Dynamical quantum phase transitions in systems with broken continuous time and space translation symmetries

Arkadiusz Kosior, Andrzej Syrwid, and Krzysztof Sacha

Phys. Rev. A 98, 023612 – Published 13 August 2018

Abstract

Spontaneous breaking of continuous time translation symmetry into a discrete one is related to time crystal formation. While the phenomenon is not possible in the ground state of a time-independent many-body system, it can occur in an excited eigenstate. Here, we concentrate on bosons on a ring with attractive contact interactions and analyze a quantum quench from the time crystal regime to the noninteracting regime. We show that dynamical quantum phase transitions can be observed where the return probability of the system to the initial state before the quench reveals a nonanalytical behavior in time. The problem we consider constitutes an example of the dynamical quantum phase transitions in a system where both time and space continuous translation symmetries are broken.

Received 19 June 2018

DOI:https://doi.org/10.1103/PhysRevA.98.023612

Physics Subject Headings (PhySH)

Cold atoms & matter waves Dynamical phase transitions Solitons Spontaneous symmetry breaking

Atomic, Molecular & Optical

Authors & Affiliations

Arkadiusz Kosior¹, Andrzej Syrwid¹, and Krzysztof Sacha^1,2

¹Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland
²Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland

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Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
The rate function of the Loschmidt echo for two values of the initial interaction strength $g_{0} N = - 10.5$ (red solid) and $g_{0} N = - 11$ (black solid). The rate function can be approximated by $λ (t) \approx min [λ_{+} (t), λ_{-} (t)]$ , where $λ_{\pm} (t)$ (red and black circles) are auxiliary functions, Eq. (21). Since the rate $λ (t)$ is given by a minimum of $λ_{\pm} (t)$ , it is not differentiable at their crossing at the critical moment of time $t_{c} = 1 / 4 π$ , indicated by vertical dotted line, which can be associate with a dynamical quantum phase transition. All quantities are dimensionless.
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Figure 2
Increase of the purity of the many-body state at the critical time $t_{c}$ . Top: The probability density $p (r, t) = {2 π r | ψ (r, t) |}^{2}$ at three different moments of time $t = 0.8 t_{c}$ (dashed black), $t = 0.94 t_{c}$ (dotted blue), and $t = t_{c}$ (solid red) for $N = 10^{4}$ . The interference fringes visible in the plot corresponding to $t = t_{c}$ result in the decrease of uncertainty of the number of particles depleted from the condensate, hence, the increase of the purity of the many-body state. Bottom: The normalized inverse participation ratio (IPR) near the critical time $t_{c} = 1 / 4 π$ , indicated by vertical dotted line, for three different total numbers of particles $N = 10^{3}$ (dashed black), $N = 5 \times 10^{3}$ (dotted blue) and $N = 2 \times 10^{4}$ (solid red). The change of IPR becomes more rapid with increasing $N$ . All quantities are dimensionless.
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Figure 3
The rate function $λ^{(N)} (t)$ for different initial values of interparticle attractions far from the critical point, i.e., from left top to right bottom $g_{0} (N - 1) = - 13, - 15, - 20, - 25$ . In each panel we compare the results obtained for different number of particles $N$ . Note that the rates $λ^{(N)} (t)$ can change quite quickly with $N$ in the vicinity of the anticipated critical time $t_{c} = 1 / 4 π$ that is marked with vertical dotted line. The rates tend to diverge for some specific values of $N$ , which has been also observed in other systems [39, 85]. All quantities are dimensionless.
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Figure 4
The plots of the rate function $λ_{SB} (t)$ , its first derivate ${\dot{λ}}_{SB} (t)$ , and the absolute value of the second $| {\ddot{λ}}_{SB} (t) |$ for two interactions strengths $g_{0} N = - 13$ and $g_{0} N = - 25$ (left and right column, respectively). The singularity at critical moments of time are distinguished with a cusp of $λ_{SB} (t)$ , discontinuity of $\dot{λ} (t)$ and a delta-kick of the second derivate ${\ddot{λ}}_{SB} (t)$ . For $g_{0} N = - 13$ , we obtain a single critical time $t_{c} = 1 / 4 π$ , while for $g_{0} N = - 25$ we observe a cascade of singular points, see the discussion in the main text. The presented mean-field results correspond to $N \to \infty$ . Many-body results for the system prepared in the symmetry-preserving state but for a finite $N$ are shown in Fig. 3. All quantities are dimensionless.
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Figure 5
Singular points of the absolute value of the second derivative of the rate function (25), i.e., $| {\ddot{λ}}_{SB} (t) |$ , in the space spanned by the interaction strength $g_{0} N$ and time $t$ . All quantities are dimensionless.
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