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Oskar Vafek

Publications and source records attributed to Oskar Vafek.

At least 37 records · Page 2Linked to original sources
arXiv2023

Fractional Chern Insulators vs. Non-Magnetic States in Twisted Bilayer MoTe$_2$

Fractionally filled Chern bands with strong interactions may give rise to fractional Chern insulator (FCI) states, the zero-field analogue of the fractional quantum Hall effect. Recent experiments have demonstrated the existence of FCIs in twisted bilayer MoTe$_2$ without external magnetic fields -- most robust at $ν=-2/3$ -- as well as Chern insulators (CIs) at $ν=-1$. Although the appearance of both of these states is theoretically natural in an interacting topological system, experiments repeatedly observe nonmagnetic states (lacking FCIs) at $ν=-1/3$ and $-4/3$, a puzzling result which has not been fully theoretically explained. In this work, we perform Hartree-Fock and exact diagonalization calculations to test whether the standard MoTe$_2$ moiré model with the (greatly varying) parameter values available in the literature can reproduce the non-magnetic states at $ν=-1/3$ and $-4/3$ in unison with the FCI at $ν=-2/3$ and CI state at $ν= -1$. We focus on the experimentally relevant twist angles and, crucially, include remote bands. We find that the parameters proposed in [Wang et al. (2023)] can nearly capture the experimental phenomena at $ν=-1/3,-2/3,-1,-4/3$ simultaneously, though the predicted ground states at $ν=-1/3$ are still mostly fully-spin-polarized and a larger dielectric constant $ε>10$ than is typical of hexagonal boron nitride (h-BN) substrate $ε\sim 6$ is required. Our results show the importance of remote bands in identifying the competing magnetic orders and lay the groundwork for further study of the realistic phase diagram.

cond-mat.mes-hall↗
arXiv2023

Triangular lattice Hubbard model physics at intermediate temperatures

Moire systems offer an exciting playground to study many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on $\text{WSe}_2/\text{WS}_2$ moire bilayers [Y. Tang et al., Nature 579, 353-358 (2020)], which realizes a triangular superlattice with a small hopping (of approximately 10 Kelvin), with tunable density of holes, we explore the Hubbard model on the triangular lattice for intermediate temperatures $t \lesssim T < U$. Employing finite temperature Lanczos calculations, and closely following the fitting protocols used in the experiment, we recover the observed trends in the reported Curie-Weiss temperature $Θ$ with filling, using the reported interaction strength $U/t=20$. We focus on the large increase of $|Θ|$ on decreasing the density below half filling and the sign change of $Θ$ at higher fillings, which signals the onset of ferromagnetism. The increase in $|Θ|$ is also seen in the $t$-$J$ model (the low energy limit of the Hubbard model) in the intermediate temperature range, which we clarify is opposite to the trend in its high temperature limit. Differences between the low, intermediate and high temperature regimes are discussed. Our numerical calculations also capture the crossover between short-range antiferromagnetic to ferromagnetic order in the intermediate temperature regime, a result broadly consistent with the experimental findings. We find that this behavior is a finite-temperature remnant of the underlying zero temperature phase transition, which we explore with ground state density matrix renormalization group calculations. We provide evidence of ferromagnetism characterized by weak (but robust) correlations that explain the small $Θ$ seen in the experiment.

cond-mat.str-el↗
arXiv2023

Twisted-bilayer FeSe and the Fe-based superlattices

We derive BM-like continuum models for the bands of superlattice heterostructures formed out of Fe-chalcogenide monolayers: (${\bf\text I}$) a single monolayer experiencing an external periodic potential, and (${\bf\text II}$) twisted bilayers with long-range moire tunneling. A symmetry derivation for the inter-layer moire tunnelling is provided for both the $Γ$ and $M$ high-symmetry points. In this paper, we focus on moire bands formed from hole-band maxima centered on $Γ$, and show the possibility of moire bands with $C=0$ or $\pm 1$ topological quantum numbers without breaking time-reversal symmetry. In the $C=0$ region for $θ\rightarrow 0$ (and similarly in the limit of large superlattice period for ${\bf\text I}$), the system becomes a square lattice of 2D harmonic oscillators. We fit our model to FeSe and argue that it is a viable platform for the simulation of the square Hubbard model with tunable interaction strength.

