Chapter 21
Symmetry Breaking in the Ising Model
Most of the theoretical wisdom on the phase transition of the ferromagnetic type and
the related symmetry breaking is based on the two-dimensional Ising model, which
also played the role of a laboratory for ideas and strategies and it is now regarded
as a cornerstone in the foundations of statistical mechanics. Anyone interested in
critical phenomena and in the functional integral approach to quantum field theory
should have a look at the model. Even if a discussion of the two-dimensional Ising
model would be very appropriate for our purposes, we refer the reader to the very good
accounts which can be found in the literature.
116 We restrict our discussion to the onedimensional version of the model, which is almost trivial, but nevertheless provides
an interesting simple example for testing the constructive strategies of symmetry
breaking discussed above.
The Ising model was invented to mimic the phenomenon of ferromagnetism and
it is a simplified version of the Heisenberg model. The algebra A, which describes
the degrees of freedom of the system, is the spin algebra generated by polynomials
of the spins in various sites (see Sect. 17.1), and the finite volume Hamiltonian is
H V = −J
i∈V
σ i σ i+1 − h
i∈V
σ i ,
(21.1)
116 For the history of the model, see S.G. Brush, Rev. Mod. Phys. 39, 883 (1967). The model
is now part of the basic knowledge in statistical mechanics and the theory of phase transitions;
for textbook accounts, see, e.g. K. Huang, Statistical Mechanics, Wiley 1987, Chap. 24, 25; G.
Gallavotti, Statistical Mechanics: A Short Treatise, Springer 1999, Sect. 6; B. Simon, The Statistical
Mechanics of Lattice Gases, Vol. I, Princeton Univ. Press 1993, Sect. II.6. An extensive treatment,
which also emphasizes the links with quantum field theory and general theoretical physics problems,
is in B.M. McCoy and T.T. Wu, The Two Dimensional Ising Model, Harvard Univ. Press 1973.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_21
139
Symmetry Breaking in the Ising Model
Most of the theoretical wisdom on the phase transition of the ferromagnetic type and
the related symmetry breaking is based on the two-dimensional Ising model, which
also played the role of a laboratory for ideas and strategies and it is now regarded
as a cornerstone in the foundations of statistical mechanics. Anyone interested in
critical phenomena and in the functional integral approach to quantum field theory
should have a look at the model. Even if a discussion of the two-dimensional Ising
model would be very appropriate for our purposes, we refer the reader to the very good
accounts which can be found in the literature.
116 We restrict our discussion to the onedimensional version of the model, which is almost trivial, but nevertheless provides
an interesting simple example for testing the constructive strategies of symmetry
breaking discussed above.
The Ising model was invented to mimic the phenomenon of ferromagnetism and
it is a simplified version of the Heisenberg model. The algebra A, which describes
the degrees of freedom of the system, is the spin algebra generated by polynomials
of the spins in various sites (see Sect. 17.1), and the finite volume Hamiltonian is
H V = −J
i∈V
σ i σ i+1 − h
i∈V
σ i ,
(21.1)
116 For the history of the model, see S.G. Brush, Rev. Mod. Phys. 39, 883 (1967). The model
is now part of the basic knowledge in statistical mechanics and the theory of phase transitions;
for textbook accounts, see, e.g. K. Huang, Statistical Mechanics, Wiley 1987, Chap. 24, 25; G.
Gallavotti, Statistical Mechanics: A Short Treatise, Springer 1999, Sect. 6; B. Simon, The Statistical
Mechanics of Lattice Gases, Vol. I, Princeton Univ. Press 1993, Sect. II.6. An extensive treatment,
which also emphasizes the links with quantum field theory and general theoretical physics problems,
is in B.M. McCoy and T.T. Wu, The Two Dimensional Ising Model, Harvard Univ. Press 1973.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_21
139