Phase transitions for \(\phi_2^4\) quantum fields (Q1594571)

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 \(\phi\rightarrow-\phi\).