a r X i v :c o n d -m a t /0307010v 1 [c o n d -m a t .s t a t -m e c h ] 1 J u l 2003Fisher zeros and Potts zeros of the Q -state Potts model for
nonzero external magnetic ?eld
Seung-Yeon Kim ?School of Computational Sciences,Korea Institute for Advanced Study,207-43Cheongryangri-dong,Dongdaemun-gu,Seoul 130-722,Korea Abstract The properties of the partition function zeros in the complex temperature plane (Fisher zeros)and in the complex Q plane (Potts zeros)are investigated for the Q -state Potts model in an arbitrary nonzero external magnetic ?eld H q ,using the exact partition function of the one-dimensional model.The Fisher zeros of the Potts model in an external magnetic ?eld are discussed for any real value of Q ≥0.The Potts zeros in the complex Q plane for nonzero magnetic ?eld are studied for the ferromagnetic and antiferromagnetic Potts models.Some circle theorems exist for these zeros.All Fisher zeros lie on a circle for Q >1and H q ≥0except Q =2(Ising model)whose zeros lie on the imaginary axis.All Fisher zeros also lie on a circle for any value of Q when H q =0(except Q =0,1and 2)or H q =?∞(except Q =1,2and 3).All Potts zeros of the ferromagnetic model lie on a circle for H q ≥0.When H q =0or ?∞,all Potts zeros lie on a circle for both the ferromagnetic and antiferromagnetic models.All Potts zeros of the antiferromagnetic model with H q <0also lie on a circle for (x +1)?1≤a <1,where a =e βJ and x =e βH q .It is found that a part of the Fisher zeros or the Potts zeros lie on a circle for the speci?c ranges of H q .It is shown that some Fisher or Potts zeros can cut the positive real axis.Furthermore,the Fisher zeros or the Potts zeros lie on the positive real axis for the speci?c ranges of H q .The densities of zeros are also calculated and discussed.The density of zeros at the Fisher edge singularity diverges,and the edge critical exponents at the singularity satisfy the scaling law.There exists the Potts edge singularity in the complex Q plane which is similar to the Fisher edge singularity in the complex temperature plane.
PACS numbers:05.50.+q;https://www.doczj.com/doc/2c12704212.html,;75.10.Hk;02.10.Ox
I.INTRODUCTION
The Q-state Potts model[1,2,3,4,5]is a generalization of the Ising(Q=2)model. The Q-state Potts model exhibits a rich variety of critical behavior and is very fertile ground for the analytical and numerical investigations of?rst-and second-order phase transitions. The Potts model is also related to other outstanding problems in physics and mathematics. Fortuin and Kasteleyn[6,7]have shown that the Q-state Potts model in the limit Q→1 de?nes the problem of bond percolation.They[7]also showed that the problem of resistor network is related to a Q=0limit of the partition function of the Potts model.In addition, the zero-state Potts model describes the statistics of treelike percolation[8],and is equivalent to the undirected Abelian sandpile model[9].The Q=1
on the negative real axis[28].
The?rst zero,which we call the edge zero,and its complex conjugate of a circular distribution of the Yang-Lee zeros of the Potts model cut the positive real axis at the physical critical point x c=1for T≤T c in the thermodynamic limit.However,for T>T c the edge zero does not cut the positive real axis in the thermodynamic limit,that is,there is a gap in the distribution of zeros around the positive real axis.Within this gap,the free energy is analytic and there is no phase transition.Kortman and Gri?ths[29]carried out the?rst systematic investigation of the magnetization at the edge zero,based on the high-?eld,high-temperature series expansion for the Ising model on the square lattice and the diamond lattice.They found that above T c the magnetization at the edge zero diverges for the square lattice and is singular for the diamond lattice.For T>T c we rename the edge zero as the Yang-Lee edge singularity.The divergence of the magnetization at the Yang-Lee edge singularity means the divergence of the density of zeros,which does not occur at a physical critical point.Fisher[30]proposed the idea that the Yang-Lee edge singularity can be thought of as a new second-order phase transition with associated critical exponents and the Yang-Lee edge singularity can be considered as a conventional critical point.The critical point of the Yang-Lee edge singularity is associated with aφ3theory,di?erent from the usual critical point associated with theφ4theory.The crossover dimension of the Yang-Lee edge singularity is d c=6.
