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arXiv:1902.05350 · Physical review. B./Physical review. B

Competing phases of interacting electrons on triangular lattices in moiré heterostructures

Laura Classen, Carsten Honerkamp A, Michael M. Scherer

Published May 10, 2019• 80 citations
arXivHTMLPDF

Abstract

We study the quantum many-body instabilities of interacting electrons with $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetry in spin and orbital degrees of freedom on the triangular lattice near van-Hove filling. Our work is motivated by effective models for the flat bands in hexagonal moir\'e heterostructures like twisted bilayer boron nitride and trilayer graphene-boron nitride systems. We consider an extended Hubbard model including onsite Hubbard and Hund's couplings, as well as nearest-neighbor exchange interactions, and analyze the different ordering tendencies with the help of an unbiased functional renormalization group approach. We find three classes of instabilities controlled by the filling and bare interactions. For a nested Fermi surface at van-Hove filling, Hund-like couplings induce a weak instability towards spin or orbital density wave phases. An SU(4) exchange interaction moves the system towards a Chern insulator, which is robust with respect to perturbations from Hund-like interactions or deviations from perfect nesting. Further, in an extended range of fillings and interactions, we find topological $d\ifmmode\pm\else\textpm\fi{}id$ and (spin-singlet)-(orbital-singlet) $f$-wave superconductivity.

