ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Physics 19 May 2023

Electronic correlation effects on stabilizing a perfect Kagome lattice and ferromagnetic fluctuation in LaRu3Si2

Cite this:
https://doi.org/10.52396/JUSTC-2022-0182
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  • Author Bio:

    Yilin Wang is currently a Research Professor at School of Emerging Technology, University of Science and Technology of China. He received his Ph.D. degree from Institute of Physics, CAS in 2016. His research mainly focuses on first-principle calculations of strongly correlated electronic materials using DFT plus dynamical mean-filed theory

  • Corresponding author: E-mail: yilinwang@ustc.edu.cn
  • Received Date: 11 December 2022
  • Accepted Date: 20 March 2023
  • Available Online: 19 May 2023
  • A perfect Kagome lattice features flat bands that usually lead to strong electronic correlation effects, but how electronic correlation, in turn, stabilizes a perfect Kagome lattice has rarely been explored. Here, we study this effect in a superconducting ($T_{\rm{c}} \sim 7.8$ K) Kagome metal LaRu3Si2 with a distorted Kagome plane consisting of pure Ru ions, using density functional theory plus $ U $ and plus dynamical mean-field theory. We find that increasing electronic correlation can stabilize a perfect Kagome lattice and induce substantial ferromagnetic fluctuations in LaRu3Si2. By comparing the calculated magnetic susceptibilities to experimental data, LaRu3Si2 is found to be on the verge of becoming a perfect Kagome lattice. It thus shows moderate but non-negligible electronic correlations and ferromagnetic fluctuations, which are crucial to understand the experimentally observed non-Fermi-liquid behavior and the pretty high superconducting $T_{\rm{c}}$ of LaRu3Si2.
    Strong electronic correlations tends to stabilize a perfect Kagome lattice in superconducting Kagome metal LaRu3Si2.
    A perfect Kagome lattice features flat bands that usually lead to strong electronic correlation effects, but how electronic correlation, in turn, stabilizes a perfect Kagome lattice has rarely been explored. Here, we study this effect in a superconducting ($T_{\rm{c}} \sim 7.8$ K) Kagome metal LaRu3Si2 with a distorted Kagome plane consisting of pure Ru ions, using density functional theory plus $ U $ and plus dynamical mean-field theory. We find that increasing electronic correlation can stabilize a perfect Kagome lattice and induce substantial ferromagnetic fluctuations in LaRu3Si2. By comparing the calculated magnetic susceptibilities to experimental data, LaRu3Si2 is found to be on the verge of becoming a perfect Kagome lattice. It thus shows moderate but non-negligible electronic correlations and ferromagnetic fluctuations, which are crucial to understand the experimentally observed non-Fermi-liquid behavior and the pretty high superconducting $T_{\rm{c}}$ of LaRu3Si2.
    • Electronic correlation effects are found to stabilize a perfect Kagome lattice in superconducting Kagome metal LaRu3Si2.
    • LaRu3Si2 is found to be on the verge of becoming a perfect Kagome lattice.
    • Electronic correlations and ferromagnetic fluctuations are found to be crucial to understand the non-Fermi-liquid behavior and the high superconducting Tc in LaRu3Si2.

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    Mielke A. Ferromagnetic ground states for the Hubbard model on line graphs. Journal of Physics A: Mathematical and General, 1991, 24: L73. doi: 10.1088/0305-4470/24/2/005
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    Tasaki H. Ferromagnetism in the Hubbard models with degenerate single-electron ground states. Physical Review Letters, 1992, 69: 1608. doi: 10.1103/PhysRevLett.69.1608
    [4]
    Sachdev S. Kagomé- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons. Physical Review B, 1992, 45: 12377. doi: 10.1103/PhysRevB.45.12377
    [5]
    Tang E, Mei J W, Wen X G. High-temperature fractional quantum Hall states. Physical Review Letters, 2011, 106: 236802. doi: 10.1103/PhysRevLett.106.236802
    [6]
    Cao Y, Fatemi V, Fang S, et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature, 2018, 556: 43–50. doi: 10.1038/nature26160
    [7]
    Balents L, Dean C R, Efetov D K, et al. Superconductivity and strong correlations in moiré flat bands. Nature Physics, 2020, 16: 725–733. doi: 10.1038/s41567-020-0906-9
    [8]
    Aoki H. Theoretical possibilities for flat band superconductivity. Journal of Superconductivity and Novel Magnetism, 2020, 33: 2341–2346. doi: 10.1007/s10948-020-05474-6
    [9]
    Heikkilä T T, Volovik G E. Flat bands as a route to high-temperature superconductivity in graphite. In: Esquinazi P, editor. Basic Physics of Functionalized Graphite. Cham: Springer International Publishing, 2016: 123–143.