Fisher[31]emphasized that the partition function zeros in the complex temperature plane (Fisher zeros)are also very useful in understanding phase transitions,and showed that for the square lattice Ising model in the absence of an external magnetic?eld the Fisher zeros in
√
the complex y=e?βJ plane lie on two circles(the ferromagnetic circle y FM=?1+
2e iθ)in the thermodynamic limit.In particular, using the Fisher zeros both the ferromagnetic phase and the antiferromagnetic phase can be considered at the same time.The critical behavior of the square-lattice Potts model in both the ferromagnetic and antiferromagnetic phases has been studied using the distribution of the Fisher zeros,and the Baxter conjecture[32]for the antiferromagnetic critical temperature has been veri?ed[33,34].Recently,the Fisher zeros of the Q-state Potts model on square lattices have been studied extensively for integer Q>2[5,16,18,19,20,34,35,36,37,38, 39,40,41,42,43,44,45,46,47,48,49,50,51]and noninteger Q[33,34,48,49].Exact
conditions the Fisher zeros of the Q>1Potts models on a?nite square lattice are located on the unit circle in the complex p plane for Re(p)>0,where p=(eβJ?1)/√
investigated for general Q except few integer values of Q.
The partition function of the Q-state Potts model in the absence of an external magnetic ?eld is also known as the Tutte dichromatic polynomial[54]or the Whitney rank function[55] in graph theory and combinatorics of mathematics.One of the most interesting properties of the antiferromagnetic Potts model is that for Q>2the ground-state is highly degenerate and the ground-state entropy is nonzero.The ground states of the antiferromagnetic Potts model are equivalent to the chromatic polynomials[56]in mathematics,which play a central role in the famous four-color problem[57].The partition function zeros in the complex Q plane(Potts zeros)of the Q-state Potts model have been studied both in mathematics and in physics.The Potts zeros atβJ=?∞,corresponding to T=0for the antiferromagnetic model,have been investigated extensively to understand the chromatic polynomials and the ground states of the antiferromagnetic Potts model[34,51,58,59,60,61,62,63,64,65, 66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85].Recently,the Potts zeros at?nite temperatures have been studied for ladder graphs[48,51],L×L square lattices[34],and the model with long-range interactions[86].However,the properties of the Potts zeros for nonzero magnetic?eld have never been known until now.
In this paper,we investigate the exact results on the partition function zeros of the ferromagnetic and antiferromagnetc Q-state Potts models in one dimension,and we unveil the unknown properties of the Fisher zeros and the Potts zeros in an external magnetic?eld. In the next section we obtain two master equations to determine the partition function zeros of the one-dimensional Potts model.In Sec.III we calculate and discuss the Fisher zeros of the Q-state Potts model for any value of Q and in an arbitrary magnetic?eld.In Sec.IV we study the Potts zeros in the complex Q plane of the ferromagnetic and antiferromagnetic Potts models in an arbitrary external magnetic?eld.
II.THE Q-STATE POTTS MODEL
The Q-state Potts model in an external magnetic?eld H q on a lattice G with N s sites and N b bonds is de?ned by the Hamiltonian
H Q=?J i,j δ(σi,σj)?H q kδ(σk,q),(1)
where J is the coupling constant(ferromagnetic model for J>0and antiferromagnetic model for J<0), i,j indicates a sum over nearest-neighbor pairs,σi=1,2,...,Q,and q is a?xed integer between1and Q.The partition function of the model is
Z Q= {σn}e?βH Q,(2) where{σn}denotes a sum over Q N s possible spin con?gurations andβ=(k B T)?1.The partition function can be written as
Z(a,x,Q)=N b
E=0N s M=0?Q(E,M)a E x M,(3)
where a=y?1=eβJ,x=eβH q,E and M are positive integers0≤E≤N b and0≤M≤N s, respectively,and?Q(E,M)is the number of states with?xed E and?xed M.The states with E=0(E=N b)correspond to the antiferromagnetic(ferromagnetic)ground states. The parameter Q enters the Potts model as an integer.However,the study of the Q-state Potts model has been extended to continuous Q due to the Fortuin-Kasteleyn representation of the partition function[6,7]and its extension[87].
