Exploring dynamical quantum phase transition from pure states to mixed states through extended Su-Schrieffer-Heeger models

TL;DR

This work extends the study of dynamical quantum phase transitions (DQPTs) to multi-band SSH models, focusing on SSH-3 (with a point chiral symmetry) and SSH-4 (with genuine chiral symmetry). Using the generalized Loschmidt amplitude and rate function, along with the Pancharatnam geometric phase, the authors characterize how symmetry and temperature shape DQPTs under quenches that cross or avoid topological transitions. In SSH-3, pure-state DQPTs appear after crossing the topological point, and mixed-state DQPTs resemble pure-state behavior at low temperature but exhibit multiple critical times and dynamical vortices at high temperature. In SSH-4, pure-state DQPTs require starting from a gapless band and crossing the critical point, with mixed-state DQPTs persisting at low temperature only if the corresponding pure-state quench does, and vanishing at high temperature. Overall, the results illuminate how symmetry and finite temperature govern non-equilibrium topological dynamics in extended SSH models.

Abstract

We investigate dynamical quantum phase transitions (DQPTs) in both pure and mixed states within the extended SSH model framework, focusing on the SSH-3 and SSH-4 variants, which differ in symmetry properties. The SSH-3 model, characterized by a chiral-like point symmetry rather than true chiral symmetry, supports robust localized edge states tied to its topological nature. Our results show that for pure states, DQPTs occur after quenches crossing the topological transition, even when the energy band gap remains open. For mixed states, DQPT behavior aligns with pure states at low temperatures but undergoes significant changes at higher temperatures, including the emergence of multiple critical times. In contrast, the SSH-4 model, which possesses chiral symmetry, features four distinct energy spectrum configurations. We find that pure-state DQPTs arise only when the quench starts from a gapless initial state and crosses the critical topological point. At finite temperature, mixed-state DQPTs persist at low temperatures only if the corresponding pure-state quench induces DQPTs, but they disappear at elevated temperatures. These findings elucidate the interplay between symmetry, topology, and temperature in governing DQPTs within generalized SSH models.

Figures (13)

• Figure 1: Energy spectra as a function of the intercell coupling $t_{3}$ in the SSH-3 model for (a) with $t_{1} = t_{2} = 1$ (inversion-symmetric case), and for (b) with $t_{1} = 1, t_{2} = 1.2$ (inversion-symmetric broken case). The solid lines denote the bulk states, and the dotted lines represent the edge states. In (a), the energy gaps close at point $t_{3} = t_{1} = t_{2} = 1$.
• Figure 2: (a)-(c) The Pancharatnam geometric phases in the SSH-3 model with inversion symmetry from different initial states ($|u_{k1}^{i}\rangle$, $|u_{k2}^{i}\rangle$, and $|u_{k3}^{i}\rangle$ ) are shown in momentum-time space, respectively. The quench paths are from $t_{3}^{i}=0.2$ to $t_{3}^{f}=1.1$ with fixed $t_{1}=t_{2}=1$. The dynamical vortices marked by in white circles exhibit the critical wave vectors and times. (d)-(f) The corresponding transition probabilities $|p_{k\nu}|^{2}$ for different initial states.
• Figure 3: The rate functions in the SSH-3 models with inversion symmetry for the quenches from different initial states, where the quenches in (a) are from $t_{3}^{i} = 0.2$ to $t_{3}^{f} = 0.8$, and in (b) are from $t_{3}^{i} = 0.2$ to $t_{3}^{f} = 1.1$.
• Figure 4: The rate functions in the SSH-3 models with broken inversion symmetry for the quenches from different initial states, where the quench in (a) is from $t_{3}^{i} = 0.2$ to $t_{3}^{f} = 1.1$ and in (b) is from $t_{3}^{i} = 0.2$ to $t_{3}^{f} = 1.3$.
• Figure 5: The value of $\mathrm{max}(|p_{k2}|^{2}) - \mathrm{min}(|p_{k1}|^{2})$ in the SSH-3 model with broken inversion symmetry for quenches from the initial state $|u_{k1}^{i}\rangle$. Here, we set the intracell coupling $t_{1} = 1$ and the initial intercell coupling $t_{3}^{i} = 0.2$. The critical line of the couplings $t_{2}$ and $t_{3}^{f}$ is denoted by the line $\mathrm{max}(|p_{k2}|^{2}) - \mathrm{min}(|p_{k1}|^{2}) = 0$.