Nonanalyticity of circuit complexity across topological phase transitions

Zijian Xiong, Dao-Xin Yao, and Zhongbo Yan

Phys. Rev. B 101, 174305 – Published 7 May 2020

Abstract

The presence of nonanalyticity in observables is a manifestation of phase transitions. Through the study of two paradigmatic topological models in one and two dimensions, in this work we show that the circuit complexity based on our specific quantification can reveal the occurrence of topological phase transitions, both in and out of equilibrium, by the presence of nonanalyticity. By quenching the system out of equilibrium, we find that the circuit complexity grows linearly or quadratically in the short-time regime if the quench is finished instantaneously or in a finite time, respectively. Notably, we find that for both the sudden quench and the finite-time quench, a topological phase transition in the pre-quench Hamiltonian will be manifested by the presence of nonanalyticity in the first-order or second-order derivative of circuit complexity with respect to time in the short-time regime, and a topological phase transition in the post-quench Hamiltonian will be manifested by the presence of nonanalyticity in the steady value of circuit complexity in the long-time regime. We also show that the increase of dimension does not remove, but only weakens the nonanalyticity of circuit complexity. Our findings can be tested in quantum simulators and cold-atom systems.

Received 28 June 2019
Revised 7 March 2020
Accepted 20 April 2020

DOI:https://doi.org/10.1103/PhysRevB.101.174305

Physics Subject Headings (PhySH)

Dynamical phase transitions Geometric & topological phases Topological phase transition

General Physics

Authors & Affiliations

Zijian Xiong, Dao-Xin Yao ^*, and Zhongbo Yan ^†

State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

^*yaodaox@mail.sysu.edu.cn
^†yanzhb5@mail.sysu.edu.cn

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Vol. 101, Iss. 17 — 1 May 2020

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Figure 1
(a) Phase diagram. $ν$ denotes the winding number, and the horizontal ( $μ / t_{1}$ =0.2) and vertical ( $t_{2} / t_{1} = 1.2$ ) red dashed lines are two paths that we choose to vary the parameters. (b) Circuit complexity (in units of system size, below this is implicitly assumed) for several reference states. Parameters are $t_{1} = 1$ , $t_{2} = 1.2$ , and $Δ_{1} = Δ_{2} = 1$ . (c) The derivative of circuit complexity with respect to $μ^{T}$ . Parameters are the same as in (b). $| d C / d μ^{T} |$ turns out to be independent of $μ^{R}$ , and is equal to $γ (μ)$ . (d) The derivative of circuit complexity with respect to $t_{2}^{T}$ . Parameters are $t_{1} = 1$ , $μ = 0.2$ , and $Δ_{1} = Δ_{2} = 1$ . $| d C / d t_{2}^{T} |$ is also found to be independent of $t_{2}^{R}$ and coincide with $γ (t_{2})$ . Throughout this work, vertical pink dashed lines [see (b)–(d)] correspond to the critical points at which topological phase transitions take place.
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Figure 2
Common parameters are $t_{1} = 1$ , $t_{2} = 1.2$ , and $Δ_{1} = Δ_{2} = 1$ . (a) The evolution of post-quench circuit complexity. $μ^{i} = - 0.9$ . (b) Steady value of post-quench circuit complexity shown in (a). (c) Dynamical growth rate. $μ^{f} = 0.3$ is fixed.
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Figure 3
(a) Common parameters are $t_{1} = 1$ , $t_{2} = 1.2$ , and $Δ_{1} = Δ_{2} = 1$ . (a) The evolution of circuit complexity under a finite-time quench. The quench duration time $t_{q}$ is fixed to 1 and $μ^{i} = - 0.9$ . (b) The dependence of steady value on $μ^{f}$ , $t_{q}$ , and $μ^{i}$ keep the same as in (a). (c) $\partial_{t} C$ , the dynamical growth rate of circuit complexity, in the short-time regime, $μ^{i} = - 0.9$ , $μ^{f} = 1.8$ . The slope of $\partial_{t} C$ is found to increase with the quench speed. (d) $\partial_{t}^{2} C$ in the limit $t \to 0^{+}$ . The concrete value of $\partial_{t}^{2} C$ depends on the quench speed, but the presence of nonanalyticity at the critical points does not rely on the quench speed.
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Figure 4
(a) Phase diagram with $n$ denoting the Chern number. The horizontal ( $μ / t_{1} = 0.5$ ) and vertical ( $t_{2} / t_{1} = 1.5$ ) red dashed lines are two paths that we choose to vary the parameters. (b) Circuit complexity for several reference states. Parameters are $t_{1} = 1.0$ and $t_{2} = 1.5$ . (c) The derivative of circuit complexity with respect to $μ^{T}$ . Parameters are the same as in (b). (d) The derivative of circuit complexity with respect to $t_{2}^{T}$ . Parameters are $t_{1} = 1$ and $μ = 0.5$ .
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Figure 5
Common parameters are $t_{1} = 1$ , $t_{2} = 1.5$ . (a) The evolution of post-quench circuit complexity. $μ^{i} = 0.9$ . (b) Steady value of post-quench circuit complexity shown in (a). (c) Dynamical growth rate. $μ^{f} = 0.7$ is fixed.
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Figure 6
$θ_{k} − k$ curves near critical points. Common parameters are $t_{1} = 1$ , $t_{2} = 1.2$ , and $Δ_{1} = Δ_{2} = 1$ . (a) Across the critical point $μ_{c} = - 0.2$ , one can see that $θ_{k}$ has a sudden jump at $k = π$ . (b) Across the critical point $μ_{c} = 1.1$ , $θ_{k}$ has a sudden jump at $k = 2 π / 3$ .
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