Abstract
We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain T→T c , H→0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang–Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose “extended analyticity;” roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated “extended dispersion relation.”
Similar content being viewed by others
REFERENCES
M. Caselle and M. Hasenbusch, Critical amplitudes and mass spectrum of the 2D Ising model in a magnetic field, Nucl. Phys. B 579:667–703 (2000), (preprint hep-th/9911216).
M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, Irrelevant Operators in the Two-Dimensional Ising Model, DFTT 17/2001, DESY 01–074, IFUP-TH 99/2001, Roma1–1963/01, June 2001 (preprint cond-mat/0106372).
M. Caselle, P. Grinza, and N. Magnoli, Correction induced by irrelevant operators in the correlators of the 2-d Ising model in a magnetic field, J. Phys. A 34:8733–8750 (2001) (preprint hep-th/0103263).
L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65:117–149 (1944).
B. Kaufman, Crystal statistics. II. Partition function evaluated by spinor analysis, Phys. Rev. 76:1232–1243 (1949).
C. N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85:808–816 (1952).
E. Barouch, B. M. McCoy, and T. T. Wu, Zero-field susceptibility of the two-dimensional Ising model near Tc, Phys. Rev. Lett. 31:1409–1411 (1973).
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).
A. B. Zamolodchikov, Integrals of motion and S-matrix of the (scaled) T=Tc Ising model with magnetic field, Int. J. Mod. Phys. A 4:4235–4248 (1989).
V. A. Fateev, The exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324:45–51 (1994).
G. Delfino, Universal amplitude ratios in the two-dimensional Ising model, Phys. Lett. B 419:291–295 (1998), (preprint hep-th/9710019).
J. W. Essam and D. L. Hunter, Classical behaviour of the Ising model above and below the critical temperature, J. Phys. C 1:392–407 (1968).
S. Zinn, S.-N. Lai, and M. E. Fisher, Renormalized coupling constants and related amplitude ratios for Ising systems, Phys. Rev. E 54:1176–1182 (1996).
M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, High-precision estimate of g4 in the 2D Ising model, J. Phys. A 33:8171–8180 (2000), (preprint hep-th/0003049).
M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, The critical equation of state of the 2D Ising model, J. Phys. A 34:2923–2948 (2001), (preprint cond-mat/0011305).
V. P. Yurov and Al. B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod.Phys. A 5:3221–3245 (1990).
V. P. Yurov and Al. B. Zamolodchikov, Truncated-fermionic-space approach to the critical 2D Ising model with magnetic field, Int. J. Mod. Phys. A 6:4557–4578 (1991).
C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev. 87:404–409 (1952).
T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87:410–419 (1952).
J. S. Langer, Theory of the condensation point, Ann. Phys. 41:108–157 (1967)
J. S. LangerAnn. Phys. 281:941–990 (2000).
M. B. Voloshin, Decay of a metastable vacuum in (1+1) dimensions, Yad. Fiz. 42:1017–1026 (1985)
M. B. VoloshinSov. J. Nucl. Phys. 42:644–649 (1985).
T. T. Wu and B. M. McCoy, The Two-Dimensional Ising Model (Harvard University Press, 1973).
G. Mussardo, Off-critical statistical models: factorized scattering theories and bootstrap program, Phys. Rep. 218:215–379 (1992).
B. Berg, M. Karowski, and P. Weisz, Construction of Green's functions from an exact S matrix, Phys. Rev. D 19:2477–2479 (1979).
L. P. Kadanoff and H. Ceva, Determination of an operator algebra for the two-dimensional Ising model, Phys. Rev. B 3:3918–3939 (1971).
S. Sachdev, Universal, finite temperature, crossover functions of the quantum transition in the Ising chain in a transverse field, Nucl. Phys. B 464:576–595 (1996), (preprint cond-mat/9509147).
A. I. Bugrij, The Correlation Function in Two Dimensional Ising Model on the Finite Size Latice. I. (preprint hep-th/0011104)
A. I. BugrijForm Factor Representation of the Correlation Function of the Two Dimensional Ising Model on a Cylinder (preprint hep-th/0107117).
J. Balog, M. Niedermaier, F. Niedermayer, A. Patrascioiu, E. Seiler, and P. Weisz, The intrinsic coupling in integrable quantum field theories, Nucl. Phys. B 583:614–670 (2000), (preprint hep-th/0001097).
M. E. Fisher, Yang-Lee edge singularitry and φ3 field theory, Phys. Rev. Lett. 40:1610–1613 (1978).
J. L. Cardy, Conformal invariance and the Yang-Lee edge singularity in two dimensions, Phys. Rev. Lett. 54:1354–1356 (1985).