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arXiv2023

Phase diagram of twisted bilayer graphene at filling factor $ν=\pm3$

We study the correlated insulating phases of twisted bilayer graphene (TBG) in the absence of lattice strain at integer filling $ν=\pm3$. Using the self-consistent Hartree-Fock method on a particle-hole symmetric model and allowing translation symmetry breaking terms, we obtain the phase diagram with respect to the ratio of $AA$ interlayer hopping $(w_0)$ and $AB$ interlayer hopping $(w_1)$. When the interlayer hopping ratio is close to the chiral limit ($w_0/w_1 \lesssim 0.5$), a quantum anomalous Hall state with Chern number $ν_c = \pm 1$ can be observed consistent with previous studies. Around the realistic value $w_0/w_1 \approx 0.8$, we find a spin and valley polarized, translation symmetry breaking, state with $C_{2z}T$ symmetry, a charge gap and a doubling of the moiré unit cell, dubbed the $C_{2z}T$ stripe phase. The real space total charge distribution of this $C_{2z}T$ stripe phase in the flat band limit does not have modulation between different moiré unit cells, although the charge density in each layer is modulated, and the translation symmetry is strongly broken. Other symmetries, including $C_{2z}$, $C_{2x}$ and particle-hole symmetry $P$, and the topology of the $C_{2z}T$ stripe phase are also discussed in detail. We observed braiding and annihilation of the Dirac nodes by continuously turning on the order parameter to its fully self-consistent value, and provide a detailed explanation of the mechanism for the charge gap opening despite preserving $C_{2z}T$ and valley $U(1)$ symmetries. In the transition region between the quantum anomalous Hall phase and the $C_{2z}T$ stripe phase, we find an additional competing state with comparable energy corresponding to a phase with a tripling of the moiré unit cell.

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arXiv2023

Pseudo-magnetic fields, particle-hole asymmetry, and microscopic effective continuum Hamitonians of twisted bilayer graphene

Using the method developed in the companion paper, we construct the effective continuum theories for two different microscopic tight binding models of the twisted bilayer graphene at the twist angle of $1.05^\circ$, one Slater-Koster based and the other ab-initio Wannier based. The energy spectra obtained from the continuum theory -- either for rigid twist or including lattice relaxation -- are found to be in nearly perfect agreement with the spectra from the tight binding models when the gradient expansion is carried out to second order, demonstrating the validity of the method. We also analyze the properties of the Bloch states of the resulting narrow bands, finding non-negligible particle-hole symmetry breaking near the $Γ$ point in our continuum theory constructed for the ab-initio based microscopic model due to a term in the continuum theory that was previously overlooked. This reveals the difference with all existing continuum models where the particle-hole symmetry of the narrow band Hilbert space is nearly perfect.

cond-mat.str-el↗
arXiv2023

Continuum effective Hamiltonian for graphene bilayers for an arbitrary smooth lattice deformation from microscopic theories

We provide a systematic real space derivation of the continuum Hamiltonian for a graphene bilayer starting from a microscopic lattice theory, allowing for an arbitrary inhomogeneous smooth lattice deformation, including a twist. Two different microscopic models are analyzed: first, a Slater-Koster like model and second, ab-initio derived model. We envision that our effective Hamiltonian can be used in conjunction with an experimentally determined atomic lattice deformation in twisted bilayer graphene in a specific device to predict and compare the electronic spectra with scanning tunneling spectroscopy measurements. As a byproduct, our approach provides electron-phonon couplings in the continuum Hamiltonian from microscopic models for any bilayer stacking. In the companion paper we analyze in detail the continuum models for relaxed atomic configurations of magic angle twisted bilayer graphene.

cond-mat.str-el↗
arXiv2022

Electronic anisotropy in magic-angle twisted trilayer graphene

Due to its potential connection with nematicity, electronic anisotropy has been the subject of intense research effort on a wide variety of material platforms. The emergence of spatial anisotropy not only offers a characterization of material properties of metallic phases, which cannot be accessed via conventional transport techniques, but it also provides a unique window into the interplay between Coulomb interaction and broken symmetry underlying the electronic order. In this work, we utilize a new scheme of angle-resolved transport measurement (ARTM) to characterize electron anisotropy in magic-angle twisted trilayer graphene. By analyzing the dependence of spatial anisotropy on moiré band filling, temperature and twist angle, we establish the first experimental link between electron anisotropy and the cascade phenomenon, where Coulomb interaction drives a number of isospin transitions near commensurate band fillings. Furthermore, we report the coexistence between electron anisotropy and a novel electronic order that breaks both parity and time reversal symmetry. Combined, the link between electron anisotropy, cascade phenomenon and PT-symmetry breaking sheds new light onto the nature of electronic order in magic-angle graphene moiré systems.

cond-mat.mes-hall↗
arXiv2022

Unusual magnetotransport in twisted bilayer graphene from strain-induced open Fermi surfaces

Anisotropic hopping in a toy Hofstadter model was recently invoked to explain a rich and surprising Landau spectrum measured in twisted bilayer graphene away from the magic angle. Suspecting that such anisotropy could arise from unintended uniaxial strain, we extend the Bistritzer-MacDonald model to include uniaxial heterostrain. We find that such strain strongly influences band structure, shifting the three otherwise-degenerate van Hove points to different energies. Coupled to a Boltzmann magnetotransport calculation, this reproduces previously-unexplained non-saturating $B^2$ magnetoresistance over broad ranges of density near filling $ν=\pm 2$, and predicts subtler features that had not been noticed in the experimental data. In contrast to these distinctive signatures in longitudinal resistivity, the Hall coefficient is barely influenced by strain, to the extent that it still shows a single sign change on each side of the charge neutrality point -- surprisingly, this sign change no longer occurs at a van Hove point. The theory also predicts a marked rotation of the electrical transport principal axes as a function of filling even for fixed strain and for rigid bands. More careful examination of interaction-induced nematic order versus strain effects in twisted bilayer graphene could thus be in order.

