Yang-Lee zeros of the two- and three-state Potts model defined on π3 Feynman diagrams

Yang-Lee zeros of the two- and three-state Potts model defined on

␾

3

Feynman diagrams

Luiz C. de Albuquerque*

Faculdade de Tecnologia de Sa˜o Paulo – CEETEPS-UNESP, Prac¸a Fernando Prestes, 30, 01124-060 Sa˜o Paulo, Sa˜o Paulo, Brazil D. Dalmazi†

UNESP—Campus de Guaratingueta´—DFQ, Avenida Dr. Ariberto Pereira da Cunha, 333, 12516-410 Guaratingueta´, Sa˜o Paulo, Brazil 共Received 15 July 2002; published 18 June 2003兲

We present both analytical and numerical results on the position of partition function zeros on the complex magnetic field plane of the q⫽2 state 共Ising兲 and the q⫽3 state Potts model defined on␾3_{Feynman diagrams} 共thin random graphs兲. Our analytic results are based on the ideas of destructive interference of coexisting

phases and low temperature expansions. For the case of the Ising model, an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the q⫽3 state Potts model, our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.

DOI: 10.1103/PhysRevE.67.066108 PACS number共s兲: 05.50.⫹q, 05.70.Fh, 64.60.Cn, 75.10.Hk

I. INTRODUCTION

The Ising model is known to be exactly solvable in one dimension and also in two dimensions in the absence of magnetic field. The magnetic field breaks the Z2 symmetry, which complicates a possible exact solution of the model. However, it is still possible to prove, for an arbitrary tem-perature, an exact theorem共see 共Ref. 关1兴兲 for the location of the zeros of the partition function on the complex fugacity u⫽e2Hplane, where H stands for a complex magnetic field measured in units of 1/␤⫽T. Henceforth, these zeros will be called Yang-Lee zeros. The theorem establishes that the Yang-Lee zeros are located on the unit circle 兩uk兩⫽1, k ⫽1, . . . ,n. Here, n is the number of sites of the lattice

whose details such as coordination number and topology are not important for the proof. The symmetry under a change of sign of the magnetic field u→1/u is a key factor but it is not sufficient for the proof. The proof is a bit technical and strongly depends on the form of the polynomials Pn(u) in the partition function. A simpler proof would certainly be welcome. The authors of Ref.关2兴 have suggested an analytic way to find the location of the partition function zeros. It is assumed that for a system defined in a periodic volume V

⫽Ld_{with r different phases, it is possible to write down the} partition function of the system in terms of some functions fl(l⫽1, . . . ,r) as follows:

Z⫽

兺

l_⫽1 r

qle⫺␤V fl⫹O共e⫺L/L0e⫺␤V f兲, 共1兲 where ql is the degeneracy of the corresponding phase l and flis interpreted as its metastable free energy. The quantity L0

is of the order of correlation length, while f⫽minRe( fk).

Following Ref. 关2兴, we can use a destructive interference condition in order to find the zeros ofZ within O(e⫺L/L0_):

Re f_l,Leff⫽Re f_m,Leff ⬍Re f_k,Leff for all k⫽l,m, 共2兲 ␤V共Im fl,L

eff_{⫺Im f}

m,L

eff _兲⫽␲

mod共2␲兲, 共3兲 where f_leff⫽ fl⫺(␤V)⫺1ln ql. It is assumed that the meta-stable free energies are functions of some complex parameter z which in our case is the magnetic field. The continuity of the real part of the free energy across the line of zeros al-ready appeared in Ref.关3兴. It is clear that Eqs. 共2兲 and 共3兲 are useful whenever we have a closed expression for the free energies of the different phases of the system. In the case of the two-dimensional Ising model in the presence of a mag-netic field, although we do not have exact expressions for the free energies, we can use low temperature expansions

共LTE’s兲 and the symmetry H→⫺H as in Ref. 关2兴 to furnish

a simple proof of the Lee-Yang theorem valid in the thermo-dynamic limit and for low temperatures. In this work, we apply the same ideas for a particular case of fluctuating ran-dom lattice. For such lattices, no circle theorem has been proven. The spin degrees of freedom are placed on the ver-tices of Feynman diagrams that are themselves dynamical degrees of freedom and need to be summed over in the par-tition function with appropriate combinatorial factors. In the case where the diagrams have double lines and are generated by a random N⫻N matrix model 共see Refs. 关4,5兴兲, their sum mimics, in the continuum limit, the Ising model on random surfaces 共two-dimensional gravity兲. Early 关6,7兴 and recent