DOI
10.1103/physrevb.99.195120

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We study the quantum many-body instabilities of interacting electrons with SU(2)\ifmmode×\else\texttimes\fiSU(2)\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)SU(2)\ifmmode×\else\texttimes\fiSU(2) symmetry in spin and orbital degrees of freedom on the triangular lattice near van-Hove filling. Our work is motivated by effective models for the flat bands in hexagonal moir'e heterostructures like twisted bilayer boron nitride and trilayer graphene-boron nitride systems. We consider an extended Hubbard model including onsite Hubbard and Hund's couplings, as well as nearest-neighbor exchange interactions, and analyze the different ordering tendencies with the help of an unbiased functional renormalization group approach. We find three classes of instabilities controlled by the filling and bare interactions. For a nested Fermi surface at van-Hove filling, Hund-like couplings induce a weak instability towards spin or orbital density wave phases. An SU(4) exchange interaction moves the system towards a Chern insulator, which is robust with respect to perturbations from Hund-like interactions or deviations from perfect nesting. Further, in an extended range of fillings and interactions, we find topological d\ifmmode±\else\textpm\fiidd\ifmmode\pm\else\textpm\fi{}idd\ifmmode±\else\textpm\fiid and (spin-singlet)-(orbital-singlet) fff-wave superconductivity.
A simple yet paradigmatic model for the interplay of strong electronic correlations and geometric frustration is the triangular lattice Hubbard model. Recently it was proposed that moir\'e structures of transition metal dichalcogenides can be used to simulate extended versions that include nonlocal density-density interactions. We study competing instabilities of interacting electrons in such an extended Hubbard model on the triangular lattice near a filling where the density of states has a Van Hove singularity. We employ a truncated-unity functional renormalization group approach to investigate two cases: a paradigmatic minimally extended Hubbard model and a specific model with parameters that are applicable to heterobilayers of transition metal dichalcogenides. We unravel rich phase diagrams, including tendencies to spin-density-wave order and unconventional pairing, which can give rise to topological superconductivity. We classify the symmetry of the superconducting instabilities according to their irreducible representations and show that higher lattice harmonics are dominant when the nearest-neighbor interaction is sizable indicating pair formation between second-nearest neighbors. The phenomenological consequences can be enhanced spin and thermal quantum Hall responses in a topological superconductor.
Abstract We study competing orders of spinless fermions in the triangular-lattice Hubbard model with nearest-neighbor interaction. We calculate the effective, momentum-resolved two-particle vertex in an unbiased way in terms of the functional renormalization group method and compare two different schemes for the momentum discretization, one based on dividing the Fermi surface into patches and one based on a channel decomposition. We study attractive and repulsive nearest-neighbor interaction and find a competition of pairing and charge instabilities. In the attractive case, a Pomeranchuk instability occurs at Van Hove filling and f-wave and p-wave pairing emerge when the filling is reduced. In the repulsive case, we obtain a charge density wave at Van Hove filling and extended p-wave pairing with reduced filling. The p-wave pairing solution is doubly degenerate and can realize chiral p+ipp+ipp+ip p + i p superconductivity with different Chern numbers in the ground state. We discuss implications for strongly correlated spin-orbit coupled hexagonal electron systems such as moiré heterostructures. Graphic Abstract
Several new classes of compounds can be modeled in first approximation by electrons on the triangular lattice that interact through on-site repulsion UUU as well as nearest-neighbor repulsion VVV. This extended Hubbard model on a triangular lattice has been studied mostly in the strong coupling limit for only a few types of instabilities. Using the extended two-particle self-consistent approach (ETPSC), that is valid at weak to intermediate coupling, we present an unbiased study of the density and interaction dependent crossover diagram for spin- and charge-density wave instabilities of the normal state at arbitrary wave vector. When UUU dominates over VVV and electron filling is large, instabilities are chiefly in the spin sector and are controlled mostly by Fermi surface properties. Increasing VVV eventually leads to charge instabilities. In the latter case, it is mostly the wave vector dependence of the vertex that determines the wave vector of the instability rather than Fermi surface properties. At small filling, nontrivial instabilities appear only beyond the weak coupling limit. There again, charge-density wave instabilities are favored over a wide range of dopings by large VVV at wave vectors corresponding to $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ superlattice in real space. Commensurate fillings do not play a special role for this instability. Increasing UUU leads to competition with ferromagnetism. At negative values of UUU or VVV, neglecting superconducting fluctuations, one finds that charge instabilities are favored. In general, the crossover diagram presents a rich variety of instabilities. We also show that thermal charge-density wave fluctuations in the renormalized-classical regime can open a pseudogap in the single-particle spectral weight, just as spin or superconducting fluctuations.
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Interacting fermions on the half-filled honeycomb lattice with short-range repulsions have been suggested to host a variety of interesting many-body ground states, e.g., a topological Mott insulator. A number of recent studies of the spinless case in terms of exact diagonalization, the infinite density matrix renormalization group, and the functional renormalization group, however, indicate a suppression of the topological Mott insulating phase in the whole range of interaction parameters. Here, we complement the previous studies by investigating the quantum many-body instabilities of the physically relevant case of spin-1/2 fermions with onsite, nearest-neighbor, and second-nearest-neighbor repulsion. To this end, we employ the multipatch functional renormalization group for correlated fermions with refined momentum resolution observing the emergence of an antiferromagnetic spin-density wave and a charge-density wave for dominating onsite and nearest-neighbor repulsions, respectively. For dominating second-nearest neighbor interaction our results favor an ordering tendency towards a charge-modulated ground state over the topological Mott insulating state. The latter evades a stabilization as the leading instability by the additional onsite interaction.
The phase diagram of the two-dimensional extended one-band $U\ensuremath{-}V\ensuremath{-}J$ Hubbard model is considered within a mean-field approximation and two- and many-patch renormalization-group (RG) approaches near the Van Hove band fillings. At small ${t}^{\ensuremath{'}}$ and J>0,J>0,J>0, mean-field and many-patch RG approaches give similar results for the leading spin-density-wave (SDW) instability, while the two-patch RG approach that predicts a wide region of charge-flux (CF) phase becomes unreliable due to nesting effect. There is a complex competition between SDW, CF phases, and d-wave superconductivity at small ${t}^{\ensuremath{'}}$ in two- and many-patch RG approaches. The spin-flux phase, which is not stable at the mean-field level, is identified as a possible ground state at J<0J<0J<0 in both RG approaches. With increasing ${t}^{\ensuremath{'}}$ the results of all three approaches merge: d-wave superconductivity at J>0J>0J>0 and ferromagnetism at J<0J<0J<0 become the leading instabilities. For large enough V, the charge-density-wave state occurs.