Critical Behavior of a Correlated Impurity on Graphene
Main Article Content
Abstract
The single-orbital Anderson impurity model using graphene as the host material is considered for the case where the impurity is placed on top of a carbon atom of the graphene lattice. This is an excellent setup for the pseudogap impurity model, where there exists a quantum phase transition from the free local moment phase to the Kondo screening phase. In this work, the scaling behavior of the spin-spin correlator at quantum critical points is numerically investigated. It shows signatures of the logarithmic correction to scaling to the lowest temperature in use. Furthermore, the result suggests that the scaling dimension might vanish as , thus the widely-accepted scaling behavior for might be destroyed at , signifying that is the upper critical dimension for the class of pseudogap impurity problem.
Keywords:
Kondo model, Anderson impurity model, pseudogap, graphene, heavy fermion systems.
References
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[40] M. C. Aronson, R. Osborn, R. A. Robinson, J. W. Lynn, R. Chau, C. L. Seaman, M. B. Maple, Non-Fermi-Liquid Scaling of the Magnetic Response in UCu {5 - X}Pd x (x=1,1.5), Physical Review Letters, Vol. 75, 1995,
pp. 725-728, https://doi.org/10.1103/PhysRevLett.75.725.
[41] H. V. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, Fermi-Liquid Instabilities at Magnetic Quantum Phase Transitions, Reviews of Modern Physics, Vol. 79, 2007, pp. 1015-1075, https://doi.org/10.1103/RevModPhys.79.1015.
[42] K. Ingersent, Q. Si, Critical Local-Moment Fluctuations, Anomalous Exponents, and ω / T Scaling in the Kondo Problem with a Pseudogap, Physical Review Letters, Vol. 89, 2002, pp. 076403, https://doi.org/10.1103/PhysRevLett.89.076403.
[43] Q. Si, F. Steglich, Heavy Fermions and Quantum Phase Transitions, Science, Vol. 329, 2010, pp. 1161, https://doi.org/10.1126/science.1191195.
[2] J. A. Hertz, Quantum Critical Phenomena, Physical Review B, Vol. 14, 1976, pp. 1165-1184, https://doi.org/10.1103/PhysRevB.14.1165.
[3] A. J. Millis, Effect of a Nonzero Temperature on Quantum Critical Points in Itinerant Fermion Systems, Physical Review B, Vol. 48, 1993, pp. 7183-7196, https://doi.org/10.1103/PhysRevB.48.7183.
[4] H. V. Löhneysen, T. Pietrus, G. Portisch, H. G. Schlager, A. Schröder, M. Sieck, and T. Trappmann, Non-Fermi-Liquid Behavior in a Heavy-Fermion Alloy at a Magnetic Instability, Physical Review Letters, Vol. 72, 1994,
pp. 3262-3265, https://doi.org/10.1103/PhysRevLett.72.3262.
[5] G. R. Stewart, Non-Fermi-Liquid Behavior in d- and f-electron Metals, Reviews of Modern Physics, Vol. 73, 2001, pp. 797-855, https://doi.org/10.1103/RevModPhys.73.797.
[6] P. Gegenwart, Q. Si, F. Steglich, Quantum Criticality in Heavy-Fermion Metals, Nature Physics, Vol. 4, 2008,
pp. 186-197, https://doi.org/10.1038/nphys892.
[7] T. Senthil, Deconfined Quantum Critical Points, Science, Vol. 303, 2004, pp. 1490-1494, https://doi.org/10.1126/science.1091806.
[8] Q. Si, S. Rabello, K. Ingersent, J. L. Smith, Locally Critical Quantum Phase Transitions in Strongly Correlated Metals, Nature, Vol. 413, 2001, pp. 804-808, https://doi.org/10.1038/35101507.
[9] S. Paschen, Q. Si, Quantum Phases Driven by Strong Correlations, Nature Reviews Physics, Vol. 3, 2021,
pp. 9-26, https://doi.org/10.1038/s42254-020-00262-6.
[10] S. Doniach, The Kondo Lattice and Weak Antiferromagnetism, Physica B+C, Vol. 91, 1977, pp. 231-234, https://doi.org/10.1016/0378-4363(77)90190-5.
