Effective field theory for ${}^4\mathrm{He}$

We introduce an effective scalar field theory to describe the ${}^4\mathrm{He}$ phase diagram, which can be considered as a generalization of the $XY$ model which gives the usual $\lambda$ transition. This theory results from a Ginzburg-Landau Hamiltonian with higher order derivatives, which allow one to produce transitions between the superfluid, normal liquid, and solid phases of ${}^4\mathrm{He}$. Mean field and Monte Carlo analyses suggest that this model is able to reproduce the main qualitative features of ${}^4\mathrm{He}$ phase transitions.

Figure 1

Experimental pressure-temperature phase diagram for ${}^4\mathrm{He}$ in the low temperature region. Two liquid phases, He I and superfluid He II, and a solid phase can be distinguished.

Figure 2

${\mathcal{M}}^2$ as a function of ${\widehat{P}}^2$ for different values of $(c_2,c_4)$ corresponding to the cases signaled in the text (from left to right, top to bottom): 1, 2(a), 2(b), 3(a), and 3(b).

Figure 3

(Color online) Mean field phase diagram in the $(c_2,c_4)$ plane, supposing that $m^2+{\mathcal{M}}^2⩽0$ at every point of the plane. The black zone represents the AF phase, the grey zone is the MOD phase, and the white region, the FM phase.

Figure 4

Values of ${\widehat{P}}^2$ which minimize ${\mathcal{M}}^2$ at $c_2=0.5$ as a function of $c_4$. They define the different vacua of the mean field action, as explained in the text.

Figure 5

Mean field phase diagram in the $(m^2,c_2)$ plane, where at every point $c_4=0.25c_2$. Here the phases are PM (white region), FM (grey region), and MOD (black region).

Figure 6

(Color online) Surfaces in $K$ space with the same ${\widehat{P}}^2$, Eq. (17), for three different values ${\widehat{P}}^2=1$,3,6.

Figure 7

Monte Carlo phase diagram for an $L=16$ lattice (continuous line and dots) versus mean field phase diagram (dashed line). Error bars which are not shown are smaller than the size of the points.

Figure 8

(Color online) $\xi ∕L$ for $c_2=0.5$ and lattice sizes $L=8$, 12, 16, 24.

Figure 9

(Color online) Top: Hysteresis cycle in $c_2$. Figure shows $S_c$, Eq. (39), versus $c_2$ for $m^2=-0.55$ in an $L=8$ lattice. Inset: $\overline{{\widehat{P}}^2}$ distribution, Eq. (37), at both sides of the transition for the same run as in the main plot. Inside the FM phase $F\left(\overline{{\widehat{P}}^2}\right)$ has a peak in $\overline{{\widehat{P}}^2}=0$, while when crossing to the MOD phase the peak changes its position to $\overline{{\widehat{P}}^2}\sim 2$. Bottom: Histogram of $S_c$ for a run with fixed parameters $c_2=1.0,m^2=-0.55$, in an $L=8$ lattice. Dotted lines mark the position of the peaks of the histogram in the hysteresis plot. The double-peak form of the histogram shows clearly the first order character of the FM-MOD transition.

Figure 10

(Color online) Top: Hysteresis cycle in $m^2$. Figure shows $S_m$, Eq. (38), versus $m^2$ for $c_2=1.5$ for lattice sizes of $L=8$,12,16. Middle: $\overline{{\widehat{P}}^2}$ distribution at both sides of the transition for the same run as the one in the top part for the $L=8$ and $L=16$ lattices. The dotted line at $\overline{{\widehat{P}}^2}=2.27$ is the value of $\overline{{\widehat{P}}^2}$ predicted by mean field for this value of the parameter $c_2$. Bottom: Histogram of $S_m$ for a run with fixed parameters $c_2=1.5,m^2=0.57$ in an $L=12$ lattice. The histogram shows clearly the first order character of the MOD-PM transition.