Successive phase transitions and kink solutions in ${\varphi }^8$, ${\varphi }^{10}$, and ${\varphi }^{12}$ field theories

We obtain exact solutions for kinks in ${\varphi }^8$, ${\varphi }^{10}$, and ${\varphi }^{12}$ field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higher-order field theories have kink solutions with algebraically decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order ${\varphi }^4$ and ${\varphi }^6$ theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the coexistence of up to three distinct kinks (for a ${\varphi }^{12}$ potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higher-order field theories have specific cases in which only nonlinear phonons are allowed. For the ${\varphi }^{10}$ field theory, which is a quasiexactly solvable model akin to ${\varphi }^6$, we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit.

Figure 1

(a) Example potentials of the form (7) for various illustrative values of the coefficient of the quadratic term, ${\alpha }_2$, showing the various phases and phase transitions in the ${\varphi }^8$ theory. (b) Magnification of (a) near the origin.

Figure 2

${\varphi }^8$ field theory at the first-order phase transition, $T=T_c^I$. (a) Example ${\varphi }^8$ potential with four degenerate minima (3) and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ superimposed as the dotted curve. (b) Kink solution (11) connecting $-a$ to $+a$ and the generic ${\varphi }^4$ kink $\varphi \left(x\right)=\frac{1}{2}({\varphi }_{+\infty }+{\varphi }_{-\infty })+\frac{1}{2}({\varphi }_{+\infty }-{\varphi }_{-\infty })tanh\left(\lambda x\right)=atanh\left(\lambda x\right)$ for ${\varphi }_{\pm \infty }=\pm a$ superimposed as the dotted curve. (c) Kink solution (15) connecting $a$ to $b$. In all panels, $a=(-1+\sqrt{3})/2$ and $b=(1+\sqrt{3})/2$.

Figure 3

Away from the first-order phase transition, $T\ne T_c^I$, in ${\varphi }^8$ field theory. (a) The potentials (19) (solid curve, $T>T_c^I$) and (21) (dashed curve, $T<T_c^I$). (b) The corresponding kinks computed by solving the equation of motion $d\varphi /dx=\sqrt{2V\left(\varphi \right)}$ numerically subject to the symmetry condition $\varphi \left(0\right)=0$.

Figure 4

${\varphi }^8$ field theory with three degenerate minima. (a) The potentials (22) (bottom curve, blue), (26) (top curve, red), and a representative ${\varphi }^6$ potential $V\left(\varphi \right)={\lambda }^2{\varphi }^2{({\varphi }^2-a^2)}^2$ (dotted curve). (b) The kink solutions (23) (top curve, blue) and (28) (bottom curve, red) connecting 0 to $a$, and the corresponding ${\varphi }^6$ kink $\varphi \left(x\right)=a/\sqrt{1+e^{-2\sqrt{2}{a}^2\lambda x}}$ (dotted curve). In all panels, $a=3/4$ and $b=1$.

Figure 5

${\varphi }^8$ field theory with two degenerate minima. (a) The potentials (31) (bottom curve, blue), (35) (top curve, red), and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ (dotted curve). (b) The corresponding kinks (32) (inner curve, blue) and (37) (outer curve, red) connecting $-a$ to $+a$, and the corresponding ${\varphi }^4$ kink $\varphi \left(x\right)=atanh\left(\lambda x\right)$ (dotted curve). In all panels, $a=4/5$ and $b=1$.

Figure 6

(a) Example potentials of the form (52) for various illustrative values of the coefficient of the quadratic term, ${\alpha }_2$, showing the various phases and phase transitions in the ${\varphi }^{10}$ theory. (b) Magnification of (a) near the origin.

