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Critical behaviour of the 1D q-state Potts model

arXiv:hep-lat/9303016v1 31 Mar 1993Critical behaviour of the 1D q-state Potts modelwith long-range interactionsZ Glumac and K UzelacInstitute of Physics, University of Zagreb, Bijeniˇcka 46, POB 304 , 41000 Zagreb,CroatiaAbstractThe critical behaviour of the one-dimensional q-state Potts model with long-rangeinteractions decaying with distance r as r−(1+σ) has been studied in the wide rangeof parameters 0 < σ ≤1 and116 ≤q ≤64. A transfer matrix has been constructedfor a truncated range of interactions for integer and continuous q, and finite rangescaling has been applied.

Results for the phase diagram and the correlation lengthcritical exponent are presented.Short title: Critical behaviour of the 1D LR q-state Potts modelPhysics Abstracts classification number: 0550

1IntroductionRecently, finite range scaling (FRS) appeared as an efficient method to deal withvarious discrete one-dimensional models with long range (LR) interactions (Glumacand Uzelac, 1989, 1991). The aim of the present study is to apply it to the one-dimensional q-state Potts model with LR interactions, defined by the hamiltonian−βH = KXi

In the two dimensional SR case for example, besides integer-q modelswhich are equivalent to the classical (q-1)/2-spin models, there are also non-integerq models which are connected with other critical problems such as percolation (q=1limit), resistor network (q=0 limit), dilute spin glass (q=1/2 limit), branched poly-mers (0≤q ≤1) (see review by Wu, 1982).It would be interesting to extend those concepts to LR interactions. However, incontrast to the rich literature on the SR Potts model, the LR Potts model has beenmuch less explored.

The best explored member of the q-state Potts models is the q=2model where both, analytical (Dyson 1969; Fisher, Ma and Nickel, 1972; Kosterlitz,1976) and numerical (Nagle and Bonner, 1970; Glumac and Uzelac 1989,1991) resultsare available. Dyson has shown that, as a consequence of the LR forces, there is acritical temperature different from zero as long as 0 < σ ≤1.

Fisher et al. haveperformed ǫ = 2σ−d and 1/n expansions in the context of the renormalization group(RG) for all dimensions d and σ ̸= 2 on a more general system with an n-component2

order parameter. They obtained the correlation function exponent η = 2 −σ to allorders in ǫ, ν = 1/σ when ǫ < 0 (classical regime), ν = 1/σ + n+2n+8ǫσ2 + .

. .

whenǫ > 0, and leading irrelevant exponent y3 = −|ǫ|. A similar expansion, in 1-σ > 0,has been done by Kosterlitz (1976), who obtained : 1/ν = −y3 =q2(1 −σ) whenσ →1.

Very recently, a new method, called cycle expansion, has been demonstratedby Mainieri (1992) on the example of the s=1/2 Ising model. Mainieri obtained veryaccurate estimates of the critical temperature and other thermodynamic quantitiesin the critical region.The literature on q ̸= 2 LR Potts models is relatively poor.

By using the RGǫ-expansion, Priest and Lubensky (1976) have found a fixed point and critical ex-ponent ν to the first order in ǫ = 3σ −d. More recently, Theumann and Gusm˜ao(1985), performed similar calculation to the second order of ǫ = 3σ −d.Theyobtained the following results for q< 3: η = 2 −σ to all orders of ǫ, andν−1 = σ −α2βǫ + αβ(α −β)S(σ) + 12α(β2 −γ)G(σ)8β3ǫ2 + 0(ǫ3)(2)where α = q2(q −2), β = q2(q −3), γ = q4(q2 −6q + 10), S(σ) ≡ψ( 32σ) −ψ(σ) −ψ(σ/2) + ψ(1), and G(σ) ≡Γ( 32 σ)Γ3(σ/2)Γ3(σ).