关8兴 numerical results on graphs of planar topology indicate

that the Yang-Lee zeros lie on the unit circle although no circle theorem exists in this case. Notice that the form of polynomials used in the proof of the Lee-Yang theorem is not preserved by their linear combinations. However, we be-lieve that there might be a generalization of the circle theo-rem for the new class of polynomials obtained by linear combinations with positive coefficients of partition functions of the Ising type. In fact, further support to this conjecture *Email address: lclaudio@fatecsp.br

has been given in Ref. 关9兴, where we studied the effect of topology of the graphs on the position of the Yang-Lee zeros. We have noticed that changing from the planar to the torus topology, the Yang-Lee zeros remained on the unit circle with a small change in their positions, which is in agreement with the existence of an underlying robust theorem. Further-more, in order to take the continuum limit we take large matrices N→⬁ that allows one to find 共see Ref. 关4兴兲 with the help of the orthogonal polynomial technique a closed expres-sion for the free energy even in the presence of magnetic field. Thus, an analytic study of the free-energy singularities is possible and leads one to pure imaginary values for the magnetic field 共see Ref. 关6兴兲, i.e., 兩u_k兩⫽1. Of course, this is only correct in the thermodynamic limit. On the other hand, the same issue of Yang-Lee zeros on a dynamical lattice can be studied in a simplified model of thin graphs where matri-ces become numbers (N⫽1) and our final partition function becomes a linear combination of partition functions on single-line Feynman diagrams. In this case, no random sur-face interpretation exists and the model exhibits a ferromag-netic phase transition with mean field exponents, see Ref.

关10兴, and references therein. Once again numerical results for

finite number of vertices n indicate the unit circle as the locus of the Yang-Lee zeros 关9兴. However, since we do not have a random surface interpretation, the results of Ref. 关6兴 do not go through the case of thin graphs. In particular, the location of the Yang-Lee zeros is not known even in the thermodynamic limit. In this work, we analyze this question starting from a closed expression for the free energy of the model obtained in the thermodynamic limit by means of a saddle point approximation both for the Ising and the Potts model. Then, we use the destructive interference equations of Ref. 关2兴 to find out the location of the Yang-Lee zeros. Un-fortunately, the presence of a magnetic field complicates the form of the resulting expression for the free energy, but an argument based on low temperature expansions and a sym-metry (H→⫺H) between different saddle point solutions will allow us to prove that the Yang-Lee zeros of the Ising model on thin graphs should be on the unit circle in the thermodynamic limit. The proof is exact for the whole range of temperatures for which the LTE’s converge. For the case

of the Potts model, we have not been able to promote our low temperature results to an exact level, but perturbatively we show that the Yang-Lee zeros should lie outside 共but close to兲 the unit circle for low temperatures. The precise location of the zeros seem to change with the temperature in a complex way. We also display numerical results in agree-ment with our analytic findings.

II. THE PARTITION FUNCTION ON THIN GRAPHS We can define the q-state Potts model as a generalization of the Ising model (q⫽2), where on each site of the lattice there is a spin degree of freedom ␴ that can take q different values ␴⫽1,2, . . . ,q. By summing over all spin configura-tions兵␴_i其on a given static lattice G_nof n vertices, we obtain the partition function Zq(Gn). In the presence of a static magnetic field H in the ␴⫽1 direction, we have

Zq共Gn兲⫽

兺

兵␴i其 exp

冉

␤

兺

具i, j典

冊

␦␴i,␴j⫹2Hi

兺

⫽1 n ␦1,␴_i, 共4兲

where 兺_具i, j典 is a sum over all bonds 共propagators兲 of Gn including loop bonds which connect a site 共vertex兲 to itself. Here, we are interested in the case where the lattice is a dynamical degree of freedom and the final partition function is obtained by summing over all Gn with n vertices:

Zq (n)_⫽

兺

k⫽1 K(n) Zq共Gn (k)_{兲. 共5兲}

In our case, K(n) will be the total number of Feynman graphs G_n(k) with n cubic vertices ␾3 _{and no external legs.}

