Global phase diagram of a spin–orbit-coupled Kondo lattice model on the honeycomb lattice
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摘要
Motivated by the growing interest in the novel quantum phases in materials with strong electron correlations and spin–orbit coupling, we study the interplay among the spin–orbit coupling, Kondo interaction, and magnetic frustration of a Kondo lattice model on a two-dimensional honeycomb lattice.We calculate the renormalized electronic structure and correlation functions at the saddle point based on a fermionic representation of the spin operators.We find a global phase diagram of the model at half-filling, which contains a variety of phases due to the competing interactions.In addition to a Kondo insulator, there is a topological insulator with valence bond solid correlations in the spin sector, and two antiferromagnetic phases.Due to the competition between the spin–orbit coupling and Kondo interaction, the direction of the magnetic moments in the antiferromagnetic phases can be either within or perpendicular to the lattice plane.The latter antiferromagnetic state is topologically nontrivial for moderate and strong spin–orbit couplings.
Motivated by the growing interest in the novel quantum phases in materials with strong electron correlations and spin–orbit coupling, we study the interplay among the spin–orbit coupling, Kondo interaction, and magnetic frustration of a Kondo lattice model on a two-dimensional honeycomb lattice.We calculate the renormalized electronic structure and correlation functions at the saddle point based on a fermionic representation of the spin operators.We find a global phase diagram of the model at half-filling, which contains a variety of phases due to the competing interactions.In addition to a Kondo insulator, there is a topological insulator with valence bond solid correlations in the spin sector, and two antiferromagnetic phases.Due to the competition between the spin–orbit coupling and Kondo interaction, the direction of the magnetic moments in the antiferromagnetic phases can be either within or perpendicular to the lattice plane.The latter antiferromagnetic state is topologically nontrivial for moderate and strong spin–orbit couplings.
Motivated by the growing interest in the novel quantum phases in materials with strong electron correlations and spin–orbit coupling, we study the interplay among the spin–orbit coupling, Kondo interaction, and magnetic frustration of a Kondo lattice model on a two-dimensional honeycomb lattice.We calculate the renormalized electronic structure and correlation functions at the saddle point based on a fermionic representation of the spin operators.We find a global phase diagram of the model at half-filling, which contains a variety of phases due to the competing interactions.In addition to a Kondo insulator, there is a topological insulator with valence bond solid correlations in the spin sector, and two antiferromagnetic phases.Due to the competition between the spin–orbit coupling and Kondo interaction, the direction of the magnetic moments in the antiferromagnetic phases can be either within or perpendicular to the lattice plane.The latter antiferromagnetic state is topologically nontrivial for moderate and strong spin–orbit couplings.
引文
[1]L?hneysen H 2010 J.Low Temp.Phys.161 1
[2]Sachdev S 1999 Quantum Phase Transitions(New York:Cambridge University Press)
