To be more precise, the critical mode is a mixture between the $\sigma$, the ${\omega }^0$ meson, and the Polyakov loops, and the relevant mass that vanishes at the CEP is the spacelike screening mass [15].

This also includes models of magnetic materials [37, 38, 39, 43], elastic stressed systems [44], liquid crystals [45, 46], chemical mixtures and membranes [39], and general Ginzburg-Landau theories describing competing orders [47]. In these and numerous other arenas, the interplay between the varying order gradient terms leads to low-energy states which may obtain their minima at finite wave vectors. The resulting moat-type structures underlie basic aspects of crystallization [48, 49], superconductors in external magnetic fields [50, 51, 52], quantum Hall systems [53, 54, 55, 56, 57, 58, 59], and countless others.

In perturbative QCD with massless quarks, Friedel oscillations arise as an oscillatory function times a power-law falloff [90].

Recently, Ref. [101] appeared and analyzed dilepton production from a specific inhomogeneous phase. Reference [101] includes the modified dispersion relation of a dual chiral density wave but not the changes to the photon-moaton vertices. This is discussed in Supplemental Material [103].

On a lattice, the continuum limit $k^4$ term is replaced by a squared lattice Laplacian corresponding to competing nearest- and next-nearest-neighbor interactions.

If higher derivative terms dominate with negative coefficients, then, even if $Z>0$, it is possible that $E(p)$ about $p=0$ is a local minimum but that the global minimum of $E(p)$ is at $p\ne 0$.

This occurs, e.g., for a roton, where $p=0$ is the global minimum, but there is a local minimum for one or more $p_{\text{roton}}\ne 0$.

This can be derived in a matrix form, where $\mathrm{\Phi }=\sigma +i{\pi }^a{\sigma }^a$, with ${\sigma }^a$ the Pauli matrices. Under a gauge transformation, $\mathrm{\Phi }\to {\mathrm{e}}^{ie\theta {\sigma }^3}\mathrm{\Phi }{\mathrm{e}}^{-ie\theta {\sigma }^3}$, with $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }[{\sigma }^3,]$. For example, charged pions contribute to the usual kinetic term as $|({\partial }_{\mu }-ie{A}_{\mu }){\pi }^{+}{|}^2$.

There is also a term which is odd under charge conjugation, $\sim F_{0i}(D_0\overrightarrow{\varphi }·D_i\overrightarrow{\varphi })$. Unlike the other electromagnetic couplings, this term couples a photon to both charged and neutral (the $\sigma$ and ${\pi }^3$) fields. If $\mu \ne 0$ implies net electrical charge, then the coefficient of this term must vanish at $\mu =0$.