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Machine learning for many-body physics: The case of the Anderson impurity model

Louis-François Arsenault, Alejandro Lopez-Bezanilla, O. Anatole von Lilienfeld, Andrew J. Millis

arXiv:1408.1143v2cond-mat.str-elstat.ML

TL;DR

Accurately solving interacting impurity models is computationally demanding, motivating efficient approximations. This paper applies machine learning to predict the Anderson impurity model’s Green’s function using alternative function representations. The authors report that direct imaginary-time learning is promising, while Matsubara-frequency representations may be problematic for general DMFT calculations.

  • Problem

    Accurate solutions of interacting impurity models require substantial numerical effort, motivating efficient approximate methods.

    Source: §I · p. 1

  • Method

    The authors develop a machine-learning scheme for learning the Green’s function and test continued-fraction, Matsubara-frequency, imaginary-time, and Legendre-polynomial representations.

    Source: §VI · p. 12

  • Results

    The authors report that directly learning the Green’s function in imaginary time is promising for learning correlation functions for real materials.

    Source: §VI · p. 12

  • Takeaways & Limitations

    The results indicate that machine learning may be feasible for DMFT, although the Matsubara-frequency representation may not work efficiently in general cases.

    Source: §VI · p. 12

    Matsubara-frequency Green’s functions change qualitatively across the metal-to-insulator transition, making the dataset less homogeneous at low frequencies and potentially limiting general DMFT use.

    Source: §VI · p. 12

Abstract

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Machine learning methods are applied to finding the Green's function of the Anderson impurity model, a basic model system of quantum many-body condensed-matter physics. Different methods of parametrizing the Green's function are investigated; a representation in terms of Legendre polynomials is found to be superior due to its limited number of coefficients and its applicability to state of the art methods of solution. The dependence of the errors on the size of the training set is determined. The results indicate that a machine learning approach to dynamical mean-field theory may be feasible.

I. INTRODUCTION

The paper explores machine learning as an alternative for approximating costly quantum many-body calculations, focusing on predicting the Anderson impurity model’s Green’s function. This function-to-function task requires choosing a compact representation that can be learned from solved examples.

I. INTRODUCTION

  • Accurate solutions of interacting impurity models require substantial numerical effort, motivating efficient approximations for material surveys and preliminary analysis.Source: §I · p. 1
  • A machine-learning approach to DMFT must interpolate from solved examples while mapping an input function to an output function, making efficient function representations a key issue.Source: §I · p. 1
  • For the Anderson impurity model, the input hybridization is specified by a few parameters, and the learning target is the electron Green’s function.Source: §I · p. 2

A. Representation of the function

  • The general learning setup infers a function from model descriptors and a database of solved examples, representing the continuous function with discrete values or defining coefficients.Source: A. Representation of the function · pp. 2–3
  • Independent prediction of function components can violate their correlations and yield a spectral function with negative regions, although the authors report no practical difficulty in their cases.Source: A. Representation of the function · p. 2

B. Machine learning: kernel approach

The paper uses kernel ridge regression to predict discretized components of a function from training examples. Components are learned independently, with a shared kernel and regularization parameter in the calculations.

C. Quality of machine learning

The paper assesses function predictions with an average relative difference and checks specific physical quantities because a single aggregate error can hide narrow inaccuracies. It studies the Anderson impurity model using a descriptor based on its interaction, hybridization, and filling.

C. Quality of machine learning

  • The average relative difference compares predicted and exact function values over selected arguments, using low Matsubara frequencies for the reported comparisons.Source: C. Quality of machine learning · p. 4
  • A single aggregate error can conceal badly predicted narrow regions, so the authors also inspect selected Green’s-function, density-of-states, and low-frequency-renormalization predictions.Source: C. Quality of machine learning · p. 4

III. QUANTUM IMPURITY MODEL

  • With fixed bath density of states and constant hybridization, the Anderson impurity model is described by three parameters and used to predict correlation functions and other physical properties.Source: Figure 1 · p. 4
  • The model’s impurity is a localized level embedded in a noninteracting-electron bath, with on-site energy, interaction strength, and bath hybridization specifying its terms.Source: §III · p. 4
  • The authors use exact diagonalization, representing the hybridization function with a small number of poles and weights.Source: §III · p. 5

A. Methodology of the calculations

The study builds a database of solved impurity models and evaluates machine-learning predictions using training and test subsets. It also notes a practical restriction of the continued-fraction representation.

