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(a) (b)
Figure 2: Schematic pictures of contours (gray) separating ground state regions (white): (a) possible
contours, (b) typical contours at low T
The Pirogov‐Sinai theory also enables one to construct low temperature phase diagrams of the
models. It is able to introduce “meta‐stable” free energy densities fn for each phase n = 1, …, N.
By comparing the densities fn the phase diagram can be obtained (analogously to a ground state
diagram that is obtained by comparing energy densities en). Thus, if fn is minimal of all N free
energy densities, the n‐th phase is stable (for given values of thermodynamic parameters). The
coexistence of two or more phases occurs at those points or lines/surfaces, where several phases
are stable (two or more free energy densities are minimal). In addition, since free energy
densities fn are very close to the ground state energy densities en (the difference being of the
order e −β ), the phase diagram is only a small (~e −β ) deformation of the ground state diagram.
Finally, the theory enables us to detect phase transitions. Starting from the observation that the
true free energy density f is equal to the minimum of all “meta‐stable” free energy densities, f =
min{f1,…, fN}, it immediately follows that, for example, across a coexistence line of phases n and
m the one‐sided derivatives of f do not coincide, f ′− = fn′ ≠ fm′ = f ′+, implying that a first‐order
phase transition takes place here.
By now there are several puzzling problems to which the Pirogov‐Sinai theory has been
successfully applied, such as the finite‐size effects (the details of the rounding of the phase
transition discontinuities and singularities) or the Lee‐Yang zeros of partition functions (see
references given in [18] for more details). The models that are covered by the theory
include, besides the standard Ising and Potts model, many other models with a finite‐range
potential and even quantum perturbations of classical models, systems with continuous spins,
and continuous systems. There are actually only three basic conditions under which the theory
is applicable: (i) sufficiently low temperatures, (ii) finite number of ground states, and (iii)
contours (i.e., the boundaries between ground states) represent sufficiently high energetic
barriers so that the energy of a contour is proportional to the contour size (the Peierls
condition). The last condition can be verified directly or sometimes by the method of an m‐
potential [19]. Although the theory applies rather generally to a large number of systems, the
introduction of contours is basically a model dependent problem (even for a given model one
may use several types of contours according to the properties to be studied). Nevertheless, there
are canonical prescriptions how contours may be constructed.
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