid:80) di = µ x i as well as a discrete Z symmetry: σ + → − σ + (1.5) ψ a → i(cid:15) ( x ) ψ a . (1.6)Both of the Z and SO(4) symmetries prohibit local bilinear fermion mass terms from appear-ing as a result of quantum corrections. While non-local SO(4)-symmetric bilinear terms can beconstructed by coupling fields at different sites in the unit hypercube, such terms break the shiftsymmetry. Further detailed discussion of possible bilinear mass terms is presented in Ref. [3].
2. Numerical Results σ + - / GL=8 κ =-0.05 κ =0.0 κ =0.05 (a) Four fermion condensate vs G χ s t ag G κ =0.0L=4L=8L=12 (b) Susceptibility at κ = 0 Despite these constraints fermions can nevertheless pick up a mass via interaction with a sym-metric four fermion condensate. The latter is shown in fig. 1a which plots the vev of the SO(4)symmetric operator O = (cid:15) abcd ψ a ψ b ψ c ψ d for three values of κ = − . , . − . as G is varied.A strong coupling expansion of the κ = 0 theory results in a prediction for the momentum spacefermion propagator of the form F ( p ) = √ G i sin p µ (cid:80) µ sin p µ + m F (2.1)where a non-zero fermion mass m F = 24 G − is generated.However, a fermion mass can also arise via a spontaneous breaking of one of the exact sym-metries and the formation of a bilinear condensate. This appears to happen at κ = 0 where anintermediate antiferromagnetically ordered phase appears around G ∼ . - see fig. 1b whichshows a staggered fermion susceptibility that scales linearly with volume as expected in the pres-ence of a condensate. This susceptibility is defined by χ = 1 V (cid:88) x,y (cid:15) ( x ) (cid:15) ( y ) ψ a ( x ) ψ b ( x ) ψ a ( y ) ψ b ( y ) . (2.2)2 iggs-Yukawa Model Simon Catterall
To study this question more carefully we have added an SO(4) breaking source term to the actionand studied the condensate as a function of the magnitude of the source as the thermodynamic limitis taken. The source terms take the form δS = (cid:88) x,a,b ( m + m (cid:15) ( x ))[ ψ a ( x ) ψ b ( x )]Σ ab + (2.3)where the SO(4) symmetry breaking source Σ ab + is Σ ab + = (cid:32) iσ iσ (cid:33) . (2.4) ε ( x ) ψ a ( x ) ψ b ( x ) m κ =0.0,G=1.05L=8L=12L=16 (a) Condensate vs m at κ = 0 ε ( x ) ψ a ( x ) ψ b ( x ) m κ =0.05, G=1.05L=6L=8L=12 (b) Condensate vs m at κ = 0 . The results for the vev of the antiferromagnetic bilinear ( m = 0 ) as a function of source areshown in fig. 2a at κ = 0 and G = 1 . corresponding to the peak of the susceptibility. Clearly forsmall values of the source the signal grows with increasing volume. This effect trades off againstthe general result that the vev must vanish for zero source on a finite volume. As a result a plateaudevelops that strengthens and moves to smaller source values as the thermodynamic limit is taken.This provides unequivocal evidence for spontaneous symmetry breaking at this value of G andconfirms the presence of a condensate at this point in the phase diagram.Contrast this with fig. 2b which shows the behavior of the same quantity at κ = 0 . and G = 1 . . In the latter plot the bilinear shows no significant volume dependence and falls smoothlyto zero as the source is reduced. Thus in this region of the phase diagram there is no evidence ofsymmetry breaking. Of course one should worry that perhaps there is no phase transition at all orthat the transition is shifted away from G ∼ . . Fig. 3 shows the corresponding susceptibilitywhich reveals a peak at that same value of the Yukawa coupling G . However, notice that thispeak is independent of the volume. While this is consistent with the absence of a condensate itleaves open the possibility that only a crossover rather than a true phase transition survives for κ > . One way to test this is to examine the number of conjugate gradient iterations required toinvert the fermion operator close to the transition. Fig. 4 shows a striking peak close to G ∼ . similar to that seen at κ = 0 and confirms the system still appears to undergo a phase transitionfor ( G, κ ) = (1 . , . . Indeed the peak is higher for positive κ indicating a faster growth of3 iggs-Yukawa Model Simon Catterall χ s t ag G κ =0.05 L=6L=8L=12 Figure 3: Susceptibility at κ = 0 .