A. Methodology of the calculations

B. Representations of the Green’s function

1. Continued fraction representation

  • Accurate numerical continued-fraction coefficients are available only in the context of exact-diagonalization calculations.Source: §1 · p. 6

2. Matsubara frequency representation

The section describes sampled Matsubara and imaginary-time representations, then motivates Legendre coefficients as a compact, smoothing target for learning.

2. Matsubara frequency representation

  • The Matsubara representation evaluates G at ωn = (2n + 1)πT and arranges values across examples as a learned vector.Source: §2 · p. 6

3. Imaginary time representation

  • G(τ) is real and smooth; it is sampled at Nc discrete points, with the β endpoint derived separately and the τ = 0+ derivative added for reverse Fourier transformation.Source: §3 · p. 6

Green’s function

  • Legendre polynomials [30] provide a simple Matsubara transformation and filter high-frequency noise, while their coefficient vector keeps the learning target compact.Source: §4 · pp. 6–7
  • A fast Chebyshev–Legendre transform [31] is used to calculate the Legendre coefficients from the Green’s function.Source: §4 · p. 7
  • Odd Legendre coefficients set lmax at their first sign change, which varies by example; the expansion keeps 121 terms and pads higher odd coefficients with a small constant.Source: §4 · p. 7

V. RESULTS

The section specifies two AIM examples for calculating the full Green’s function and Z, and describes averaging predictions over random learning sets for short datasets.

  • The calculations report full G and, for two examples, the renormalization factor Z.Source: §V · p. 7
  • The first case uses U = 3.04, V = 0.5105, and nd = 1; the infinite-U case uses U = 4, V = 0.3053, and nd = 0.95.Source: §V · p. 7
  • For short datasets, many random learning-set combinations are predicted and their results averaged.Source: §V · p. 7

A. Continued fraction representation

Across four Green’s-function representations, the reported ARDs are of the same order, while Legendre polynomials use far fewer coefficients and make analytical continuation less problematic. Prediction quality improves with larger training sets, with the continued-fraction representation’s small-set performance limited at low frequencies.

A. Continued fraction representation

  • ARD falls from about 6.5% with 500 examples to about 0.016% with 4,900 for continued-fraction coefficients.For the continued-fraction representation, the figure plots ARD against training-set size; the other markers show Matsubara-frequency, imaginary-time, and Legendre representations.Source: A. Continued fraction representation · p. 7, Figure 2 · p. 8

B. Matsubara frequency representation

  • Matsubara-frequency learning has systematically lower ARD than continued-fraction learning, while remaining the same order of magnitude.For the two examples, direct learning of G(iωn) performs better on the Matsubara axis, but analytical continuation from a 500-example set gives poor-quality real-frequency results; large-set predictions are very good.Source: B. Matsubara frequency representation · p. 8

C. Imaginary time representation

  • Imaginary-time learning has ARDs below continued-fraction learning, of the same order as the other representations, and is reported as generally applicable.The reported real-frequency results are similar to those for Matsubara-frequency learning.Source: C. Imaginary time representation · p. 8

D. Legendre orthogonal polynomial representation of the Green’s function

  • Legendre expansion is identified as the most efficient representation for learning many-body Green’s functions.Its ARDs are similar to those of imaginary-time and Matsubara-frequency learning; it uses 121 coefficients rather than 2,048 imaginary-time slices, and its coefficient-based reconstruction makes Padé continuation less problematic.Source: D. Legendre orthogonal polynomial… · p. 8

E. Prediction as a function of learning set length

  • For the half-filled density of states, convergence to the correct prediction occurs before the training set reaches its maximum length of 4,999.The comparison covers four representations and random training-set lengths from 500 to 3,000.Source: E. Prediction as a function of… · p. 9

F. Prediction of the mass enhancement

For the mass enhancement, three representations converge to the correct answer with learning sets around 2000–3000, whereas continued fractions do not fully converge. Localized Legendre learning also predicts two new systems with sets of 8 and 4 examples, and the conclusion favors Legendre coefficients while warning that Matsubara-frequency learning may not generalize efficiently to DMFT.