[3]Si Q and Steglich F 2010 Science 329 1161
[4]Gegenwart P, Si Q and Steglich F 2008 Nat.Phys.4 186
[5]L?hneysen H von, Rosch A, Vojta M and Wolfle P 2007 Rev.Mod.Phys.79 1015
[6]Tsunetsugu H, Sigrist M and Ueda K 1997 Rev.Mod.Phys.69 809
[7]Yang Y F and Yu L 2015 Acta Phys.Sin.64 217401(in Chinese)
[8]Hewson A C 1993 The Kondo Problem to Heavy Fermions(Cambridge:Cambridge University Press)
[9]Doniach S 1977 Physica B+C 91 231
[10]Custers J, Gegenwart P, Wilhelm H, Neumaier K, Tokiwa Y, Trovarelli O, Geibel C, Steglich F, Pépin C and Coleman P 2003 Nature 424 524
[11]Schr?der A, Aeppli G, Coldea R, Adams M, Stockert O, L?hneysen H v, Bucher E, Ramazashvili R and Coleman P 2000 Nature 407 351
[12]Paschen S, Luhmann T, Wirth S, Gegenwart P, Trovarelli O, Geibel C,Steglich F, Coleman P and Si Q 2004 Nature 432 881
[13]Si Q, Rabello S, Ingersent K and Smith J L 2001 Nature 413 804
[14]Coleman P, Pépin C, Si Q and Ramazashvili R 2001 J.Phys.:Condens.Matter 13 R723
[15]Hertz J A 1976 Phys.Rev.B 14 1165
[16]Millis A J 1993 Phys.Rev.B 48 7183
[17]Si Q 2006 Physica B 378 23
[18]Si Q 2010 Phys.Stat.Solid.B 247 476
[19]Pixley J H, Yu R and Si Q 2014 Phys.Rev.Lett.113 176402
[20]Si Q and Paschen S 2013 Phys.Stat.Solid.(b)250 425
[21]Mong R S K, Essin A M and Moore J E 2010 Phys.Rev.B 81 245209
[22]Nakatsuji S, Machida Y, Maeno Y, Tayama T, Sakakibara T, Duijn J v,Balicas L, Millican J N, Macaluso R T and Chan J Y 2006 Phys.Rev.Lett.96 087204
[23]Chen G 2017 Phys.Rev.B 94 205107
[24]Dzero M, Sun K, Galitski V and Coleman P 2010 Phys.Rev.Lett.104106408
[25]Barla A, Derr J, Sanchez J P, Salce B, Lapertot G, Doyle B P, Rüffer R, Lengsdorf R, Abd-Elmeguid M M and Flouquet J 2005 Phys.Rev.Lett.94 166401
[26]Yamamoto S J and Si Q 2010 J.Low Temp.Phys.161 233
[27]Lai H H, Grefe S E, Paschen S and Si Q 2018 Pro.Natl.Acad.Sci.USA115 93
[28]Dzsaber S, Prochaska L, Sidorenko A, Eguchi G, Svagera R, Waas M,Prokofiev A, Si Q and Paschen S 2017 Phys.Rev.Lett.118 246601
[29]Dzsaber S, Yan X, Eguchi G, Prokofiev A, Shiroka T, Blaha P, Rubel O, Grefe S E, Lai H H, Si Q and Paschen S 2018 arXiv:1811.02819
[30]Feng X Y, Chung C H, Dai J and Si Q 2013 Phys.Rev.Lett.111 016402
[31]Kane C L and Mele E J 2005 Phys.Rev.Lett.95 226801
[32]Feng X Y, Zhong H, Dai J and Si Q 2016 arXiv:1605.02380
[33]Haldane F D M 1988 Phys.Rev.Lett.61 2015
[34]Zhong Y, Wang Y F, Wang Y Q and Luo H G 2013 Phys.Rev.B 87035128
[35]Lacroix C and Cyrot M 1979 Phys.Rev.B 20 1969
[36]Li H, Liu Y, Zhang G M and Yu L 2015 J.Phys.:Condens.Matter 27425601
[37]Li H, Song H F and Liu Y 2016 Euro.Phys.Lett.116 37005
[38]Ganesh R, Brink J van den and Nishimoto S 2013 Phys.Rev.Lett.110127203
[39]Clark B K, Abanin D A and Sondhi S L 2011 Phys.Rev.Lett.107087204
[40]Hasan M Z and Kane C L 2010 Rev.Mod.Phys.82 3045
[41]Qi X L and Zhang S C 2011 Rev.Mod.Phys.83 1057
[42]Pixley J H, Yu R, Paschen S and Si Q 2018 Phys.Rev.B 98 085110
[43]Zhou Y, Wu Q, Rosa P F S, Yu R, Guo J, Yi W, Zhang S, Wang Z, Wang H, Cai S, Yang K, Li A, Jiang Z, Zhang S, Wei X, Huang Y, Yang Y F,Fisk Z, Si Q, Sun L and Zhao Z 2017 Sci.Bull.62 1439
[44]Kasaya M, Tani T, Iga F and Kasuya T 1988 J.Magn.Magn.Mater.76&77 278
[45]Malik S K, Adroja D T, Dhar S K, Vijayaraghavan R and Padalia B D1989 Phys.Rev.B 40 2414
[46]Adroja D T and Rainford B D 1994 Physica B 194–196 363
[47]Hu J, Alicea J, Wu R, and Franz M 2012 Phys.Rev.Lett.109 266801