Figure 7

${\varphi }^{10}$ field theory with five degenerate minima. (a) The potential (44) (solid curve), a representative ${\varphi }^8$ potential with four degenerate minima (3) (gray dotted curve), and a representative ${\varphi }^6$ potential with three degenerate minima $V\left(\varphi \right)={\lambda }^2{\varphi }^2{({\varphi }^2-a^2)}^2$ (black dotted curve). (b) The kink solution (54) (solid curve) connecting 0 to $a$ and the corresponding ${\varphi }^6$ kink $\varphi \left(x\right)=a/\sqrt{1+e^{-2\sqrt{2}{a}^2\lambda x}}$ (black dotted curve). (c) The kink solution (57) (solid curve) connecting $a$ to $b$ and the corresponding ${\varphi }^8$ kink (15) (gray dotted curve). In all panels, $a=1/2$ and $b=1$.

Figure 8

${\varphi }^{10}$ field theory at the second first-order phase transition, $T=T_c^I\left(II\right)$. (a) The potential (45) (solid curve), a representative ${\varphi }^8$ potential with four degenerate minima (3) (gray dotted curve), and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ (black dotted curve). (b) The kink solution (60) (solid curve) connecting $-a$ to $+a$ and the corresponding ${\varphi }^4$ kink $\varphi \left(x\right)=atanh\left(\lambda x\right)$ (black dotted curve). (c) The kink solution (63) (solid curve) connecting $a$ to $b$ and the corresponding ${\varphi }^8$ kink (15) (gray dotted curve). In all panels, $a=1/2$, $b=1$, and $c=3/4$.

Figure 9

Kink solutions between the first and second first-order phase transition [$T_c^I\left(II\right)<T<T_c^I\left(I\right)$] and below the second first-order phase transition [$T<T_c^I\left(II\right)$] in ${\varphi }^{10}$ field theory. (a) The potentials (67) (solid curve) and (69) (dashed curve). (b) The corresponding kinks computed by solving the equation of motion $d\varphi /dx=\sqrt{2V\left(\varphi \right)}$ numerically subject to the symmetry condition $\varphi \left(0\right)=0$.

Figure 10

${\varphi }^{10}$ field theory with three degenerate minima. (a) The potentials: (70) (top curve, blue), (75) (second curve from top, red), (80) (fourth curve from top, yellow), (84) (third curve from top, green), and a representative ${\varphi }^6$ potential $V\left(\varphi \right)={\lambda }^2{\varphi }^2{({\varphi }^2-a^2)}^2$ (dotted curve). (b) The kink solutions connecting 0 to $a$: (72) (fourth curve from top, blue), (77) (third curve from top, red), (81) (first curve from top, yellow), (86) (second curve from top, green), and the corresponding ${\varphi }^6$ kink $\varphi \left(x\right)=a/\sqrt{1+e^{-2\sqrt{2}{a}^2\lambda x}}$ (dotted curve). In all panels, $a=4/5$, $b=1$, and $c=1$.

Figure 11

${\varphi }^{10}$ field theory with two degenerate minima. (a) The potentials: (89) (top curve, blue), (94) (middle curve, red), (99) (bottom curve, yellow), and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ (dotted curve). (b) The kink solutions connecting $-a$ to $a$: (91) (outer curve, blue), (96) (inner curve, red), (100) (middle curve, yellow), and the corresponding ${\varphi }^4$ kink $\varphi \left(x\right)=atanh\left(\lambda x\right)$ (dotted curve). In all panels, $a=9/10$ and $b=1$.

Figure 12

${\varphi }^{12}$ field theory with six degenerate minima. (a) The potential (113) (solid curve), a representative ${\varphi }^8$ potential (3) (gray dotted curve), and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ (black dotted curve). (b) Kink solutions connecting $-a$ to $+a$ (114) (bottom curve, blue), $a$ to $b$ (117) (middle curve, red), $b$ to $c$ (120) (top curve, yellow), the corresponding ${\varphi }^8$ kinks connecting $-a$ to $+a$ (11) and $a$ to $b$ (15) (both gray dotted curves), and the corresponding ${\varphi }^4$ kink $\varphi \left(x\right)=atanh\left(\lambda x\right)$ (black dotted curve). In all panels, $a=1/4$, $b=2/3$, and $c=1$.

