Contents

Matrix product states for real materials

In this talk, I will present recent advances on how to combine MPS-based and similar algorithms to provide impurity solvers for dynamical mean-field theory (DMFT) in combination with density functional theory. Several physical applications, in particular to strongly correlated materials with Hund’s coupling and spin-orbit coupling (transition metal and rare earth oxides) will be presented, showing how this method can solve physical questions that are not easily accessible to competing methods.

Notes

Note

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The main problem with the Hamiltonian is the electron electron interactions. Where the system cannot be framed in terms of a delocalized interaction (e.g. where the valence electrons are bound to sites) then the many body problem needs to be solved. This is the realm of strongly correlated systems where model Hamiltonians are used to study such interactions.

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Slide1

Strong correlations matter, and their study in every dimension is important:

  • 0 dimensional systems related to magnetic systems, impurity physics, and quantum dots

  • 1 dimensional systems model spin chains and ladders, along with the Luttinger liquid

  • 2 dimensional systems are relevant for modeling frustrated magnets and high-T_C superconductors

  • 3 dimensional models have the capacity to study realistic systems, and can describe transition metals and rare earth compounds

    • Focus area for talk

Dynamical mean field theory is the main bridging method.

Attention

Main results are derived in [LZH+20] with applications described in [WGM+15,BMG+21,KBG+20].

Essentially, the system is simplified in multiple stages.

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Slide2

  • Some transitions have no order parameter, atleast, not a symmetry breaking simple order parameter.

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Slide3

For the Mott Hubbard transition there is no symmetry breaking, and the transition is due to energetic blocking

DMFT

The dynamical mean field theory aims to recreate the temporal dependence by certain key aspects:

  • interacting model \(\to\) interacting impurity in a non-interacting effictive bath

  • impurity dynamically exchanges electrons with bath

  • dynamical self-consistency implies that the local lattice GF/SE is the same as the impurity GF/SE

This is exact in the limit of infinite coordination number or dimension.

It can be made more realistic by virtue of:

  • Having more bands

  • Using cluster methods

  • Combinations with DFT

The method is originally described in [GK92] with more details in [KSH+06,GKKR96].

Green’s function is required. This is done with an impurity solver like:

  • Hirsch-Fye MC (Action) [Obsolete]

  • continuous time QMC (Action)

    • analytic continuation

    • sign problem?

  • exact diagnolization (Hamiltonian)

    • analytic continuation

    • computational cost (exponential)

  • numerical renormalization group (Hamiltonian)

    • logarithmic discretization

    • computational cost (exponential)

MPS Formulation

  • Good for 1D

  • No sign issues (unlike CT-QMC)

  • Real frequency axis

  • No logarihmic discretization of the spectral range

  • Computational tractable

  • Works best at zero temperature (unlike CT-QMC)

On the imaginary axis:

  • smaller bath sizes

  • no entanglement growth in time evolution

  • more orbitals

  • however, needs analytic continuation of spectral function

  • worked upto 6 orbitals (in [WGM+15])

Case study

Properties of strontium ruthenate are described in [MM03]. Overview:

  • 2D, 3 relevant d-orbitals

  • Strongly correlated

  • Fermi liquid (Hund’s metal)

  • Unocnventional superconductor (p-wave)

  • Appreciable spin-orbit coupling (SOC)

Good system for integrating MPS-DMFT with DFT band structure, since without SOC CTQMC-DMFT is very accurate and can be used for benchmarks, with SOC there is a sign problem.

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Page4

Key questions were:

  • does SOC affect the Fermi liquid behaviour

  • interplay of Coulomb interaction and SOC

  • calculation of new topology

  • CTQMC has a sign problem at low temp.

Existing literature [TZR+19,ZGSP16,KMF+18] suggested that Fermi liquid was a little affected with a strong SOC enhancement via a fitting process.

Effective steps for imaginary time DMFT-MPS [time=18m10s]:

  • From Green’s function and self energy

  • Derive an effective Hamiltonian where the impurity is

  • Time evolution and then feed into the SCF loop

The tensor related steps are the calculation of the ground state, excitation, and then time evolution. Without spin orbit coupling there are no interactions and the calculation is fairly straightforward.

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Page5

Issues

However, the mapping of a bath to a 1D MPS is non-trivial and was disputed [WMSchollwock14].

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Page6

The star geometry is an order of magnitude more efficient. The star geometry has a growing potential, and the sites have a full/empty product state, and the impurity hopping causes some entanglement but is offset by the fact that the hoppings are energetically unfavorable. With the chain model, the potentials are constant, hopping is therefore cheap (intrabath) and effectively there is much more entanglement.

Time Scaling

In order to capture the behavior of the anticommutator, a higher resolution is needed for lower times.

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Page7 Page8

[time=23:16 for benchmarks]

Linear predictions for the time work as well, empirically [BSchollwockW09].

[time=24:00 for benchmarks]

Spin Orbit coupling

The bath fitting becomes much more difficult, due to interactions. From two self energies to 21 self energies. However, the diagonal self-energies are essentially unchanged, that is when the impurity jumps in and out of the bath, does not change the Fermi liquid behavior, as expected.

The off diagonals show agreement with ARPES, angular resolved photo electron spectroscopy signals. This now allows for a fit free comparison for real systems.

