Flux tubes and the type-I/type-II transition in a superconductor coupled to a superfluid

We analyze magnetic-flux tubes at zero temperature in a superconductor that is coupled to a superfluid via both density and gradient (“entrainment”) interactions. The example we have in mind is high-density nuclear matter, which is a proton superconductor and a neutron superfluid, but our treatment is general and simple, modeling the interactions as a Ginzburg-Landau effective theory with four-fermion couplings, including only $s$-wave pairing. We numerically solve the field equations for flux tubes with an arbitrary number of flux quanta and compare their energies. This allows us to map the type-I/type-II transition in the superconductor, which occurs at the conventional $\kappa \equiv \lambda /\xi =1/\sqrt{2}$ if the condensates are uncoupled. We find that a density coupling between the condensates raises the critical $\kappa$ and, for a sufficiently high neutron density, resolves the type-I/type-II transition line into an infinite number of bands corresponding to “$\text{type-II}\left(n\right)$” phases, in which $n$, the number of quanta in the favored flux tube, steps from 1 to infinity. For lower neutron density, the coupling creates spinodal regions around the type-I/type-II boundary, in which metastable flux configurations are possible. We find that a gradient coupling between the condensates lowers the critical $\kappa$ and creates spinodal regions. These exotic phenomena may not occur in nuclear matter, which is thought to be deep in the type-II region but might be observed in condensed-matter systems.

Figure 1

(Color online) Profile of flux tube with $n=1$ units of flux (top) and $n=100$ units of flux (bottom) showing the effect of density coupling $\beta$ between neutron and proton condensates. The plot shows the deviation $\delta \rho$ of the condensates from their vacuum values (18). With no coupling between the condensates $\left(\beta =\sigma =0\right)$, the neutrons are undisturbed $\left(\delta {\rho }_n=0\right)$. With a nonzero density coupling $\beta$, the neutron condensate (broken lines) is significantly perturbed by the flux tube. Note that the neutron $\delta {\rho }_n$’s are multiplied by 10 (not by 100 as in Fig. 2) to make them visible. The other parameters are $\kappa =3.0$, $\sigma =0.0$, and ${⟨{\varphi }_p⟩}^2/{⟨{\varphi }_n⟩}^2=0.05$.

Figure 2

(Color online) Profile of flux tube with $n=1$ units of flux (top) and $n=100$ units of flux (bottom) showing the effect of gradient coupling $\sigma$ between neutrons and protons. The plot shows the deviation $\delta \rho$ of the condensates from their vacuum values (18). With no coupling between the condensates $\left(\beta =\sigma =0\right)$, the neutrons are undisturbed $\left(\delta {\rho }_n=0\right)$. With a nonzero gradient coupling $\sigma$, the neutron condensate (broken lines) is slightly perturbed by the flux tube. Note that the neutron $\delta {\rho }_n$’s are multiplied by 100 to make them visible. The other parameters are $\kappa =3.0$, $\beta =0.0$, and ${⟨{\varphi }_p⟩}^2/{⟨{\varphi }_n⟩}^2=0.05$.

Figure 3

(Color online) The energy per flux quantum $E_n/n$, in units of $E_{\text{Bog}}$ [see Eq. (19)], as a function of the number $n$ of units of flux in the flux tube. Top left, simple proton superconductor with neutrons completely decoupled $\left(\beta =\sigma =0\right)$; bottom left, density coupling between condensates $\left(\beta =0.5,\sigma =0\right)$; top right, gradient coupling between condensates $\left(\beta =0,\sigma =0.5\right)$; and bottom right, both couplings $\left(\beta =\sigma =0.5\right)$.

Figure 4

(Color online) Effect on the superconductor of density coupling $\beta$ to a superfluid, displayed as a phase diagram in the $\kappa -\beta$ plane, with no gradient coupling $\left(\sigma =0\right)$ and ${⟨{\varphi }_p⟩}^2/{⟨{\varphi }_n⟩}^2=0.05$. The top panel shows how nonzero $\beta$ causes an increase in ${\kappa }_{\text{critical}}$. In the bottom panel we magnify the transition region near $\beta =0.5$, illustrating that on the type-II side there is a sequence of $\text{type-II}\left(n\right)$ bands in which the number of flux quanta in the favored flux tube rises, reaching infinity when the superconductor becomes type I.

Figure 5

(Color online) Effect on the superconductor of gradient coupling $\sigma$ to a superfluid, displayed as a phase diagram in the $\kappa \text{−}\sigma$ plane, with no density coupling $\left(\beta =0\right)$ and ${⟨{\varphi }_p⟩}^2/{⟨{\varphi }_n⟩}^2=0.05$. The gradient coupling causes a decrease in ${\kappa }_{\text{critical}}$ and creates metastable states on either side of the transition with spinodal lines as shown.

Figure 6

(Color online) Phase diagram for combined density and gradient interactions: the $\kappa \text{−}\beta$ plane for $\sigma =0.5$ and ${⟨{\varphi }_p⟩}^2/{⟨{\varphi }_n⟩}^2=0.05$. The type-I/type-II boundary is no longer symmetric under $\beta \to -\beta$. In the bottom panel we magnify the transition region near $\beta =0.5$, illustrating that on the type-II side as $\kappa$ decreases the number of flux quanta in the favored flux tube jumps from 1 to a finite value (in this case $n=5$) and then there is a sequence of bands in which $n$ rises to infinity, at which point the superconductor becomes type I.

Figure 7

(Color online) Phase diagrams in the $\kappa$ vs ${⟨{\varphi }_n⟩}^2/{⟨{\varphi }_p⟩}^2$ plane. The vertical dashed lines show ${⟨{\varphi }_n⟩}^2/{⟨{\varphi }_p⟩}^2=20$, the value used for other figures in this paper. The top panel is for density coupling $\beta =-0.1$, but no gradient coupling $\left(\sigma =0\right)$. The bottom panel is for gradient coupling $\sigma =0.1$ but no density coupling $\left(\beta =0\right)$. In both cases, we see that the type-I/type-II transition converges to $\kappa =1/\sqrt{2}$ as the neutron condensate disappears. For the case of a density coupling, as the neutron condensate decreases, the type-I/type-II boundary changes at ${⟨{\varphi }_n⟩}^2/{⟨{\varphi }_p⟩}^2\sim 10$ from a narrow region of $\text{type-II}\left(n\right)$ phase bands (thick line) to wider metastable regions.