For the one-dimensional Potts model in an external?eld the eigenvalues of the transfer matrix were found by Glumac and Uzelac[27].The eigenvalues areλ±=(A±iB)/2,where A=a(1+x)+Q?2and B=?i
(a?1)(a+Q?1)x, thenλ±=C exp(±iψ),and the partition function is
Z N=2C N cos Nψ.(6)
The zeros of the partition function are then given by
2k+1
In the thermodynamic limit the locus of the partition function zeros is determined by the solution of
A=2C cosψ,(8) where0≤ψ≤π.In the special case Q=2the contribution by the eigenvalueλ0disappears from the partition function,Eq.(4),and the equation(8)determines all the locus even for ?nite systems.
On the other hand,whenλ+andλ0are two dominant eigenvalues,we have
Z N?λN++(Q?2)λN0(9) for large N.The partition function zeros are then determined by
λ+
N
,k=0,1,2,...,N?1.(11) In the thermodynamic limit the locus of the partition function zeros is determined by the solution of
a2x2+(Q?1)x?axA+(a?1)Ae iφ?(a?1)2e2iφ=0,(12) where0≤φ≤2π.The equation(12)also determines the locus of the zeros whenλ?and λ0are two dominant eigenvalues.
III.FISHER ZEROS
From the equation(8)the locus of the Fisher zeros is obtained to be
y1(ψ)=(Q?2)(x cos2ψ?1)±i2
√
(Q?2)2+4(Q?1)x cos2ψ
,(13)
where f=x cos2ψ[(Q+x?1)(Qx?x+1)?Q2x cos2ψ].The edge zeros of y1(ψ)are given by
y±=y1(0)=(Q?2)(x?1)±i2|x?1|
(Q?2)2+4(Q?1)x
.(14)
In the absence of an external?eld,H q=0(x=1),these edge zeros cut the real axis at the origin,
which corresponds to the T=0ferromagnetic transition point.If f<0,the zeros of y1(ψ) lie on the real axis.However,if f>0,the zeros of y1(ψ)lie on a circle
y1(ψ)=y c+De±iθ(ψ)(16)
in the complex y plane,where y c(the center of the circle)and D are given by
y c=1
(Q?2)2(x?1)+E(x+1)
(17)
and
D=1
(Q?2)2(x?1)+E(x+1)
.(18)
E is de?ned by
E=(Q?2)2+4(Q?1)x,(19) the argumentθis given by
cosθ(ψ)=1
(Q?2)2+4(Q?1)x cos2ψ?y c
,(20)
and the radius r of the circle is
r=|D|.(21) The one point of the circle y1(ψ),
y1 ψ=π2?Q,(22) always lies on the real axis.The point y1(π
e i2φ+(Q?2)e iφ?(Q?1)x.(23) The two points o
f y2(φ),
y2(0)=0(24) and
y2(π)=
2(x+1)
A.Q>1
In the special case Q=2all the Fisher zeros lie on the imaginary axis,and they meet the real axis at the origin only for x=1.At x=1(H q=0),the loci y1(ψ)and y2(φ)become the identical locus as a circle[48]
y(ρ)=?1
Q?2
e iρ(27)
for any value of Q except Q=0,1,and2.On this circle three eigenvalues have the same magnitude
|λ+|2=|λ?|2=|λ0|2=(Q?1)(Q?1?2cosρ)+1
2?Q
.(30) The point y(ρ=0)is the T=0ferromagnetic transition point.The point y(ρ=π)has no physical meaning for Q>2,but it may correspond to an antiferromagnetic transition point for Q<2because the physical interval is
1≤y≤∞(∞≥T≥0)(31) for antiferromagnetic interaction J<0.For x>1(H q>0)all the Fisher zeros lie on the circle y1(ψ)(f>0for Q>1and x>1),whereas for x<1(H q<0)they are located on the loop y2(φ).
