Multipartite entanglement at dynamical quantum phase transitions with nonuniformly spaced criticalities

Stav Haldar, Saptarshi Roy, Titas Chanda, Aditi Sen(De), and Ujjwal Sen

Phys. Rev. B 101, 224304 – Published 12 June 2020

Abstract

We report a dynamical quantum phase transition portrait in the alternating field transverse XY spin chain with Dzyaloshinskii-Moriya interaction by investigating singularities in the Loschmidt echo and the corresponding rate function after a sudden quench of system parameters. Unlike the Ising model, the analysis of the Loschmidt echo yields nonuniformly spaced transition times in this model. Comparative study between the equilibrium and the dynamical quantum phase transitions in this case reveals that there are quenches where one occurs without the other and reveals the regimes where they coexist. However, such transitions happen only when quenching is performed across at least a single gapless or critical line. Contrary to equilibrium phase transitions, bipartite entanglement measures do not turn out to be useful for the detection, while multipartite entanglement emerges as a good identifier of this transition when the quench is done from a disordered phase of this model.

Received 28 March 2019
Revised 10 May 2020
Accepted 11 May 2020

DOI:https://doi.org/10.1103/PhysRevB.101.224304

Physics Subject Headings (PhySH)

Dynamical phase transitions Fermions Phase diagrams Quantum entanglement Quantum phase transitions

1-dimensional spin chains

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Stav Haldar^1,2, Saptarshi Roy¹, Titas Chanda ^3,*, Aditi Sen(De)¹, and Ujjwal Sen¹

¹Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India
²Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
³Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland

^*titas.chanda@uj.edu.pl

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Vol. 101, Iss. 22 — 1 June 2020

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Images

Figure 1
Phase diagrams of the DATXY model in different parameter spaces. Here, we choose $γ = 0.8$ . Unless otherwise stated, we use $γ = 0.8$ throughout the paper for demonstration purposes. All quantities plotted are dimensionless.
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Figure 2
Rate function $F (t)$ vs $t / t_{1}^{*}$ . The dynamics occurs due to quenches from the point $g_{0} = {1.5, 0, 0}$ , which lies in the PM-I phase, to three different points, namely, (i) $g_{1} = {0, 0.2, 0}$ (AFM phase), (ii) $g_{1} = {- 0.5, 1.5, 0}$ (PM-II phase), and (iii) $g_{1} = {0.4, 0.2, 1}$ (CH phase), in the parameter space of the DATXY model. All the quenches are across EQPT lines. Clearly, in all three cases, $F (t)$ becomes nonanalytic for different critical times $t^{*}$ . Here, we have normalized the time axis by the first critical time $t_{1}^{*}$ in all three cases to highlight the fact that, in the DATXY model, the critical times are not uniformly spaced with each other. All quantities plotted are dimensionless.
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Figure 3
DQPT regions in the parameter space of the DATXY model for the quenches from the points marked by $\times$ . When a quantum quench is performed from a point, marked by the symbol $\times$ , to any other point on the parameter space, only the shaded region shows nonanalyticity in the rate function given in Eq. (12). All the axes are dimensionless.
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Figure 4
Time variation of $G$ after various quenches within and across equilibrium phases. We observe distinctly high amounts of fluctuations during the transient period of dynamics for specific regions (corresponding to DQPT, see Fig. 3) of disorder to order (PM-I to AFM and PM-I to CH) quenches and disorder to disorder (PM-I to PM-II) quenches, irrespective of the size of quench. Apart from these, other quenches show relatively lower amounts of fluctuations. Both the axes are dimensionless.
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Figure 5
Variation of the time-averaged standard deviation $〈 σ_{G} (t) 〉$ , for quenches in the ( $λ_{1}, λ_{2}, d$ ) space with initial choices indicated by $\times$ as in Fig. 3. After the quench, we observe the emergence of a specific region in the ( $λ_{1}, λ_{2}$ ) space in panel (a) and in the ( $λ_{1}, d$ ) space in panel (b), which are distinctly characterized by high fluctuations, i.e., higher values of $〈 σ_{G} (t) 〉$ . This region possess a high overlap with that depicted in Figs. 3, 3, and 3, which correspond to DQPTs, obtained by analyzing nonanalyticities in the rate function. This substantial overlap establishes multipartite entanglement as a good detector of DQPT. All the axes are dimensionless.
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Figure 6
The GGM phase portrait of the ground state of the DATXY model for (a) $d = 0.5$ and (b) $d = 1.2$ , with respect to the uniform and alternate fields $λ_{1}$ and $λ_{2}$ . The anisotropy parameter $γ$ is fixed to 0.8. All axes are dimensionless.
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