2000 character limit reached
Fast Fourier-Chebyshev approach to real-space simulations of the Kubo formula (2307.09690v2)
Published 19 Jul 2023 in cond-mat.mes-hall
Abstract: The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over $107$ sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.
- J. C. Wheeler and C. Blumstein, Modified moments for harmonic solids, Phys. Rev. B 6, 4380 (1972).
- H. Tal‐Ezer and R. Kosloff, An accurate and efficient scheme for propagating the time dependent Schrödinger equation, The Journal of Chemical Physics 81, 3967 (1984).
- G. Schubert, A. Weiße, and H. Fehske, Localization effects in quantum percolation, Phys. Rev. B 71, 045126 (2005).
- J. H. Pixley, D. A. Huse, and S. Das Sarma, Rare-region-induced avoided quantum criticality in disordered three-dimensional Dirac and Weyl semimetals, Phys. Rev. X 6, 021042 (2016).
- R. N. Silver and H. Röder, Calculation of densities of states and spectral functions by chebyshev recursion and maximum entropy, Phys. Rev. E 56, 4822 (1997).
- T. Iitaka and T. Ebisuzaki, Algorithm for linear response functions at finite temperatures: Application to esr spectrum of s=12𝑠12s=\frac{1}{2}italic_s = divide start_ARG 1 end_ARG start_ARG 2 end_ARG antiferromagnet cu benzoate, Phys. Rev. Lett. 90, 047203 (2003).
- A. Ferreira and E. R. Mucciolo, Critical delocalization of chiral zero energy modes in graphene, Phys. Rev. Lett. 115 (2015).
- G. D. Mahan, Many Particle Physics, Third Edition (Plenum, New York, 2000).
- H. U. Baranger and A. D. Stone, Electrical linear-response theory in an arbitrary magnetic field: A new Fermi-surface formation, Phys. Rev. B 40, 8169 (1989).
- F. D. M. Haldane, Berry curvature on the Fermi surface: Anomalous Hall effect as a topological Fermi-liquid property, Phys. Rev. Lett. 93, 206602 (2004).
- J. H. García, L. Covaci, and T. G. Rappoport, Real-space calculation of the conductivity tensor for disordered topological matter, Phys. Rev. Lett. 114, 116602 (2015).
- See Supplemental Material at [https://journals.aps.org/prl/] for the derivation of energy-space vectors in the Kubo-Bastin formulation, and for additional details on the simulations and the partitioning scheme employed to mitigate the memory cost.
- D. Jackson, On approximation by trigonometric sums and polynomials, Trans Am. Math, Soc. 13 (1912).
- km=(M−m+1)M+1cos(πmM+1)+sin(πmM+1)cot(πmM+1)subscript𝑘𝑚𝑀𝑚1𝑀1cos𝜋𝑚𝑀1sin𝜋𝑚𝑀1cot𝜋𝑚𝑀1k_{m}=\frac{(M-m+1)}{M+1}\textrm{cos}\left(\frac{\pi m}{M+1}\right)+\textrm{% sin}\left(\frac{\pi m}{M+1}\right)\textrm{cot}\left(\frac{\pi m}{M+1}\right)italic_k start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = divide start_ARG ( italic_M - italic_m + 1 ) end_ARG start_ARG italic_M + 1 end_ARG cos ( divide start_ARG italic_π italic_m end_ARG start_ARG italic_M + 1 end_ARG ) + sin ( divide start_ARG italic_π italic_m end_ARG start_ARG italic_M + 1 end_ARG ) cot ( divide start_ARG italic_π italic_m end_ARG start_ARG italic_M + 1 end_ARG ).
- J. W. Cooley, P. Lewis, and P. Welch, The fast fourier transform and its applications, IEEE Trans on Education 12, 28 (1969).
- M. Frigo and S. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE 93, 216 (2005).
- S. G. de Castro, A. Ferreira, and D. A. Bahamon, Efficient Chebyshev polynomial approach to quantum conductance calculations: Application to twisted bilayer graphene, Phys. Rev. B 107, 045418 (2023).
- J. C. Slater and G. F. Koster, Simplified LCAO method for the periodic potential problem, Phys. Rev. 94, 1498 (1954).
- D. A. Bahamon, G. Gómez-Santos, and T. Stauber, Emergent magnetic texture in driven twisted bilayer graphene, Nanoscale 12, 15383 (2020).
- P. Moon and M. Koshino, Energy spectrum and quantum hall effect in twisted bilayer graphene, Phys. Rev. B 85, 195458 (2012).
- B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993).
- K. Slevin and T. Ohtsuki, Critical exponent for the quantum Hall transition, Phys. Rev. B 80, 041304 (2009).
- H. Obuse, I. A. Gruzberg, and F. Evers, Finite-size effects and irrelevant corrections to scaling near the integer quantum Hall transition, Phys. Rev. Lett. 109, 206804 (2012).
- K. Slevin and T. Ohtsuki, Irrelevant corrections at the quantum Hall transition, physica status solidi (RRL) – Rapid Research Letters n/a, 2300080.
- V. P. Gusynin and S. G. Sharapov, Unconventional integer quantum hall effect in graphene, Phys. Rev. Lett. 95, 146801 (2005).
- F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
- E. J. Dresselhaus, B. Sbierski, and I. A. Gruzberg, Scaling collapse of longitudinal conductance near the integer quantum Hall transition, Phys. Rev. Lett. 129, 026801 (2022).
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