Figure1shows the locus of the Fisher zeros in the complex y plane of the three-state Potts model for x=1,2and3.For Q=3the center y c and the radius r of the circle y1(ψ) are given by
y c=?1,r=1(x=1),(32)
y c=?19
13
(x=2),(33)
and
y c=?33
(x=3).(34)
For x=1two edge zeros y±cut the real axis at the origin which is the T=0ferromagnetic transition point.However,as x increases,they move away from the origin,and there is a gap in the distribution of zeros,centered atθ=0,that is,the density of zeros is zero, g(θ)=0,for|θ|<θ0.The edge angleθ0(=θ(ψ=0))is given by
cosθ0=1
E?y c
,(35)
and the edge zeros are expressed as
y±=y c+De±iθ0.(36) For example,for the three-state Potts model,the edge zeros are located at
y±=0(x=1),(37)
y±=
1
25(2±4
√
2π [(Q?2)?2(Q?1)(y c+D cosθ)]
2π
Q|Q?2|
2π
Q
that is,the density of zeros diverges at the Fisher edge zeros y±for x>1.In this case, the Fisher edge zero is called the Fisher edge singularity because of the divergence of the density of zeros.The edge critical exponentsαe,βe andγe associated with the Fisher edge singularity are de?ned in the usual way,
C e～(y?y±)?αe,(45)
m e～(y?y±)βe,(46) and
χe～(y?y±)?γe,(47) where C e,m e,andχe are the singular parts of the speci?c heat,magnetization,and sus-ceptibility,respectively.The density of zeros near the Fisher edge singularity is also given by
g(θ～θ0)～(y?y±)1?αe.(48) The Fisher edge singularity is characterized by the edge critical exponentsαe=3
2
, andγe=3
2
state Potts model for x=1and x=3.For Q=3
17,r=
70
25
(?1±2
√
2
orθ=π),Eq.(22),of the circle y1(θ(ψ)).
The point y1(π)lies on the negative real axis for Q>2,whereas it lies on the negative real axis for1 y1(π)>2,(52) In the limit H q→∞(x→∞)the positive?eld H q favors the state q for every site and the Q-state Potts model is transformed into the one-state model.The Fisher zeros are determined by Z(y,x→∞,Q)～N b E=0?Q(E,M=N s)y?E=0(53) for any Q.Because?Q(E,M=N s)=1for E=N b and0otherwise,Eq.(53)is y?N b=0.(54) As x increases,|y|for all the zeros increases without bound[47]. Now we turn to the distributions of Fisher zeros for x<1(H q<0)which are completely determined by the loop y2(φ).The locus cuts the real axis at two points y2(0)and y2(π). The point y2(0)=0(55) is the T=0ferromagnetic transition point.For Q≥3the point y2(π)lies on the negative real axis,and it has no physical meaning.Figure3shows the locus of the Fisher zeros in the complex y plane of the three-state Potts model for x=1 2 state Potts model for x=1 2 and x=1 5 .(58) For2 ˉx1= 3?Q 2 In the limit H q→?∞(x→0)the partition function of the Q-state Potts model,Eq.(4), becomes Z(a,x=0,Q)=(a+Q?2)N+(Q?2)(a?1)N.(60) For an external?eld H q<0,one of the Potts states is supressed relative to the others,and the symmetry of the Hamiltonian is that of the(Q?1)-state Potts model in zero external ?eld.Therefore,the partition function Z(a,x=0,Q+1)of the(Q+1)state Potts model in the limit H q→?∞is the same as the partition function Z(a,x=1,Q)=(a+Q?1)N+(Q?1)(a?1)N.(61) of the Q-state Potts model in the absence of an external magnetic?eld.As x decreases from 1to0,the Q-state Potts model is transformed into the(Q?1)-state Potts model in zero external?eld[47].At x=0,the Fisher zeros of the Q-state Potts model lie on a circle y(ρ)=?1 Q?3 e iρ(62) for any value of Q except Q=1,2,and3.The zeros lie on the imaginary axis for the three-state Potts model at x=0. B.Q<1and x>1 For x<ˉx2,where ˉx2= 1 2 and x=3 At x=ˉx2,y?=?∞,and the line y1(ψ)lies on the real axis between?∞and y?(<0). In the regionˉx2 ˉx3=1+√1?√ 2 and x=3,from which we obtain y?