Ψ(z) is the logarithmic derivative of theΓ function. In the q →1 limit, their results are applicable to the description ofpercolation critical behaviour for a random Ising ferromagnet with a LR power-lawinteraction.The advantage of the FRS method is the capability to cover, unlike the ǫ-expansion, a wide range of parameter space.

In the present article the phase diagramand the correlation length critical exponent will be studied in the region 0 < σ ≤1and116 ≤q ≤64.The main idea of our approach is the following: we truncate the originally infinite-range interaction to the L first neighbours and solve exactly (although numerically)the finite-range version of (1)−βH =NXi=1LXj=1Kj[δ(si, si+j) −1q]p.b.c. (3)3

where Kj denotesKj1+σ . Then, by using the FRS method (section 2.1) with the ap-propriate extrapolating procedure (section 3), the L→∞behaviour will be deduced.For this purpose a transfer matrix for a reduced model defined by eq.

(3) willbe constructed. Two different formalisms will be applied: one for integer q values(section 2.2) and the other for non-integer q values ( section 2.3).The results are presented in section 3, while section 4 contains a short discussionand some open questions.2Method2.1FRSThe FRS method has been constructed in analogy with the FSS (Fisher and Barber,1972), where instead of finite-size, the finite-range of interactions is scaled (Glumacand Uzelac, 1989,1991).

The basic idea of FRS is that by studying the sequence ofsystems with their long-range interactions truncated above certain range, one canobtain, by using scaling properties, the information on the critical behaviour of thetrue infinite system.Let A∞(t) be the physical quantity of the true long-range system, which alge-braically diverges, in the vicinity of the critical point t=0A∞(t) ≃A0t−ρ(4)where t=(T-Tc)/Tc, Tc is the critical temperature, ρ is the related critical exponent,and A0 is a constant. Then, analogous to the FSS hypothesis one can assume thatfor large finite range M and small t, AM(t) can be written in the formAM(t) = A∞(t)·f( Mξ∞)(5)where f is a homogeneous function with following propertieslimx→∞f(x) = 1,limx→0 f(x) = const.·xρν(6)By applying the equation (5) to the correlation length ξ∞(t) = ξ0t−ν, the stan-dard scaling procedure gives the condition for critical temperature through the fixed4

point equationξM(t∗)M= ξM′(t∗)M′(7)and the expression for correlation length critical exponent νν−1 =ln ξ′M(t∗)ξ′M′(t∗)ln MM′−1(8)where M’=M-1 in all calculations, and ξ’ is temperature derivative of ξ.There are two important facts concerning the above method which are of interest.First, since the critical behaviour is essentially dependent on the range of inter-action, one can (unlike in the FSS (Br´ezin, 1982)) expect applicability of FRS inmean-field as well as in the non-trivial region (Glumac and Uzelac 1989, 1991). Sec-ond, the correlation length calculated from the transfer matrix presents the averagedistance between domain walls for an infinitely long strip so that relation (7) givesthe transition temperature for both first and second-order phase transitions.

Thisfact is common with FSS (Binder, 1987).2.2Transfer matrix: integer-q formalismIt is straightforward to construct the transfer matrix for model (3) with an integernumber of Potts states.The chain with a range of interaction L, can be represented as a strip withcolumns of height L (Uzelac and Glumac, 1988). Each column then can be consid-ered as an object with qL possible states, interacting only with its first neighbour.The transfer matrix T is given by:⟨i|T|j⟩= exp( LXk=1Kk"L−kXn=1δ(in, in+k) +kXn=1δ(iL+n−k, jn)#)(9)where jm = 0, 1, .

. .

, q −1 are the elements of the L-component vector of states |j⟩of a column of height L. The matrix T can be further decomposed into a productof L matrices Tn, each one describing the addition of one more site to the column(Temperley and Lieb, 1971).T = T1 · . .

