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Phase transitions for $$\phi_2^4$$ quantum fields. (English) Zbl 0956.82501
The authors prove the existence of a phase transition for the quantum field interaction $$\lambda\phi^4+m_0{}^2\phi^2$$ in two-dimensional space-time. Technically the following two results (along with a related estimate) are shown to hold for sufficiently large $$\lambda$$: (1) Let $$\Delta_j$$ be a unit square lattice with center at $$j\in{Z}^2$$, let $$\phi(\Delta)$$ denote a smeared out field $$\int_\Delta\phi(x) dx$$, let $$\chi_\pm$$ be the characteristic functions for the half-lines $$(0,+\infty)$$ and $$(-\infty,0)$$, respectively, and let $$\chi_\pm(\Delta_j)$$ denote $$\chi_\pm(\phi(\Delta_j))$$. Then $$|\langle\chi_+(\Delta_i)\chi_-(\Delta_j)\rangle-\langle\chi_+(\Delta_i)\rangle\langle\chi_-(\Delta_j)\rangle|>a>0$$, which shows the existence of long-range (large $$i-j$$) correlation. (2) If a linear term $$-\mu\phi$$ is added to the interaction and the limit $$\mu\rightarrow+0$$ is taken, then $$\langle\chi_+(\Delta)\rangle_\mu$$ has a limit greater than $${\textstyle\frac 1{2}}+\surd a$$, showing asymmetry under the transformation $$\phi\rightarrow-\phi$$.
Reviewer: H.Araki (Kyoto)

##### MSC:
 82B10 Quantum equilibrium statistical mechanics (general) 82B26 Phase transitions (general) in equilibrium statistical mechanics
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##### References:
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