    [10]
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    [12]
    Jiang Y X, Yin J X, Denner M M, et al. Unconventional chiral charge order in kagome superconductor KV3Sb5. Nature Materials, 2021, 20: 1353–1357. doi: 10.1038/s41563-021-01034-y
    [13]
    Teng X, Chen L, Ye F, et al. Discovery of charge density wave in a kagome lattice antiferromagnet. Nature, 2022, 609: 490–495. doi: 10.1038/s41586-022-05034-z
    [14]
    Kang M, Ye L, Fang S, et al. Dirac fermions and flat bands in the ideal kagome metal FeSn. Nature Materials, 2020, 19: 163–169. doi: 10.1038/s41563-019-0531-0
    [15]
    Lin Z, Wang C, Wang P, et al. Dirac fermions in antiferromagnetic FeSn kagome lattices with combined space inversion and time-reversal symmetry. Physical Review B, 2020, 102: 155103. doi: 10.1103/PhysRevB.102.155103
    [16]
    Huang L, Lu H. Signatures of hundness in kagome metals. Physical Review B, 2020, 102: 125130. doi: 10.1103/PhysRevB.102.125130
    [17]
    Ye L, Kang M, Liu J, et al. Massive Dirac fermions in a ferromagnetic kagome metal. Nature, 2018, 555: 638–642. doi: 10.1038/nature25987
    [18]
    Lin Z, Choi J H, Zhang Q, et al. Flatbands and emergent ferromagnetic ordering in Fe3Sn2 kagome lattices. Physical Review Letters, 2018, 121: 096401. doi: 10.1103/PhysRevLett.121.096401
    [19]
    Yin J X, Zhang S S, Li H, et al. Giant and anisotropic many-body spin-orbit tunability in a strongly correlated kagome magnet. Nature, 2018, 562: 91–95. doi: 10.1038/s41586-018-0502-7
    [20]
    Kang M, Fang S, Ye L, et al. Topological flat bands in frustrated kagome lattice CoSn. Nature Communications, 2020, 11: 4004. doi: 10.1038/s41467-020-17465-1
    [21]
    Liu Z, Li M, Wang Q, et al. Orbital-selective Dirac fermions and extremely flat bands in frustrated kagome-lattice metal CoSn. Nature Communications, 2020, 11: 4002. doi: 10.1038/s41467-020-17462-4
    [22]
    Yin J X, Shumiya N, Mardanya S, et al. Fermion-boson many-body interplay in a frustrated kagome paramagnet. Nature Communications, 2020, 11: 4003. doi: 10.1038/s41467-020-17464-2
    [23]
    Meier W R, Du M H, Okamoto S, et al. Flat bands in the CoSn-type compounds. Physical Review B, 2020, 102: 075148. doi: 10.1103/PhysRevB.102.075148
    [24]
    Huang H, Zheng L, Lin Z, et al. Flat-band-induced anomalous anisotropic charge transport and orbital magnetism in kagome metal CoSn. Physical Review Letters, 2022, 128: 096601. doi: 10.1103/PhysRevLett.128.096601
    [25]
    Yin J X, Ma W, Cochran T A, et al. Quantum-limit Chern topological magnetism in TbMn6Sn6. Nature, 2020, 583: 533–536. doi: 10.1038/s41586-020-2482-7
    [26]
    Yang T Y, Wan Q, Song J P, et al. Fermi-level flat band in a kagome magnet. Quantum Frontiers, 2022, 1: 14. doi: 10.1007/s44214-022-00017-7
    [27]
    Vandenberg J M, Barz H. The crystal structure of a new ternary silicide in the system rare-earth-ruthenium-silicon. Materials Research Bulletin, 1980, 15 (10): 1493–1498. doi: 10.1016/0025-5408(80)90108-7
    [28]
    Li B, Li S, Wen H H. Chemical doping effect in the LaRu3Si2 superconductor with a kagome lattice. Physical Review B, 2016, 94: 094523. doi: 10.1103/PhysRevB.94.094523
    [29]