This clearly includes graphs of different topologies. Each

Zq(Gn

(k)_{) as well as Z}

q

(n) _{will be proportional to a}

polyno-mial P_n(u) in the fugacity u⫽e2H. We take cubic vertices, since these are the simplest ones after the trivial quadratic case␾2, which corresponds to a D⫽1 lattice. The partition functionsZ_q(n) can be generated by integrating over q zero-dimensional fields X1, . . . ,Xq representing the q different spin states ␴⫽1,2, . . . ,q, respectively,

Zq (2n)_⫽ Tn 2␲i

冖

dg g2n⫹1

冉

冕

D␮exp

再

⫺1 2

冋

i

兺

⫽1 q X_i2⫺2c

兺

i⬎ j XiXj⫺ 2g 3

冉

e 2H_X 1 3_⫹

兺

i⫽2 q X_i3

冊

册冎

冕

D␮exp

冋

⫺1 2

冉

兺

i_⫽1 q X_i2⫺2c

兺

i_{⬎ j} XiXj

冊

册

冊

, 共6兲

whereD␮⫽兿_iq_⫽1dXi and Tn is a numerical factor indepen-dent of the magnetic field and the temperature, which is nec-essary to get rid of the bad asymptotic behavior of the un-decorated ␾3 graphs. The contour integral is necessary to single out the term g2n containing 2n cubic vertices. Notice thatZ_q(n)vanishes for odd number of vertices; therefore, we

can write the total number of vertices as 2n. The constant c will be related to the temperature. To each cubic vertex (X₁3) representing a spin in the direction of the magnetic field, there will be a u⫽e2Hfactor in agreement with the paramag-netic interaction term in Eq.共4兲. The quadratic terms in the argument of the exponentials in Eq. 共6兲 are responsible for

the free propagators

具

XiXj

典

共or bonds

具

␴i␴j

典

) between the q different vertices that associate with the states ␴_i. Taking ratios of the propagators and comparing with the ratios of the corresponding Boltzmann weights of Eq.共4兲, we identify the parameter c with a function of the temperature as follows. The action in the numerator of Eq.共6兲 can be written as

Sg⫽ 1 2 i, j

兺

⫽1 q XiKi jXj⫺ g 3

冉

e 2H_X 1 3_⫹

兺

i⫽2 q X_i3

冊

, 共7兲 where we have introduced the kinetic q⫻q matrix Ki j⫽ ⫺c, if i⫽ j and Kii⫽1. We can check that det K⫽(1 ⫹c)q⫺1_{关1⫺(q⫺1)c兴. The propagators are given by the} el-ements of the inverse matrix K_{i j}⫺1⫽(det K)⫺1c if i⫽ j and K_ii⫺1⫽(det K)⫺1关1⫺(q⫺2)c兴. Therefore, using the Boltz-mann weights from Eq. 共4兲 at vanishing magnetic field we have

具

XiXj

典

具

X_iX_i

典

⫽ c 关1⫺共q⫺2兲c兴⫽ eEi j(H⫽0) eEii(H⫽0)⫽e ⫺␤ _共8兲 and, consequently, c⫽共e␤⫹q⫺2兲⫺1. 共9兲 Therefore, corresponding to 0⭐T⭐⬁ we have, respectively, the compact range 0⭐c⭐1/(q⫺1).

In the thermodynamic limit n→⬁, one can evaluate the free energy

f_q(2n)⫽⫺ 1 2nlnZq

(2n) _共10兲

by means of a saddle point approximation as follows. The fist step is to rescale the zero-dimensional fields Xi

→(1/g)X˜i such that Sg(Xi,u,c)→(1/g2) Sg_⫽1(X˜i,u,c). Next, we decouple the contour integral over the coupling g from the integral over the zero-dimensional fields by a change of variables g→g˜

冑

S_g⫽1. Finally, we have

fq⫽ lim n_→⬁

冋

⫺ 1 2nln

冕

D␮e ⫺n ln Sg⫽1⫹O

冉

1 n

冊册

. 共11兲 Therefore, the free energy per site is given in the thermody-namic limit by关11兴

fq⫽

1

2ln S˜g_⫽1, 共12兲

where S˜g_⫽1 corresponds to Sg_⫽1 at a solution of the saddle point equations ⳵X_iSg⫽1⫽0. Henceforth, it is always as-sumed that g⫽1 and we will treat the q⫽2 共Ising兲 state and the q⫽3 state Potts model separately.

