Quantum spin liquid emerging in two-dimensional correlated dirac fermions

ABSTRACT At sufficiently low temperatures, condensed-matter systems tend to develop order. A notable exception to this behaviour is the case of quantum spin liquids, in which quantum

fluctuations prevent a transition to an ordered state down to the lowest temperatures. There have now been tentative observations of such states in some two-dimensional organic compounds,

yet quantum spin liquids remain elusive in microscopic two-dimensional models that are relevant to experiments. Here we show, by means of large-scale quantum Monte Carlo simulations of

correlated fermions on a honeycomb lattice (a structure realized in, for example, graphene), that a quantum spin liquid emerges between the state described by massless Dirac fermions and an

antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence-bond liquid, akin to the one proposed for

high-temperature superconductors: the possibility of unconventional superconductivity through doping therefore arises in our system. We foresee the experimental realization of this model

system using ultra-cold atoms, or group IV elements arranged in honeycomb lattices. Access through your institution Buy or subscribe This is a preview of subscription content, access via

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* Learn about institutional subscriptions * Read our FAQs * Contact customer support SIMILAR CONTENT BEING VIEWED BY OTHERS QUANTUM LOOP STATES IN SPIN-ORBITAL MODELS ON THE HONEYCOMB

LATTICE Article Open access 21 May 2021 TRIMER QUANTUM SPIN LIQUID IN A HONEYCOMB ARRAY OF RYDBERG ATOMS Article Open access 14 December 2023 MULTINODE QUANTUM SPIN LIQUIDS IN EXTENDED

KITAEV HONEYCOMB MODELS Article Open access 26 November 2024 REFERENCES * Weinberg, S. _The Quantum Theory of Fields Vol. 1 Foundations_ 1–48 (Cambridge Univ. Press, 2005) MATH  Google

Scholar  * Novoselov, K. et al. Two-dimensional gas of massless Dirac fermions in graphene. _Nature_ 438, 197–200 (2005) Article  ADS  CAS  Google Scholar  * Zhang, Y., Tan, Y.-W., Stormer,

H. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. _Nature_ 438, 201–204 (2005) Article  ADS  CAS  Google Scholar  * Zhang, H. et al.

Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. _Nature Phys._ 5, 438–442 (2009) Article  ADS  CAS  Google Scholar  * Chen, Y. L. et al.

Experimental realization of a three-dimensional topological insulator, Bi2Te3 . _Science_ 325, 178–181 (2009) Article  ADS  CAS  Google Scholar  * Castro Neto, A. H., Guinea, F., Peres, N.

M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. _Rev. Mod. Phys._ 81, 109–162 (2009) Article  ADS  CAS  Google Scholar  * Herbut, I. F. Interactions and

phase transitions on graphene’s honeycomb lattice. _Phys. Rev. Lett._ 97, 146401 (2006) Article  ADS  Google Scholar  * Drut, J. E. & Lähde, T. A. Is graphene in vacuum an insulator?

_Phys. Rev. Lett._ 102, 026802 (2009) Article  ADS  Google Scholar  * Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. _Rev.

Mod. Phys._ 78, 17–85 (2006) Article  ADS  CAS  Google Scholar  * Jördens, R., Strohmaier, N., Günter, K., Moritz, H. & Esslinger, T. A Mott insulator of fermionic atoms in an optical

lattice. _Nature_ 455, 204–207 (2008) Article  ADS  Google Scholar  * Schneider, U. et al. Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice.

_Science_ 322, 1520–1525 (2008) Article  ADS  CAS  Google Scholar  * Lee, S.-S. & Lee, P. A. U(1) gauge theory of the Hubbard model: spin liquid states and possible application to

_κ_-(BEDT-TTF)2Cu2(CN)3 . _Phys. Rev. Lett._ 95, 036403 (2005) Article  ADS  Google Scholar  * Hermele, M. SU(2) gauge theory of the Hubbard model and application to the honeycomb lattice.