    Li S, Zeng B, Wan X, et al. Anomalous properties in the normal and superconducting states of LaRu3Si2. Physical Review B, 2011, 84: 214527. doi: 10.1103/PhysRevB.84.214527
    [30]
    Mielke C, Qin Y, Yin J X, et al. Nodeless kagome superconductivity in LaRu3Si2. Physical Review Materials, 2021, 5: 034803. doi: 10.1103/PhysRevMaterials.5.034803
    [31]
    Gong C, Tian S, Tu Z, et al. Superconductivity in kagome metal YRu3Si2 with strong electron correlations. Chinese Physics Letters, 2022, 39: 087401. doi: 10.1088/0256-307X/39/8/087401
    [32]
    Anisimov V I, Zaanen J, Andersen O K. Band theory and Mott insulators: Hubbard U instead of Stoner I. Physical Review B, 1991, 44: 943. doi: 10.1103/PhysRevB.44.943
    [33]
    Georges A, Kotliar G, Krauth W, et al. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Reviews of Modern Physics, 1996, 68: 13. doi: 10.1103/RevModPhys.68.13
    [34]
    Lichtenstein A I, Katsnelson M I, Kotliar G. Finite-temperature magnetism of transition metals: An ab initio dynamical mean-field theory. Physical Review Letters, 2001, 87: 067205. doi: 10.1103/PhysRevLett.87.067205
    [35]
    Kotliar G, Savrasov S Y, Haule K, et al. Electronic structure calculations with dynamical mean-field theory. Reviews of Modern Physics, 2006, 78: 865. doi: 10.1103/RevModPhys.78.865
    [36]
    Momma K, Izumi F. VESTA: A three-dimensional visualization system for electronic and structural analysis. Journal of Applied Crystallography, 2008, 41: 653–658. doi: 10.1107/S0021889808012016
    [37]
    Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B, 1996, 54: 11169. doi: 10.1103/PhysRevB.54.11169
    [38]
    Blöchl P E. Projector augmented-wave method. Physical Review B, 1994, 50: 17953. doi: 10.1103/PhysRevB.50.17953
    [39]
    Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Europace, 1996, 77: 3865. doi: 10.1103/PhysRevLett.77.3865
    [40]
    Liechtenstein A I, Anisimov V I, Zaanen J. Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators. Physical Review B, 1995, 52: R5467. doi: 10.1103/PhysRevB.52.R5467
    [41]
    Haule K, Yee C H, Kim K. Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn5, CeCoIn5, and CeRhIn5. Physical Review B, 2010, 81: 195107. doi: 10.1103/PhysRevB.81.195107
    [42]
    Haule K, Birol T. Free energy from stationary implementation of the DFT+DMFT functional. Physical Review Letters, 2015, 115: 256402. doi: 10.1103/PhysRevLett.115.256402
    [43]
    Blaha P, Schwarz K, Tran F, et al. WIEN2k: An APW+lo program for calculating the properties of solids. The Journal of Chemical Physics, 2020, 152 (7): 074101. doi: 10.1063/1.5143061
    [44]
    Gull E, Millis A J, Lichtenstein A I, et al. Continuous-time Monte Carlo methods for quantum impurity models. Reviews of Modern Physics, 2011, 83: 349. doi: 10.1103/RevModPhys.83.349
    [45]
    Haule K. Exact double counting in combining the dynamical mean field theory and the density functional theory. Physical Review Letters, 2015, 115: 196403. doi: 10.1103/PhysRevLett.115.196403
    [46]
    Haule K, Pascut G L. Forces for structural optimizations in correlated materials within a DFT+embedded DMFT functional approach. Physical Review B, 2016, 94: 195146. doi: 10.1103/PhysRevB.94.195146
    [47]
    Haule K. DFT+embedded DMFT Functional. [2022-10-11] http://hauleweb.rutgers.edu/tutorials/index.html.