=4 6)=?0.506,y2(π)=2,y?=4.(67) At x=ˉx3,φ?=π,the other edge zero y+appears,and the loop y2(φ)shrinks to the point y+=y?=y2(π)=?2√ Q+(Q?2) √ x 2 ) (y? 2 )are given by y01=y1(ψ0)=y1(π?ψ0)=y c+D= 2(Q?2)x(x?1) x+1 respectively.Similarly,the loop also cuts the real axis at two points y2(0)and y?(0= y2(0) 10 .For these values we obtain y?=6 11 ,y?= 17 2 =17 165 ,r= 26 Q Q ,(75) the locus consists of the loop y2(φ)and the line y1(ψ)on the real axis between y?and y?(y2(0) real axis at the point y2(0)(=0).At x=ˉx4,the line Q1(φ)shrinks to the point y?=y?=y2(π)= 2 √ Q+(2?Q) √ IV.POTTS ZEROS From the equation(8)the locus of the Potts zeros is obtained to be Q1(ψ)=1?a+ √g2 2,(79) where g1=(a?1)x and g2=g1sin2ψ+x?1.The edge zeros of Q1(ψ)are given by Q±=(a?2)(x?1)±2 cos2ψ?1.(86) ax?1 The one point of the circle Q1(ψ), Q1 ψ=π The two points of Q2(φ), Q2(0)=1(89) and x+3 Q2(π)= .(91) x A.Three special cases:x=1,x=0,and a=1 In the absence of an external?eld(H q=0or x=1),the locus Q1(ψ)becomes Q1(ψ)=1?a+(a?1)e iθ,(92) where the argumentθ(ψ)is simply given by θ(ψ)=2ψ.(93) At the same time,the locus Q2(φ)reduces to Q2(φ)=(a?1)(e iφ?1),(94) which is the identical circle to the locus Q1(ψ).The equation(93)means that the Potts zeros are uniformly distributed on this circle.In particular,the Potts zeros lie on the unit circle at a=0[68,69].The circle Q1(ψ)or Q2(φ)cuts the real axis at two points Q=0 and2(1?a).For ferromagnetic interaction J>0the physical interval is 1≤a≤∞(∞≥T≥0),(95) whereas for antiferromagnetic interaction J<0the physical interval 0≤a≤1(0≤T≤∞).(96) It should be noted that the point2(1?a)lies on the positive real axis for the Potts an-tiferromagnet.For1 it does in1<2(1?a)≤2.The locus of the Potts zeros consists of the circle and 2 In the limit H q→?∞(x→0),the partition function of the Q-state Potts model is again given by Eq.(60),and the symmetry of the Hamiltonian is that of the(Q?1)-state Potts model in zero external?eld.At x=0,the locus Q2(φ)(Eq.(88))of the Q-state Potts model reduces to Q2(φ)=2?a+(a?1)e iφ,(97) which is a circle with center Q c=2?a and radius r=|a?1|.The equation(97)is also obtained by replacing Q in Eq.(94)with Q?1.The circle Q2(φ)cuts the real axis at two points Q=1and3?2a.For1 2 it does in2<3?2a≤3.The locus of the Potts zeros consists of the circle Q2(φ) and the isolated point at Q=2for0≤a<1 |λ0|= a?1 >1,(99) which implies that the locus Q2(φ)does not appear.The circle Q1(ψ)always cuts the real axis at the point Q1(π for a=3.For x=1two edge zeros Q±cut the real axis at the origin,whereas they move away from the origin as x increases.The Potts edge zeros are determined by the edge angle θ0=θ(ψ=0)=cos?1 (a?2)x+1 3i(x=3).(105) On the circle Q1(ψ),the density of zeros g(θ(ψ))is given by g(θ)= |sinθ 2π 2?sin2θ0 2π .(108) However,for x>1the density of zeros at the Potts edge zeros Q±diverges,that is, g(θ～θ0)～ 1 Q(θ)?Q±(θ0) .(109) In this case,the Potts edge zero can be called the Potts edge singularity with the edge critical exponentμe(=?1 and ˉa2= 1 2(x+1) ,(114) and negative forˉa3 from which we obtain Q2(π)=?2 2 ,Q+= 1 √ x+1 2.(116) For a<ˉa1,the only locus is the line Q1(ψ)on the positive real axis between Q?and Q+ (0 Q+=Q?=Q2(0)=1.(117) For a>ˉa2,all the Potts zeros lie on the loop Q2(φ)which again cuts the real axis at two points Q2(π)and Q2(0)(Q2(π) D.a<1and x>1 Because g1is always negative,the zeros of Q1(ψ)lie on a circle if g2<0(ψ0<ψ<π?ψ0) and on the real axis if g2>0(0≤ψ<ψ0orπ?ψ0<ψ≤π).For0≤a<ˉa4,where 1
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