Fermion loop simulation of the lattice Gross-Neveu model
aa r X i v : . [ h e p - l a t ] A p r Fermion loop simulation of the lattice Gross-Neveu model
Christof Gattringer a , Verena Hermann a,b , and Markus Limmer a a Institut f¨ur Physik, FB Theoretische Physik, Universit¨at Graz8010 Graz, Austria b Department of Earth and Environmental Sciences, Geophysics, Munich University80333 Munich, Germany
We present a numerical simulation of the Gross-Neveu model on the lattice using a new rep-resentation in terms of fermion loops. In the loop representation all signs due to Pauli statisticsare eliminated completely and the partition function is a sum over closed loops with only positiveweights. We demonstrate that the new formulation allows to simulate volumes which are two ordersof magnitude larger than those accessible with standard methods.
PACS numbers: 11.15.Ha, 11.10.Kk
I. INTRODUCTION
Numerical simulations with fermions are notoriouslydifficult. The reason is that the minus signs due to Paulistatistics give rise to cancellation effects. In quantumfield theories the fermions are usually integrated out andthe fermion determinant appears as a weight factor. Evenin cases where the fermion determinant is real and posi-tive its numerical treatment is very costly, since it essen-tially couples all degrees of freedom with each other andthe individual contributions have changing signs.Finding alternative strategies for dealing with fermionswould considerably improve the quality of numerical sim-ulations. Such strategies could either be new algorithms(see, e.g., [1] for a prominent example) or a reformulationof the problem. Here we discuss the latter: In [2] an al-ternative representation was given for a two-dimensionalfermionic quantum field theory, the Gross-Neveu model.The partition function was rewritten as a sum over closedloops where each contribution has a real positive weight.This allows for a new approach to simulate the modelwhich avoids dealing with the fermion determinant.This paper presents the first test of the loop approachfor the Gross-Neveu model in a numerical simulation,and we explore the prospects and limitations of usingloop-type representations in a numerical simulation ofa fermionic system. Our results demonstrate that themethod is promising and it is worthwhile to pursue itin higher dimensions. In higher dimensions loop rep-resentations of quantum field theories are known, butso far have exclusively been used in the strong couplinglimit [3]. Our study here, although 2-dimensional, is per-formed at arbitrary coupling. We are currently exploringthe generalization to higher dimensions and believe thatfor certain four-fermi interactions, representations simi-lar to the one used here can be found and successfullyapplied in numerical simulations.For gauge theories the situation is more complicated.Since gauge fields are oriented one has to use orientedloops dressed with the gauge links and complex phasesappear. This was seen in the Schwinger model, where a loop representation exists [4], but a numerical simulationsuffers from the fermion sign problem. Upon going tothe strong coupling limit, the sign problem disappears[5] and a numerical simulation with loops again unleashesits power [6]. Four-fermi interactions may be generatedwith a Hubbard-Stratonovich transformation using scalarfields. These do not introduce complex phases and a looprepresentation without signs is possible.
II. THE LOOP REPRESENTATION
We begin with discussing the Gross-Neveu model [7]and its loop representation. In the continuum the actionof the Gross-Neveu model is given by S = S F + S S with S F = Z d x ψ ( x ) (cid:2) γ µ ∂ µ + ϕ ( x ) + m (cid:3) ψ ( x ) ,S S = 12 g Z d x ϕ ( x ) . (1)Here ψ and ψ are Grassmann valued vectors of N fla-vors of 2-spinors and we use vector/matrix notation forboth, spinor and flavor indices. The Euclidean partitionfunction is defined by integrating over all fields, Z = Z Y x dϕ ( x ) dψ ( x ) ψ ( x ) exp( − S [ ϕ, ψ, ψ ] ) . (2)Upon integrating out the scalar fields ϕ , the model turnsinto a purely fermionic theory with a four fermi interac-tion given by − g/ R d x ( ψ ( x ) ψ ( x )) . The Gross-Neveumodel is well understood analytically (see e.g. [8]) andhas been analyzed on the lattice in various settings [9].As it stands, the path integral (2) is only formally de-fined and a cutoff needs to be introduced. Here we uselattice regularization, which replaces the Euclidean spacetime R by a finite regular lattice Λ. The path integral(2) is well defined when the measure is understood asthe product over individual measures over the fields liv-ing on the lattice points. The action is discretized usingthe Wilson formulation such that it reads S F = X x ∈ Λ ψ ( x ) (cid:18) − ± X µ = ± ∓ γ µ ψ ( x ± ˆ µ )+ ϕ ( x ) ψ ( x ) + [2 + m ] ψ ( x ) (cid:19) ,S S = 12 g X x ∈ Λ ϕ ( x ) . (3)For the scalar field ϕ we use periodic boundary condi-tions for both directions, the fermions are periodic in thespatial direction and anti-periodic in time.Using hopping expansion techniques, the N -flavor lat-tice Gross-Neveu model (3) can be mapped into a modelof 2 N sets of loops [2]. For convenience we will oftenrefer to the loops in different sets as blue, red etc. loops.Within each set the loops are non-oriented, closed andself-avoiding. However, when loops belong to differentsets, e.g., a red and a blue loop, they may touch or crosseach other. The partition function of the lattice Gross-Neveu model is then a sum over all possible configura-tions of the loops in the 2 N sets. Each configuration hasa