[2]Sachdev S 1999 Quantum Phase Transitions(New York:Cambridge University Press)
[3]Si Q and Steglich F 2010 Science 329 1161
[4]Gegenwart P, Si Q and Steglich F 2008 Nat.Phys.4 186
[5]L?hneysen H von, Rosch A, Vojta M and Wolfle P 2007 Rev.Mod.Phys.79 1015
[6]Tsunetsugu H, Sigrist M and Ueda K 1997 Rev.Mod.Phys.69 809
[7]Yang Y F and Yu L 2015 Acta Phys.Sin.64 217401(in Chinese)
[8]Hewson A C 1993 The Kondo Problem to Heavy Fermions(Cambridge:Cambridge University Press)
[9]Doniach S 1977 Physica B+C 91 231
[10]Custers J, Gegenwart P, Wilhelm H, Neumaier K, Tokiwa Y, Trovarelli O, Geibel C, Steglich F, Pépin C and Coleman P 2003 Nature 424 524
[11]Schr?der A, Aeppli G, Coldea R, Adams M, Stockert O, L?hneysen H v, Bucher E, Ramazashvili R and Coleman P 2000 Nature 407 351
[12]Paschen S, Luhmann T, Wirth S, Gegenwart P, Trovarelli O, Geibel C,Steglich F, Coleman P and Si Q 2004 Nature 432 881
[13]Si Q, Rabello S, Ingersent K and Smith J L 2001 Nature 413 804
[14]Coleman P, Pépin C, Si Q and Ramazashvili R 2001 J.Phys.:Condens.Matter 13 R723
[15]Hertz J A 1976 Phys.Rev.B 14 1165
[16]Millis A J 1993 Phys.Rev.B 48 7183
[17]Si Q 2006 Physica B 378 23
[18]Si Q 2010 Phys.Stat.Solid.B 247 476
[19]Pixley J H, Yu R and Si Q 2014 Phys.Rev.Lett.113 176402
[20]Si Q and Paschen S 2013 Phys.Stat.Solid.(b)250 425
[21]Mong R S K, Essin A M and Moore J E 2010 Phys.Rev.B 81 245209
[22]Nakatsuji S, Machida Y, Maeno Y, Tayama T, Sakakibara T, Duijn J v,Balicas L, Millican J N, Macaluso R T and Chan J Y 2006 Phys.Rev.Lett.96 087204
[23]Chen G 2017 Phys.Rev.B 94 205107
[24]Dzero M, Sun K, Galitski V and Coleman P 2010 Phys.Rev.Lett.104106408
[25]Barla A, Derr J, Sanchez J P, Salce B, Lapertot G, Doyle B P, Rüffer R, Lengsdorf R, Abd-Elmeguid M M and Flouquet J 2005 Phys.Rev.Lett.94 166401
[26]Yamamoto S J and Si Q 2010 J.Low Temp.Phys.161 233
[27]Lai H H, Grefe S E, Paschen S and Si Q 2018 Pro.Natl.Acad.Sci.USA115 93
[28]Dzsaber S, Prochaska L, Sidorenko A, Eguchi G, Svagera R, Waas M,Prokofiev A, Si Q and Paschen S 2017 Phys.Rev.Lett.118 246601
[29]Dzsaber S, Yan X, Eguchi G, Prokofiev A, Shiroka T, Blaha P, Rubel O, Grefe S E, Lai H H, Si Q and Paschen S 2018 arXiv:1811.02819
[30]Feng X Y, Chung C H, Dai J and Si Q 2013 Phys.Rev.Lett.111 016402
[31]Kane C L and Mele E J 2005 Phys.Rev.Lett.95 226801
[32]Feng X Y, Zhong H, Dai J and Si Q 2016 arXiv:1605.02380
[33]Haldane F D M 1988 Phys.Rev.Lett.61 2015
[34]Zhong Y, Wang Y F, Wang Y Q and Luo H G 2013 Phys.Rev.B 87035128
[35]Lacroix C and Cyrot M 1979 Phys.Rev.B 20 1969
[36]Li H, Liu Y, Zhang G M and Yu L 2015 J.Phys.:Condens.Matter 27425601
[37]Li H, Song H F and Liu Y 2016 Euro.Phys.Lett.116 37005
[38]Ganesh R, Brink J van den and Nishimoto S 2013 Phys.Rev.Lett.110127203
[39]Clark B K, Abanin D A and Sondhi S L 2011 Phys.Rev.Lett.107087204
[40]Hasan M Z and Kane C L 2010 Rev.Mod.Phys.82 3045
[41]Qi X L and Zhang S C 2011 Rev.Mod.Phys.83 1057
[42]Pixley J H, Yu R, Paschen S and Si Q 2018 Phys.Rev.B 98 085110
[43]Zhou Y, Wu Q, Rosa P F S, Yu R, Guo J, Yi W, Zhang S, Wang Z, Wang H, Cai S, Yang K, Li A, Jiang Z, Zhang S, Wei X, Huang Y, Yang Y F,Fisk Z, Si Q, Sun L and Zhao Z 2017 Sci.Bull.62 1439
[44]Kasaya M, Tani T, Iga F and Kasuya T 1988 J.Magn.Magn.Mater.76&77 278
[45]Malik S K, Adroja D T, Dhar S K, Vijayaraghavan R and Padalia B D1989 Phys.Rev.B 40 2414
[46]Adroja D T and Rainford B D 1994 Physica B 194–196 363
[47]Hu J, Alicea J, Wu R, and Franz M 2012 Phys.Rev.Lett.109 266801