. · TL(10)5

⟨i|Tn|j⟩=LYl=1l̸=nδ(il, jl) exp(n−1Xk=1Kkδ(jn−k, jn) +L−1Xk=nKkδ(jL+n−k, jn) + KLδ(in, jn))(11)There is a simple relation between neighbouring one-site matrices:UTTn+1U = Tn(12)where TL+1=T1 and U is a matrix of the translation operator in the vertical stripdirection, with matrix elements given by⟨i|U|j⟩= δ(i1, jL)δ(i2, j1)δ(i3, j2), . .

. , δ(iL−1, jL−2)δ(iL, jL−1).

(13)The matrix Usatisfies the relations UL = 1 and UL−1 = UT = U−1. Conse-quently, the transfer matrix can be written as the L-th power of a single matrix asa peculiarity of the present long-range model:T = (U · TL)L = ˜TL(14)⟨i| ˜T|j⟩= δ(j1, i2)δ(j2, i3) .

. .

δ(jL−1, iL) exp( LXm=1KL+1−mδ(jL, im))(15)Notice that ˜T has only q nonzero elements per row which greatly reduces computermemory.Further reduction is obtained by using the symmetry properties of Potts interac-tion. In the present calculation, only the translation invariance in the space of Pottsstates has been used, which decomposes ˜T into q submatrices (of order qL−1).

Theq-1 of them have the identical eigenvalues as a consequence of Kronecker-δ type ofinteraction. Thus we have to diagonalize only two submatrices or, more precisely,we have to find only the largest eigenvalue of each of the two submatrices, µ1,L andµ2,L respectively, to be able to construct the correlation length ξLξL =Lln λ1,Lλ2,L=1ln µ1,Lµ2,L(16)where λ1 > λ2 and µ1 > µ2 are the eigenvalues of T and ˜T respectively.For this purpose, the method of direct iteration with a single vector is suitable(Wilkinson, 1965).

If one defines a set of vectors bj in a following wayAbj = bj+1(17)6

for arbitrary initial value b0, where A is a real nonsymmetric matrix, then thelargest eigenvalue µ of matrix A is given byµ = limj→∞bj(n)bj−1(n)(18)for any vector’s component n. By applying this method for cases with q= 2, 3, 4and 5, we obtained the two largest eigenvalues of ˜T for maximum ranges L= 20,13, 10 and 9, respectively. A few ten to a few hundred iterations were required toobtain the leading eigenvalue with an accuracy of 10−15 for both submatrices.2.3Transfer matrix: continuous-q formalismAs mentioned in Introduction, there are some interesting cases (q=0, 1/2, 1, ...) towhich the described integer-q formalism cannot be applied or becomes inefficient(limitation to small ranges for large q models).

Thus, it would be advantageousto formulate a q-independent transfer matrix, which would allow to treat q as acontinuous parameter. Such a type of transfer matrix has already been constructedfor the 2D Potts model with SR interaction (Bl¨ote, Nightingale, Derrida 1981; Bl¨oteand Nightingale, 1982).

We will shortly explain the application of that method tothe considered LR model.Let us translate the partition function of model (3), into the graph-theory lan-guage. First, we rewrite the basic expression exp{Kj · δ(si, si+j)} aseKjδ(si,si+j) = 1 + vjδ(si, si+j),(19)where vj = eKj−1, so that the partition function, with periodic boundary conditions,becomesZN,L =qXs1=1qXs2=1.

. .qXsN=1LYj=1(20)[1 + vjδ(s1, s1+j)][1 + vjδ(s2, s2+j)] .

. .

[1 + vjδ(sN, sN+j)]If we perform the multiplications of the above terms, the one-to-one correspon-dence between every term of that expansion and the particular graph on the m×Llattice (N=m·L) can be established in the standard way (Bl¨ote and Nightingale,1982).7

Now, the partition function becomesZN,L = qN ·XGub1(G)1·ub2(G)2· . .

. ·ubL(G)L·ql(G).

(21)The above summation goes over all possible graphs on the lattice with L differenttypes of bonds described by uj = vj/q , bj(G) is the number of bonds of type j, andl(G) is the number of independent loops on the graph G.A certain number of different bonds in every single graph, produce connectionsbetween sites. If one considers a strip of height L, then all possible interactionsbetween all sites in that strip produce a particular state of connectivity α of the sitesin the last column.

