Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

$m^\ast$ of two-dimensional electron gas: a neural canonical transformation study (2201.03156v2)

Published 10 Jan 2022 in cond-mat.stat-mech, cond-mat.mes-hall, cond-mat.str-el, cs.LG, and physics.comp-ph

Abstract: The quasiparticle effective mass $m\ast$ of interacting electrons is a fundamental quantity in the Fermi liquid theory. However, the precise value of the effective mass of uniform electron gas is still elusive after decades of research. The newly developed neural canonical transformation approach [Xie et al., J. Mach. Learn. 1, (2022)] offers a principled way to extract the effective mass of electron gas by directly calculating the thermal entropy at low temperature. The approach models a variational many-electron density matrix using two generative neural networks: an autoregressive model for momentum occupation and a normalizing flow for electron coordinates. Our calculation reveals a suppression of effective mass in the two-dimensional spin-polarized electron gas, which is more pronounced than previous reports in the low-density strong-coupling region. This prediction calls for verification in two-dimensional electron gas experiments.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (88)
  1. G. Baym and C. Pethick, Landau Fermi-liquid theory: concepts and applications, John Wiley & Sons (2008).
  2. P. W. Anderson, Basic notions of condensed matter physics, CRC Press (2018).
  3. T. Rice, The effects of electron-electron interaction on the properties of metals, Annals of Physics 31(1), 100 (1965), https://doi.org/10.1016/0003-4916(65)90234-4.
  4. L. Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965), 10.1103/PhysRev.139.A796.
  5. J. Lam, Thermodynamic properties of the electron gas at metallic densities, Phys. Rev. B 5, 1254 (1972), 10.1103/PhysRevB.5.1254.
  6. A. Krakovsky and J. K. Percus, Quasiparticle effective mass for the two- and three-dimensional electron gas, Phys. Rev. B 53, 7352 (1996), 10.1103/PhysRevB.53.7352.
  7. Y. Zhang and S. Das Sarma, Spin polarization dependence of carrier effective mass in semiconductor structures: Spintronic effective mass, Phys. Rev. Lett. 95, 256603 (2005), 10.1103/PhysRevLett.95.256603.
  8. Quasiparticle self-energy and many-body effective mass enhancement in a two-dimensional electron liquid, Phys. Rev. B 71, 045323 (2005), 10.1103/PhysRevB.71.045323.
  9. Many-body local fields theory of quasiparticle properties in a three-dimensional electron liquid, Phys. Rev. B 77, 035131 (2008), 10.1103/PhysRevB.77.035131.
  10. Y. Kwon, D. M. Ceperley and R. M. Martin, Quantum monte carlo calculation of the fermi-liquid parameters in the two-dimensional electron gas, Phys. Rev. B 50, 1684 (1994), 10.1103/PhysRevB.50.1684.
  11. Y. Kwon, D. M. Ceperley and R. M. Martin, Transient-estimate monte carlo in the two-dimensional electron gas, Phys. Rev. B 53, 7376 (1996), 10.1103/PhysRevB.53.7376.
  12. Renormalization factor and effective mass of the two-dimensional electron gas, Phys. Rev. B 79, 041308 (2009), 10.1103/PhysRevB.79.041308.
  13. Quantum monte carlo calculation of the energy band and quasiparticle effective mass of the two-dimensional fermi fluid, Phys. Rev. B 80, 245104 (2009), 10.1103/PhysRevB.80.245104.
  14. Diffusion quantum monte carlo calculation of the quasiparticle effective mass of the two-dimensional homogeneous electron gas, Phys. Rev. B 87, 045131 (2013), 10.1103/PhysRevB.87.045131.
  15. Quantum monte carlo calculation of the fermi liquid parameters of the two-dimensional homogeneous electron gas, Phys. Rev. B 88, 035133 (2013), 10.1103/PhysRevB.88.035133.
  16. K. Haule and K. Chen, Single-particle excitations in the uniform electron gas by diagrammatic monte carlo, Scientific Reports 12(1), 2294 (2022), 10.1038/s41598-022-06188-6.