III. ISING MODEL

Aligning the magnetic field in the direction of␴1, which

is represented by the field X1, and using the notation X1

⫽x and X2⫽y, we have, respectively, the saddle point

equa-tions ⳵xS⫽0⫽⳵yS:

x⫺cy⫽ux2,

y⫺cx⫽y2. 共13兲

Using these equations we can write down the action at the saddle point关12兴 in a quadratic form:

S ˜_⫽1

6关x

2_⫹y2_{⫺2cxy兴. 共14兲}

From Eq.共13兲 we have

x⫽共1/c兲共y⫺y2兲, 共15兲 u y共1⫺y兲2⫹c共y⫺1兲⫹c3⫽0. 共16兲 In the absence of magnetic field (u⫽1), the solutions of Eq.

共16兲 simplify 共see Ref. 关13兴兲 and we have two low

tempera-ture solutions below the critical value c_cr⫽1/3 and a high temperature 共HT兲 solution valid for the entire range 0⭐c

⭐1:

yLT_⫾⫽12关1⫹c⫿

冑

共1⫺3c兲共1⫹c兲兴, 共17兲

yHT⫽1⫺c. 共18兲

Defining the magnetization

m⫽

冓

ux

3

共ux3_⫹y3_兲

冔

, 共19兲

we can check that yLT_⫹ and yLT_⫺ correspond, respectively, to m⬎1/2 and m⬍1/2. Both solutions collapse at m(xHT, yHT)⫽1/2 for c⫽1/3. Substituting Eqs. 共17兲, 共18兲, and共15兲 back in Eq. 共14兲, we have

S ˜_LT⫽共1⫹c兲 2_共1⫺2c兲 6 , 共20兲 S ˜_HT⫽共1⫺c兲 3 3 . 共21兲

Notice that both low temperature solutions have furnished the same action S˜LT.

The presence of a nonvanishing magnetic field (u⫽1) makes the solution of the cubic equation 共16兲 awkward and not very informative. We have solved Eq. 共16兲 instead as a Taylor expansion around c⫽0 共LTE兲. We found three expan-sions yLT⫾ and yHT which reduce to Eqs. 共17兲 and 共18兲 at u⫽1. They furnish the following actions :

S ˜ LT⫹⫽ 1 6u2⫺ c2 2u2⫺ c3 3u3⫹ 共u2_⫺1兲c4 2u4 ⫹O共c 5_{兲, 共22兲} S ˜_LT⫺⫽1 6⫺ c2 2 ⫺ uc3 3 ⫺ 共u2_⫺1兲c4 2 ⫹O共c 5_{兲, 共23兲}

S ˜ HT⫽ 1⫹u2 6u2 ⫺ c u⫹ 共1⫹u2_兲c2 2u2 ⫹ 共u4_⫺3u2_⫹1兲c3 3u3 ⫹共u 2_⫺1兲2_共1⫹u2_兲c4 2u4 ⫹O共c 5_{兲. 共24兲}

Clearly, S˜LT⫹ and S˜LT⫺ collapse into Eq.共20兲 at u⫽1. For u⬎1(H⬎0), fLT_⫹⬍ fLT_⫺ and the system is magnetized in the direction of the magnetic field (x direction兲, while the situation is reversed for u⬍1(H⬍0) as expected. In order to locate the Yang-Lee zeros of Z_q(2n)_⫽2 in the thermodynamic limit, we first assume u⫽␳ei␪ and then solve Eqs. 共2兲 and

共3兲:

Re共 fLT_⫹⫺ fLT_⫺兲⫽0, 共25兲 Im fLT_⫺⫽Im fLT_⫹⫹

共2k⫺1兲

␤V␲ . 共26兲 There are two basic factors that will allow us to locate ex-actly the Yang-Lee zeros for the whole convergence range of the LTE’s without really worrying about the numerical de-tails of the coefficients of expansions共22兲–共24兲. First of all, notice that for H→⫺⬁ 共or u→0) the cubic equation 共16兲 has a unique solution y (u→0)⫽1⫺c2 and, therefore, there must be at least one LTE that is well behaved for u→0 and terminate at the term proportional to c2. Of course, this can only be identified with S˜LT⫺ which implies that their coeffi-cients can only involve positive powers of u, i.e.,