Figure 13

${\varphi }^{12}$ field theory with five degenerate minima. (a) The potentials: (123) (top curve, blue), (130) (bottom curve, red), a representative ${\varphi }^8$ potential with four degenerate minima (3) (gray dotted curve), and a representative ${\varphi }^6$ potential with three degenerate minima $V\left(\varphi \right)={\lambda }^2{\varphi }^2{({\varphi }^2-a^2)}^2$ (black dotted curve). (b) The kink solutions connecting 0 to $a$: (124) (outer curve, blue), (132) (inner curve, red), and the corresponding ${\varphi }^6$ kink $\varphi \left(x\right)=a/\sqrt{1+e^{-2\sqrt{2}{a}^2\lambda x}}$ (black dotted curve). (c) The kink solutions connecting $a$ to $b$: (127) (outer curve, blue), (135) (inner curve, red), and the corresponding ${\varphi }^8$ kink (15) (gray dotted curve). In all panels, $a=1/2$, $b=1$, and $c=2$.

Figure 14

${\varphi }^{12}$ field theory with four degenerate minima. (a) The potentials: (138) (second curve from top, blue), (145) (third curve from top, red), (153) (top curve, yellow), a representative ${\varphi }^8$ potential with four degenerate minima (3) (gray dotted curve), and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ (black dotted curve). (b) The kink solutions connecting $-a$ to $+a$: (139) (outer curve, blue), (147) (inner curve, red), (155) (middle curve, yellow), the corresponding ${\varphi }^8$ kink (11) (gray dotted curve), and the corresponding ${\varphi }^4$ kink $\varphi \left(x\right)=atanh\left(\lambda x\right)$ (black dotted curve). (c) The kink solutions connecting $a$ to $b$: (142) (outer curve, blue), (150) (inner curve, red), (158) (middle curve, yellow), and the corresponding ${\varphi }^8$ kink (15) (gray dotted curve). In all panels, $a=1/2$, $b=1$, and $c=3/4$.

Figure 15

${\varphi }^{12}$ field theory with three degenerate minima. (a) The potentials: (161) (fourth curve from top, blue), (165) (fifth curve from top, red), (169) (second curve from top, yellow), (174) (top curve, green), (179) (third curve from top, black), and a representative ${\varphi }^6$ potential $V\left(\varphi \right)={\lambda }^2{\varphi }^2{({\varphi }^2-a^2)}^2$ (gray dotted curve). (b) The kink solutions connecting 0 to $a$: (162) (top curve, blue), (166) (third curve from top, red), (171) (fourth curve from top, yellow), (176) (fifth curve from top, green), (181) (second curve from top, black), and the corresponding ${\varphi }^6$ kink $\varphi \left(x\right)=a/\sqrt{1+e^{-2\sqrt{2}{a}^2\lambda x}}$ (gray dotted curve). In all panels, $a=4/5$ and $b=1$.

Figure 16

${\varphi }^{12}$ field theory with two degenerate minima. (a) The potentials: (184) (top curve, blue), (189) (middle curve, red), (194) (bottom curve, yellow), and a representative ${\varphi }^4$ potential $V\left(\varphi \right)={\lambda }^2{({\varphi }^2-a^2)}^2$ (dotted curve). (b) The kink solutions connecting $-a$ to $+a$: (186) (outer curve, blue), (191) (middle curve, red), (195) (inner curve, yellow), and the corresponding ${\varphi }^4$ kink $\varphi \left(x\right)=atanh\left(\lambda x\right)$ (dotted curve). In all panels, $a=9/10$ and $b=1$.

Figure 17

Kink solutions (201) of the $n\to \infty$ limiting field theories $V\left(\varphi \right)={\lambda }^2(1-cos\varphi )$ on the left (blue) and $V\left(\varphi \right)={\lambda }^2(1+cos\varphi )$ on the right (red).