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A connection between complex-temperature properties of the 1D and 2D spin s Ising model

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A connection between complex-temperature properties of the 1D and 2D spin s Ising model
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    a  r   X   i  v  :   h  e  p  -   l  a   t   /   9   5   0   5   0   2   2  v   1   2   7   M  a  y   1   9   9   5 ITP-SB-95-11May, 1994 A Connection Between Complex-TemperatureProperties of the 1D and 2D Spin  s  Ising Model Victor Matveev ∗ and Robert Shrock ∗∗ Institute for Theoretical PhysicsState University of New YorkStony Brook, N. Y. 11794-3840 Abstract Although the physical properties of the 2D and 1D Ising models are quite different, wepoint out an interesting connection between their complex-temperature phase diagrams. Wecarry out an exact determination of the complex-temperature phase diagram for the 1DIsing model for arbitrary spin  s  and show that in the  u s  =  e − K/s 2 plane (i) it consists of  N  c, 1 D  = 4 s 2 infinite regions separated by an equal number of boundary curves where thefree energy is non-analytic; (ii) these curves extend from the srcin to complex infinity, andin both limits are oriented along the angles  θ n  = (1 + 2 n ) π/ (4 s 2 ), for  n  = 0 ,..., 4 s 2 − 1; (iii)of these curves, there are  N  c,NE, 1 D  =  N  c,NW, 1 D  = [ s 2 ] in the first and second (NE and NW)quadrants; and (iv) there is a boundary curve (line) along the negative real  u s  axis if andonly if   s  is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition functionzeros for the 2D spin  s  Ising model. ∗ email: vmatveev@max.physics.sunysb.edu ∗∗ email: shrock@max.physics.sunysb.edu  Recently, we presented calculations of the complex-temperature (CT) zeros of the parti-tion functions for square-lattice Ising models with several higher values of spin,  s  = 1 ,  3 / 2,and 2 [1]. In the thermodynamic limit, these zeros merge to form curves across which the freeenergy is non-analytic, and thus calculations for reasonably large finite lattices give insightinto the complex-temperature phase diagrams of these models. These phase diagrams consistof the complex-temperature extensions of the Z 2  –symmetric, paramagnetic (PM) phase; of the two phases in which the Z 2  symmetry is spontaneously broken with long-range ferromag-netic (FM) and antiferromagnetic (AFM) order; and, in addition, certain phases which haveno overlap with any physical phase (denoted “O” for other). Some of the zeros lie alongcurves which, in the thermodynamic limit, separate the various phases. In addition, thereare zeros lying along various curves or arcs which terminate in the interiors of the FM andAFM phase. Physical and CT singularities of the magnetization, susceptibility, and specificheat obtained from analysis of low-temperature series have been discussed recently for thesquare lattice Ising model with the higher spin values  s  = 1 [2] and  s  = 1, 3/2, 2, 5/2, and3 [3].Here we give some further insight into the complex-temperature phase diagrams forhigher-spin Ising models. We first report an exact determination of the CT phase diagramsof the 1D Ising model for arbitrary spin  s . We then point out a very interesting connectionbetween features of these 1D phase diagrams and certain properties of the phase diagramsof the Ising model on the square lattice inferred from our calculation of partition functionzeros for higher spin values. This connection is useful because, unlike the 2D spin 1/2 case,no exact closed–form solution has ever been found for the 2D Ising model with spin  s  ≥ 1,and hence further elucidation of its properties is of continuing value, especially insofar asthese constrain conjectures for such a solution. Of course, the physical properties of a spinmodel at its lower critical dimensionality (here  d ℓ.c.d.  = 1) are quite different from those for d > d ℓ.c.d. . However, as we shall discuss, some of the properties of the CT phase diagram for d  = 2 exhibit simple relations with the  d  = 1 case.  1 There are several reasons why CT properties of statistical mechanical models are of interest. First, one can understand more deeply the behavior of various thermodynamicquantities by seeing how they behave as analytic functions of complex temperature. Second,one can see how the physical phases of a given model generalize to regions in appropriateCT variables. Third, a knowledge of the CT singularities of quantities which have not beencalculated exactly helps in the search for exact expressions for these quantities. Fourth, onecan see how CT singularities in functions such as the magnetization and susceptibility are 1 Indeed, from the  d  = 1+  ǫ  and  d  = 2+  ǫ  expansions for the Ising and O( N  ) models [4], one knows thatexpansions above  d ℓ.c.d.  can even give useful information about physical critical behavior. 1  associated with the boundaries of the phases and with other points where the free energy isnon-analytic. Such CT properties were first considered (for the 2D,  s  = 1 / 2 square-latticeIsing model) in Ref. [5] and for higher-spin (2D and 3D) Ising models in Ref. [6].The spin  s  (nearest-neighbor) Ising model is defined, for temperature  T   and externalmagnetic field  H  , by the partition function  Z   =   { S  n } e − β  H where, in a commonly usednormalization, H = − ( J/s 2 )  <nn ′ > S  n S  n ′ − ( H/s )  n S  n  (1)where  S  n  ∈{− s, − s +1 ,...,s − 1 ,s } and  β   = ( k B T  ) − 1 .  H   = 0 unless otherwise indicated. Wedefine  K   =  βJ   and  u s  =  e − K/s 2 .  Z   is then a generalized (i.e. with negative as well as positivepowers) polynomial in  u s . The (reduced) free energy is  f   =  − βF   = lim N  s →∞ N  − 1 s  ln Z   inthe thermodynamic limit.For  d  = 1, one can solve this model by transfer matrix methods. One has Z   =  Tr ( T   N  ) = 2 s +1   j =1 λ N s,j  (2)where the  λ s,j ,  j  = 1 ,... 2 s  + 1 denote the eigenvalues of the transfer matrix  T    defined by T   nn ′  = < n | exp(( K/s 2 ) S  n S  n ′ ) | n ′ >  (we assume periodic boundary conditions for definite-ness). It is convenient to analyze the phase diagram in the  u s  plane. For physical temper-ature, phase transitions are associated with degeneracy of leading eigenvalues [7]. There isan obvious generalization of this to the case of complex temperature: in a given region of  u s , the eigenvalue of   T    which has maximal magnitude,  λ max , gives the dominant contribu-tion to  Z   and hence, in the thermodynamic limit,  f   receives a contribution only from  λ max : f   = ln( λ max ). For complex  K  ,  f   is, in general, also complex. The CT phase boundaries aredetermined by the degeneracy, in magnitude, of leading eigenvalues of  T   . As will be evidentin our 1D case, as one moves from a region with one dominant eigenvalue  λ max  to a regionin which a different eigenvalue  λ ′ max  dominates, there is a non-analyticity in  f   as it switchesfrom  f   = ln( λ max ) to  f   = ln( λ ′ max ). The boundaries of these regions are defined by thedegeneracy condition  | λ max | = | λ ′ max | . These form curves in the  u s  plane. 2 Of course, a 1D spin model with finite-range interactions has no non-analyticities for any(finite) value of   K  , so that, in particular, the 1D spin  s  Ising model is analytic along thepositive real  u s  axis. For a bipartite lattice,  Z   and  f   are invariant under  K   → − K  , i.e., u s  →  1 /u s . It follows that the CT phase diagram also has this symmetry, i.e., is invariantunder inversion about the unit circle in the  u s  plane. This symmetry also holds for a finitebipartite lattice; for  d  = 1, the lattice is bipartite iff   N   is even, and for our comments 2 By “curves” we include also the special case of a line segment. 2  about finite-lattice results, we thus make this restriction. Further, since the  λ s,j  are analyticfunctions of   u s , whence  λ s,j ( u ∗ s ) =  λ s,j ( u s ) ∗ , it follows that the solutions to the degeneracyequations defining the boundaries between different phases, | λ s,j | = | λ s,ℓ | , are invariant under u s  → u ∗ s . Hence, the complex-temperature phase diagram is invariant under  u s  → u ∗ s .We shall present results for a few  s  values explicitly. For  s  = 1 / 2, one has ( u 1 / 2 ) 1 / 4 λ 1 / 2 ,j  =1 ± ( u 1 / 2 ) 1 / 2 .  