S ˜ LT⫺共u,c兲⫽

兺

n⫽0 ⬁ an共u兲cn, 共27兲 where each a_n(u) is a polynomial in u. The next important factor is a symmetry of the saddle point equations共13兲 which relate solutions with H⬎0 to solutions with H⬍0. Namely, Eq. 共13兲 is invariant under

u→1/u, 共28兲

xi共1/u,c兲→uyj共u,c兲, 共29兲 yi共1/u,c兲→uxj共u,c兲, 共30兲 where (xi, yi) and (xj,yj) are, in principle, distinct solutions of Eq.共13兲. Back in Eq. 共14兲 we are led to the relation

S ˜_i

冉

1

u,c

冊

⫽u

2_˜S

j共u,c兲. 共31兲

The labels i, j indicate that the saddle point solutions on both sides of Eq.共31兲 do not need to be the same. From the first terms of the LTE’s in Eqs.共22兲–共24兲 and Eq. 共31兲, we have the identification below

S ˜ LT⫹共u,c兲⫽ 1 u2S ˜ LT⫺

冉

1 u,c

冊

, 共32兲 S ˜ HT共u,c兲⫽ 1 u2S ˜ HT

冉

1 u,c

冊

. 共33兲 Putting back in the definition of the free energy and using Eq. 共27兲, we obtain 2共 fLT_⫺⫺ fLT_⫹兲⫽ln u2⫹

冋

ln

兺

n⫽0 ⬁ an共u兲cn⫺ln

兺

n⫽0 ⬁ an

冉

1 u

冊

c n

册

_. 共34兲

Using a₀⫽1/6 and u⫽␳ei␪_{, after expanding the logarithms} we deduce 2 Re共 f_LT⫺⫺ f_LT⫹兲⫽2 ln␳⫹

兺

n⫽1 ⬁ cn

兺

k⫽1 B(n) A_k,n ⫻cos共k␪兲

冉

␳k_⫺1 ␳k

冊

, 共35兲 where B(n) and Ak,n are pure numerical factors. Thus, the condition Re( f_LT⫺⫺ f_LT⫹)⫽0 imply that the zeros must be on the unit circle uk⫽ei␪k in the thermodynamic limit. The condition on the imaginary parts of the free energies will locate the corresponding angles␪kon the circle as a function of the temperature. We emphasize that our proof of the circle theorem for thin␾3 graphs hold inside the convergence re-gion of low temperature expansions.

IV. qÄ3 STATE POTTS MODEL

Using the notation X1⫽x,X2⫽y,X3⫽z, and fixing the

magnetic field once again along the x direction, we derive the saddle point equations, ⳵_X

iS⫽0:

x⫺c共y⫹z兲⫽ux2, 共36兲 y⫺c共x⫹z兲⫽y2, 共37兲 z⫺c共x⫹y兲⫽z2. 共38兲 From the difference of Eqs. 共37兲 and 共38兲, we deduce that there are two groups of solutions, either y⫽z or y⫹z⫽1

⫹c: z⫽y, 共39兲 x⫽y c共1⫺c⫺y兲, 共40兲 u y共y⫹c⫺1兲2⫹c共y⫹c⫺1兲⫹2c3⫽0; 共41兲 z⫽1⫹c⫺y, 共42兲 x_⫾⫽1⫾关1⫺4uc共1⫹c兲兴 1/2 2u , 共43兲 y2⫺共1⫹c兲y⫹c共1⫹c⫹x兲⫽0. 共44兲

As in the Ising case, it is instructive to first look at the solu-tions for vanishing magnetic field (u⫽1). In this case, we have from the first set of solutions two low temperature so-lutions and one high temperature solution:

yLT_⫾⫽zLT_⫾⫽ 1⫿关1⫺4c共1⫹c兲兴1/2 2 , xLT⫾⫽ 1⫹2c⫾关1⫺4c共1⫹c兲兴1/2 2 , 共45兲 xHT⫽yHT⫽zHT⫽1⫺2c. 共46兲 Defining the magnetization

m共q⫽3兲⫽

冓

ux

3

ux3_⫹y3_⫹z3

冔

, 共47兲

we identify the labels LT⫹ and LT⫺ with solutions such that m⬎1/3 and m⭐1/3, respectively. The solution LT⫺ meets the high temperature solution with m⫽1/3 at c⫽1/5. The second group of solutions共43兲 does not furnish for u⫽1 any new physical solution. Explicitly, for 0⭐c⭐(