The LR interactions produce a larger number of connectivitiesthan in the 2D SR models.The connectivity is represented by an integer, and it will be used as an indexof the transfer matrix. The partition function ZN can be written as a vector withcomponents ZN(α) , where each component ZN(α) describes the contribution to thepartition function of all graphs that produce one particular state of connectivityα in the last column.

By adding a new column, we get ZN+L which again can bewritten as a vector ZN+L(β). The connection between those two partition functionsis given by the transfer matrix in the following wayZN+L(β) =XαT(β, α)ZN(α)(22)where the summation goes over all graphs, produced by the added column, thatlead from connectivity α to β.

A transfer matrix decomposition analogous to thatof section 2.1, leads toT = qT1 · . .

. · qTL = (qT1VT)L = ˜TL(23)where Tj describes the addition of the j-th site to the column, and V is the matrixof the translation operator in the vertical strip direction.

Further decomposition ofT1 is possibleT1 = T1,L(uL) ·1Yj=L−1T1,j(uj)(24)8

Each of the T1,j(uj) matrices describes the presence of an uj type bond on the firstsite and has at most two nonzero elements per row. T1,L(uL) is an upper triangularmatrix, while the other matrices are lower triangular.

It is easy to construct a matrixM which relates the lower triangular matricesMT T1,j(x)M = T1,j+1(x)(25)The matrix ˜T then becomes˜T = qT1,L(uL)1Yj=L−1(T1,L−1M)ujVT(26)By using these decompositions, it is sufficient to keep in computer memory onlyfour matrices: T1,L, T1,L−1, M and V, with at most two nonzero elements per row,which strongly reduces amount of data to be stored.Since we want to get information about LR correlations, we should also take careof the connectivity between sites in the last column and the sites in the first column.In order to do so, one can imagine the top site in the last column of ZN to be somekind of ”ghost” site representing the connection between the first site of last columnof ZN+L and the sites of the first column. Thus we need a column of height L todescribe all connectivities in the model with the range of interaction equal to L-1.In order to keep the ”ghost” site fixed, the matrix of the translation operator V willbe constructed by acting with operator V on non-”ghost” sites only.

Consequently,if we perform the following set of transformationsuL →uL−1, uL−1 →uL−2, . .

. , u2 →u1, u1 →0,(27)we will obtain ˜T for the model with range L-1, with LR correlations included.With this redefinition, ˜T has the following structure˜T = ˜T−X0˜T+where ˜T+ and ˜T−contain the largest µ1,L and the second largest µ2,L eigenvalueof the whole matrix ˜T, respectively.

The correlation length is given by the relation(16) again.9

The order of ˜T for a model with range L is given by NL, and the order of ˜T+is given by NL−1, whereNL =L+1Xi=1ϕLi(28)ϕLi = i · ϕL−1i+ ϕL−1i−1(29)for L≥2, and ϕ22=3, ϕn1 = ϕnn+1 = 1. Some values of NL are presented in table 1.The present calculations were performed for the maximum range L=8 withq=1/16, 1/2, 1, 8, 16, 32 and 64.

The largest eigenvalues of the matrices ˜T±,are extracted by using the method explained earlier (section 2.2) with a similaraccuracy and number of iterations.3Results3.1Convergence and extrapolation proceduresThe critical temperature and exponent ν, given by the equations (7) and (8), still de-pend on the range M and it is important to apply the right extrapolation procedure.A simple analysis done within FSS (Privman and Fisher, 1983) can be reproducedin the case of the FRS (Glumac and Uzelac, 1989) and shows that the convergenceis dominantly governed by the leading irrelevant field and the corresponding criticalexponent y3 < 0. On the basis of such analysis, we expect power-law convergenceof Tc and ν in the large-M limitTc,M = Tc + a · My3−1ν(30)ν−1M = ν−1 + b(Tc,M −Tc) · M1ν + c · My3 = ν−1 + b′ · My3(31)where a, b, b′ and c are constants.

