Probing an excited-state quantum phase transition in a quantum many-body system via an out-of-time-order correlator

Abstract

As a measure of information scrambling and quantum chaos, out-of-time-ordered correlator (OTOC) plays more and more important role in many different fields of physics. In this work, we verify that the OTOC can also be used as a prober of the excited-state quantum phase transition (ESQPT) in a quantum many body system. By using the exact diagonalization method, we examine the dynamical properties of OTOC in the Lipkin model, which undergoes an ESQPT. We demonstrate that the OTOC exhibits a remarkable distinct evolution behaviors in different phases of ESQPT. Therefore, the presence of an ESQPT in the quantum many body system can be clearly signaled by the different dynamical behaviors of the OTOC. In particular, we show that the steady state value of the OTOC serves as the order parameter of the ESQPT. Our results highlight the connections between the OTOC and ESQPT, which enable one to use OTOC for experimental tests ESQPTs in quantum many body systems.

Out-of-time-order correlator in coupled harmonic oscillators

“…The numerical result C T (τ ) of (5.7) at each fixed T is fit by an exponential function αe Lτ , see figure 9. We find 22 the exponent L as a function of T , see figure 10. Then finally we numerically fit this set of L's by a power function of T , γT c with constants c and γ, as shown in figure 10.…”

“…21 We discretize the variable τ by units of 0.01: τ = 0.01m (m = 0, 1, • • • , mMax = 120), and the temperature T by units of 0.1: T = 0.1n (n = 0, 1, • • • , 50). 22 Note that this L is different from what we call the quantum Lyapunov exponent by the time-rescaling factor a. However, since our target is just the power of T in the quantum Lyapunov exponent, we don't need to worry about the overall rescaling factor.…”

Does Scrambling Equal Chaos?

“…In the classical limit, its Hamiltonian H = 1 2 (q 2 + p 2 ) + x + γzq (A1) describes an SU (2) (pseudo)-spin (x, y, z) and a harmonic oscillator (p, q), interacting with a coupling constant γ > 0. A quantum phase transition occurs at γ = 1, and the super-radiant γ > 1 phase is characterized by a degenerate pair of ground states, and a saddle at (q = p = 0, x = 1) with H = −1, associated with an excited state transition [59]. The saddle has a single unstable exponent ω 1 = √ γ − 1, which turns out to be larger than the Lyapunov exponent of a typical classical trajectory with comparable energy, see Fig.…”

Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator

“…Various kinds of OTOCs in quantum maps were also studied[20][21][22][23]. The cases with large N are found in[24][25][26][27][28][29][30][31][32][33][34][35].…”