Pushing the frontier of simulations of two-dimensional multi-orbital fermionic lattice and Heisenberg models by fully exploiting non-abelian symmetries in tensor network states

• Applicant: Andreas Weichselbaum, Ph.D.
• Subject Area: Theoretical Condensed Matter Physics
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 336042529

• Final Report Year: 2021

Final Report Abstract

The project focused on strongly-correlated physical lattice models with nonabelian symmetries. By using the framework of tensor network states (TNSs) this allowed us to tackle quantum many-body effects in a numerically exact manner. Simulations were carried out on the proven machinery of the QSpace tensor library developed by the PI in prior years. QSpace can deal with arbitrary combinations of abelian and generic non-abelian symmetries including all classical simple Lie algebras. Tensor network (TN) simulations generally follow two strategies. (i) a variational approach to obtain low-energy or ground state properties. (ii) real-time or imaginary time evolution to simulate the dynamics (or thermodynamics) of a well-prepared state. At the start of the project, we reexamined the standard approach for thermal simulations, which are non-variational but can be expressed as imaginary time evolution in terms of the inverse temperature β = 1/T , starting from β = 0, i.e., infinite temperature. The widely used approach in this context is based on Trotterization which requires a small Trotter step dβ, giving rise to a Trotter error on a linear β grid. However, we realized that there exists a much more efficient and direct approach to reach low-temperatures in an exponential fashion (i.e., on a logarithmic β grid) by systematically doubling a given density matrix with itself. This point of view was strongly motivated also by analytical arguments in the literature on entanglement scaling, as well as the analogies it resembled to the numerical renormalization group (NRG) which also deals with exponential energy scales. Overall, this led us to introduce the exponential thermal renormalization group (XTRG) early on in the project and which we then also used extensively throughout the project later on. Considerable effort was attributed to the thermal properties of the triangular spin-half Heisenberg lattice. Here we addressed contradictory statements in the literature based on quantum Monte Carlo simulations that found it difficult to extrapolate to the ground state 120◦ phase when lowering temperature. By using our XTRG approach with the SU(2) spin symmetry fully incorporated, this permitted us to reconcile these discrepancies. In particular, we argued in favor of a second low-energy scale that we linked to massive quasiparticles also referred to as rotons in spin wave theory. This second energy scales considerably delays the onset of the incipient 120◦ in the ground state and thus must be taken into account in the extrapolation towards zero temperature. A major part of the project also focussed on the low-energy properties of the 2D Hubbard model at half-filling. There we targeted the intermediate range for the Hubbard interaction U/t ∼ 9 where it was argued that a chiral topological phase may exist. This was supported by infinite density matrix renormalization group (iDMRG) simulations, yet contradicted by finite size DMRG. Based on our finite-size DMRG fully exploiting the SU(2) spin symmetry, we showed that surprisingly large system sizes were needed to eventually see the low-energy gapped chiral phase emerge. This reconciled the earlier findings with strong support for a chiral topological spin liquid. The project also dealt with fermionic projected entangled pair states (fPEPS) based on earlier exploratory work. We focused on the ground state of the tJ model, with our numerical approach also summarized in a SciPost lecture note. We performed a systematic study of the effects of turning on and off the SU(2) spin symmetry in the PEPS simulation which permitted us to shed new light on the spontaneous symmetry breaking towards antiferromagnetic order at low doping.

Publications

• “Exponential thermal tensor network approach for quantum lattice models”, Phys. Rev. X 8, 031082 (2018)
  B.-B. Chen, L. Chen, Z. Chen, W. Li, and A. Weichselbaum
  
• “Nontopological Majorana zero modes in inhomogeneous spin ladders”, Phys. Rev. Lett. 122, 027201 (2019)
  N. J. Robinson, A. Altland, R. Egger, N. M. Gergs, W. Li, D. Schuricht, A. M. Tsvelik, A. Weichselbaum, and R. M. Konik
  
• “Thermal tensor renormalization group simulations of square-lattice quantum spin models”, Phys. Rev. B 100, 045110 (2019)
  H. Li, B.-B. Chen, Z. Chen, J. von Delft, A. Weichselbaum, and W. Li
  
• “Two-temperature scales in the triangular-lattice Heisenberg antiferromagnet”, Phys. Rev. B 99, 140404(R) (2019)
  L. Chen, D.-W. Qu, H. Li, B.-B. Chen, S.-S. Gong, J. von Delft, A. Weichselbaum, and W. Li
  
• “A beginner’s guide to non-abelian iPEPS for correlated fermions”, SciPost Phys. Lect. Notes, 25 (2021)
  B. Bruognolo, J.-W. Li, J. v. Delft, and A. Weichselbaum
  
• “Quantum many-body simulations of the two-dimensional Fermi-Hubbard model in ultracold optical lattices”, Phys. Rev. B 103, L041107 (2021)
  B.-B. Chen, C. Chen, Z. Chen, J. Cui, Y. Zhai, A. Weichselbaum, J. von Delft, Z. Y. Meng, and W. Li
  
• “Study of spin symmetry in the doped t-j model using infinite projected entangled pair states”, Phys. Rev. B 103, 075127 (2021)
  J.-W. Li, B. Bruognolo, A. Weichselbaum, and J. von Delft