Theoretical Solid State Physics

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Real-frequency dynamical-mean-field-theory treatment of correlated lattice models

A major long-term challenge in condensed matter theory is the development of reliable computational tools for  quantitative materials research -- the ultimate goal being to start from ab initio methods for computing band structure  properties, while accurately accounting for correlation effects induced by interactions.

A widely-used method for dealing with interactions in strongly-correlated electron systems and electronic structure calculations is dynamical mean-field theory (DMFT) [1,2]. It treats the interplay between a given lattice site (the “impurity”) and the rest of the lattice (the “bath”) as a quantum impurity model with a self-consistently determined hybridization function. Since DMFT’s performance depends on that of the method used to solve this impurity model, much effort has been invested over the years to develop ever more powerful impurity solvers. For multi-band models, continuous-time quantum Monte Carlo (CTQMC) methods have long been the leading impurity solvers in terms of versatility and performance [3]. However, they are not without limitations: sign problems can occur, low-temperature calculations are costly, and obtaining real-frequency spectra requires analytic continuation of imaginary (Matsubara) frequency QMC data, which is notoriously difficult. Thus, there is a continued need for real frequency impurity solvers suitable for multi-band DMFT applications.

We have recently demonstrated that the numerical renormalization group (NRG) [4-6] is a viable multi-band impurity solver for DMFT, offering unprecedented real-frequency spectral resolution at arbitrarily low energies and temperatures [7]. Using DMG+NRG, we were able to clarify numerous open questions regarding a minimal model of a three-band “Hund metal” (relevant for studies of iron pnictide superconductors), which has both a Hubbard interaction U and ferromagnetic Hund coupling J [7-11]. Moreover, this project demonstrated that having access to real-frequency information is truly valuable for understanding the relevant physical processes at various different energy scales.

Our next step was to study more realistic multi-band models, involving less orbital symmetry [12] and more realistic modeling of material properties [13]. Though such models are more challenging than that studied in [7-11], their treatment was feasible using several recent  refinements in NRG methodology developed by our group [14-16].

I am currently looking for Master's and PhD students to continue this very promising line of research. In the medium term, we will aim to integrate our DMFT+NRG tools with ab initio band structure methods to calculate ac and dc transport properties in strongly correlated materials, such as high-temperature superconductors, iron pnictides, chalcogenides, ruthenates and other 4d transition metal oxides.

An important part of this endeavor will be to improve the efficiency of NRG to such an extent that models with 4 or 5 bands become accessible. Possible strategies for improving efficiency include mode optimization [17], or adopting techniques from two-dimensional tensor networks for encoding the entanglement more efficiently (isometric PEPS) [18], or using methods from machine learning for devising more compact effective Hamiltonians [19].

A Master's or PhD project in this field of research typically involves using our well-established DMFT+NRG technology to study the physics of an interesting multiband model with experimental relevance, while possibly also engaging in method development to improve the efficiency or scope of our methods. For Master's projects, the emphasis would be predominantly on model studies; for Ph.D. projects, method development would play a significant role, too.

Suitable candidates should have a good understanding of many-body physics and enjoy doing numerical work.

Master's project currently on offer: Quantum criticality in the 2D Hubbard model: a study via plaquette-DMFT+NRG

Guide to reading: to get a feel for how we do NRG, read [5]; for typical model studies, read [7,12]; for combining numerics with conformal field theory arguments, read [10]; for a realistic material study, read [13]; for method development studies, read [7,14]; for a lecture-style introduction to NRG (including lecture notes and videos), see my course on Tensor Networks.

[1] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg
Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions
Rev. Mod. Phys. 68, 13 (1996).

[2] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet and C. A. Marianetti
Electronic structure calculations with dynamical mean-field theory
Rev. Mod. Phys. 78, 865-951 (2006).

[3] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer and P. Werner
Continuous-time Monte~Carlo methods for quantum impurity models
Rev. Mod. Phys. 83, 349-404 (2011).

[4] A. Weichselbaum and J. von Delft
Sum-Rule Conserving Spectral Functions from the Numerical Renormalization Group
Phys. Rev. Lett. 99, 076402 (2007).

[5] A. Weichselbaum
Tensor networks and the numerical renormalization group
Phys. Rev. B 86, 245124 (2012).

[6] A. Weichselbaum
Non-abelian symmetries in tensor networks: A quantum symmetry space approach
Annals of Physics 327, 2972-3047 (2012).

[7] K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar and A. Weichselbaum
Dynamical Mean-Field Theory Plus Numerical Renormalization-Group Study of Spin-Orbital Separation in a Three-Band Hund Metal
Phys. Rev. Lett. 115, 136401 (2015).

[8] K. Stadler, G. Kotliar, A. Weichselbaum and J. von Delft
Hundness versus Mottness in a three-band Hubbard-Hund model: On the origin of strong correlations in Hund metals
Annals of Physics 405, 365 - 409 (2019).

[9] X. Deng, K. M. Stadler, K. Haule, A. Weichselbaum, J. von Delft and G. Kotliar
Signatures of Mottness and Hundness in archetypal correlated metals
Nature Communications 10, 2721 (2019).

[10] E. Walter, K. M. Stadler, S.-S. B. Lee, Y. Wang, G. Kotliar, A. Weichselbaum and J. von Delft
Uncovering non-Fermi-liquid behavior in Hund metals: conformal field theory analysis of a SU(2) × SU(3) spin-orbital Kondo model
arXiv:1908.04362 [cond-mat.str-el] (2019)

[11] Y. Wang, E. Walter, S.-S. B. Lee, K. M. Stadler, J. von Delft, A. Weichselbaum and G. Kotliar
Global Phase Diagram of a Spin-Orbital Kondo Impurity Model and the Suppression of Fermi-Liquid Scale
Phys. Rev. Lett. 124, 136406 (2020).

[12] F. B. Kugler, S.-S. B. Lee, A. Weichselbaum, G. Kotliar and J. von Delft
Orbital differentiation in Hund metals
Phys. Rev. B 100, 115159 (2019).

[13] F. B. Kugler, M. Zingl, H. U. R. Strand, S.-S. B. Lee, J. von Delft and A. Georges
Strongly Correlated Materials from a Numerical Renormalization Group Perspective: How the Fermi-Liquid State of $Sr_2RuO_4$ Emerges
Phys. Rev. Lett. 124, 016401 (2020).

[14] K. M. Stadler, A. K. Mitchell, J. von Delft and A. Weichselbaum
Interleaved numerical renormalization group as an efficient multiband impurity solver Phys. Rev. B 93, 235101 (2016).

[15] S.-S. B. Lee and A. Weichselbaum
Adaptive broadening to improve spectral resolution in the numerical renormalization group
Phys. Rev. B 94, 235127 (2016).

[16] B. Bruognolo, N.-O. Linden, F. Schwarz, S.-S. B. Lee, K. Stadler, A. Weichselbaum, M. Vojta, F. B. Anders and J. von Delft
Open Wilson chains for quantum impurity models: Keeping track of all bath modes
Phys. Rev. B 95, 121115 (2017).

[17] C. Krumnow, L. Veis, J. Eisert and Ö. Legeza
Dimension reduction with mode transformations: Simulating two-dimensional fermionic condensed matter systems

[18] M. P. Zaletel and F. Pollmann
Isometric Tensor Network States in Two Dimensions
Phys. Rev. Lett. 124, 037201 (2020).

[19] J. B. Rigo and A. K. Mitchell
Machine learning effective models for quantum systems