0 0.5 1 1.5 2 2.5 A v e r ge o f C G i t e r a t i on pe r t r a j GL=8 κ =0.0 κ =0.05 Figure 4: Number of CG iterations vs G for κ = 0 and κ = 0 . at L = 8 iggs-Yukawa Model Simon Catterall the correlation length as compared to the κ = 0 situation. Thus the numerical evidence favors acontinuous phase transition at small, positive κ and no intermediate, broken phase.We have explored this model over a wider range of couplings and find results consistent withthe phase diagram sketched in fig. 5. For κ < the antiferromagnetic (AFM) phase observed for κ = 0 widens and still separates a massless symmetric phase (the so-called paramagnetic weak orPMW phase) from a massive symmetric phase (the paramagnetic strong coupling or PMS phase).If κ is sufficiently large and positive we see signs of a ferromagnetic (FM) phase, which for large G is separated from the PMS phase by first order phase transitions. The most interesting regionis the one we focused on above, namely small positive κ and Yukawa couplings close to unity. Inthis region there appears to be a continuous transition between the two symmetric phases. It isimportant to recognize that some sort of phase transition must be present at this point since on oneside the fermion is massless while on the other it is gapped.
3. Summary
Our numerical results, which were obtained with a Rational Hybrid Monte Carlo algorithm,appear to reveal the existence of a continuous phase transition separating a massless symmetric(PMW) from a massive symmetric (PMS) phase in the plane of Yukawa and kinetic couplings. Wehave concentrated the bulk of our efforts on the region near ( G, κ ) = (1 . , . . In the originalfour fermi model with κ = 0 , this value of G = 1 . lies close to the center of the antiferromag-netic phase. The transition between massless and massive phases appears to be continuous around ( G, κ ) = (1 . , . and there is no sign of a bilinear fermion condensate either antiferromagneticor ferromagnetic in nature at that point. If the evidence for this transition holds up on larger latticesit suggests the existence of a new fixed point for strongly coupled fermions in four dimensionsand should allow for the possibility of giving fermions mass in the continuum without breakingsymmetries. This is already a very interesting possibility.It also may offer some hope for formulating chiral fermions on the lattice following an oldidea by Eichten and Preskill in which strong four fermion interactions are used to gap out wouldbe doubler states that arise in a naive approach to formulating Weyl fermions on the lattice. Thebasic idea is to modify the Yukawa terms by including additional derivative operators so that theinteractions become large for doubler states but remain small for the primary, naive Weyl fermions.The spirit of the approach is rather similar to the inclusion of a Wilson term for vector like theoriesin which lattice derivatives are inserted into a bare fermion mass term to generate large doublermasses. For a chiral theory there are no invariant mass terms but there are invariant four fermionor Yukawa couplings and provided these break any anomalous symmetries it is possible to imaginea scenario where the doublers can be decoupled. It would be very interesting to repeat the workdescribed here for such a model based on reduced staggered fermions to see whether such a scenariocan in fact be realized. Acknowledgments
SMC acknowledges the support of the DOE through the grant DE-SC0009998. DS was sup-ported by UK Research and Innovation Future Leader Fellowship MR/S015418/1.5 iggs-Yukawa Model
Simon Catterall st order transition PMWPMW PMSPMS . . .
95 1 . . . AFM FM G κ = κ = κ = κ = 0 κ = 0 κ = 0 κ = − κ = − κ = − Figure 5: Sketch of the phase diagram in the ( κ, G ) plane References [1] E. Eichten and J. Preskill, “Chiral Gauge Theories on the Lattice,” Nucl. Phys. B , 179 (1986).doi:10.1016/0550-3213(86)90207-5[2] V. Ayyar and S. Chandrasekharan, “Fermion masses through four-fermion condensates,” JHEP ,058 (2016) doi:10.1007/JHEP10(2016)058 [arXiv:1606.06312 [hep-lat]].[3] S. Catterall and D. Schaich, “Novel phases in strongly coupled four-fermion theories,” Phys. Rev. D , no. 3, 034506 (2017) doi:10.1103/PhysRevD.96.034506 [arXiv:1609.08541 [hep-lat]].[4] D. Schaich and S. Catterall, “Phases of a strongly coupled four-fermion theory,” EPJ Web Conf. ,03004 (2018) doi:10.1051/epjconf/201817503004 [arXiv:1710.08137 [hep-lat]].[5] S. Catterall and N. Butt, “Topology and strong four fermion interactions in four dimensions,” Phys.Rev. D , no. 9, 094502 (2018) doi:10.1103/PhysRevD.97.094502 [arXiv:1708.06715 [hep-lat]].[6] N. Butt, S. Catterall and D. Schaich, “SO(4) invariant Higgs-Yukawa model with reduced staggeredfermions,” Phys. Rev. D , no. 11, 114514 (2018) doi:10.1103/PhysRevD.98.114514[arXiv:1810.06117 [hep-lat]].[7] V. Ayyar and S. Chandrasekharan, “Origin of fermion masses without spontaneous symmetrybreaking,” Phys. Rev. D , no. 8, 081701 (2016) doi:10.1103/PhysRevD.93.081701[arXiv:1511.09071 [hep-lat]]., no. 8, 081701 (2016) doi:10.1103/PhysRevD.93.081701[arXiv:1511.09071 [hep-lat]].