Collective modes and Kosterlitz-Thouless transition in a magnetic field in the planar Nambu–Jona-Lasinio model

It is known that a constant magnetic field is a strong catalyst of dynamical chiral symmetry breaking in $2+1$ dimensions, leading to generating dynamical fermion mass even at weakest attraction. In this work we investigate the collective modes associated with the dynamical chiral symmetry breaking in a constant magnetic field in the ($2+1$)-dimensional Nambu–Jona-Lasinio model with continuous U(1) chiral symmetry. We introduce a self-consistent scheme to evaluate the propagators of the collective modes at the leading order in $1/N$. The contributions from the vacuum and from the magnetic field are separated such that we can employ the well-established regularization scheme for the case of vanishing magnetic field. The same scheme can be applied to the study of the next-to-leading order correction in $1/N$. We show that the sigma mode is always a lightly bound state with its mass being twice the dynamical fermion mass for arbitrary strength of the magnetic field. Since the dynamics of the collective modes is always $2+1$ dimensional, the finite temperature transition should be of the Kosterlitz-Thouless (KT) type. We determine the KT transition temperature $T_{\mathrm{KT}}$ as well as the mass melting temperature $T^{*}$ as a function of the magnetic field. It is found that the pseudogap domain $T_{\mathrm{KT}}<T<T^{*}$ is enlarged with increasing strength of the magnetic field. The influence of a chiral imbalance or axial chemical potential ${\mu }_5$ is also studied. We find that even a constant axial chemical potential ${\mu }_5$ can lead to inverse magnetic catalysis of the KT transition temperature in $2+1$ dimensions. The inverse magnetic catalysis behavior is actually the de Haas–van Alphen oscillation induced by the interplay between the magnetic field and the Fermi surface.

Figure 1

(a) The dynamical fermion mass $M$ scaled by $M_0$ as a function of $g_{\mathrm{B}}=E_{\mathrm{B}}/M_0$ for the supercritical case $G>G_c$ and subcritical case $G<G_c$. The dashed line denote the universal limit $M/E_{\mathrm{B}}=0.4460$ for $B\to \infty$. (b) The velocity of Goldstone mode ${\upsilon }_{\pi }$ as a function of $g_{\mathrm{B}}$ for the supercritical case $G>G_c$ and subcritical case $G<G_c$. The dashed line denotes the universal limit ${\upsilon }_{\pi }=0.5875$ for $B\to \infty$.

Figure 2

The KT transition temperature $T_{\mathrm{KT}}$ and the mass melting temperature $T^{*}$ as a function of $g_B=E_{\mathrm{B}}/M_0$ for $G>G_c$ (a) and $G<G_c$ (b). For KT transition temperature, the results for $N=3$ and $N=10$ are shown with blue dot-dashed lines and red dashed lines, respectively.

Figure 3

The dynamical fermion mass $M$ (scaled by $M_0$) as a function of ${\mu }_5/M_0$ for $G>G_c$ and $G<G_c$.

Figure 4

Dependence of the mass melting temperature $T^{*}$ (a), KT transition temperature $T_{\mathrm{KT}}$ (b), and the effective mass $M_{\mathrm{KT}}$ at $T=T_{\mathrm{KT}}$ (c) on the magnetic field. $T^{*}$, $T_{\mathrm{KT}}$, and $M_{\mathrm{KT}}$ are all scaled by their values at $B=0$. The magnetic field strength is denoted by $\sqrt{eB}/{\mu }_5$. The black solid lines and the red dashed lines show the results for ${\mu }_5/M_0=5$ and ${\mu }_5/M_0=10$, respectively.