Exciton condensation in bilayer spin-orbit insulator

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Exciton condensation in bilayer spin-orbit insulator

Hidemaro Suwa, Shang-Shun Zhang, and Cristian D. Batista

Phys. Rev. Research 3, 013224 – Published 9 March 2021

Abstract

We investigate the nature of the magnetic excitations of a bilayer single-orbital Hubbard model in the intermediate-coupling regime. This model exhibits a quantum phase transition (QPT) between a paramagnetic (PM) and an insulating antiferromagnetic (AFM) phase at a critical value of the coupling strength. By using the random phase approximation, we show that the QPT is continuous when the PM state is a band insulator and that the corresponding quantum critical point (QCP) arises from the condensation of preformed excitons. These low-energy excitons re-emerge on the other side of the QCP as the transverse and longitudinal modes of the AFM state. In particular, the longitudinal mode remains sharp for the model parameters relevant to ${Sr}_{3} {Ir}_{2} O_{7}$ because of the strong easy-axis anisotropy of this material.

Received 5 October 2020
Revised 26 January 2021
Accepted 11 February 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013224

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)

Excitons Quantum phase transitions Spin dynamics Spin-orbit coupling

Higgs bosons Iridates Mott insulators

Hubbard model Random phase approximation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Hidemaro Suwa ^1,2, Shang-Shun Zhang², and Cristian D. Batista ^2,3

¹Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
²Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
³Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

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Issue

Vol. 3, Iss. 1 — March - May 2021

Subject Areas

Condensed Matter Physics

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Images

Figure 1
(a) Schematic picture for the band structure of noninteracting metal and band insulator. (b) Phase boundaries in the mean field approximation for $α = 0, 10^{- 4}, 10^{- 3}, 10^{- 2}, 10^{- 1}$ , 0.5, and 1.45 (from top to bottom). For $α = 0$ and $U = 0$ , the system is a metal for $| t_{z} / t | < 4$ and a band insulator for $| t_{z} / t | > 4$ . The metal becomes an AFM insulator for an infinitesimally small $U$ . For $α \neq 0$ , the present model (1) exhibits a continuous phase transition from the band insulator to the AFM insulator at a finite $U / t$ . The solid circle symbol indicates the estimated parameter set for the bilayer iridate ${Sr}_{3} {Ir}_{2} O_{7}$ with $α = 1.45$ . (c) Energy diagram as a function of $U$ for the parameter set relevant to ${Sr}_{3} {Ir}_{2} O_{7}$ . The red dashed line indicates the energy of the $S^{z} = \pm 1$ exciton that becomes the transverse mode for $U > U_{c}$ . The blue dotted line indicates the energy of the $S^{z} = 0$ exciton that becomes a longitudinal mode for $U > U_{c}$ . The gray area indicates the particle-hole continuum. From a comparison with RIXS data [24, 25, 26, 27], we estimate that $U = 0.33$ eV for ${Sr}_{3} {Ir}_{2} O_{7}$ .
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Figure 2
Dynamical spin structure factors of the out-of-phase mode: (a) in-plane or transverse $S^{x x} (q, ω) = S^{y y} (q, ω)$ component and (b) out-of-plane or longitudinal component $S^{z z} (q, ω)$ for $U = 0.33$ eV; (c) $S^{x x} (q, ω) = S^{y y} (q, ω)$ and (d) $S^{z z} (q, ω)$ for $U = U_{c} \approx 0.28$ eV; and (e) $S^{x x} (q, ω) = S^{y y} (q, ω)$ and (f) $S^{z z} (q, ω)$ for $U = 0.23$ eV. The hopping parameters were chosen to reproduce the RIXS data of the bilayer iridate ${Sr}_{3} {Ir}_{2} O_{7}$ [24, 25, 26, 27]: $t = 0.11$ eV, $t_{z} = 0.09$ eV, $α = 1.45$ , and the intralayer next nearest neighbor hopping 0.012 eV. The broadening factor is $η = 10^{- 4}$ eV. Intensity larger than the maximum value in the scale (color) bar is plotted in the same color.
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Figure 3
(a) Free fermion propagator in the background of the magnetic order: $- G_{0} (q)$ , where $q \equiv (q, i ω_{n})$ . (b) Bare interaction vertex $- \frac{1}{N_{u} β} V_{σ_{1} σ_{4}; σ_{2} σ_{3}}$ . (c) Bare and (d) RPA spin-charge susceptibility.
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Figure 4
The particle-hole Green's function and the $T$ matrix under the RPA.
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Figure 5
Probability distribution $| ψ_{↓ ↓} (0, r_{2}) |^{2}$ for several strengths of the on-site coupling $U$ . A hole with spin $↓$ is assumed located at $r_{1} = 0$ on layer 1. The critical coupling is $U_{c} \approx 0.277$ eV for the hopping parameters relevant to ${Sr}_{3} {Ir}_{2} O_{7}$ , while the on-site Hubbard interaction is estimated to be $U = 0.33$ eV $(= 1.19 U_{c})$ for the material. $M$ is the magnitude of the local ordered magnetic moment. The distribution function $| ψ_{↓ ↓} (0, r_{2}) |^{2}$ is normalized for each value of $U$ , so that the sum over $r_{2}$ is equal to one.
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