Long-range orders and spin/orbital freezing in the two-band Hubbard model

Karim Steiner, Shintaro Hoshino, Yusuke Nomura, and Philipp Werner

Phys. Rev. B 94, 075107 – Published 3 August 2016

Abstract

We solve the orbitally degenerate two-band Hubbard model within dynamical mean field theory and map out the instabilities to various symmetry-broken phases based on an analysis of the corresponding lattice susceptibilities. Phase diagrams as a function of the Hund coupling parameter $J$ are obtained both for the model with rotationally invariant interaction and for the model with Ising-type anisotropy. For negative $J$ , an intraorbital spin-singlet superconducting phase appears at low temperatures, while the normal state properties are characterized by an orbital-freezing phenomenon. This is the $negative- J$ analog of the recently discovered fluctuating-moment induced $s -wave$ spin-triplet superconductivity in the spin-freezing regime of multiorbital models with $J > 0$ .

Received 20 May 2016
Revised 16 July 2016

DOI:https://doi.org/10.1103/PhysRevB.94.075107

Physics Subject Headings (PhySH)

Antiferromagnetism Superconductivity

Dynamical mean field theory Extended Hubbard model Hubbard model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Karim Steiner¹, Shintaro Hoshino², Yusuke Nomura³, and Philipp Werner¹

¹Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland
²Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan
³Centre de Physique Théorique, École Polytechnique, Centre National de la Recherche Scientifique, Université Paris-Saclay, F-91128 Palaiseau, France

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Issue

Vol. 94, Iss. 7 — 15 August 2016

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Images

Figure 1
Phase diagrams of the half-filled system in the space of $J$ and $T$ for fixed $U = 2$ (left panels) and $U = 4$ (right panels). Both the results for Ising anisotropy (top panels) and the rotationally invariant interaction (bottom panels) are shown. The following phases are found: paired Mott insulator (PM), intraorbital spin-singlet pairing (SC), metal phase (M), interorbital spin-triplet pairing (SC'), conventional Mott insulator (MI), and double paired state (DP). Antiferromagnetic order (AFM), anti-ferro-orbital order (AOO), and charge order (CO) appear in the hashed regions of the phase diagram. The solid metal-insulator boundaries have been obtained by starting from a metallic initial solution and thus correspond to $J_{c 2}$ . On the $J < 0$ side, we indicate $J_{c 1}$ (stability region of the insulator) by a dashed line with empty circles. Dashed lines emanating from the end points of the metal-insulator transition lines indicate a metal-insulator crossover. The thin panels below the phase diagrams show the $J$ dependence of the local spin and orbital fluctuations for $β = 75$ (see text).
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Figure 2
Inverse susceptibilities for different pairings, AFM, AOO, and CO as a function of $J$ for $U = 2$ and $β = 50$ . The top panel shows the result for the model with Ising anisotropy, and the bottom panel shows the result for the model with rotationally invariant Hund coupling. Negative values of inverse susceptibilities indicate a possible symmetry breaking. Also plotted is the estimate $- β G (β / 2)$ for the density of states at the Fermi energy.
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Figure 3
Top panels: Phase diagrams for filling $n_{tot} = 1.5$ and density-density interactions. The left panel shows the result for $U = 2$ and the right panel shows the result for $U = 4$ . Solid and dotted red lines indicate the maxima of the local spin fluctuations, while solid blue lines indicate the maxima of the orbital fluctuations. In the absence of long-range order, the solid lines mark the crossover from a Fermi-liquid metal to a spin-frozen $(J > 0)$ and orbital-frozen $(J < 0)$ bad metal, respectively. The green hashed region corresponds to ferro-orbital order (FOO) and the yellow hashed region corresponds to antiferromagnetic order (AFM). Bottom panels: Long-time values of the spin-spin and orbital-orbital correlators, $〈 S_{z} (0) S_{z} (β / 2) 〉$ and $\frac{1}{4} 〈 (n_{1} - n_{2}) (0) (n_{1} - n_{2}) (β / 2) 〉$ , for indicated temperatures.
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Figure 4
Orbital freezing in the Ising anisotropic model with $U = 2$ . Left panel: Local orbital fluctuations measured by the correlation function (8) for indicated values of the inverse temperature. The maxima of these curves define the orbital-freezing crossover line displayed in Fig. 3. Right panel: Ratio of two orbital correlation functions measured at inverse temperature $β = 30$ and 60, indicating a crossover from Fermi-liquid metal to orbitally frozen metal with increasing negative $J$ .
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