Pirogov–Sinai theory (low-temperature phase diagrams with multiple ground states)

Context

Many lattice models (see lattice Hamiltonians ) have several distinct ground states at zero temperature. Pirogov–Sinai theory provides a systematic way to analyze the corresponding low-temperature equilibrium structure: existence of multiple Gibbs measures , stability under perturbations, and precise first-order transition lines.

It builds on contour representations and convergent expansions (see cluster expansion ).

Theorem (Pirogov–Sinai; standard qualitative form)

Consider a finite-range lattice spin system with:

• a finite set of periodic ground states $\mathcal{G}=\{g_1,\dots,g_m\}$ at zero temperature,
• a Peierls-type condition: excitations above each ground state have energy cost proportional to the size of their boundary (contour cost).

Then there exists $\beta_0<\infty$ such that for all $\beta\ge \beta_0$:

1. Phase coexistence and pure phases: for each stable ground state $g_k$ there exists a corresponding translation-invariant pure infinite-volume Gibbs state $\mu_k$ solving the DLR equations .

2. Uniqueness away from coexistence: in regions of parameter space where a single ground state is stable, the infinite-volume Gibbs measure is unique and correlations decay exponentially (finite correlation length ).

3. First-order transitions: coexistence surfaces (where two or more ground states have equal free-energy competition) correspond to first-order transitions: multiple Gibbs states exist and macroscopic observables (e.g. an order parameter ) exhibit discontinuities across the surface.

4. Interfaces and surface tension: at coexistence, mixed boundary conditions generate stable interfaces with positive surface tension , giving control of droplet formation and providing a rigorous foundation for metastability at low temperature.

What you can do with it

• Prove low-temperature phase diagrams for models like Potts-type systems, Ising models with fields or competing interactions, lattice gases, and small perturbations of exactly solvable ground-state structures.
• Establish sharp statements about stable phases, coexistence lines, and interface costs in terms of contour weights and convergent expansions.

Prerequisites and connections (cross-links)

• Gibbs formalism on lattices: finite-volume Gibbs measures , DLR equations , infinite-volume Gibbs measures .
• Thermodynamic potentials: free energy , pressure (log-partition density) .
• Phase-transition diagnostics: phase transitions in Gibbs measures , equivalent Gibbs characterizations .
• Expansion technology: cluster expansion theorem .