f   is an analytic function of   u 1 / 2  except at points which constitute the solution to | λ 1 / 2 , 1 | = | λ 1 / 2 , 2 | ; these comprise the negative real axis, −∞≤ u 1 / 2  ≤ 0. Apart from this line,the dominant eigenvalue of   T    is  λ 1 / 2 , 1 . For  s  = 1, the eigenvalues of   T  s  are  λ 1 , 1  =  u − 11  − u 1 and λ 1 ,j =2 , 3  = (1 / 2)  u − 11  + 1 + u 1 ± ( u − 21  − 2 u − 11  + 11 − 2 u 1  +  u 21 ) 1 / 2   (3)As shown in Fig. 1(a), the phase diagram consists of four phases, the complex-temperatureextension of the PM phase, together with three O phases. The curves separating thesephases are the solutions of  | λ 1 , 1 | = | λ 1 , 2 | . The third eigenvalue,  λ 1 , 3 , is always subdominant.In the two phases containing the real  u s  axis,  λ 1 , 2  has maximal magnitude, while in thetwo containing the imaginary  u s  axis,  λ 1 , 1  is dominant. The CT zeros of   Z   calculated forfinite lattices lie on or close to these curves, starting a finite distance from the srcin andbeing distributed in a manner symmetric under the inversion  u s  → 1 /u s . As the lattice sizeincreases, the zeros spread out, the one with smallest (largest) magnitude moving closer to(farther from) the srcin. For  s  = 3 / 2, the eigenvalues are given by2 u 9 / 4 λ 3 / 2 ,j  = (1 + u 2 )(1 + ηu 5 / 2 ) ζ   (1 − 2 u 2 + 4 u 3 +  u 4 +  u 5 + 4 u 6 − 2 u 7 +  u 9 ) − 2 ηu 5 / 2 (1 − 6 u 2 +  u 4 )  1 / 2 (4)where here  u ≡ u 3 / 2  and ( η,ζ  ) = (+ , +) ,  ( − , +) ,  (+ , − ) ,  ( − , − ) for  j  = 1 , 2 , 3 , 4. The phasediagram is shown in Fig. 1(b) and consists of nine regions separated by the curves where | λ 3 / 2 , 1 |  =  | λ 3 / 2 , 2 | . In the region containing the positive real  u s  axis,  λ 3 / 2 , 1  is dominant, andas one makes a circle around the srcin, each of the nine times that one crosses a boundary,there is an alternation between  λ 3 / 2 , 1  and  λ 3 / 2 , 2  as the dominant eigenvalue.We find the following results for general  s : (i) the complex-temperature phase diagramconsists of  N  c, 1 D  = 4 s 2 (5)(infinite) regions separated by an equal number of boundary curves where the free energy isnon-analytic; (ii) these curves extend from the srcin to complex infinity, and in both limitsare oriented along the angles θ n  = (1 + 2 n ) π 4 s 2  (6)3  for  n  = 0 ,... 4 s 2 − 1; (iii) of these curves, there are N  c,NE, 1 D  =  N  c,NW, 1 D  = [ s 2 ] (7)in the first and second (NE and NW) quadrants, where [ x ] denotes the integral part of   x ;and (iv) there is a boundary curve (which in this case is a straight line) along the negativereal  u s  axis if and only if   s  is half-integral. ( N  c,SE, 1 D  =  N  c,NE, 1 D  and  N  c,SW, 1 D  =  N  c,NW, 1 D by the  u → u ∗ symmetry.) To derive these results, we carry out a Taylor series expansion of the  λ s,j  in the vicinity of   u s  = 0. The dominant eigenvalues have the form u s 2 s  λ s,j  = 1 + ... +  a s,j u 2 s 2 s  +  ...  (8)where the first  ...  dots denote terms which are independent of   j  and the second  ...  dotsrepresent higher order terms which are dependent upon  j . Setting  u s  =  re iθ and solving theequation  | λ s,j | = | λ s,ℓ |  yields cos(2 s 2 θ ) = 0, whence2 s 2 θ  =  π 2 +  nπ , n  = 0 ,... 4 s 2 − 1 (9)Each of these solutions yields a curve across which  f   is non-analytic, corresponding to aswitching of dominant eigenvalue. These curves cannot terminate since if they did, onecould analytically continue from a region where the expression for  f   depends on a givendominant eigenvalue, to a region where it depends on a different eigenvalue. Owing to the u s  → 1 /u s  symmetry of the model, a given curve labelled by  n  approaches complex infinityin the same direction  θ n  as it approaches the srcin. This yields (i) and (ii). From (6), itfollows that0  < θ n  < π/ 2 for 0 ≤ n <  [ s 2 − 1 / 2] (10)which comprises [ s 2 ] values, and similarly, π/ 2  < θ n  < π  for [ s 2 − 1 / 2]  < n <  [2 s 2 − 1 / 2] (11)which again comprises [ s 2 ] values. Finally, if and only if   s  is half-integral, then the equation θ n  =  π  has a solution (for integral  n ), viz.,  n  = (2 s  + 1)(2 s − 1) / 2. From the  u s  →  u ∗ s symmetry of the phase diagram, the curve corresponding to this solution must lie on thenegative real axis for all  r , not just  r → 0 and  r →∞ . This yields (iii)-(iv).  ✷ .We note that from (6), the angular size of each region near the srcin (or infinity) is∆ θ  =  π 2 s 2  (12)Also, from the  u s  →  u ∗ s  symmetry of the phase diagram, it follows in particular, that foreach curve starting out from the srcin at  θ n , there is a complex conjugate curve at − θ n . As4
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