positive weight computed from the loops.Although [2] gives the partition function for arbitrary N , we here only quote the one-flavor expression which weuse in our simulation. For N = 1 we need two sets of self-avoiding loops, red and blue, denoted by r and b . Theone-flavor partition function in the loop representationreads (up to an overall normalization factor) Z = X r,b (cid:16) / √ (cid:17) c ( r,b ) f n ( r,b )1 f n ( r,b )2 . (4)In this formula c ( r, b ) is the total number of corners forboth red and blue loops. Thus every corner contributes afactor of 1 / √ n ( r, b ) isthe number of lattice sites which are singly occupied byeither r or b and n ( r, b ) is the number of doubly occupiedsites. We remark that, since the loops in the two sets areself avoiding, double occupation can appear only when ared and a blue loop cross our run alongside each other.The weight factors f and f are related to the mass m and the coupling g through f = (2 + m )[(2 + m ) + g ] − , f = [(2 + m ) + g ] − . (5)We stress that the mapping (4), (5) is exact in thethermodynamic limit. For finite volume different typesof boundary conditions in the two representations leadto finite size effects: In the loop representation we needto have closed loops and in a finite volume the loopscan wind around the compact lattice. The loop configu-rations fall into three equivalence classes, C ee , C eo , C oo ,depending on the numbers of red and blue non-triviallywinding loops (see also [6]): C ee (even-even): The totalnumber of windings for both, red and blue loops is evenfor both directions. C eo (even-odd): One of the colors has an odd number of windings for one of the directions. C oo (odd-odd): Both colors have an odd number of windingsin one of the directions. These equivalence classes cannotbe linked in a simple way to the boundary conditions inthe standard representation which we discussed above.However, below we will demonstrate that the boundaryeffects vanish as 1 / √ V , with V denoting the volume. III. NUMERICAL SIMULATION
The numerical simulation of the Gross-Neveu modelnow is performed directly in the loop representation (4)using a local Metropolis update (see e.g. [10]). We updatethe red and the blue loops alternately, by performing afull sweep through the lattice for one of the colors andtreating the other one as a background field. A sweepconsists of visiting all plaquettes of the lattice. For eachplaquette we generate a trial configuration by invertingthe occupation of the color we want to update for allthe links in the plaquette. This guarantees that all loopsremain closed. Furthermore new loops may be generatedwhen all links of the plaquette are empty. When thetrial configuration violates the self-avoiding condition itis rejected immediately and the algorithm tries the nextplaquette. Otherwise the trial configuration is acceptedwith the Metropolis probability p = (1 / √ ∆ c f ∆ n f ∆ n ,where ∆ c is the change in the number of corners and∆ n , ∆ n are the changes in the occupation numbers.The initial configuration can either be the empty lattice(for C ee ) or has one or two winding loops ( C eo and C oo ).The observables we discuss here are all first and sec-ond derivatives of the free energy F = − ln Z , and canbe written as moments of the occupation numbers. Inparticular for the chiral condensate and its susceptibil-ity, which in the standard language are given by χ = 1 V X x ∈ Λ h ψ ( x ) ψ ( x ) i = − V ∂ ln Z∂m , C χ = ∂χ∂m , (6)we quote the corresponding expressions in terms of occu-pation numbers and their fluctuations, χ = − V f (cid:16) f h n i + 2 f h n i (cid:17) ,C χ = − V f (cid:16) [4 f − f f ] D ( n − h n i ) E + [ f − f f ] D ( n − h n i ) E + 2 f f D ( n + n − h n + n i ) E − [4 f − f f ] h n i − f h n i (cid:17) . (7)Here we have introduced n , the total number of emptysites, i.e, sites visited by neither a red nor a blue loop.Equivalent formulas can be derived for the internal en-ergy, the heat capacity as well as for derivatives of the freeenergy with respect to the coupling g . n -point functions -0.2 0.0 0.2 m -0.80-0.75-0.70-0.65 χ exact result512 exact result32 fermion loops512 fermion loops FIG. 1: The chiral condensate χ for g = 0 as a function of m for 2 different lattice sizes. We compare the simulation inthe loop representation (symbols with error bars) to the exactresult from Fourier transformation (curves). may be treated as usually by introducing source fields anddifferentiating with respect to them. This gives expres-sions involving correlators of local occupation numbers.Finally, the generalization of the above formulas to anarbitrary number of flavors is straightforward. IV. RESULTS
In this section we present some selected results whichserve to illustrate the advantages of the loop approach,but also allow to assess its limitations. In order to com-pare with traditional methods, we performed a refer-ence simulation of the Gross-Neveu model using standardmethods. The fermions were integrated out giving rise tothe fermion determinant in a background configuration ofthe scalar field ϕ . These background configurations werecomputed according to the Gaussian distribution of S S ,and the determinant was used as a factor for reweight-ing. This is possible, since the eigenvalues of the Diracmatrix come in complex conjugate pairs, and the scalarfield does not have topological modes which could giverise to zero eigenvalues. Thus the fermion determinant isalways strictly positive.We stress that the reweighting in the standard formu-lation works only in two dimensions due to the numeri-cal