_Phys. Rev. B_ 76, 035125 (2007) Article  ADS  Google Scholar  * Raghu, S., Qi, X.-L., Honerkamp, C. & Zhang, S.-C. Topological Mott insulators. _Phys. Rev. Lett._ 100, 156401 (2008)

Article  ADS  CAS  Google Scholar  * Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. _Phys. Rev. Lett._ 95, 226801 (2005) Article  ADS  CAS  Google Scholar  * Uchoa, B.

& Castro Neto, A. H. Superconducting states of pure and doped graphene. _Phys. Rev. Lett._ 98, 146801 (2007) Article  ADS  Google Scholar  * Anderson, P. W. Resonating valence bonds: a

new kind of insulator? _Mater. Res. Bull._ 8, 153–160 (1973) Article  CAS  Google Scholar  * Fazekas, P. & Anderson, P. W. On the ground state properties of the anisotropic triangular

antiferromagnet. _Phil. Mag._ 30, 423–440 (1974) Article  ADS  CAS  Google Scholar  * Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. _Science_ 235,

1196–1198 (1987) Article  ADS  CAS  Google Scholar  * Kivelson, S. A., Rokhsar, D. S. & Sethna, J. P. Topology of the resonating valence-bond state: solitons and high-_T_ c

superconductivity. _Phys. Rev. B_ 35, 8865–8868 (1987) Article  ADS  CAS  Google Scholar  * Rokhsar, D. S. & Kivelson, S. A. Superconductivity and the quantum hard-core dimer gas. _Phys.

Rev. Lett._ 61, 2376–2379 (1988) Article  ADS  CAS  Google Scholar  * Moessner, R., Sondhi, S. L. & Fradkin, E. Short-ranged resonating valence bond physics, quantum dimer models, and

Ising gauge theories. _Phys. Rev. B_ 65, 024504 (2001) Article  ADS  Google Scholar  * Moessner, R. & Raman, K. S. Quantum dimer models. Preprint at 〈http://arxiv.org/abs/0809.3051〉

(2008) * Cahangirov, S., Topsakal, M., Aktürk, E., Şahin, H. & Ciraci, S. Two- and one-dimensional honeycomb structures of silicon and germanium. _Phys. Rev. Lett._ 102, 236804 (2009)

Article  ADS  CAS  Google Scholar  * Duan, L.-M., Demler, E. & Lukin, M. Controlling spin exchange interactions of ultracold atoms in optical lattices. _Phys. Rev. Lett._ 91, 090402

(2003) Article  ADS  Google Scholar  * Sorella, S. & Tosatti, E. Semi-metal-insulator transition of the Hubbard model in the honeycomb lattice. _Europhys. Lett._ 19, 699–704 (1992)

Article  ADS  Google Scholar  * Paiva, T., Scalettar, R. T., Zheng, W., Singh, R. R. P. & Otimaa, J. Ground-state and finite-temperature signatures of quantum phase transitions in the

half-filled Hubbard model on a honeycomb lattice. _Phys. Rev. B_ 72, 085123 (2005) Article  ADS  Google Scholar  * Ryu, S., Mudry, C., Hou, C.-Y. & Chamon, C. Masses in graphene-like

two-dimensional electronic systems: topological defects in order parameters and their fractional exchange statistics. _Phys. Rev. B_ 80, 205319 (2009) Article  ADS  Google Scholar  *

Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. _Phys. Rev. Lett._ 61, 2015–2018 (1988) Article  ADS  CAS 

Google Scholar  * Assaad, F. F., Hanke, W. & Scalapino, D. J. Flux quantization in the two-dimensional repulsive and attractive Hubbard models. _Phys. Rev. Lett._ 71, 1915–1918 (1993)