F. Prediction of the mass enhancement

  • Around 2000 examples for Fig. 8(a) and 3000 for Fig. 8(b), the other three representations systematically converge to the correct Z, unlike continued fractions.For nd = 1, continued-fraction predictions are inaccurate at 500 and 1000 examples; the better result at 500 for nd = 0.95 is described as accidental.Source: F. Prediction of the mass enhancement · pp. 9–10

G. Prediction from a minimal learning set

  • With localized sets of 8 and 4 examples, respectively, Legendre predictions were quite precise for two new systems.The first system has no shared parameters with the database and uses 8 examples; the half-filled second system uses 4.Source: G. Prediction from a minimal learning… · p. 10
  • The effective Γ curve is smooth from quite low frequency, opening interpolation from Γ curves as another efficient machine-learning route.For Legendre learning, the effective α-like parameter is reconstructed from predicted coefficients under the fixed-kernel-parameter approximation.Source: G. Prediction from a minimal learning… · p. 11

VI. SUMMARY AND CONCLUSION

  • Learning Legendre coefficients is superior to directly learning G(τ), because the smaller number of coefficients improves accuracy.The paper also identifies learning these coefficients as promising for real-material correlation functions measured by CTQMC algorithms.Source: §VI · p. 12
  • Matsubara-frequency learning may be inefficient for general DMFT because the low-frequency Green’s function changes qualitatively across the metal–insulator transition; G(τ) changes less drastically.The continued-fraction representation is limited to AIM solutions by exact diagonalization.Source: §VI · p. 12
  • The next steps are learning DMFT for model Hamiltonians and constraining learned function values, potentially through moments in the kernel-ridge minimization.Source: §VI · p. 13

Appendix A: Kernel ridge regression

The appendices outline kernel-ridge coefficient solving and the finite-bath exact-diagonalization setup used to generate Green’s functions.

Appendix A: Kernel ridge regression

  • Kernel-space expansion coefficients are obtained by minimizing the cost function with respect to α.For fixed m, the coefficients and targets can be written as vectors of length NL, with the kernel represented by an NL × NL matrix.Source: Appendix A · pp. 13–14
  • Figure 8 plots Z against training-set size for two parameter settings, comparing exact results with predictions from four Green’s-function representations.Figure 9 shows density-of-states predictions from Legendre polynomial representations for learning sets of length 8 and 4, alongside exact results.Source: Figure 8 · p. 14, Figure 9 · p. 14

Appendix B: Exact Diagonalization

  • The continuous bath is approximated by a finite set of poles and weights, yielding a Hamiltonian with Ns = 1 + Nb sites.The pole parameters are chosen to reproduce the bath effect as closely as possible; the finite-bath Hamiltonian has a Hilbert space of size 4^Ns.Source: Appendix B · p. 14
  • The bath parameters are chosen to minimize a distance defined from the Matsubara-axis representation, using the inverse noninteracting Green’s function.The calculation uses a fictitious inverse temperature β = 200 and typically Nmax = 400 frequencies.Source: Appendix B · p. 15

1. Representation of the Green’s function

The Green’s function is obtained from the ground state and represented as a continued fraction, whose coefficients are calculated with a second Lanczos procedure.

1. Representation of the Green’s function

  • For the fixed-hybridization target Hamiltonian, the ground state is nondegenerate and lies in the N = Ns, Sz = 0 sector.The Green’s function is then calculated from the electron and hole parts of the ground state.Source: §1 · p. 16
  • The Green’s function is represented as a continued fraction, with its coefficients obtained through a second Lanczos procedure.The procedure constructs a Lanczos space by recursion and then obtains the coefficients from that space.Source: §1 · p. 16
  • In double precision, the three-vector recursion typically loses orthogonality after about 50 steps, though this has little effect on the Green’s function.Partial orthogonalization is available if needed, and proper numerical calculation requires normalizing the states in the recursion.Source: §1 · p. 16

Appendix C: Legendre polynomial expansion

The appendix describes the Legendre expansion of a function on [−1, 1] and its application to representing the Green’s function.

Appendix C: Legendre polynomial expansion

  • A function on [−1, 1] is expanded in Legendre polynomials, and its coefficients are obtained using polynomial orthogonality.The appendix then changes the integration variable to obtain the stated Green’s-function expansion.Source: Appendix C · p. 17

Appendix D: Effective α in the Machine Learning of the Legendre polynomial expansion

  • The Green’s function on the Matsubara axis is represented using the Legendre polynomial expansion.The passage identifies this representation through Eq. (24).Source: Appendix D · p. 17
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