cost of evaluating the determinant. However, for ourproblem where the scalar fields are independent Gaus-sians at each site, reweighting has the big advantage, thatautocorrelation is avoided. Alternative strategies such asHybrid Monte Carlo, cannot make use of that advantage.Another important conceptual point has to be ad-dressed: For the free case, g = 0, the standard repre-sentation allows for an exact solution with the help ofFourier transformation. In the loop formulation, how-ever, the case g = 0 is not special at all. Thus g = 0 is theoptimal point for testing the power of the loop approach -0.2 0.0 0.2 m -2.0-1.5-1.0-0.50.0 C χ exact result512 exact result32 fermion loops512 fermion loops FIG. 2: Same as Fig. 1, now for the chiral susceptibility C χ . to the limits because we have exact results on almost ar-bitrary large volumes, which we use to compare with thedata of the loop simulation. Since the weight factors f and f of Eq. (5) are smooth functions of g and m it isreasonable to transfer the experience obtained with theloop approach at g = 0 to nearby values of g .Thus we begin our assessment of the loop approach at g = 0. In Figs. 1 and 2 we compare the loop results in the C ee sector (symbols) with those from Fourier transforma-tion (curves). We use two volumes for the comparison,a relatively small lattice of size 32 ×
32 and a consid-erably larger one, 512 × m we typically performed10000 sweeps of our local update for both colors for equi-libration and used 50000 measurements of the observ-ables separated by 10 pairs of sweeps. The observableswere calculated using the occupation number representa-tion (7) and the statistical error was computed with thejackknife method.Fig. 1 shows that already on the small lattice the datapoints are very close to the exact result. The largest dis-crepancy is seen near m = 0, the chiral point where thefermions become massless. For the larger lattice the datapoints fall exactly on top of the analytic result. The sit-uation is similar for the susceptibility in Fig. 2. For thesmall lattice we find a clear finite size effect, a shift of thesusceptibility curve. On the larger lattice the agreementis almost perfect and only at the chiral point m = 0 westill see a slight discrepancy. We remark, that the de-crease of the minimum of C χ with increasing volume V does not signal a phase transition (at N = 1 there is nodiscrete symmetry that could be broken spontaneously).The minima decrease only logarithmically with V . For g = 0 it can be shown exactly that C χ diverges loga-rithmically when removing the IR cutoff. For g > C χ as obtained from the simu-lation very reliably with ln V . Also the comparison withthe results from the standard approach shows that thelargest discrepancy is found near the chiral point, which,however, vanishes quickly with increasing volume. V -3 -2 -1 ∆ χ ∆ χ (eo-ee) ∆ χ (oo-ee) y = 0.72 x -0.96 y = 1.47 x -0.97 FIG. 3: Splitting of the results for the chiral condensate inthe different equivalence classes as a function of the volume.
An important part of comparing the standard and theloop approach is to test how the different types of bound-ary effects scale with the volume and at what rate theperfect equivalence of the two representations is reachedwith increasing V . We assess this question directly in theloop approach: At a fixed point ( m, g ) in parameter spacewe compute the chiral condensate χ for the three differ-ent equivalence classes C ee , C eo , C oo introduced above.This is repeated for several volumes V and in Fig. 3 weplot the discrepancy of the results as a function of √ V .The data shown in the plot are for g = 0 . m = 0 . χ ( eo − ee ) denotes the discrepancy betweenthe C eo and C ee results and ∆ χ ( oo − ee ) is the splittingbetween C oo and C ee . Also a comparison of χ to the re-sults from the standard approach with mixed boundaryconditions shows a 1/ √ V behavior.The splitting ∆ χ can be analyzed with the help ofmean field theory. One finds that it should behave as∆ χ ∝ / √ V . We performed a fit of our data to the form ∆ χ ∝ cV α and found values of α which are close to − / m, g ) (see also Fig. 3). Thusmean field arguments as well as our numerical findingsindicate, that the finite volume effects scale as 1 / √ V . V. DISCUSSION
In this letter we have explored an alternative formu-lation for fermionic systems using the example of a 2-dimensional quantum field theory. The representation interms of fermion loops allows one to simulate the systemwithout having to use fermion determinants. An impor-tant aspect is that in the loop formulation used here weare not restricted to the case of strong coupling but canwork at arbitrary g . In this exploratory study we simu-late the model using a simple local update and comparethe outcome to analytic results and the data from a sim-ulation in the standard approach. Many observables canbe expressed in terms of occupation numbers and theircorrelators. We show that finite size effects decrease like1 / √ V and thus the thermodynamic limit, where the looprepresentation becomes exact is approached rapidly.An important issue is of course the assessment of thegain in numerical efficiency when using the loop algo-rithm. Already with the local algorithm used here a con-siderable increase of the accessible volumes was found.Using the same small cluster of PC’s the standard ap-proach could be used on lattices with a maximum volumeof 32 ×
64, while in the loop formulation we were able tosimulate systems up to 700 × Acknowledgments:
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