We consider the diagonal susceptibility of the isotropic 2D Ising model for temperatures below the critical temperature. For a parameter k related to temperature and the interaction constant, we extend the diagonal susceptibility to complex k inside the unit disc, and prove the conjecture that the unit circle is a natural boundary.
REFERENCES
1.
C.
Andréief
, “Note sur une relation les intégrales définies des produits des fonctions
,” in Mém. de la Soc. Sci., Bordeause
(1883
), Vol. 2
, pp. 1
–14
.2.
M.
Assis
, S.
Boukraa
, S.
Hassani
, M.
Van Hoeij
, J.-M.
Maillard
, and B. M.
McCoy
, “Diagonal Ising susceptibility: Elliptic integrals, modular forms and Calabi-Yau equations
,” J. Phys. A: Math. Theor.
45
, 075205
(2012
).3.
E.
Basor
and H.
Widom
, “On a Toeplitz determinant identity of Borodin and Okounkov
,” Integral Equ. Oper. Theory
37
, 397
–401
(2000
).4.
R. J.
Baxter
, “Onsager and Kaufman's calculation of the spontaneous magnetization of the Ising model
,” J. Stat. Phys.
145
, 518
–548
(2011
).5.
A.
Borodin
and A.
Okounkov
, “A Fredholm determinant formula for Toeplitz determinants
,” Integral Equ. Oper. Theory
37
, 386
–396
(2000
).6.
A.
Böttcher
, “One more proof of the Borodin-Okounkov formula for Toeplitz determinants
,” Integral Equ. Oper. Theory
41
, 123
–125
(2001
).7.
S.
Boukraa
, S.
Hassani
, J.-M.
Maillard
, B. M.
McCoy
, and N.
Zenine
, “The diagonal Ising susceptibility
,” J. Phys. A: Math. Theor.
40
, 8219
–8236
(2007
).8.
A. I.
Bugrii
, “Correlation function of the two-dimensional Ising model on a finite lattice: I
,” Theor. Math. Phys.
127
, 528
–548
(2001
).9.
A. I.
Bugrii
and O. O.
Lisovyy
, “Correlation function of the two-dimensional Ising model on a finite lattice: II
,” Theor. Math. Phys.
140
, 987
–1000
(2004
).10.
Y.
Chan
, A. J.
Guttmann
, B. G.
Nickel
, and J. H. H.
Perk
, “The Ising susceptibility scaling function
,” J. Stat. Phys.
145
, 549
–590
(2011
).11.
P.
Deift
, A.
Its
, and I.
Krasovsky
, “Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: Some history and some recent results
,” Commun. Pure Appl. Math.
66
, 1360
–1438
(2013
).12.
J. S.
Geronimo
and K. M.
Case
, “Scattering theory and polynomials orthogonal on the unit circle
,” J. Math. Phys.
20
, 299
–310
(1979
).13.
A. J.
Guttmann
and I. G.
Enting
, “Solvability of some statistical mechanical systems
,” Phys. Rev. Lett.
76
, 344
–347
(1996
).14.
I.
Lyberg
and B. M.
McCoy
, “Form factor expansion of the row and diagonal correlation functions of the two-dimensional Ising model
,” J. Phys. A: Math. Theor.
40
, 3329
–3346
(2007
).15.
B. M.
McCoy
and T. T.
Wu
, The Two-Dimensional Ising Model
(Harvard University Press
, 1973
).16.
17.
B. M.
McCoy
, M.
Assis
, S.
Boukraa
, S.
Hassani
, J.-M.
Maillard
, W. P.
Orrick
, and N.
Zenine
, “The saga of the Ising susceptibility
,” in New Trends in Quantum Integrable Systems: Proceedings of the Infinite Analysis 09
, edited by B. L.
Feigin
, M.
Jimbo
, and M.
Okado
(World Scientific
, 2010
), pp. 287
–306
; e-print arXiv:1003.0751.18.
B.
Nickel
, “On the singularity structure of the 2D Ising model
,” J. Phys. A
32
, 3889
–3906
(1999
).19.
B.
Nickel
, “Addendum to ‘On the singularity structure of the 2D Ising model’
,” J. Phys. A
33
, 1693
–1711
(2000
).20.
L.
Onsager
, “Crystal statistics. I. A two-dimensional model with an order-disorder transition
,” Phys. Rev.
65
, 117
–149
(1944
).21.
22.
W. P.
Orrick
, B.
Nickel
, A. J.
Guttmann
, and J. H. H.
Perk
, “The susceptibility of the square lattice Ising model: New developments
,” J. Stat. Phys.
102
, 795
–841
(2001
).23.
J.
Palmer
, Planar Ising Correlations
Progress in Mathematical Physics
Vol. 49
(Birkhäuser
, Boston
, 2007
).24.
J.
Stephenson
, “Ising-model spin correlations on the triangular lattice
,” J. Math. Phys.
5
, 1009
–1024
(1964
).25.
C. A.
Tracy
and H.
Widom
, “Correlation functions, cluster functions, and spacing distributions for random matrices
,” J. Stat. Phys.
92
, 809
–835
(1998
).26.
N. S.
Witte
and P. J.
Forrester
, “Fredholm determinant evaluations of the Ising model diagonal correlations and their λ generalization
,” Stud. Appl. Math.
128
, 183
–223
(2012
).27.
T. T.
Wu
, B. M.
McCoy
, C. A.
Tracy
, and E.
Barouch
, “Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region
,” Phys. Rev. B
13
, 316
–374
(1976
).28.
C. N.
Yang
, “The spontaneous magnetization of the two-dimensional Ising model
,” Phys. Rev.
85
, 808
–816
(1952
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