冑

2⫺1)/2, we have from the second group two real solutions

x1⫽y1⫽ 1⫺关1⫺4c共1⫹c兲兴1/2 2 , 共48兲 z1⫽ 1⫹关1⫺4c共1⫹c兲兴1/2 2 , 共49兲 and x2⫽ 1⫹关1⫺4c共1⫹c兲兴1/2 2 , 共50兲 y2⫽ 1⫹c⫹兩关1⫺4c共1⫹c兲兴1/2⫺c兩 2 , 共51兲 z₂⫽1⫹c⫺兩关1⫺4c共1⫹c兲兴 1/2_⫺c兩 2 . 共52兲

Notice that solution (x1,y1,z1) corresponds to a permutation

(x,z)→(z,x) of the solution LT⫹ given in Eq. 共45兲, while the solution (x2, y2,z2) coincides for 0⭐c⭐1/5 with the

permutation (x,z)→(z,x) of the solution LT⫺. However, its derivative is discontinuous at c⫽1/5 and the solution devi-ates from LT⫺ for 1/5⬍c⬍(

冑

2⫺1)/2. Concluding, we can safely neglect both solutions of the second group.

Turning on the magnetic field (u⫽1), the solutions of the cubic equation in Eq.共40兲 become cumbersome. Once again we make a low temperature expansion around c⫽0, which we display below after the substitution in the action. From the first group of solutions (y⫽z), we have three possibili-ties S ˜ 0⫽ 1⫹2u2 6u2 ⫺c 2⫹u u2 ⫹ c2 u3共1⫹2u⫹3u 2_兲 ⫹ c 3 3u4共8u

4_⫺13u3_⫺12u2_{⫹2u⫹2兲⫹O共c}4_{兲, 共53兲}

S ˜_1⫽ 1 6u2⫺ c2 u2⫺ c3共2⫹3u兲 3u3 ⫹O共c 4_{兲, 共54兲} S ˜_2⫽1 3⫺c⫺c 2_⫹c 3_{共11⫺8u兲} 3 ⫹O共c 4_{兲. 共55兲}

Expanding the second group of solutions (y⫹z⫽1⫹c), we have two possibilities

S ˜_3⫽1 6⫺c 2_⫺c 3_共u⫹4兲 3 ⫹O共c 4_{兲, 共56兲} S ˜_4⫽1⫹u 2 6u2 ⫺ c u⫺ c2 u ⫺ c3共2⫹u兲 3 ⫹O共c 4_{兲. 共57兲}

Next, we analyze the physical content of all those solutions. First, we notice that in the limit of vanishing magnetic field, the only solution that correctly reproduces the LTE of S

˜

HT(u⫽1,c) is S˜0and, therefore, we will identify it with the

LTE of S˜HT(u,c). However, the limit u→1 cannot be used to identify the remaining solutions, since the solutions of the second group S˜3and S˜4become identical, respectively, to S˜1

and S˜2 for u→1 which on their turn correspond to the LTE

of the physical solutions S˜LT⫹and S˜LT⫺of the H⫽0 case.

1

In order to understand the meaning of S˜₁⫺S˜₄, we have to look at the limits H→⫹⬁ (u→⬁) and H→⫺⬁ (u→0). The only LTE solution well behaved for u→⬁ is S˜1 which will be therefore identified with a strongly magnetized low temperature solution, i.e., for this solution we expect lim_c

→0m⫽1. For u→0, both S˜2 and S˜3 are well behaved

and good candidates, but S˜3 possesses the lowest free energy

共recall f ⬃ln S) as one can check numerically. Therefore, S˜3

will represent the weakly magnetized phase which satisfies lim

c→0m⫽0. Thus, we have for the free energies of the physical low temperature solutions:

1_{Incidentally, we notice that the discontinuous behavior of}

solu-tion共22兲 only appears for 1/5⬍c⬍(

冑

2⫺1)/2 which is outside the convergence region of the LTE of Eq.共22兲. Remember that for 0

