Driven Critical Dynamics in Tricitical Point

TL;DR

This work addresses nonequilibrium critical dynamics near a one-dimensional Ising tricritical point with two independent relevant directions, where the adiabatic-impulse scenario of the Kibble-Zurek mechanism can fail along the Ising-critical line. It employs time-dependent variational principle with matrix product states to drive the system across the tricritical point and unveils two distinct finite-time scaling regimes: along the Ising-critical direction the dynamics obey KZ scaling with $r_\mu = z + 1/\nu_\mu$ and $\nu_\mu=5/4$, while along the perpendicular direction the scaling is governed by $r_p = z + 1/\nu_p$ with $\nu_p=5/9$. Observables such as the order parameter $M_2$, half-chain fermion correlation $C_f$, and entanglement entropy $S$ exhibit clear FTS forms and data collapses, despite AIS breakdown in the gapless initial state. The results offer a fundamental perspective on nonequilibrium dynamics near tricritical points and suggest potential realizations in programmable Rydberg-atom quantum simulators.

Abstract

The conventional Kibble-Zurek (KZ) mechanism, describing driven dynamics across critical points based on the adiabatic-impulse scenario (AIS), have attracted broad attentions. However, the driven dynamics in tricritical point with two independent relevant directions has not been adequately studied. Here, we employ time dependent variational principle to study the driven critical dynamics at a one-dimensional supersymmetric Ising tricritical point. For the relevant direction along the Ising critical line, the AIS apparently breaks down. Nevertheless, we find that the critical dynamics can still be described by the KZ scaling in which the driving rate has the dimension of $r=z+1/ν_μ$ with $z$ and $ν_μ$ being the dynamic exponent and correlation length exponent in this direction, respectively. For driven dynamics along other direction, the driving rate has the dimension $r=z+1/ν_p$ with $ν_p$ being the other correlation length exponent. Our work brings new fundamental perspective into the nonequilibrium critical dynamics near the tricritical point, which could be realized in programmable quantum processors in Rydberg atomic systems.

Figures (8)

• Figure 1: Sketch of the phase diagram near tricritical point of the Ising model and the protocol for driven dynamics. Around the tricritical dimension, there are two independent relevant directions. One is the $h_\mu$-direction for $h_\sigma=1$. Although the conventional AIS breaks down for driven dynamics from the Ising critical phase along this direct, we show that the dynamic scaling behaviors satisfy the FTS form in which the driving rate $R$ has the dimension of $r_\mu=z+1/\nu_\mu$. The driven dynamics along other relevant direction is explored by tuning $h_\sigma$ to cross the tricritical point, which exhibits a distinct FTS form in which $R$ has the dimension of $r_p=z+1/\nu_p$.
• Figure 2: Scaling behavior of order parameter $M_2$ at the tricritical point when the system is driven from the gapless Ising critical phase by decreasing $h_{\mu}$. The solid line in (a) shows the power law $M_2\propto R^{-(1-\eta_b)/r_\mu}$ fitted to the $L=60$ data in large $R$ region. The rescaled curves in (b) collapse onto each other.
• Figure 3: Scaling behavior of the half-chain fermion correlation $C_f$ at the tricritical point when the system is driven from the gapless Ising critical phase by decreasing $h_{\mu}$. The solid line in (a) shows the power law $C_f\propto R^{\eta_f/r_\mu}$ fitted to the $L=60$ data in large $R$ region. The rescaled curves in (b) collapse onto each other.
• Figure 4: Scaling behavior of the half-chain entanglement entropy $S$ at the tricritical point when the system is driven from the gapless Ising critical phase by decreasing $h_{\mu}$. The solid line in (a) shows the logarithmic law $S-(c_I/3)\propto [(c_I-c_T)/3r_\mu] \log R$ fitted to the $L=60$ data in large $R$ region. The rescaled curves in (b) collapse onto each other.
• Figure 5: Scaling behavior of $M_2$ at the tricritical point for increasing $h_{\mu}$. The solid line in (a) shows the logarithmic law $M_2\propto R^{\eta_b/r_\mu}$ fitted to the $L=40$ data in large $R$ region. The rescaled curves in (b) collapse onto each other.