Article  ADS  CAS  Google Scholar  * Byers, N. & Yang, C. N. Theoretical considerations concerning quantized magnetic flux in superconducting cylinders. _Phys. Rev. Lett._ 7, 46–49

(1961) Article  ADS  Google Scholar  * Pauling, L. _The Nature of the Chemical Bond_ 183–220 (Cornell Univ. Press, 1960) Google Scholar  * Rantner, W. & Wen, X. G. Spin correlations in

the algebraic spin liquid: implications for high-_T_ c superconductors. _Phys. Rev. B_ 66, 144501 (2002) Article  ADS  Google Scholar  * Hermele, M. et al. On the stability of U(1) spin

liquids in two dimensions. _Phys. Rev. B_ 70, 214437 (2004) Article  ADS  Google Scholar  * Assaad, F. F. Phase diagram of the half-filled two-dimensional SU(N) Hubbard-Heisenberg model: a

quantum Monte Carlo study. _Phys. Rev. B_ 71, 075103 (2005) Article  ADS  Google Scholar  * Mizusaki, T. & Imada, M. Gapless quantum spin liquid, stripe, and antiferromagnetic phases in

frustrated Hubbard models in two dimensions. _Phys. Rev. B_ 74, 014421 (2006) Article  ADS  Google Scholar  * Moessner, R. & Sondhi, S. L. Resonating valence bond phase in the triangular

lattice quantum dimer model. _Phys. Rev. Lett._ 86, 1881–1884 (2001) Article  ADS  CAS  Google Scholar  * Fradkin, E., Huse, D. A., Moessner, R., Oganesyan, V. & Sondhi, S. L. Bipartite

Rokhsar-Kivelson points and Cantor deconfinement. _Phys. Rev. B_ 69, 224415 (2004) Article  ADS  Google Scholar  * Wen, X. G. Mean-field theory of spin-liquid states with finite energy gap

and topological orders. _Phys. Rev. B_ 44, 2664–2672 (1991) Article  ADS  CAS  Google Scholar  * Lieb, E. H. Two theorems on the Hubbard model. _Phys. Rev. Lett._ 62, 1201–1204 (1989)

Article  ADS  MathSciNet  CAS  Google Scholar  * Shimizu, Y., Miyagawa, K., Kanoda, K., Maesato, M. & Saito, G. Spin liquid state in an organic Mott insulator with a triangular lattice.

_Phys. Rev. Lett._ 91, 107001 (2003) Article  ADS  CAS  Google Scholar  * Yamashita, M. et al. Thermal-transport measurements in a quantum spin-liquid state of the frustrated triangular

magnet _κ_-(BEDT-TTF)2Cu2(CN)3 . _Nature Phys._ 5, 44–47 (2009) Article  ADS  CAS  Google Scholar  * Chayes, J. T., Chayes, L. & Kivelson, S. A. Valence bond ground states in a

frustrated two-dimensional spin-1/2 Heisenberg antiferromagnet. _Commun. Math. Phys._ 123, 53–83 (1989) Article  ADS  MathSciNet  Google Scholar  * Nam, M.-S., Ardavan, A., Blundell, S. J.

& Schlueter, J. A. Fluctuating superconductivity in organic molecular metals close to the Mott transition. _Nature_ 449, 584–587 (2007) Article  ADS  CAS  Google Scholar  * Kopnin, N. B.

& Sonin, E. B. BCS superconductivity of Dirac electrons in graphene layers. _Phys. Rev. Lett._ 100, 246808 (2008) Article  ADS  CAS  Google Scholar  * Nakano, H. et al. Soft synthesis

of single-crystal silicon monolayer sheets. _Angew. Chem._ 118, 6451–6454 (2006) Article  ADS  Google Scholar  * Sugiyama, G. & Koonin, S. E. Auxiliary field Monte-Carlo for quantum

many-body ground states. _Ann. Phys._ 168, 1–26 (1986) Article  ADS  Google Scholar  * Furukawa, N. & Imada, M. Optimization of initial state vector in the ground state algorithm of

lattice fermion simulations. _J. Phys. Soc. Jpn_ 60, 3669–3674 (1991) Article  ADS  MathSciNet  Google Scholar  * Assaad, F. F. & Evertz, H. G. _Computational Many-Particle Physics_