⭐c⭐1/5, both S˜LT⫺and Eq.共22兲 are identical, which explains why

S

2共 fLT_⫺⫺ fLT_⫹兲⫽ln S˜3⫺ln S˜1⫽ln u2⫹

冉

4 u⫺2u⫺2

冊

c 3 ⫹3c4

冉

2 u2⫺u 2_⫹4 u⫺2u⫺3

冊

⫹O共c 5_兲. 共58兲

We can find the position of the Yang-Lee zeros plugging u

⫽␳ei␪ in Eq. 共58兲 and solving the destructive interference equations共25兲 and 共26兲. First, if we look at the leading terms

共zero temperature兲, we will see from Eqs. 共25兲 and 共26兲 that

the Yang-Lee zeros, in the thermodynamic limit, will be lo-cated on the unit circle␳⫽1 at T⫽0. Our finite size numeri-cal numeri-calculations are in agreement with these expectations共see Fig. 1兲. Adding the next to leading terms of the order of c2, we will still have the zeros on the unit circle, but if we further truncate the LTEs at c3 level the zeros will slightly move out the unit circle. Indeed, from Eqs.共25兲 and 共58兲 we have at the order c3,

cos␪⫽␳关c

3_⫺ln␳兴

c3共2⫺␳2兲 . 共59兲 Imposing that ⫺1⭐cos␪⭐1, we find numerically for 0⭐c

⭐0.5 such that 1⭐␳⭐␳*, where␳*increases with tempera-ture. It is worth commenting that the right hand side of Eq.

共59兲 is a monotonically decreasing function of ␳. Conse-quently, we have a closed curve outside the unit circle that is nonsymmetric across the imaginary axis with the farthest point from the origin being (␳⫽␳*,␪⫽␲). In Fig. 2, both our numerical and analytical results are overlapped. The fi-nite size results seem to tend to the共full line兲 analytic one in the thermodynamic limit. In Fig. 2, we have used the trun-cation at the order c4_{instead of Eq.共59兲, although they differ}

by a small amount, including the c4terms makes the analytic results closer to the numerical ones. The c4 terms are quite complex to be displayed here explicitly.

V. T\ⴥ

The results of the previous two sections hold for a range of temperatures for which the LTE’s converge关14兴. In gen-eral, we have not been able to go beyond these temperatures analytically. One exception is the case T→⬁ 关or c→1/(q

⫺1)] for which a subtle decoupling of the zero-dimensional

fields take place for arbitrary q similar to what happens for the Ising model (q⫽2), see Ref. 关9兴. The key point is to take a particular scaling for the coupling g appearing in action共7兲 as c→1/(q⫺1). Namely, the matrix Ki jappearing in Eq.共7兲 has q⫺1 degenerate eigenvalues ␭_j⫽1⫹c( j⫽1,2, . . . ,q

⫺1), and the nondegenerate one ␭q⫽1⫺(q⫺1)c. In order to decouple the fields Xjin the action, it is natural to diago-nalize the matrix Ki j through an orthogonal transformation

2

Xi→Ai jXj and rescale Xi→X˜ /i

冑

␭i. The Jacobian is can-celed out in the calculation of Z_q(2n). After these changes, the action becomes 共repeated indices are summed over兲

S_g⫽

兺

i⫽1 q X ˜ i 2 2 ⫺ g 3

冋

u

冉

A1kX˜k

冑

␭k

冊

3 ⫹

兺

j⫽2 q

冉

AjkX˜k

冑

␭k

冊

3

册

. 共60兲 Apparently, we have just transferred the mixing terms to the higher powers of the potential, however in the limit T→⬁

2_{Explicitly, the orthogonal matrix reads A}

i j⫽0 if i⭓ j⫹2, Aiq

⫽1/冑q, and Ai j⫽⫺1/冑j⫹ j2 if i⭐ j⭐q⫺1 or j/冑j⫹ j2 if i⫽ j

⫹1⭐q.

FIG. 1. Yang-Lee zeros ofZ_q(2n)_⫽3 at zero temperature, c⫽0, for graphs with 2n⫽20 (〫), 2n⫽40 (⫻), and 2n⫽200 (䊊) vertices. The closed curve is the unit circle.