Thus, we chose to fit the obtained results forKc,M and νM to the formyM = ye + A · M−xy(32)in the least-squares approximation (LSA), where ye denotes the extrapolated quan-tity. The calculations are performed by taking the five largest values of M.10

It should be noticed, however, that Privman and Fisher (1983) analysis doesnot apply to the MF region and, consequently, the extrapolation form (32) becomesarbitrary there. On the other hand, in our previous analysis for the Ising model(Glumac and Uzelac, 1989) it turns out that the extrapolation formyM = B + A(M −1M)xy(33)gives a better agreement with the exactly known results for ν in the MF region thenthe function of the form (32).The error bars of the above extrapolations can be estimated only roughly, bylooking for the remaining L dependence of the results.

In tables (2) and (3) arepresented results which last digit is modified under the change of maximal rangefrom L-1 to L. The more careful examination of the errors, shows that the size ofthe error estimate changes with q and σ, being less then ten percents at the most ofq and σ region. Exception is the small q and σ region where the error is estimatedto be a few times larger.3.2Critical temperatureThe inverse critical temperature Kc,M, defined by equation (7), is calculated withan accuracy higher than 10−10 for different values of Potts states and with σ as anparameter.When increasing the exponent of LR interaction, there will be a value of σbeyond which the SR critical behaviour (with Tc = 0) takes place.

For the Isingmodel (q=2) it was analytically shown (Dyson, 1969) that this σ is equal to 1. Asargued more generally (Fisher et al.

1972, Sak 1973) this exchange of regimes shouldoccur when the correlation function exponent η of the LR system becomes smalleror equal to the SR one, ηSR. Thus in present case, we expect that σc = 1 for allvalues of q.

This expectation is confirmed by our numerical results. In our earlierworks (Glumac and Uzelac, 1989, 1991), we have been able to detect the appearanceof SR forces governed critical behaviour by the change of Kc,M from descending toascending sequence and by the sudden increase of convergence exponent.

In the11

present calculations, both of these phenomena were observed for any q consideredat σ ≈1. The extrapolated values for the critical temperature are presented infigures (1a) and (1b).We have used the LSA extrapolation method with the form (32) to calculateKc,e and the leading convergence exponent xK as a function of q and σ which arepresented in tables (2a) and (2b) respectively.

The only point where this could notbe applied is σ = 1, where the simple convergence expression does not apply anymore. In order to avoid non-monotonic behaviour, we have used only the data forthe three largest values of M fitting a linear relation to them (xK = 1).

The errorshould not be important, since for σ = 1 the variation of data with M is ratherweak.3.3Critical exponent νThe correlation length critical exponents νM has been calculated by using the equa-tion (8). The accuracy of results is reduced (with respect to that of the criticaltemperature) to the order of 10−6 due to the numerical differentiation.Following the form (31) for the scaling correction in non-MF region, one canexpect that the correction terms to ν−1M would be smaller while calculating the νMon the temperature Kc,e instead of Kc,M.This is indeed the case for the q=2(s=1/2 Ising) model, where detailed comparison between νM(Kc,M) and νM(Kc,e)has been given (Glumac and Uzelac, 1989).

In the present model we observe betterconvergence of the data calculated from Kc,e in the LR σ-region, for all q.According to the discussion at the end of section 3.1, the extrapolations in thelow σ region where MF behaviour is expected were performed at the temperatureKc,M using the extrapolation function of the form (33). Since the σMF border is stillan open question for q > 2 (see later in text), we have used eq.

(33) for σ < .3, q ≤1,and for σ < .5, q = 2.The extrapolated results are presented in figure (2). In tables (3a) and (3b) areresults for ν−1eand the corresponding convergence exponents respectively.