277–356 (Lect. Notes Phys. 739, Springer, 2008) Book  Google Scholar  * Feldbacher, M. & Assaad, F. F. Efficient calculation of imaginary-time-displaced correlation functions in the

projector auxiliary-field quantum Monte Carlo algorithm. _Phys. Rev. B_ 63, 073105 (2001) Article  ADS  Google Scholar  Download references ACKNOWLEDGEMENTS We thank L. Balents, S. Capponi,

A. H. Castro Neto, A. Georges, M. Hermele, A. Läuchli, E. Molinari, Y. Motome, S. Sachdev, K. P. Schmidt and S. Sorella for discussions. We are grateful to S. A. Kivelson for thoroughly

reading our manuscript and providing important suggestions. F.F.A. is grateful to the Kavli Institute for Theoretical Physics of the University of California, Santa Barbara, for hospitality

and acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through grants AS120/4-3 and FG1162. A.M. thanks the Aspen Center for Physics for hospitality and acknowledges partial

support by the DFG through grant SFB/TRR21. S.W. acknowledges support by the DFG through grants SFB/TRR21 and WE3649. We thank the John von Neumann Institute for Computing, Jülich; the

Hochleistungsrechenzentrum, Stuttgart; the BW Grid; and the Leibniz-Rechenzentrum, München, for the allocation of CPU time. AUTHOR CONTRIBUTIONS F.F.A. developed the simulation codes; Z.Y.M.

and T.C.L. performed the simulations and analyses and prepared the figures; F.F.A., A.M. and S.W. directed the investigation and wrote the paper. The manuscript reflects the contributions

of all authors. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Institut für Theoretische Physik III, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany , Z. Y. Meng, S.

Wessel & A. Muramatsu * Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany , T. C. Lang & F. F. Assaad Authors * Z. Y. Meng

View author publications You can also search for this author inPubMed Google Scholar * T. C. Lang View author publications You can also search for this author inPubMed Google Scholar * S.

Wessel View author publications You can also search for this author inPubMed Google Scholar * F. F. Assaad View author publications You can also search for this author inPubMed Google

Scholar * A. Muramatsu View author publications You can also search for this author inPubMed Google Scholar CORRESPONDING AUTHOR Correspondence to Z. Y. Meng. ETHICS DECLARATIONS COMPETING

INTERESTS The authors declare no competing financial interests. SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION This file contains Supplementary Discussions which comprises: 1

Green's function and single particle gap; 2 Spin correlations and SAF; 3 Spin excitation gaps; 4 Density correlations; 5 Dimer-dimer correlations - charge sector; 6 Dimer-dimer

correlations - spin sector, 7 Flux quantization measurement for superconductivity; 8 Order parameters for superconductivity and it also includes Supplementary Figures 1-11 with legends. (PDF

438 kb) POWERPOINT SLIDES POWERPOINT SLIDE FOR FIG. 1 POWERPOINT SLIDE FOR FIG. 2 POWERPOINT SLIDE FOR FIG. 3 POWERPOINT SLIDE FOR FIG. 4 RIGHTS AND PERMISSIONS Reprints and permissions

ABOUT THIS ARTICLE CITE THIS ARTICLE Meng, Z., Lang, T., Wessel, S. _et al._ Quantum spin liquid emerging in two-dimensional correlated Dirac fermions. _Nature_ 464, 847–851 (2010).

https://doi.org/10.1038/nature08942 Download citation * Received: 30 October 2009 * Accepted: 17 February 2010 * Published: 08 April 2010 * Issue Date: 08 April 2010 * DOI:

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