Al. B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models, Nucl. Phys. B 342:695–720 (1990).
A. F. Andreev, Singularity of thermodynamic quantities at a first order phase transition point, Sov. Phys. JETP 18:1415–1416 (1964).
M. E. Fisher, University of Colorado, Boulder, Summer School Lectures, 1964 (University of Colorado Press, Boulder, 1965)
M. E. FisherProc. Centennial Conf. Phase Transformations (Univ. Kentucky, 1965).
N. J. Günther, D. A. Nicole, and D. J. Wallace, Goldstone model in vacuum decay and first-order phase transitions, J. Phys. A 13:1755–1767 (1980).
M. J. Lowe and D. J. Wallace, Instantons and the Ising model below Tc, J. Phys. A 13:L381-L385 (1980).
C. K. Harris, The Ising model below Tc: calculation of non-universal amplitudes using a primitive droplet model, J. Phys. A 17:L143-L419 (1984).
B. M. McCoy and T. T. Wu, Two-dimensional Ising model near Tc: Approximation for small magnetic field, Phys. Rev. B 18:4886–4901 (1978).
J. S. Langer, Statistical theory of the decay of metastable states, Annals Phys. 54:258–275 (1969).
I. Yu. Kobzarev, L. B. Okun, and M. B. Voloshin, Bubbles in metastable vacuum, Yad. Fiz. 20:1229–1234 (1974)
I. Yu. Kobzarev, L. B. Okun, and M. B. Voloshin, Bubbles in metastable vacuumSov. J. Nucl. Phys. 20:644–646 (1975).
S. Coleman, Fate of the false vacuum: Semiclassical theory, Phys. Rev. D 15:2929–2936 (1977).
C. G. Callan and S. Coleman, Fate of the false vacuum. II. First quantum corrections, Phys. Rev. D 16:1762–1768 (1977).
S. B. Rutkevich, Decay of the metastable phase in d=1 and d=2 Ising models, Phys. Rev. B 60:14525–14528 (1999), (preprint cond-mat/9904059).
V. Fateev, S. Lukyanov, A. Zamolodchikov, and Al. Zamolodchikov, Expectation values of local fields in Bullough-Dodd model and integrable perturbed conformal field theories, Nucl. Phys. B 516:652–674 (1998), (preprint hep-th/9709034).
G. Delfino and G. Mussardo, The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T=Tc, Nucl. Phys. B 455:724–758 (1995), (preprint hep-th/9507010).
J. L. Cardy and G. Mussardo, S-matrix of the Yang-Lee edge singularity in two dimensions, Phys. Lett. B 225:275–278 (1989).
Al. B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358:497–523 (1991).
Al. B. Zamolodchikov, Mass scale in the Sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10:1125–1150 (1995).
B. M. McCoy and T. T. Wu, Two-dimensional Ising field theory in a magnetic field: Breakup of the cut in the two-point function, Phys. Rev. D 18:1259–1267 (1978).
G. Delfino, G. Mussardo, and P. Simonetti, Non-integrable quantum field theories as perturbations of certain integrable models, Nucl. Phys. B 473:469–508 (1996), (preprint hep-th/9603011).
V. Privman and L. S. Shulman, Analytic continuation at first-order phase transitions, J. Stat. Phys. 29:205–229 (1982).
J. S. Langer, Metastable states, Physica 73:61–72 (1974).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, 1989).
B. G. Nickel, On the singularity structure of the 2D Ising model susceptibility, J. Phys. A 32:3889–3906 (1999); Addendum to "On the singularity structure of the 2D Ising model susceptibility," J. Phys. A 33:1693–1711 (2000).
W. P. Orrick, B. G. Nickel, A. J. Guttmann, and J. H. H. Perk, Critical behaviour of the two-dimensional Ising susceptibility, Phys. Rev. Lett. 86:4120–4123 (2001), (preprint cond-mat/0009059).
W. P. Orrick, B. G. 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), (preprint cond-mat/0103074).
P. Fonseca, S. Lukyanov, and A. Zamolodchikov, to be published.
A. LeClair, Spectrum generating affine Lie algebras in massive field theory, Nucl. Phys. B 415:734–780 (1994), (preprint hep-th/9305110).
V. P. Yurov and Al. B. Zamolodchikov, Correlation functions of integrable 2D models of relativistic field theory; Ising model, Int. J. Mod. Phys. A 6:3419–3440 (1991).
G. von Gehlen, Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field, J. Phys. A 24:5371–5400 (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fonseca, P., Zamolodchikov, A. Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy. Journal of Statistical Physics 110, 527–590 (2003). https://doi.org/10.1023/A:1022147532606
Issue Date:
DOI: https://doi.org/10.1023/A:1022147532606