[time=25:00 for benchmarks]

Conclusions

  • MPS for realistic non-1D problems

  • Advanced complementary time evolution

  • Krylov+2TDVP (time stepper combinations)

  • MPS-DMFT allows a first principles treatment of SOC, reproduces CTQMC, is cheaper

References

BSchollwockW09

Thomas Barthel, Ulrich Schollwöck, and Steven R. White. Spectral functions in one-dimensional quantum systems at finite temperature using the density matrix renormalization group. Physical Review B, 79(24):245101, June 2009. doi:10.1103/PhysRevB.79.245101.

BMG+21

Max Bramberger, Jernej Mravlje, Martin Grundner, Ulrich Schollwöck, and Manuel Zingl. BaOsO\$_3\$: A Hund's metal in the presence of strong spin-orbit coupling. Physical Review B, 103(16):165133, April 2021. arXiv:2012.05597, doi:10.1103/PhysRevB.103.165133.

GK92

Antoine Georges and Gabriel Kotliar. Hubbard model in infinite dimensions. Physical Review B, 45(12):6479–6483, March 1992. doi:10.1103/PhysRevB.45.6479.

GKKR96

Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Reviews of Modern Physics, 68(1):13–125, January 1996. doi:10.1103/RevModPhys.68.13.

KBG+20

Jonathan Karp, Max Bramberger, Martin Grundner, Ulrich Schollwöck, Andrew J. Millis, and Manuel Zingl. \$\\mathrm\\vphantom \\Sr\vphantom \\\vphantom \\_\2\\\mathrm\\vphantom \\MoO\vphantom \\\vphantom \\_\4\\$ and \$\\mathrm\\vphantom \\Sr\vphantom \\\vphantom \\_\2\\\mathrm\\vphantom \\RuO\vphantom \\\vphantom \\_\4\\$: Disentangling the Roles of Hund's and van Hove Physics. Physical Review Letters, 125(16):166401, October 2020. doi:10.1103/PhysRevLett.125.166401.

KMF+18

Minjae Kim, Jernej Mravlje, Michel Ferrero, Olivier Parcollet, and Antoine Georges. Spin-Orbit Coupling and Electronic Correlations in \$\\mathrm\\vphantom \\Sr\vphantom \\\vphantom \\_\2\\\mathrm\\vphantom \\RuO\vphantom \\\vphantom \\_\4\\$. Physical Review Letters, 120(12):126401, March 2018. doi:10.1103/PhysRevLett.120.126401.

KSH+06

G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti. Electronic structure calculations with dynamical mean-field theory. Reviews of Modern Physics, 78(3):865–951, August 2006. doi:10.1103/RevModPhys.78.865.

LZH+20

Nils-Oliver Linden, Manuel Zingl, Claudius Hubig, Olivier Parcollet, and Ulrich Schollwöck. Imaginary-time matrix product state impurity solver in a real material calculation: Spin-orbit coupling in \$\mathrm\\vphantom \Sr\vphantom \\\\_\2\\mathrm\\vphantom \RuO\vphantom \\\\_\4\\$. Physical Review B, 101(4):041101, January 2020. doi:10.1103/PhysRevB.101.041101.

MM03

Andrew Peter Mackenzie and Yoshiteru Maeno. The superconductivity of \$\\mathrm\\vphantom \\Sr\vphantom \\\vphantom \\_\2\\\mathrm\\vphantom \\RuO\vphantom \\\vphantom \\_\4\\$ and the physics of spin-triplet pairing. Reviews of Modern Physics, 75(2):657–712, May 2003. doi:10.1103/RevModPhys.75.657.

TZR+19

A. Tamai, M. Zingl, E. Rozbicki, E. Cappelli, S. Riccò, A. de la Torre, S. McKeown Walker, F. Y. Bruno, P. D. C. King, W. Meevasana, M. Shi, M. Radović, N. C. Plumb, A. S. Gibbs, A. P. Mackenzie, C. Berthod, H. U. R. Strand, M. Kim, A. Georges, and F. Baumberger. High-Resolution Photoemission on \$\\mathrm\\vphantom \\Sr\vphantom \\\vphantom \\_\2\\\mathrm\\vphantom \\RuO\vphantom \\\vphantom \\_\4\\$ Reveals Correlation-Enhanced Effective Spin-Orbit Coupling and Dominantly Local Self-Energies. Physical Review X, 9(2):021048, June 2019. doi:10.1103/PhysRevX.9.021048.

WGM+15(1,2)

F. Alexander Wolf, Ara Go, Ian P. McCulloch, Andrew J. Millis, and Ulrich Schollwöck. Imaginary-Time Matrix Product State Impurity Solver for Dynamical Mean-Field Theory. Physical Review X, 5(4):041032, November 2015. doi:10.1103/PhysRevX.5.041032.

WMSchollwock14

F. Alexander Wolf, Ian P. McCulloch, and Ulrich Schollwöck. Solving nonequilibrium dynamical mean-field theory using matrix product states. Physical Review B, 90(23):235131, December 2014. doi:10.1103/PhysRevB.90.235131.

ZGSP16

Guoren Zhang, Evgeny Gorelov, Esmaeel Sarvestani, and Eva Pavarini. Fermi Surface of \$\\mathrm\\vphantom \\Sr\vphantom \\\vphantom \\_\2\\\mathrm\\vphantom \\RuO\vphantom \\\vphantom \\_\4\\$: Spin-Orbit and Anisotropic Coulomb Interaction Effects. Physical Review Letters, 116(10):106402, March 2016. doi:10.1103/PhysRevLett.116.106402.

Multigrid in Hierarchical Low Rank Tensor Formats Code