[11] M. T. Glossop, S. Kirchner, J. H. Pixley, Q. Si, Critical Kondo Destruction in a Pseudogap Anderson Model: Scaling and Relaxational Dynamics, Physical Review Letters, Vol. 107, 2011, pp. 076404, https://doi.org/10.1103/PhysRevLett.107.076404.
[12] J. H. Pixley, Stefan Kirchner, Kevin Ingersent, and Qimiao Si, Kondo Destruction and Valence Fluctuations in an Anderson Model, Physical Review Letters, Vol. 109, 2012, pp. 086403, https://doi.org/10.1103/PhysRevLett.109.086403.
[13] Alexander Cyril Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge,
UK, 1993.
[14] D. Withoff, E. Fradkin, Phase Transitions in Gapless Fermi Systems with Magnetic Impurities, Physical Review Letters, Vol. 64, 1990, pp. 1835-1838, https://doi.org/10.1103/PhysRevLett.64.1835.
[15] R. Bulla, T. Pruschke, A. C. Hewson, Anderson Impurity in Pseudo-Gap Fermi Systems, Journal of Physics: Condensed Matter, Vol. 9, 1997, pp. 10463-10474, https://doi.org/10.1088/0953-8984/9/47/014.
[16] C. G. Buxton, K. Ingersent, Renormalization-Group Study of Anderson and Kondo Impurities in Gapless Fermi Systems, Physical Review B, Vol. 57, 1998, pp. 14254-14293, https://doi.org/10.1103/PhysRevB.57.14254.
[17] L. Fritz, M. Vojta, The Physics of Kondo Impurities in Graphene, Reports on Progress in Physics, Vol. 76, 2013, pp. 032501, https://doi.org/10.1088/0034-4885/76/3/032501.
[18] Tathagata Chowdhury and Kevin Ingersent, Critical Charge Fluctuations in a Pseudogap Anderson Model, Physical Review B, Vol. 91, 2015, pp. 035118, https://doi.org/10.1103/PhysRevB.91.035118.
[19] C. Wagner, T. Chowdhury, J. H. Pixley, K.Ingersent, Long-Range Entanglement near a Kondo-Destruction Quantum Critical Point, Physical Review Letters, Vol. 121, 2018, pp. 147602, https://doi.org/10.1103/PhysRevLett.121.147602.
[20] M. Vojta, L. Fritz, Upper Critical Dimension in A Quantum Impurity Model: Critical Theory Of The Asymmetric Pseudogap Kondo Problem, Physical Review B, Vol. 70, 2004, pp. 094502, https://doi.org/10.1103/PhysRevB.70.094502.
[21] L. Fritz, M. Vojta, Phase Transitions in The Pseudogap Anderson and Kondo Models: Critical Dimensions, Renormalization Group, and Local-Moment Criticality, Physical Review B, Vol. 70, 2004, pp. 214427, https://doi.org/10.1103/PhysRevB.70.214427.
[22] C. R. Cassanello, E. Fradkin, Kondo Effect in Flux Phases, Physical Review B, Vol. 53, 1996, pp. 15079–15094.
[23] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science, Vol. 306, 2004, pp. 666-669, https://doi.org/10.1126/science.1102896.
[24] A. H. C. Neto, N. M. R. Peres, K. S. Novoselov, A. K. Geim, The Electronic Properties of Graphene. Rev. Mod. Phys., Vol. 81, 2009, pp. 109-162, https://doi.org/10.1103/RevModPhys.81.109.
[25] P. W. Anderson, Localized Magnetic States in Metals. Physical Review, Vol. 124, 1961, pp. 41-53, https://doi.org/10.1103/PhysRev.124.41.
[26] H. T. Dang, N. T. M. Hoa, Phase Diagram of a Pseudogap Anderson Model with Application to Graphene, arXiv:2107.09854 [cond-mat], 2021, https://doi.org/10.48550/arXiv.2107.09854.
[27] Philipp Werner and Andrew J. Millis, Hybridization Expansion Impurity Solver: General Formulation and Application to Kondo Lattice and Two-Orbital Models, Physical Review B, Vol. 74, 2006, pp. 155107, https://doi.org/10.1103/PhysRevB.74.155107.
[28] P. Werner, A. Comanac, Luca de’ Medici, Matthias Troyer, and Andrew J. Millis, Continuous-Time Solver for Quantum Impurity Models, Physical Review Letters, Vol. 97, 2006, pp. 076405, https://doi.org/10.1103/PhysRevLett.97.076405.