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    Figure  1.  Three possible crystal structures of LaRu3Si2 with a Kagome plane consisting of pure Ru atoms. (a), (c) Perfect Kagome structure. (b), (d), and (e) Two possible distorted Kagome structures with a doubling of the $ c $-axis. (c)–(e) Top view of the Kagome planes. The angles $\alpha, \;\beta$ and side lengths $l_1,\; l_2$ of the hexagon, the fractional coordinates of Ru atoms $x,\; y$, and the space group are shown, respectively. The crystal structures are constructed by VESTA[36].

    Figure  2.  Electronic correlation effects on stabilizing a perfect Kagome lattice. (a), (d) Fractional coordinates x of Ru (Fe) relaxed by LDA+U and GGA+U as functions of Hubbard U, for different lattice parameter ratio c/a with the fixed crystal volume of experiment. c = 7.12 Å is the experimental value. x = 0.5 for a perfect Kagome lattice. (b), (e) x relaxed by LDA+DMFT as functions of U. The yellow area mark that LaRu3Si2 is on the verge of becoming a perfect Kagome lattice. (c), (f) LDA+DMFT calculated mass-enhancement of Ru-4d (Fe-3d) orbitals due to electronic correlations, as functions of U. (a)–(c) for LaRu3Si2 and (d)–(f) for LaFe3Si2.

    Figure  3.  Electronic correlation induced magnetism. (a), (d) Local magnetic susceptibilities calculated by LDA+DMFT as functions of temperate $ T $ for different $ U $ values. (b), (e) The energies (per unit cell) of FM and AFM orders with respect to the non-magnetic state as functions of $ U $, calculated by GGA+$ U $. (c), (f) Ordered magnetic moment in the FM and AFM states. (a)–(c) for LaRu3Si2, the crystal structures relaxed at the corresponding $ U $ are used. (d)–(f) for LaFe3Si2. The experimental lattice parameters, $ a=5.676 $Å and $ c=7.12 $Å, are used for both compounds.

    Figure  4.  Flat band near Fermi level. (a), (d) LDA calculated band structures. The flat bands (FB) with $ d_{x^2-y^2} $ character are shown in red. (b), (e) LDA+DMFT calculated spectrum function $ A(k, \omega) $ at $ U=4 $ eV, $J_{\rm{H}}=0.782$ eV and $ T=290 $ K. (c), (f) The corresponding density of states. (a)–(c) for LaRu3Si2, the crystal structure relaxed by LDA+DMFT at $ U=4 $ eV, $J_{\rm{H}}=0.782$ eV is used. (d)–(f) for LaFe3Si2. The experimental lattice parameters, $ a=5.676 $Å and $ c=7.12 $Å, are used for both compounds.

    [1]
    Syôzi I. Statistics of Kagomé lattice. Progress of Theoretical Physics, 1951, 6 (3): 306–308. doi: 10.1143/ptp/6.3.306
    [2]
    Mielke A. Ferromagnetic ground states for the Hubbard model on line graphs. Journal of Physics A: Mathematical and General, 1991, 24: L73. doi: 10.1088/0305-4470/24/2/005
    [3]
    Tasaki H. Ferromagnetism in the Hubbard models with degenerate single-electron ground states. Physical Review Letters, 1992, 69: 1608. doi: 10.1103/PhysRevLett.69.1608
    [4]
    Sachdev S. Kagomé- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons. Physical Review B, 1992, 45: 12377. doi: 10.1103/PhysRevB.45.12377
    [5]
    Tang E, Mei J W, Wen X G. High-temperature fractional quantum Hall states. Physical Review Letters, 2011, 106: 236802. doi: 10.1103/PhysRevLett.106.236802
    [6]
    Cao Y, Fatemi V, Fang S, et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature, 2018, 556: 43–50. doi: 10.1038/nature26160
    [7]
    Balents L, Dean C R, Efetov D K, et al. Superconductivity and strong correlations in moiré flat bands. Nature Physics, 2020, 16: 725–733. doi: 10.1038/s41567-020-0906-9
    [8]
    Aoki H. Theoretical possibilities for flat band superconductivity. Journal of Superconductivity and Novel Magnetism, 2020, 33: 2341–2346. doi: 10.1007/s10948-020-05474-6
    [9]
    Heikkilä T T, Volovik G E. Flat bands as a route to high-temperature superconductivity in graphite. In: Esquinazi P, editor. Basic Physics of Functionalized Graphite. Cham: Springer International Publishing, 2016: 123–143.