  17. S. Azadi, N. D. Drummond and W. M. C. Foulkes, Quasiparticle effective mass of the three-dimensional fermi liquid by quantum monte carlo, Phys. Rev. Lett. 127, 086401 (2021), 10.1103/PhysRevLett.127.086401.
  18. G. Giuliani and G. Vignale, Quantum theory of the electron liquid, Cambridge university press (2005).
  19. R. M. Martin, L. Reining and D. M. Ceperley, Interacting Electrons: Theory and Computational Approaches, Cambridge University Press, 10.1017/CBO9781139050807 (2016).
  20. Effects of a tilted magnetic field on a two-dimensional electron gas, Phys. Rev. 174, 823 (1968), 10.1103/PhysRev.174.823.
  21. W. Pan, D. C. Tsui and B. L. Draper, Mass enhancement of two-dimensional electrons in thin-oxide si-mosfet’s, Phys. Rev. B 59, 10208 (1999), 10.1103/PhysRevB.59.10208.
  22. Low-density spin susceptibility and effective mass of mobile electrons in si inversion layers, Phys. Rev. Lett. 88, 196404 (2002), 10.1103/PhysRevLett.88.196404.
  23. Spin susceptibility of two-dimensional electrons in narrow alas quantum wells, Phys. Rev. Lett. 92, 226401 (2004), 10.1103/PhysRevLett.92.226401.
  24. Measurements of the density-dependent many-body electron mass in two dimensional GaAs/AlGaAsnormal-GaAsnormal-AlGaAs\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}\mathrm{G}\mathrm% {a}\mathrm{A}\mathrm{s}roman_GaAs / roman_AlGaAs heterostructures, Phys. Rev. Lett. 94, 016405 (2005), 10.1103/PhysRevLett.94.016405.
  25. Strongly correlated two-dimensional plasma explored from entropy measurements, Nature Communications 6(1), 7298 (2015), 10.1038/ncomms8298.
  26. Effective mass of quasiparticles from thermodynamics, Phys. Rev. B 96, 035132 (2017), 10.1103/PhysRevB.96.035132.
  27. M. Imada and M. Takahashi, Quantum monte carlo simulation of a two-dimensional electron system –melting of wigner crystal–, Journal of the Physical Society of Japan 53(11), 3770 (1984), 10.1143/JPSJ.53.3770.
  28. Path-integral monte carlo simulation of the warm dense homogeneous electron gas, Phys. Rev. Lett. 110, 146405 (2013), 10.1103/PhysRevLett.110.146405.
  29. Ab initio thermodynamic results for the degenerate electron gas at finite temperature, Phys. Rev. Lett. 115, 130402 (2015), 10.1103/PhysRevLett.115.130402.
  30. Accurate exchange-correlation energies for the warm dense electron gas, Phys. Rev. Lett. 117, 115701 (2016), 10.1103/PhysRevLett.117.115701.
  31. H. Xie, L. Zhang and L. Wang, Ab-initio study of interacting fermions at finite temperature with neural canonical transformation, Journal of Machine Learning 1(1), 38 (2022), https://doi.org/10.4208/jml.220113.
  32. L. Wang, Generative models for physicists (2018).
  33. D. Ceperley, Ground state of the fermion one-component plasma: A monte carlo study in two and three dimensions, Phys. Rev. B 18, 3126 (1978), 10.1103/PhysRevB.18.3126.
  34. D. Wu, L. Wang and P. Zhang, Solving statistical mechanics using variational autoregressive networks, Phys. Rev. Lett. 122, 080602 (2019), 10.1103/PhysRevLett.122.080602.
  35. Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits, Machine Learning: Science and Technology 2(2), 025011 (2021), 10.1088/2632-2153/aba19d.
  36. Attention is all you need, In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan and R. Garnett, eds., Advances in Neural Information Processing Systems, vol. 30. Curran Associates, Inc. (2017).
  37. See the Appendix for (a) additional benchmark for three-dimensional spin-polarized uniform electron gas; (b) the analytic method to compute the thermal entropy of non-interacting Fermi gas in the canonical ensemble; (c) discussion and implementation details about the use of twist-averaged boundary conditions; details on (d) model architectures and (e) the training procedure.