Similarlyto preceding section, for σ = 1 the LSA procedure was performed by imposing12

xν = 1 in eq. (32).The special case q=2 which corresponds to the Ising model was extensively stud-ied in our earlier work (Glumac and Uzelac, 1989) with a maximum range equal10.

The present work permits to reach the range of 20. Similar values of νe wereobtained in both works, which suggests that an increase of range from 10 to 20 doesnot change the accuracy significantly.On table 4, our results are compared to the ǫ-expansion results of Theumannand Gusm˜ao (1985).

A difference ∆of only few percent is obtained. Figure (2a)suggests that σ = 13 as the MF border and a q-independent value of ν−1MF = σ for allq ≤1.We have not found any analytical or numerical estimate for the values of σMFor ν for q ≥3.

In that situation we decided to calculate ν at the temperatureKc,e since the scaling law ξMM = const. is more accurate at that temperature thanat Kc,M.

Also, we decided to make the extrapolation using function (32) whichposesses a more transparent L-dependence.For large q (q ≥16), only the continuous formalism could be applied, whichdoes not permit to go beyond the range of L=8. Contrary to the small q case, forlarge q this range was not sufficient to find a good extrapolation.

Namely, whenapplying the LSA there, with the assumed form (31), one obtains xν →0 so thatthe correction terms become of the same order as the leading term and ν−1 diverges.This behaviour of ν suggests a change of regime of convergence and brings up thequestion of the order of transition.For all values of q the exponent ν increases as σ →1, which in the case q=2appeared (Glumac and Uzelac, 1989) as an indication of the essential singularity inthis limit (Kosterlitz 1976).4Conclusion and discussionBy using a FRS method combined with transfer matrix calculations we have beenable to study the long-range Potts model on the one-dimensional lattice in a wide13

range of values for the number of states q and the interaction exponent σ. We haveshown that for this problem the transfer matrix can be decomposed into matricesof much simpler form.

A different transfer matrix procedure can be done also in thecase of continuous q, so that our analysis could be extended to some non-integer qcases of interest.The study is concentrated on the phase diagram and the correlation length crit-ical exponent ν.The critical temperature shows monotonic decrease with q and σ and has a finitevalue at σ = 1. The change of behaviour of Kc,M at σ ≈1 indicates the exchangefrom LR to SR regime, established for any q.

The critical exponent was calculatedat the extrapolated critical temperature, and good agreement with values obtainedby other authors was observed. Generally, a good accuracy is harder to obtain inthe small-σ region due to the very long-range of interaction and in the σ →1 regionwhere the range of interaction ceases to be a good scaling variable.There are two points that require some further discussion.First is a question of mean-field border.

The present method, unlike the FSS,has the advantage that it can be applied within the mean-field region, but on theother hand, passes smoothly between the two regions (Glumac and Uzelac 1989,1991) , which does not permit to point out the mean-field border σMF. Anotherway to detect this border could be numerical, by observing the change of behaviourof ν as a function of σ.

But, our previous results on the Ising model where σMFis known, show that close to it the numerical results for ν are not sharp enough tolocate this point with precision. In the present case, we can only confirm that ourresults are consistent with σMF as known from the literature for a few particularcases: q=2 (σMF = 12 Fisher et al 1972), q=1 (σMF = 13 Priest and Lubensky 1976),q = 12 (σMF = 13 Aharony 1978, Aharony and Pfeuty 1979), but the precise locationof σMF for arbitrary q is left unestimated.A second point is the possible appearance of a first order transition for someq > qc.

Within the transfer matrix formalism this problem was considered for the2D SR case where qc exists and is exactly known (qc = 4, Baxter 1973).Igl´oi14

and S´olyom (1983) have shown that the first order transition is connected with thecrossing of the largest and the third largest eigenvalue of the transfer matrix forfinite length chain and large q. We have applied a similar calculation in our modelconsidering two groups of parameters (q=100, σ = .8, L=4 and q=300, σ = .4, L=4).Both cases give a negative result; the first and third eigenvalues do not intersect asa function of temperature.