    [10]
    Jiang K, Wu T, Yin J X, et al. Kagome superconductors AV3Sb5 (A = K, Rb, Cs). National Science Review, 2022, 10 (2): nwac199. doi: 10.1093/nsr/nwac199
    [11]
    Nie L, Sun K, Ma W, et al. Charge-density-wave-driven electronic nematicity in a kagome superconductor. Nature, 2022, 604: 59–64. doi: 10.1038/s41586-022-04493-8
    [12]
    Jiang Y X, Yin J X, Denner M M, et al. Unconventional chiral charge order in kagome superconductor KV3Sb5. Nature Materials, 2021, 20: 1353–1357. doi: 10.1038/s41563-021-01034-y
    [13]
    Teng X, Chen L, Ye F, et al. Discovery of charge density wave in a kagome lattice antiferromagnet. Nature, 2022, 609: 490–495. doi: 10.1038/s41586-022-05034-z
    [14]
    Kang M, Ye L, Fang S, et al. Dirac fermions and flat bands in the ideal kagome metal FeSn. Nature Materials, 2020, 19: 163–169. doi: 10.1038/s41563-019-0531-0
    [15]
    Lin Z, Wang C, Wang P, et al. Dirac fermions in antiferromagnetic FeSn kagome lattices with combined space inversion and time-reversal symmetry. Physical Review B, 2020, 102: 155103. doi: 10.1103/PhysRevB.102.155103
    [16]
    Huang L, Lu H. Signatures of hundness in kagome metals. Physical Review B, 2020, 102: 125130. doi: 10.1103/PhysRevB.102.125130
    [17]
    Ye L, Kang M, Liu J, et al. Massive Dirac fermions in a ferromagnetic kagome metal. Nature, 2018, 555: 638–642. doi: 10.1038/nature25987
    [18]
    Lin Z, Choi J H, Zhang Q, et al. Flatbands and emergent ferromagnetic ordering in Fe3Sn2 kagome lattices. Physical Review Letters, 2018, 121: 096401. doi: 10.1103/PhysRevLett.121.096401
    [19]
    Yin J X, Zhang S S, Li H, et al. Giant and anisotropic many-body spin-orbit tunability in a strongly correlated kagome magnet. Nature, 2018, 562: 91–95. doi: 10.1038/s41586-018-0502-7
    [20]
    Kang M, Fang S, Ye L, et al. Topological flat bands in frustrated kagome lattice CoSn. Nature Communications, 2020, 11: 4004. doi: 10.1038/s41467-020-17465-1
    [21]
    Liu Z, Li M, Wang Q, et al. Orbital-selective Dirac fermions and extremely flat bands in frustrated kagome-lattice metal CoSn. Nature Communications, 2020, 11: 4002. doi: 10.1038/s41467-020-17462-4
    [22]
    Yin J X, Shumiya N, Mardanya S, et al. Fermion-boson many-body interplay in a frustrated kagome paramagnet. Nature Communications, 2020, 11: 4003. doi: 10.1038/s41467-020-17464-2
    [23]
    Meier W R, Du M H, Okamoto S, et al. Flat bands in the CoSn-type compounds. Physical Review B, 2020, 102: 075148. doi: 10.1103/PhysRevB.102.075148
    [24]
    Huang H, Zheng L, Lin Z, et al. Flat-band-induced anomalous anisotropic charge transport and orbital magnetism in kagome metal CoSn. Physical Review Letters, 2022, 128: 096601. doi: 10.1103/PhysRevLett.128.096601
    [25]
    Yin J X, Ma W, Cochran T A, et al. Quantum-limit Chern topological magnetism in TbMn6Sn6. Nature, 2020, 583: 533–536. doi: 10.1038/s41586-020-2482-7
    [26]
    Yang T Y, Wan Q, Song J P, et al. Fermi-level flat band in a kagome magnet. Quantum Frontiers, 2022, 1: 14. doi: 10.1007/s44214-022-00017-7
    [27]
    Vandenberg J M, Barz H. The crystal structure of a new ternary silicide in the system rare-earth-ruthenium-silicon. Materials Research Bulletin, 1980, 15 (10): 1493–1498. doi: 10.1016/0025-5408(80)90108-7
    [28]
    Li B, Li S, Wen H H. Chemical doping effect in the LaRu3Si2 superconductor with a kagome lattice. Physical Review B, 2016, 94: 094523. doi: 10.1103/PhysRevB.94.094523
