-
3
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-
0031558825
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These authors use the symbol [Formula Presented] for what is denoted [Formula Presented] here. JPHAC5
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I. Jensen, A. J. Guttmann, and I. G. Enting, J. Phys. A30, 8067 (1997). These authors use the symbol u for what is denoted z here.
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(1997)
J. Phys. A
, vol.30
, pp. 8067
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Jensen, I.1
Guttmann, A.J.2
Enting, I.G.3
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6
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85037194663
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M. E. Fisher, Lectures in Theoretical Physics (University of Colorado Press, Boulder, CO, 1965), Vol. 7C, p. 1
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M. E. Fisher, Lectures in Theoretical Physics (University of Colorado Press, Boulder, CO, 1965), Vol. 7C, p. 1.
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8
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0002939085
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S. Ono, Y. Karaki, M. Suzuki, and C. Kawabata, J. Phys. Soc. Jpn. 25, 54 (1968)
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(1968)
J. Phys. Soc. Jpn.
, vol.25
, pp. 54
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Ono, S.1
Karaki, Y.2
Suzuki, M.3
Kawabata, C.4
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14
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85037182738
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The free energy also has an isolated singularity at [Formula Presented] and (see theorem 6 of Ref. c11) for the Ising model on lattices with odd coordination number, a singularity at [Formula Presented]. The latter lies on the complex-temperature phase boundary for the honeycomb lattice but is isolated for the heteropolygonal [Formula Presented] lattice
-
The free energy also has an isolated singularity at|K|=∞ and (see theorem 6 of Ref. 11) for the Ising model on lattices with odd coordination number, a singularity at z=-1. The latter lies on the complex-temperature phase boundary for the honeycomb lattice but is isolated for the heteropolygonal 3⋅122 lattice.
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16
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85037202243
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As discussed in Ref. c14, the complex-temperature extension of a physical phase is obtained by analytically continuing the free energy from the interval of physical temperature to a maximal region allowed by nonanalytic boundaries. Henceforth, we shall generally take the adjective “complex-temperature extension” as implicit when referring to phases. There are also other complex-temperatures that have no overlap with any physical phase; we shall denote these by O for “other.”
-
As discussed in Ref. 14, the complex-temperature extension of a physical phase is obtained by analytically continuing the free energy from the interval of physical temperature to a maximal region allowed by nonanalytic boundaries. Henceforth, we shall generally take the adjective “complex-temperature extension” as implicit when referring to phases. There are also other complex-temperatures that have no overlap with any physical phase; we shall denote these by O for “other.”
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23
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0002493774
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F. Y. Wu, G. Rollet, H. Y. Huang, J. M. Maillard, C. K. Hu, and C. N. Chen, Phys. Rev. Lett. 76, 173 (1996).
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(1996)
Phys. Rev. Lett.
, vol.76
, pp. 173
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Wu, F.Y.1
Rollet, G.2
Huang, H.Y.3
Maillard, J.M.4
Hu, C.K.5
Chen, C.N.6
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31
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0000605306
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R. J. Baxter, H. N. V. Temperley, and S. Ashley, Proc. R. Soc. London, Ser. A 358, 535 (1978); PRLAAZ
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(1978)
Proc. R. Soc. London, Ser. A
, vol.358
, pp. 535
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Baxter, R.J.1
Temperley, H.N.V.2
Ashley, S.3
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43
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85037203415
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For the physical critical point, one does not have to distinguish between the exponents [Formula Presented], [Formula Presented] describing the approach from the high-temperature and low-temperature sides, since these are equal
-
For the physical critical point, one does not have to distinguish between the exponents α and α′ describing the approach from the high-temperature and low-temperature sides, since these are equal.
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-
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45
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85037255817
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both the honeycomb and kagomé cases, the respective conditions (3.1) and (4.1) are the conditions that [Formula Presented], where [Formula Presented] occurs in the term [Formula Presented] in the free energy, and [Formula Presented]. The range of [Formula Presented] is then [Formula Presented], as given
-
In both the honeycomb and kagomé cases, the respective conditions (3.1) and (4.1) are the conditions that L(a, p)=0, where L occurs in the term ∝∫02πdθ1∫02πdθ2ln[L(a, p)] in the free energy, and p=cos(θ1)+cos(θ2)+cos(θ1+θ2). The range of p is then -3/2
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-
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46
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85037212574
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By duality, for the Ising model on the honeycomb lattice, [Formula Presented] in the [Formula Presented] plane is the same as [Formula Presented] in the [Formula Presented], [Formula Presented] plane for same model on the triangular lattice c31. Note that for the case of anisotropic spin-spin couplings, the partition function zeros fill out regions rather than lying on curves in the [Formula Presented] plane c34
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By duality, for the Ising model on the honeycomb lattice, B in the a plane is the same as B in the ad=(a+1)/(a-1) plane for same model on the triangular lattice 31. Note that for the case of anisotropic spin-spin couplings, the partition function zeros fill out regions rather than lying on curves in the a plane 34.
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50
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0042715204
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J. Stephenson, 148A, 107 (1988) and references therein
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J. Stephenson, 148A, 107 (1988) and references therein.
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51
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84953710796
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Princeton University Press, Princeton
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S. Lefschetz, Algebraic Geometry (Princeton University Press, Princeton, 1953);
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(1953)
Algebraic Geometry
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Lefschetz, S.1
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54
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21844510217
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J. Adler, A. Brandt, W. Janke, and S. Shmulyian, J. Phys. A 28, 5117 (1995).
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J. Phys. A
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, pp. 5117
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Adler, J.1
Brandt, A.2
Janke, W.3
Shmulyian, S.4
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55
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0003493230
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Academic, New York, C. Domb, J. Lebowitz
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A. J. Guttmann, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic, New York, 1989), Vol. 13.
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(1989)
Phase Transitions and Critical Phenomena
, vol.13
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Guttmann, A.J.1
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58
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0037923007
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Fig. 1717, Academic Press, New York, C. Domb, M. S. Green
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see also Fig. 1717 in E. H. Lieb and F. Y. Wu, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic Press, New York, 1972), p. 331.
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(1972)
Phase Transitions and Critical Phenomena
, pp. 331
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Lieb, E.H.1
Wu, F.Y.2
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59
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85037222425
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(to be published)
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H. Feldmann, A. J. Guttmann, I. Jensen, R. Shrock, and S.-H. Tsai, J. Phys. A (to be published).
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J. Phys. A
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Feldmann, H.1
Guttmann, A.J.2
Jensen, I.3
Shrock, R.4
Tsai, S.-H.5
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66
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0000996160
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C. Broholm, G. Aeppli, G. P. Espinoza, and A. S. Cooper, J. Appl. Phys. 69, 4968 (1991); JAPIAU
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(1991)
J. Appl. Phys.
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Broholm, C.1
Aeppli, G.2
Espinoza, G.P.3
Cooper, A.S.4
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