Speaker
Description
Tricritical behavior in systems with an $n$-component order parameter $\varphi = \{\varphi_k, k = 1, \ldots, n\}$ is described by the action
\begin{equation}
S(\varphi) = \frac{1}{2}\partial_i\varphi_k\partial_i\varphi_k + \frac{\tau_0}{2} \varphi_k\varphi_k + \frac{\lambda_0}{4!} (\varphi_k\varphi_k)^2 + \frac{g_0}{6!} (\varphi_k\varphi_k)^3,
\end{equation}
where the coefficients $\tau_0$, $\lambda_0$ and $g_0$ are parameters of the model [1].
Tricritical asymptotics correspond to the situation where $\tau_0 \sim \lambda_0 \rightarrow 0$. Depending on the relative smallness of the parameters $\tau_0$ and $\lambda_0$, the analysis at the canonical dimension level predicts three possible asymptotic behaviors:
- Tricritical -- the $\varphi^4$ interaction is not significant;
- Modified Critical -- the $\varphi^6$ interaction is not significant;
- Combined Tricritical -- both interactions are significant.
It is possible to enter a parameter $\alpha$, which depends on the trajectory of the approach to the tricritical point in the experiment, and at different values of which, different asymptotic behaviours will be observed. At a certain value $\alpha = \alpha_0$, the combined tricritical behavior will occur; when $\alpha > \alpha_0$, tricritical behavior is observed, and when $\alpha < \alpha_0$, modified critical behavior occurs.
Renormalization group analysis which was fully conducted by Vasil'ev [1] allows us to refine the value of the parameter $\alpha_0$. Additionally, the renormalization group analysis shows that for $\alpha = \alpha_0$, not only the combined tricritical behavior is possible, but also modified critical behavior. It depends on the parameter $a$ that occurs during the analysis. If $a$ turns out to be positive, there are two possible behaviours and the parameter determines the boundary value for the relative amplitude of $\tau_0$ and $\lambda_0$, at which the transition between different behaviours occurs. Otherwise, only the combined tricritical behaviour is possible. In the main order, the parameter $a$ was calculated by Vasil'ev [1] and it is positive. Therefore, the first situation is realized to the lowest approximation. We are investigating the effect of six-loop corrections on the parameter $a$.
The work is supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075–15–2022–287).
[1] A.N. Vasil’ev, Quantum field renormalization group in critical behavior theory and stochastic dynamics. Chapman and Hall/CRC (2004).
