Ground-state order in magic-angle graphene at filling $\ensuremath{\nu}=\ensuremath{-}3$: A full-scale density matrix renormalization group study

Ground-state order in magic-angle graphene at filling $ν = - 3$ : A full-scale density matrix renormalization group study

Tianle Wang, Daniel E. Parker, Tomohiro Soejima (副島智大), Johannes Hauschild, Sajant Anand, Nick Bultinck, and Michael P. Zaletel

Phys. Rev. B 108, 235128 – Published 8 December 2023

Abstract

We investigate twisted bilayer graphene (TBG) at filling $ν = - 3$ in the presence of realistic heterostrain. Strain amplifies the band dispersion and drives the system beyond the strong-coupling regime of previous theoretical studies. We use DMRG to conduct an unbiased, large-scale numerical calculations that include all spin and valley degrees of freedom, up to bond dimension $χ = 24 576$ . We establish a global phase diagram that unifies a number of theoretical and experimental results. Near zero strain we find an intervalley-coherent quantized anomalous Hall (QAH-IVC) state, a competitive strong-coupling order that evaded past numerical studies. A tiny strain around $0.05 %$ drives a transition into an incommensurate Kekulé spiral (IKS) phase, supporting the mean-field prediction in [Kwan et al., Phys. Rev. X 11, 041063 (2021)]. Even higher strains above $0.2 %$ favor a flavor-symmetric metallic order, which may explain metals found at $ν = - 3$ in many experiments.

Received 28 November 2022
Revised 28 September 2023
Accepted 5 October 2023

DOI:https://doi.org/10.1103/PhysRevB.108.235128

Physics Subject Headings (PhySH)

Fermi surface Flat bands Symmetry breaking states

Twisted bilayer graphene

Density matrix renormalization group

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tianle Wang^1,2, Daniel E. Parker ³, Tomohiro Soejima (副島智大)¹, Johannes Hauschild⁴, Sajant Anand ¹, Nick Bultinck^5,6, and Michael P. Zaletel^1,2

¹Department of Physics, University of California, Berkeley, California 94720, USA
²Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
³Department of Physics, Harvard University, Cambridge, Massachusetts 02139, USA
⁴Department of Physics, Technische Universität München, 85748 Garching, Germany
⁵Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
⁶Department of Physics, Ghent University, 9000 Ghent, Belgium

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Vol. 108, Iss. 23 — 15 December 2023

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Figure 1
(a) Moiré pattern from two graphene lattices with $1^{\circ}$ relative twist and $ɛ_{Gr} = 0.5 %$ uniaxial heterostrain. Strain is amplified at the superlattice scale, significantly distorting the moire unit cell. (b) Noninteracting bandstructure of TBG with $ɛ_{Gr} = 0.3 %$ heterostrain. Heterostrain shifts the Dirac nodes $D_{1, 2}$ close to the $Γ$ point (inset). (c) Phase diagram of TBG at $| ν | = 3$ obtained from DMRG up to bond dimension $χ = 24 576$ . The phase transitions are identified with their characteristic order parameters: Chern band polarization $O_{QAH}$ , ferromagnetic order magnitude $Σ_{FM}$ , intervalley coherence wave vector $q_{IVC}$ , and deviation of valley-summed electron density from uniform filling $δ N (q_{IVC})$ (see main text for the definition). (d) Valley-nested DMRG electron occupations of the first Brillouin zone in the IKS phase (parameters match Fig. 2, $L_{y} = 6$ ). The relative shift $q_{IKS}$ makes the total occupations uniform while $Γ$ is depleted in both valleys.
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Figure 2
IKS at $ɛ_{Gr} = 0.05 %$ . (a), (b) Valley-resolved electron density of TBG at $ν = - 3$ . Dashed hexagons denote the first Brillouin zone, and the dot in the middle is the $Γ$ point. (c) Total electron density $N_{q_{IKS}} (k)$ in Eq. (12), after a relative boost by $q_{IKS}$ , whereupon the density becomes uniform. (d) Fourier transform of the IKS correlation function. The peak at $q \approx 0.80 π$ matches $q_{IKS} \cdot a_{1}$ from (c). (e) Scaling analysis of correlation lengths of several order parameters, showing that IVC and FM dominate at large bond dimensions $χ > 16 384$ .
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Figure 3
FL at $ɛ_{Gr} = 0.2 %$ . (a), (b) Spin-up electron density in $K^{(')}$ valley, plotted similarly to Fig. 2. The spin-down sector is related by the spin-flip symmetry $s^{x}$ and shows identical occupations. Dotted lines in (a) circled the Fermi surfaces predicted from fully symmetric SCHF. (c) Energy $E (k)$ of the partially occupied band from SCHF. The electron only occupies the regions circled in solid lines, forming sharp Fermi surfaces. (d) The DMRG electron occupations in $(K, ↑)$ sector along the $k_{y} = \frac{π}{2}$ wire at various bond dimensions, where the filled/emptied regions closely matches that of SCHF (dashed lines).
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Figure 4
(a) DMRG energies in all four sectors as a function of heterostrain $ɛ_{Gr} \leq 0.2 %$ . The (1,0) sector was not computed beyond $ɛ_{Gr} = 0.2 %$ , but is expected to be above (0,0) and (0,1) in energy. (b) The energy of different flavor sectors as a function of $χ$ at $ɛ_{Gr} = 0.2 %$ . (c) Electron density per momentum near $Γ$ : $n (k \sim Γ) = \sum_{| k | < 1 / 5 | g_{1} |} n (k)$ . Parameters match Fig. 2.
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Physical Review B

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Ground-state order in magic-angle graphene at filling $ν = - 3$ : A full-scale density matrix renormalization group study

Tianle Wang, Daniel E. Parker, Tomohiro Soejima (副島智大), Johannes Hauschild, Sajant Anand, Nick Bultinck, and Michael P. Zaletel

Phys. Rev. B 108, 235128 – Published 8 December 2023

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Physical Review B

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Ground-state order in magic-angle graphene at filling ν=−3: A full-scale density matrix renormalization group study

Tianle Wang, Daniel E. Parker, Tomohiro Soejima (副島智大), Johannes Hauschild, Sajant Anand, Nick Bultinck, and Michael P. Zaletel

Phys. Rev. B 108, 235128 – Published 8 December 2023

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Ground-state order in magic-angle graphene at filling $ν = - 3$ : A full-scale density matrix renormalization group study