Nonequilibrium characterization of equilibrium correlated quantum phases

Long Zhang, Lin Zhang, Ying Hu, Sen Niu, and Xiong-Jun Liu

Phys. Rev. B 103, 224308 – Published 21 June 2021

Abstract

Quenching a quantum system involves three basic ingredients: the initial phase, the postquench target phase, and quantum dynamics, which may carry the information of the former two. Here we propose a dynamical theory, based on an interaction quench, to characterize both the equilibrium symmetry-breaking order and topological phases by nonequilibrium correlated quantum dynamics. We illustrate the theory with the Haldane-Hubbard model, which is quenched from an initial correlated magnetic phase to a topologically nontrivial regime. We show that the quench dynamics exhibit profound universal behaviors on the so-called band-inversion surfaces (BISs), from which both the topological phase in the weakly interacting regime and the correlated magnetic phase in the strongly interacting regime can be extracted. In particular, the topology is characterized by dynamical topological patterns emerging on BISs, which are robust against interaction-induced dephasing and heating; the symmetry-breaking order can be read out from a universal dynamical scaling behavior, which is valid beyond the mean-field theory. This work uncovers the first paradigm of nonequilibrium characterization of equilibrium symmetry-breaking and topological phases.

Received 29 March 2019
Revised 5 June 2021
Accepted 8 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.224308

Physics Subject Headings (PhySH)

Cold gases in optical lattices Synthetic gauge fields Topological phases of matter Ultracold collisions

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Long Zhang^1,2, Lin Zhang^1,2, Ying Hu^3,4, Sen Niu^1,2, and Xiong-Jun Liu ^1,2,5,*

¹International Center for Quantum Materials and School of Physics, Peking University, Beijing 100871, China
²Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
³State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China
⁴Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
⁵Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China

^*Corresponding author: xiongjunliu@pku.edu.cn

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Vol. 103, Iss. 22 — 1 June 2021

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Images

Figure 1
Interaction quench and pseudospin dynamics. (a) The system undergoes a transition from an AF phase to a topologically nontrivial phase by quenching the interaction from $U ≫ t_{1}$ to $U < t_{1}$ . (b) The first Brillouin zone (hexagon) with the reciprocal-lattice vectors $b_{1} = \frac{2 π}{3 a_{0}} (\sqrt{3}, 1)$ and $b_{2} = \frac{4 π}{3 a_{0}} (0, 1)$ ( $a_{0}$ is the lattice constant). The dashed purple line denotes the band-inversion surface of the spin-up component. (c) The pseudospin polarization $〈 τ_{z}^{σ} 〉$ oscillates after the quench for each spin $σ = ↑ ↓$ . Three points in the Brillouin zone in (b) are taken for example. Here $t_{2} = 0.3 t_{1}, M = - 0.5 t_{1}, m_{C} = 0.5 t_{1}, m_{AF} = 4 t_{1}$ , and $U = 0.3 t_{1}$ after quench.
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Figure 2
Pseudospin dynamics from the equation of motion. Time evolution of pseudospin vectors for (a) spin up and (b) spin down. Damping and heating effects modify the precession, but to different degrees in different regions with respect to the BIS. (c) The calculated distribution of damping factors $η_{1}^{σ}$ and heating factors $η_{2}^{σ}$ . The dashed purple lines denote the noninteracting BISs for each spin. Here we take $t_{2} = 0.3 t_{1}, M = - 0.5 t_{1}, m_{C} = 0.5 t_{1}, m_{AF} = 4 t_{1}$ , and the interaction $U = 0.3 t_{1}$ after quench.
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Figure 3
Emergent topology of quench dynamics. (a) Time-averaged pseudospin textures $\bar{〈 τ_{x, y, z}^{↑} (k) 〉}$ with (b) the corresponding projected dynamical field $g_{∥}^{↑} (k)$ . (c) Time-averaged pseudospin textures $\bar{〈 τ_{x, y, z}^{↓} (k) 〉}$ with (d) the corresponding projected dynamical field $g_{∥}^{↓} (k)$ . The dashed lines denote the interacting BISs. The constructed dynamical field on the BIS for either spin characterizes the topology with Chern number $C = 1$ . Here we take $t_{2} = 0.3 t_{1}, M = - 0.5 t_{1}, m_{C} = 0.5 t_{1}$ , and $m_{AF} = 4 t_{1}$ , and the postquench interaction $U = 0.3 t_{1}$ . The time average is taken over 5 times of the oscillation period for each $k$ .
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Figure 4
Characterizing symmetry-breaking order. (a) The momenta taken for measurement. Three points lie on the BIS, with $k_{1, 2}^{=}$ being in line $L_{1}, (k_{x}, k_{y}) = b_{1} + b_{2}$ , and $k_{3}^{=}$ being in line $L_{2}, k_{x} = \frac{4 π}{9 a_{0}} \sqrt{3}$ . One is chosen inside the BIS with $k_{1}^{<} = \frac{3}{5} (b_{1} + b_{2})$ , and one is outside the BIS with $k_{1}^{>} = \frac{2}{3} b_{1} + \frac{1}{3} b_{2}$ . (b) Both the half oscillation amplitude $Z_{0}$ and the oscillation period $T_{0}$ are measured for different magnetizations $m_{↑} / t_{1} = - {2, 2.5, 3, 3.5, 4}$ . (c) The numerical results are shown as $(| m_{↑} T_{0} |, \sqrt{1 - Z_{0}^{2}})$ . The data taken on the BIS all satisfy the function $f (x) = π / x$ (dashed blue curve), which verifies the relation of Eq. (4). Here we take $t_{2} = 0.3 t_{1}, M = - 0.5 t_{1}$ , and the postquench interaction $U = 0.3 t_{1}$ .
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