Ising model in a light-induced quantized transverse field

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Ising model in a light-induced quantized transverse field

Jonas Rohn, Max Hörmann, Claudiu Genes, and Kai Phillip Schmidt

Phys. Rev. Research 2, 023131 – Published 6 May 2020

Abstract

We investigate the influence of light-matter interactions on correlated quantum matter by studying the paradigmatic Dicke-Ising model. This type of coupling to a confined, spatially delocalized bosonic light mode, such as provided by an optical resonator, resembles a quantized transverse magnetic field of tunable strength. As a consequence, the symmetry-broken magnetic state breaks down for strong enough light-matter interactions to a paramagnetic state. The nonlocal character of the bosonic mode can change the quantum phase transition in a drastic manner, which we analyze quantitatively for the simplest case of the Dicke-Ising chain geometry. The results show a direct transition between a magnetically ordered phase with zero photon density and a magnetically polarized phase with superradiant behavior of the light. Our predictions are equally valid for the dual quantized Ising chain in a conventional transverse magnetic field.

Received 17 March 2020
Revised 8 April 2020
Accepted 13 April 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023131

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Light-matter interaction Magnetism Quantum criticality Quantum states of light

Dicke model Ising model

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

Authors & Affiliations

Jonas Rohn ¹, Max Hörmann¹, Claudiu Genes ^2,1, and Kai Phillip Schmidt¹

¹Department of Physics, FAU Erlangen-Nuremberg, Staudtstraße 7, D-91058 Erlangen, Germany
²Max Planck Institute for the Science of Light, Staudtstraße 2, D-91058 Erlangen, Germany

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Vol. 2, Iss. 2 — May - July 2020

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Figure 1
(a) Ground-state magnetization $M_{z}$ of the pure Ising model and the LFIM as a function of the spin-spin coupling $J$ . The change from the antiferromagnetic to the ferromagnetic ground state is at $J = 0$ ( $J = - h / 4$ ) for the pure Ising model (for the LFIM). (b) Magnetization $M_{z}$ for the TFIM as a function of $λ^{- 1} = h / 2 J$ . For small values of $h / 2 J$ the system is magnetically ordered, i.e., the eigenstate approaches the ferromagnetic state $| ⇑ 〉$ . For large values of $h / 2 J$ the system approaches the polarized state $| \Rightarrow 〉 \equiv | \to \to \to \dots 〉$ . By replacing the magnetization $M_{z}$ by the staggered magnetization $M_{z}^{s}$ and $J$ by $- J$ , the plot is also valid for an antiferromagnetic Ising interaction with $- J > 0$ . In this case the order approaches the state $| ⇓ 〉$ for $- J ≫ h$ .
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Figure 2
Ground-state energy per site $e$ for the limiting case of $N = 1$ where we compare the (i) Rabi model for $ω_{0} \in {0, 1}$ and (ii) Jaynes-Cummings model (JCM). For $g ≪ ω_{c}$ , the JCM is a good approximation of the Rabi model with finite $ω_{0}$ , whereas for large $g ≫ ω_{c}$ the solution to the Rabi model with $ω_{0} = 0$ approaches the ground-state energy of the Rabi model for $ω_{0} / ω_{c} = 1$ .
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Figure 3
Ground-state energy per site $e_{0}$ for $J / ω_{c} = 1$ : Comparison of the analytical result (blue dashed line) and numerical data obtained for $N = 4$ to $N = 10$ spins (solid lines). The quantity $g^{2} / ω_{c}$ serves as an analog of the magnetic field strength of the conventional TFIM. The vertical red line indicates the location of the quantum phase transition separating the magnetically ordered phase for small $g$ from the superradiant phase at large $g$ . The gray dotted vertical line highlights the second-order phase transition of the conventional transverse-field Ising chain [see Eq. (6)].
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Figure 4
Magnetization per spin $M_{z}$ for $J / ω_{c} = 1$ : The analytical result (blue dashed line) is approached by the numerical data for finite system sizes with fixed $ω_{0} = 0.001 ω_{c}$ (solid lines). The red vertical line marks the phase transition from the magnetically ordered phase (small $g, M_{z} = 1 / 2$ ) to the superradiant phase (large $g, M_{z} = 0$ ). The gray dotted vertical line highlights the second-order phase transition of the conventional transverse-field Ising chain [see Eq. (6)] with $h = g^{2} / ω_{c}$ .
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Figure 5
Photon number per spin $n$ : The numerical values (solid lines) indicate the point of the transition already quite well. However, the analytical solution (dashed line) is approached very slowly with increasing $g$ . The gray dotted vertical line highlights the second-order phase transition of the conventional transverse-field Ising chain [see Eq. (6)].
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Figure 6
(a) Required cutoff of the photon number in order to reach an accuracy of $δ = 0.01 ω_{c}$ depending on the number of spins for $J$ and $g$ fixed. (b) Effect of the light-matter coupling constant $g$ on the required maximum photon number for a chain of five spins and $J / ω_{c} = 0$ .
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