[29] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, P. Werner, Continuous-Time Monte Carlo Methods for Quantum Impurity Models, Reviews of Modern Physics, Vol. 83, 2011, pp. 349-404, https://doi.org/10.1103/RevModPhys.83.349.
[30] O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, P. Seth, TRIQS: A Toolbox for Research on Interacting Quantum Systems, Computer Physics Communications, Vol. 196, 2015, pp. 398-415, https://doi.org/10.1016/j.cpc.2015.04.023.
[31] P. Seth, I. Krivenko, M. Ferrero, O. Parcollet, TRIQS/CTHYB: A Continuous-Time Quantum Monte Carlo Hybridisation Expansion Solver for Quantum Impurity Problems. Computer Physics Communications, Vol. 200, 2016, pp. 274-284, https://doi.org/10.1016/j.cpc.2015.10.023.
[32] R.Bulla, T. A. Costi, T. Pruschke, Numerical Renormalization Group Method for Quantum Impurity Systems, Reviews of Modern Physics, Vol. 80, 2008, pp. 395-450, https://doi.org/10.1103/RevModPhys.80.395.
[33] A. Georges, G. Kotliar, W. Krauth, M. Rozenberg, Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and The Limit of Infinite Dimensions, Reviews of Modern Physics, Vol. 68, 1996, pp. 113-125, https://doi.org/10.1103/RevModPhys.68.13.
[34] K. Binder, Finite Size Scaling Analysis of Ising Model Block Distribution Functions, Zeitschrift für Physik B Condensed Matter, Vol. 43, 1981, pp. 119-140, https://doi.org/10.1007/BF01293604.
[35] J. L. Cardy, Conformal Invariance and Surface Critical Behavior, Nuclear Physics B, Vol. 240, 1984, pp. 514–532, https://doi.org/10.1016/0550-3213(84)90241-4.
[36] I. Affleck, A. W. W. Ludwig, Critical Theory of Overscreened Kondo Fixed Points, Nuclear Physics B, Vol. 360, 1991, pp. 641-696, https://doi.org/10.1016/0550-3213(91)90419-X.
[37] S. Kirchner, Q. Si, Scaling and Enhanced Symmetry at the Quantum Critical Point of the Sub-Ohmic Bose-Fermi Kondo Model, Physical Review Letters, Vol. 100, 2008, pp. 026403, https://doi.org/10.1103/PhysRevLett.100.026403.
[38] A. Schröder, G. Aeppli, E. Bucher, R. Ramazashvili, and P. Coleman, Scaling of Magnetic Fluctuations near a Quantum Phase Transition, Physical Review Letters, Vol. 80, 1998, pp. 5623-5626, https://doi.org/10.1103/PhysRevLett.80.5623.
[39] A. Schröder, G. Aeppli, R. Coldea, M. Adams, O. Stockert, H. v Löhneysen, E. Bucher, R. Ramazashvili, P. Coleman, Onset of Antiferromagnetism in Heavy-Fermion Metals, Nature, Vol. 407, 2000, pp. 351–355, https://doi.org/10.1038/35030039.
[40] M. C. Aronson, R. Osborn, R. A. Robinson, J. W. Lynn, R. Chau, C. L. Seaman, M. B. Maple, Non-Fermi-Liquid Scaling of the Magnetic Response in UCu {5 - X}Pd x (x=1,1.5), Physical Review Letters, Vol. 75, 1995,
pp. 725-728, https://doi.org/10.1103/PhysRevLett.75.725.
[41] H. V. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, Fermi-Liquid Instabilities at Magnetic Quantum Phase Transitions, Reviews of Modern Physics, Vol. 79, 2007, pp. 1015-1075, https://doi.org/10.1103/RevModPhys.79.1015.
[42] K. Ingersent, Q. Si, Critical Local-Moment Fluctuations, Anomalous Exponents, and ω / T Scaling in the Kondo Problem with a Pseudogap, Physical Review Letters, Vol. 89, 2002, pp. 076403, https://doi.org/10.1103/PhysRevLett.89.076403.
[43] Q. Si, F. Steglich, Heavy Fermions and Quantum Phase Transitions, Science, Vol. 329, 2010, pp. 1161, https://doi.org/10.1126/science.1191195.