Loschmidt-amplitude wave function spectroscopy and the physics of dynamically driven phase transitions

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Loschmidt-amplitude wave function spectroscopy and the physics of dynamically driven phase transitions

D. M. Kennes, C. Karrasch, and A. J. Millis

Phys. Rev. B 101, 081106(R) – Published 14 February 2020

Abstract

We introduce the Loschmidt amplitude as a powerful tool to perform spectroscopy of generic many-body wave functions and use it to interrogate the wave function obtained after ramping the transverse field quantum Ising model through its quantum critical point. Previous results are confirmed and a more complete understanding of the population of defects and of the effects of magnon-magnon interaction or finite-size corrections is obtained. The influence of quantum coherence is clarified.

Received 4 September 2018
Accepted 3 February 2020

DOI:https://doi.org/10.1103/PhysRevB.101.081106

Physics Subject Headings (PhySH)

Critical phenomena Dynamical phase transitions Magnetic phase transitions Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics Optical lattices & traps Quantum statistical mechanics

Density matrix renormalization group Matrix product states Spin lattice models Tensor network methods

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

D. M. Kennes ^1,2, C. Karrasch³, and A. J. Millis^4,5

¹Institut für Theorie der Statistischen Physik, RWTH Aachen University and JARA-Fundamentals of Future Information Technology, 52056 Aachen, Germany
²Max Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, 22761 Hamburg, Germany
³Technische Universität Braunschweig, Institut für Mathematische Physik, Mendelssohnstrae 3, 38106 Braunschweig, Germany
⁴Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA
⁵Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA

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Issue

Vol. 101, Iss. 8 — 15 February 2020

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Images

Figure 1
$S^{z}$ and $S^{x}$ for a slow linear ramp through the QCP of the quantum Ising chain ( $v / J = 0.02, t_{wait} J \to \infty, g_{end} / J = 2, N \to \infty$ ). The dashed green vertical line indicates the time when $g (t) = g_{c} = 1$ . The black dashed curves show the corresponding ground-state expectation values with respect to the instantaneous Hamiltonian. Inset: Example of a temporal profile of $g (t)$ which governs the quench ( $g_{end} = 2 J, v / J = 0.01, t_{wait} = 3 / v$ ). The QCP at $g = g_{c} = 1$ (dashed black horizontal line) separates the equilibrium ordered phase [red part of $g (t)$ ] from the disordered phase [orange part of $g (t)$ ].
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Figure 2
Wave functions spectrum $L (ω)$ for a single forward ramp with $t_{wait} J \to \infty$ and $N = 100$ . (a) $L (ω)$ for fixed $v = 0.02 J$ and different $g_{end}$ . The evolution is adiabatic for $g_{end} < g_{c} = 1$ ; the spectrum shows a single peak corresponding to the ground-state energy $ω = 0$ (dashed red line). After crossing the QCP, the wave function contains a superposition of eigenstates which are strongly localized in energy around integer multiples of the gap $Δ = 2 | g_{e} n d - J |$ (colored dashed lines). The slice for $g_{end} = g_{c} = J$ is highlighted in red. (b) $L (ω)$ for fixed $v = 0.02 J$ and $g_{end} / J = 1.4$ . Dashed vertical lines indicate $ω = 2 | g_{e} n d - J | M$ ; yellow and green horizontal bars show the width of the first and second magnon band, respectively. (c) $L (ω)$ for a ramp that stops at the QCP $g_{end} / J = 1.0$ at different $v$ . The dashed line shows a $\sim t^{- 8}$ power law.
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Figure 3
(a) The distribution $P (M)$ of the number of states at excitation energy $2 M (g - J)$ fitted to the pseudobinomial distribution $P (M) = Y! / (Y - M)! / M! p^{M} {(1 - p)}^{Y - M}$ ( $g_{end} / J = 2, J t_{wait} \to \infty$ , different speeds $v$ ). (b) Fitted values of $p$ and $Y$ as functions of ramp speed $v$ for different $g_{end}$ . Filled symbols are for $N = 100$ and open ones for $N = 50$ ( $p$ for the latter not shown, which are on top of the $N = 100$ data). The data approximately collapses on single lines. Dashed lines show a power law $Y \sim v^{1 / 2}$ (with a relative factor of 2 in the prefactor) as well as $p \sim C$ .
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Figure 4
Finite-size dependence of the wave function spectrum: Zoom-in of (a) the one-magnon as well as (b) the two- magnon energy regime. Solid vertical lines show the exact position of the lowest one- or two-magnon states. Dashed vertical lines in (b) indicate twice the one-magnon line of (a). (c) Time evolution of the spin expectation value $S^{z}$ . Wave function spectroscopy allows for an analytic estimate for the beat frequency (see [21]) and for $N = 50$ from (b) predicts that the first knot of the quantum beat should lie in the yellow shaded region (in agreement with the numerical data). The other parameters are as in Fig. 2.
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Figure 5
Results for the double ramp. (a) Dynamics of $S^{z} (t)$ for a finite $v t_{wait} = 0.4$ (and $v = 0.02 J, g_{end} = 2 J, N \to \infty$ ). After the second ramp is complete, $S^{z} (t)$ reaches a steady-state value. (b) The asymptotic value of $S^{z} (t \to \infty)$ for different waiting times $t_{wait}$ . The other parameters are as in (a). (c) Wave function spectrum $L (ω)$ after the second ramp for two $v t_{wait} = 0.0$ (red) and $v t_{wait} = 0.4$ (yellow). The other parameters are the same as in (a), but choosing $N = 100$ . Shaded regions show fits to binomial distributions as in Fig. 3. The inset shows the fitted $Y p$ in dependency of the waiting time (including more $t_{wait}$ than shown in the main panel), which is the average number of defects.
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