By analogy with Igl´oi and S´olyom, that result can beinterpreted as the absence of a first-order transitions for any q in the 1D LR Pottsmodel. Since this is opposite to the indications of the behaviour of ν for large q (atthe end of section 3.3), the question of the order of transition for large q is still leftopen.15

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Figure captions:Figure 1a:The extrapolated critical temperature as a function of σ.Figure 1b:The extrapolated critical temperature as a function of σ. Figures (a) and (b)have common q=1 line which allows the comparison of scales on both figures.solid filled circle;Figure 2a:The exponent ν−1eas a function of σ. solid filled circle.Figure 2b:The exponent ν−1eas a function of σ.

Figures (a) and (b) have common q=2 linewhich allows the comparison of scales on both figures.17

Table captions:Table 1:NL is the order of transfer matrix for continuous-q model with maximum rangeequal L.Table 2a:The extrapolated values of inverse critical temperature as a function of q and σ.For the error bars see in text.Table 2b:The critical temperature convergence exponent xK as a function of q and σ.Table 3a:The extrapolated values of ν−1 as a function of q and σ. For the error bars seein text.Table 3b:The convergence exponent xν as a function of q and σ.Table 4:The comparison between extrapolated values ν−1eand Theumann and Gusm˜aovalues ν−1TG for σ = .4.18

L23456789NL51552203877414021147115975σ\ q11612123458163264.1.004.031.061.0927.136.203.28.48.781.001.14.2.007.052.102.1831.2701.362.45.64.911.121.31.3.010.077.149.2717.3862.489.576.761.021.241.45.4.014.108.204.3625.4939.601.690.8751.141.371.59.5.020.147.270.4590.6013.713.803.991.261.531.75.6.028.198.351.5644.7143.829.9201.121.401.691.94.7.041.269.452.6833.8374.9541.0461.241.541.842.15.8.064.375.584.8231.97741.0931.1851.3841.692.002.336.9.121.552.763.99731.1441.2551.3431.5401.842.1732.5191.0.430.815.9901.2301.3481.4401.5181.6971.9972.3252.677σ\ q11612123458163264.11.51.41.3.86.81.8.91.01.11.0.8.21.51.41.31.02.991.11.11.21.21.0.9.31.61.51.41.131.131.21.31.31.21.0.9.41.61.51.41.191.211.31.31.31.21.0.9.51.61.41.31.221.251.31.41.31.21.1.9.61.61.41.31.221.261.31.41.41.31.21.0.71.51.31.31.201.261.31.41.41.41.31.2.81.41.21.31.221.311.41.51.51.51.41.4.91.11.51.71.881.701.71.71.71.61.61.61.01111111111119

σ\ q11612123458163264.1.11.12.12.101.091.12.16.21.31--.2.18.20.21.202.215.24.30.45.75--.3.32.31.31.301.323.41.531.14.1--.4.34.34.35.373.481.59.893.0---.5.33.34.37.430.577.771.25.0---.6.30.32.37.501.664.831.22.3---.7.25.29.35.518.636.781.01.44.6--.8.18.24.32.483.574.67.80.991.31.4-.9.10.21.30.405.491.56.62.76.971.1-1.0.05.18.25.309.393.46.52.63.79.921.0σ\ q11612123458163264.1----.94.89.84.83.61--.2----.75.84.76.56.24--.3.72.74.75-.78.65.50.24.04--.4.71.72.72-.52.48.30.08---.5.72.72.71.70.48.37.23.05---.6.75.74.72.49.40.36.24.12---.7.78.77.77.43.46.41.30.24.06--.8.81.92.99.43.59.57.47.42.31.32-.9.871.001.001.412.161.321.08.84.62.62-1.011111111111qν−1eν−1TG∆(%)116.340.32504.512.338.33002.51.349.338732.373.4720

This figure "fig2-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303016v1

This figure "fig3-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303016v1

This figure "fig4-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303016v1

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출처: arXiv:9303.016 • 원문 보기