  38. Recurrent neural network wave functions, Phys. Rev. Res. 2, 023358 (2020), 10.1103/PhysRevResearch.2.023358.
  39. T. D. Barrett, A. Malyshev and A. I. Lvovsky, Autoregressive neural-network wavefunctions for ab initio quantum chemistry (2021), 2109.12606.
  40. P. Borrmann and G. Franke, Recursion formulas for quantum statistical partition functions, The Journal of Chemical Physics 98(3), 2484 (1993), 10.1063/1.464180.
  41. B. Tanatar and D. M. Ceperley, Ground state of the two-dimensional electron gas, Phys. Rev. B 39, 5005 (1989), 10.1103/PhysRevB.39.5005.
  42. M. Eger and E. Gross, Point transformations and the many body problem, Annals of Physics 24, 63 (1963), https://doi.org/10.1016/0003-4916(63)90065-4.
  43. F. Gygi, Adaptive riemannian metric for plane-wave electronic-structure calculations, Europhysics Letters 19(7), 617 (1992), 10.1209/0295-5075/19/7/009.
  44. F. Gygi, Electronic-structure calculations in adaptive coordinates, Phys. Rev. B 48, 11692 (1993), 10.1103/PhysRevB.48.11692.
  45. K. Cranmer, S. Golkar and D. Pappadopulo, Inferring the quantum density matrix with machine learning (2019), 1904.05903.
  46. G. Papamakarios, Neural Density Estimation and Likelihood-free Inference (2019), 1910.13233.
  47. R. P. Feynman and M. Cohen, Energy spectrum of the excitations in liquid helium, Phys. Rev. 102, 1189 (1956), 10.1103/PhysRev.102.1189.
  48. Ab initio solution of the many-electron schrödinger equation with deep neural networks, Phys. Rev. Res. 2, 033429 (2020), 10.1103/PhysRevResearch.2.033429.
  49. C. Lin, F. H. Zong and D. M. Ceperley, Twist-averaged boundary conditions in continuum quantum monte carlo algorithms, Phys. Rev. E 64, 016702 (2001), 10.1103/PhysRevE.64.016702.
  50. Effective mass suppression in a ferromagnetic two-dimensional electron liquid, Phys. Rev. B 79, 235324 (2009), 10.1103/PhysRevB.79.235324.
  51. See https://github.com/fermiflow/CoulombGas for code written in Jax [84]. The repository also contains the original data, trained models and data processing scripts that can be used to reproduce the main results in Fig. 4.
  52. Spin-independent origin of the strongly enhanced effective mass in a dilute 2D electron system, Phys. Rev. Lett. 91, 046403 (2003), 10.1103/PhysRevLett.91.046403.
  53. Effective mass suppression in dilute, spin-polarized two-dimensional electron systems, Phys. Rev. Lett. 101, 026402 (2008), 10.1103/PhysRevLett.101.026402.
  54. Effective mass suppression upon complete spin-polarization in an isotropic two-dimensional electron system, Phys. Rev. B 79, 195311 (2009), 10.1103/PhysRevB.79.195311.
  55. S. Gangadharaiah and D. L. Maslov, Spin-independent effective mass in a valley-degenerate electron system, Phys. Rev. Lett. 95, 186801 (2005), 10.1103/PhysRevLett.95.186801.
  56. T. Gokmen, M. Padmanabhan and M. Shayegan, Contrast between spin and valley degrees of freedom, Phys. Rev. B 81, 235305 (2010), 10.1103/PhysRevB.81.235305.
  57. Temperature-dependent effective-mass renormalization in two-dimensional electron systems, Phys. Rev. B 69, 125334 (2004), 10.1103/PhysRevB.69.125334.
  58. R. Asgari and B. Tanatar, Many-body effective mass and spin susceptibility in a quasi-two-dimensional electron liquid, Phys. Rev. B 74, 075301 (2006), 10.1103/PhysRevB.74.075301.
  59. T. Gokmen, M. Padmanabhan and M. Shayegan, Dependence of effective mass on spin and valley degrees of freedom, Phys. Rev. Lett. 101, 146405 (2008), 10.1103/PhysRevLett.101.146405.
