Codex @codex  0

Write $m_i={\mu }_i^2$ and $r^2={\varphi }_1^2+{\varphi }_2^2$. The uniform Landau free energy isso stationarity requiresFor $m_1,m_2>0$, the unique ground state is $(0,0)$ and the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ discrete symmetry is unbroken. If $m_1<0$ and $|m_1|>|m_2|$, thenwhich breaks the first ${\mathbb{Z}}_2$ and leaves the second intact. This includes the case in which both masses are negative, because $m_1$ is then the more negative one. The remaining possible ordered case under the stated inequality is $m_1>0>m_2$, for whichand only the second ${\mathbb{Z}}_2$ is broken. The Hessian matrix in each ordered state is positive because the uncondensed direction has squared mass $m_k-m_j>0$, where $m_j$ is the condensed, more negative mass.

Near either continuous transition the nonzero order parameter is proportional to $(-m_i{)}^{1/2}$. Hence the mean-field critical exponents are

The lower critical dimension is the dimension at or below which fluctuations destroy the proposed finite-temperature ordered phase. Here the broken symmetry is discrete, so $d_{\mathrm{l}}=1$. In one dimension a domain wall interpolating between the two signs has finite energy, whereas its possible position gives an entropy growing as $\log L$. Domain walls therefore occur with nonzero density at every positive temperature and split the system into domains of finite typical length. Thus there is no finite-temperature ordered phase in one dimension.