    [29]
    Li S, Zeng B, Wan X, et al. Anomalous properties in the normal and superconducting states of LaRu3Si2. Physical Review B, 2011, 84: 214527. doi: 10.1103/PhysRevB.84.214527
    [30]
    Mielke C, Qin Y, Yin J X, et al. Nodeless kagome superconductivity in LaRu3Si2. Physical Review Materials, 2021, 5: 034803. doi: 10.1103/PhysRevMaterials.5.034803
    [31]
    Gong C, Tian S, Tu Z, et al. Superconductivity in kagome metal YRu3Si2 with strong electron correlations. Chinese Physics Letters, 2022, 39: 087401. doi: 10.1088/0256-307X/39/8/087401
    [32]
    Anisimov V I, Zaanen J, Andersen O K. Band theory and Mott insulators: Hubbard U instead of Stoner I. Physical Review B, 1991, 44: 943. doi: 10.1103/PhysRevB.44.943
    [33]
    Georges A, Kotliar G, Krauth W, et al. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Reviews of Modern Physics, 1996, 68: 13. doi: 10.1103/RevModPhys.68.13
    [34]
    Lichtenstein A I, Katsnelson M I, Kotliar G. Finite-temperature magnetism of transition metals: An ab initio dynamical mean-field theory. Physical Review Letters, 2001, 87: 067205. doi: 10.1103/PhysRevLett.87.067205
    [35]
    Kotliar G, Savrasov S Y, Haule K, et al. Electronic structure calculations with dynamical mean-field theory. Reviews of Modern Physics, 2006, 78: 865. doi: 10.1103/RevModPhys.78.865
    [36]
    Momma K, Izumi F. VESTA: A three-dimensional visualization system for electronic and structural analysis. Journal of Applied Crystallography, 2008, 41: 653–658. doi: 10.1107/S0021889808012016
    [37]
    Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B, 1996, 54: 11169. doi: 10.1103/PhysRevB.54.11169
    [38]
    Blöchl P E. Projector augmented-wave method. Physical Review B, 1994, 50: 17953. doi: 10.1103/PhysRevB.50.17953
    [39]
    Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Europace, 1996, 77: 3865. doi: 10.1103/PhysRevLett.77.3865
    [40]
    Liechtenstein A I, Anisimov V I, Zaanen J. Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators. Physical Review B, 1995, 52: R5467. doi: 10.1103/PhysRevB.52.R5467
    [41]
    Haule K, Yee C H, Kim K. Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn5, CeCoIn5, and CeRhIn5. Physical Review B, 2010, 81: 195107. doi: 10.1103/PhysRevB.81.195107
    [42]
    Haule K, Birol T. Free energy from stationary implementation of the DFT+DMFT functional. Physical Review Letters, 2015, 115: 256402. doi: 10.1103/PhysRevLett.115.256402
    [43]
    Blaha P, Schwarz K, Tran F, et al. WIEN2k: An APW+lo program for calculating the properties of solids. The Journal of Chemical Physics, 2020, 152 (7): 074101. doi: 10.1063/1.5143061
    [44]
    Gull E, Millis A J, Lichtenstein A I, et al. Continuous-time Monte Carlo methods for quantum impurity models. Reviews of Modern Physics, 2011, 83: 349. doi: 10.1103/RevModPhys.83.349
    [45]
    Haule K. Exact double counting in combining the dynamical mean field theory and the density functional theory. Physical Review Letters, 2015, 115: 196403. doi: 10.1103/PhysRevLett.115.196403
    [46]
    Haule K, Pascut G L. Forces for structural optimizations in correlated materials within a DFT+embedded DMFT functional approach. Physical Review B, 2016, 94: 195146. doi: 10.1103/PhysRevB.94.195146
    [47]
    Haule K. DFT+embedded DMFT Functional. [2022-10-11] http://hauleweb.rutgers.edu/tutorials/index.html.

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