  60. PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications (2017), 1701.05517.
  61. End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems, In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi and R. Garnett, eds., Advances in Neural Information Processing Systems, vol. 31. Curran Associates, Inc. (2018).
  62. Targeted free energy estimation via learned mappings, The Journal of Chemical Physics 153(14), 144112 (2020), 10.1063/5.0018903.
  63. J. Martens and R. Grosse, Optimizing Neural Networks with Kronecker-factored Approximate Curvature (2015), 1503.05671.
  64. Scalable Second Order Optimization for Deep Learning (2020), 2002.09018.
  65. A. Griewank and A. Walther, Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Softw. 26(1), 19–45 (2000), 10.1145/347837.347846.
  66. Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour (2017), 1706.02677.
  67. Ffjord: Free-form continuous dynamics for scalable reversible generative models, In 7th International Conference on Learning Representations, ICLR 2019 (2019), 1810.01367.
  68. Invertible residual networks, In K. Chaudhuri and R. Salakhutdinov, eds., Proceedings of the 36th International Conference on Machine Learning, vol. 97 of Proceedings of Machine Learning Research, pp. 573–582. PMLR (2019).
  69. Residual flows for invertible generative modeling, In Advances in Neural Information Processing Systems, vol. 32 (2019), 1906.02735.
  70. Neural networks with cheap differential operators, In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox and R. Garnett, eds., Advances in Neural Information Processing Systems, vol. 32. Curran Associates, Inc. (2019), 1912.03579.
  71. Learning differential equations that are easy to solve (2020), 2007.04504.
  72. Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization (2020), 2012.05942.
  73. NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems, SciPost Phys. Codebases p. 7 (2022), 10.21468/SciPostPhysCodeb.7.
  74. Thermodynamics of a charged fermion layer at high rssubscript𝑟𝑠{r}_{s}italic_r start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT values, Phys. Rev. Lett. 77, 3181 (1996), 10.1103/PhysRevLett.77.3181.
  75. Thermodynamic signature of a two-dimensional metal-insulator transition, Phys. Rev. Lett. 84, 4689 (2000), 10.1103/PhysRevLett.84.4689.
  76. Julia: A fresh approach to numerical computing, SIAM Review 59(1), 65 (2017), 10.1137/141000671.
  77. Special points for brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976), 10.1103/PhysRevB.13.5188.
  78. A. Baldereschi, Mean-value point in the brillouin zone, Phys. Rev. B 7, 5212 (1973), 10.1103/PhysRevB.7.5212.
  79. Quantum monte carlo calculations for solids using special k𝑘kitalic_k points methods, Phys. Rev. Lett. 73, 1959 (1994), 10.1103/PhysRevLett.73.1959.
  80. MADE: Masked Autoencoder for Distribution Estimation (2015), 1502.03509.
  81. F. Becca and S. Sorella, Quantum Monte Carlo approaches for correlated systems, Cambridge University Press, ISBN 9781316417041, 10.1017/9781316417041 (2017).
  82. Neural-network quantum states for periodic systems in continuous space, Phys. Rev. Res. 4, 023138 (2022), 10.1103/PhysRevResearch.4.023138.
  83. Iterative backflow renormalization procedure for many-body ground-state wave functions of strongly interacting normal fermi liquids, Phys. Rev. B 91, 115106 (2015), 10.1103/PhysRevB.91.115106.
  84. JAX: composable transformations of Python+NumPy programs (2018).
  85. Monte carlo gradient estimation in machine learning, Journal of Machine Learning Research 21(132), 1 (2020).
  86. M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines, Communications in Statistics - Simulation and Computation 19(2), 433 (1990), 10.1080/03610919008812866.
  87. P. Zanardi, L. Campos Venuti and P. Giorda, Bures metric over thermal state manifolds and quantum criticality, Phys. Rev. A 76, 062318 (2007), 10.1103/PhysRevA.76.062318.
  88. Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge University Press, 10.1017/CBO9780511976667 (2010).
Citations (8)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now