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arXiv:1903.10452 · Physical review. B./Physical review. B

Impact of strong correlations on a band topological insulator on the Lieb lattice

Sayan Jana, Arijit Saha, Anamitra Mukherjee

Published Jul 24, 2019• 13 citations
arXivHTMLPDF

Abstract

The Lieb lattice possesses three bands and, with intrinsic spin-orbit coupling $\ensuremath{\lambda}$, supports topologically nontrivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction $U$ on all sites. Within a slave rotor mean-field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (${U}_{\text{crit}}$), creating a correlated flat band insulator. Beyond ${U}_{\text{crit}}$, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in the presence of strong correlation. Furthermore, in the limiting case of $U$ introduced only on one sublattice where $\ensuremath{\lambda}=0$, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.

DOI
10.1103/physrevb.100.045420

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The Lieb lattice possesses three bands and, with intrinsic spin-orbit coupling \ensuremathλ\ensuremath{\lambda}\ensuremathλ, supports topologically nontrivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction UUU on all sites. Within a slave rotor mean-field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (Ucrit{U}_{\text{crit}}Ucrit​), creating a correlated flat band insulator. Beyond Ucrit{U}_{\text{crit}}Ucrit​, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in the presence of strong correlation. Furthermore, in the limiting case of UUU introduced only on one sublattice where \ensuremathλ=0\ensuremath{\lambda}=0\ensuremathλ=0, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.
The Lieb lattice possesses three bands and with intrinsic spin orbit coupling λ\lambdaλ, supports topologically non-trivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction UUU on all sites. Within a slave rotor mean field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (UcritU_{crit}Ucrit​), creating a correlated flat band insulator. Beyond UcritU_{crit}Ucrit​, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in presence of strong correlation. Furthermore, in the limiting case of UUU introduced only on one sublattice where λ=0\lambda=0λ=0, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.
The single-orbital Hubbard model is often a good description of the low energy dynamics of strongly correlated electron systems. Importantly, the single-orbital Hubbard model at half-filling is able to capture the Mott metal-insulator transition that occurs in many materials. However, there are insulating phases (as well as other exotic phases) that occur due to strong electronic correlations in materials that are away from half-filling. In many cases, these additional phases occur because the constituent electrons of the material come from multi-orbital atomic environments that require more complicated descriptions such as the multi-orbital Hubbard model. In other cases, as we show in this thesis, the rich phase diagram arises from the underlying lattice because of special geometric features.For example, decorated lattices, which often occur in organometallics (e.g., metal-organic frameworks (MOFs)), have geometrical complexity due to the molecular structures that decorate the lattice. We argue that the effective description for many of these materials is the single-orbital Hubbard model. Yet, these materials still have rich phase diagrams because the molecular structures of the material cause effective multi-orbital interactions generated from the local Coulomb interaction.We investigate the single-orbital Hubbard model on the decorated honeycomb lattice at zero temperature using rotationally invariant slave-boson (RISB) theory in the mean-field approximation, at a range of fillings. Numerous metal-insulator transitions occur - the insulating phases include a real-space Mott insulator, a dimer Mott insulator, a trimer Mott insulator, flat-band ferromagnets, and a spin-1 Mott insulator, dominated by spin-triplet fluctuations. This menagerie of phases is caused by effective multi-orbital interactions arising from the interplay of electronic correlations, lattice structure, and band filling effects. For example, with only on-site Coulomb interactions, the spin-1 Mott insulator is driven by an effective Hund's coupling on a three-site ring that favors the formation of triplets when the three-site ring has four electrons. We argue that this exotic phase may be realized in layers of the organometallic molecular crystal Mo3S7(dmit)3.
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction UUU is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``CuO2{\mathrm{CuO}}_{2}CuO2​ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``ddd orbitals'' occupy the vertices of the squares, while ``ppp orbitals'' lie halfway between two ddd orbitals; both ddd and ppp orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of UUU and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up{U}_{d}={U}_{p}Ud​=Up​, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all UUU. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0,Ud=U{U}_{p}=0,{U}_{d}=UUp​=0,Ud​=U and Up=U,Ud=0{U}_{p}=U,{U}_{d}=0Up​=U,Ud​=0. We found that the different environments of ddd and ppp sites lead to a ferromagnetic insulator when Ud=0{U}_{d}=0Ud​=0; by contrast, Up=0{U}_{p}=0Up​=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each ddd site.
The physics of strongly correlated quantum particles within a flat band was originally explored as a route to itinerant ferromagnetism and, indeed, a celebrated theorem by Lieb rigorously establishes that the ground state of the repulsive Hubbard model on a bipartite lattice with an unequal number of sites in each sublattice must have nonzero spin SSS at half filling. Recently, there has been interest in Lieb geometries due to the possibility of topological insulator, nematic, and Bose-Einstein condensed (BEC) phases. In this paper, we extend the understanding of the attractive Hubbard model on the Lieb lattice by using determinant quantum Monte Carlo to study real space charge and pair correlation functions not addressed by the Lieb theorems. Specifically, our results show unusual charge and charge transfer signatures within the flat band, and a reduction in pairing order at $\ensuremath{\rho}=2/3$ and $\ensuremath{\rho}=4/3$, the points at which the flat band is first occupied and then completely filled. We compare our results to the case of flat bands in the Kagome lattice and demonstrate that the behavior observed in the two cases is rather different.
We investigate the metal-insulator transition driven by the onsite repulsive interaction UUU in an altermagnetic Hubbard model defined on a Lieb lattice. Using the slave-rotor approach at half-filling, we find that the system exhibits a cascade of interaction-driven phase transitions. As UUU increases, the system evolves from a normal metal to an altermagnetic metal, then to an altermagnetic insulator, and eventually to an altermagnetic Mott insulator characterized by the complete suppression of the quasiparticle weight. These phases are supported by the calculation of the electronic spectral function, which features spin-split bands in both the metallic and insulating regimes. However, the spin splitting becomes substantially suppressed in the Mott-insulating phase. Our results suggest that the observation of spin splitting in the spectral function of ddd-wave altermagnets with a Lieb-lattice-like structure may be limited to the weak-to-moderate correlation regime.
We study the effects of electronic correlations on fragile topology using dynamical mean-field theory. Fragile topological insulators (FTIs) offer obstruction to the formation of exponentially localized Wannier functions, but they can be trivialized by adding certain trivial degrees of freedom. For the same reason, FTIs do not host symmetry-protected flow of edge states between bulk bands in cylindrical boundary conditions but are expected to have a spectral flow between the fragile bands and other bands under certain twisted boundary conditions. We here analyze commonly observed effects of strong correlations, such as the Mott insulator transition and magnetism, on a known model hosting fragile topology. We show that in the nonmagnetic case, fragile topology, along with the twisted boundary states, is stable with interactions below a critical interaction strength. Above this interaction strength, a transition to the Mott insulating phase occurs, and the twisted boundary states disappear. Furthermore, by applying a homogeneous magnetic field, the fragile topology is destroyed. However, we show that a magnetic field can induce a topological phase transition which converts a fragile topological insulator to a Chern insulator. Finally, we study ferromagnetic solutions of the fragile topological model.