FIG. 2. Yang-Lee zeros ofZq⫽3

(2n)

at c⫽0.2 for graphs with 2n

⫽20 (〫), 2n⫽40 (⫻), and 2n⫽200 (䊊) vertices, compared to

the saddle point result 共solid line兲 and to the unit circle 共dashed line兲.

关c→1/(q⫺1)兴 we have ␭q→0, while all other eigenvalues remain finite. Thus, after the redefinition g→g¯␭_q3/2, we have in the above limit,

Sg共T→⬁兲⫽

兺

i⫽1 q⫺1 X ˜ i 2 2 ⫹ X ˜ q 2 2 ⫺ g ¯ 3

冉

uA1q 3 _⫹

兺

j⫽2 q A_jq3

冊

X˜_q3. 共61兲

Thus, we have decoupled the zero-dimensional fields on a specific point in the parameter space, while preserving infor-mation about nontrivial interacting terms. Clearly, this pro-cedure is only useful if we do not care about the singular behavior of the original variable Xq. This is precisely the case of Eq. 共6兲. Indeed, the Jacobian from X_i to X˜_i, is sin-gular in the limit T→⬁ but it cancels out because of the ratio in Eq.共6兲 Since Aiq⫽1/

冑

q, we obtain

S共T→⬁兲⫽

兺

i_⫽1 q X_i2 2 ⫺ g ¯ 3q3/2共y⫹q⫺1兲Xq 3 . 共62兲

Substituting this action in Eq. 共6兲, the integrations over X1, . . . ,Xq⫺1cancel out leaving us with an expression that is finite in the limit c→1/(q⫺1). From the g¯2n term, we have

Zq

(2n)_{共T→⬁兲⫽T˜}

n共u⫹q⫺1兲2n, 共63兲 where T˜n is a numerical factor independent of the magnetic field and the temperature. Therefore, we conclude that the Yang-Lee zeros of the q-state Potts model on thin graphs coalesce exactly at the point u⫽1⫺q as T→⬁. The same result is valid on a static lattice 关15兴. Again, our numerical calculations confirm this analytic proof.

VI. CONCLUSION

We have proved that in the thermodynamic limit, the ze-ros of the partition function of the Ising model on thin graphs lie exactly on the unit circle in the complex fugacity plane. Our proof is exact in the range of temperatures for which the

low temperature expansions converge. For the case of the q

⫽3 Potts model, we do not have the H→⫺H symmetry

anymore and our results were fairly perturbative in the tem-perature. In this case, the zeros lie on closed curves that depend on the temperature. These curves lie outside the unit circle and tend to the circle as T→0. Our numerical results for a small number of vertices seem to be in agreement with the analytic ones derived by means of the destructive inter-ference formulas of Ref.关2兴. The numerical results were very similar to the static lattice case treated in Ref. 关15兴 for both the Ising and the q⫽3 Potts model. We should mention that a part of the motivation for this work came from a recent progress共see Refs. 关10,8兴兲 on using the destructive interfer-ence formulas of Ref.关2兴 to find analytically the exact curves formed by the Fisher zeros 共complex temperatures兲 of the q-state Potts model on thin graphs. However, the results of Refs.关10,8兴 were derived for H⫽0, and the presence of the magnetic field complicates the form of the solutions such that we had to use low temperature expansions. As a further work, one might look at the Yang-Lee zeros for complex temperatures as in Ref. 关16兴 as well as other types of verti-ces. Finally, we mention that, as in Ref. 关9兴, we have also looked at connected partition functions obtained by first tak-ing the logarithm of the ratio of integrals in Eq.共6兲 and then developing the contour integral. The results for the position of the Yang-Lee zeros were qualitatively similar to the ones reported here. We hope to return in the future to the open problem of proving the circle theorem for the Ising model on random lattices with finite number of vertices.

ACKNOWLEDGMENTS

The authors would like to thank Nelson A. Alves for use-ful discussions and collaboration in early stages of this work and also Professor D. J. Johnston for some useful email ex-change. L.C.A. would like to thank the Mathematical Phys-ics Department of USP at Sa˜o Paulo for their kind hospital-ity. This work was partially supported by FAPESP, Grant Nos. 00/03277-3共L.C.A.兲 and 00/12661-1 共D.D.兲, and CNPq

共D.D.兲.

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