cond-mat.mes-hall↗
arXiv2022

Spectroscopy of Twisted Bilayer Graphene Correlated Insulators

We analytically compute the scanning tunneling microscopy (STM) signatures of integer-filled correlated ground states of the magic angle twisted bilayer graphene (TBG) narrow bands. After experimentally validating the strong-coupling approach at $\pm 4$ electrons/moiré unit cell, we consider the spatial features of the STM signal for 14 different many-body correlated states and assess the possibility of Kekulé distortion (KD) emerging at the graphene lattice scale. Remarkably, we find that coupling the two opposite graphene valleys in the intervalley-coherent (IVC) TBG insulators does not always result in KD. As an example, we show that the Kramers IVC state and its nonchiral $\mathrm{U} \left( 4 \right)$ rotations do not exhibit any KD, while the time-reversal-symmetric IVC state does. Our results, obtained over a large range of energies and model parameters, show that the STM signal and Chern number of a state can be used to uniquely determine the nature of the TBG ground state.

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arXiv2022

Trions in Twisted Bilayer Graphene

The strong coupling phase diagram of magic angle twisted bilayer graphene (TBG) predicts a series of exact one particle charge $\pm 1$ gapped excitations on top of the integer-filled ferromagnetic ground-states. Finite-size exact diagonalization studies showed that these are the lowest charge $\pm 1$ excitations in the system (for $10$nm screening length), with the exception of charge $+1$ at filling $-1$ in the chiral limit. In the current paper we show that this "trion bound state", a $3$-particle, charge $1$ excitation of the insulating ferromagnetic ground-state of the projected Hamiltonian of TBG is the lowest charge $+1$ overall excitation at $ν=-1$, and also for some large ($\approx 20$nm) screening lengths at $ν=-2$ in the chiral limit and with very small binding energy. At other fillings, we show that trion bound states do exist, but only for momentum ranges that do not cover the entire moiré Brillouin zone. The trion bound states (at different momenta) exist for finite parameter range $w_0/w_1$ but they all disappear in the continuum far below the realistic values of $w_0/w_1= 0.8$. We find the conditions for the existence of the trion bound state, a good variational wavefunction for it, and investigate its behavior for different screening lengths, at all integer fillings, on both the electron and hole sides.

cond-mat.str-el↗
arXiv2022

Narrow bands in magnetic field and strong-coupling Hofstadter spectra

We develop a new, efficient, and general method to determine the Hofstadter spectrum of isolated narrow bands. The method works for topological as well as for trivial narrow bands by projecting the zero $\mathbf{B}$-field hybrid Wannier states -- which are localized in one direction and Bloch extended in another direction -- onto a representation of the magnetic translation group in the Landau gauge. We then apply this method to find, for the first time, the Hofstadter spectrum for the exact single particle charged excitations in the strong coupling limit of the magic angle twisted bilayer graphene at the charge neutrality point and at $|ν|=2$ down to low magnetic fields when the flux through the moiré unit cell is only $\sim 1/25$ of the electronic flux quantum i.e. $\sim 1$T at the first magic angle. The resulting spectra provide a means to investigate Landau quantization of the quasiparticles even if their dispersion is interaction induced.

cond-mat.str-el↗
arXiv2021

Realization of Topological Mott Insulator in a Twisted Bilayer Graphene Lattice Model

Magic-angle twisted bilayer graphene has recently become a thriving material platform realizing correlated electron phenomena taking place within its topological flat bands. Several numerical and analytical methods have been applied to understand the correlated phases therein, revealing some similarity with the quantum Hall physics. In this work, we provide a Mott-Hubbard perspective for the TBG system. Employing the large-scale density matrix renormalization group on the lattice model containing the projected Coulomb interactions only, we identify a first-order quantum phase transition between the insulating stripe phase and the quantum anomalous Hall state with the Chern number of $\pm 1$. Our results not only shed light on the mechanism of the quantum anomalous Hall state discovered at three-quarters filling, but also provide an example of the topological Mott insulator, i.e., the quantum anomalous Hall state in the strong coupling limit.

cond-mat.str-el↗
arXiv2021

Lattice model for the Coulomb interacting chiral limit of the magic angle twisted bilayer graphene: symmetries, obstructions and excitations

We revisit the localized Wannier state description of the twisted bilayer graphene, focusing on the chiral limit. We provide a simple method for constructing such 2D exponentially localized -- yet valley polarized -- Wannier states, centered on the sites of the honeycomb lattice, paying particular attention to maintaining all the unobstructed symmetries. This includes the unitary particle-hole symmetry, and the combination of $C_2\mathcal{T}$ and the chiral particle-hole symmetry. The $C_2\mathcal{T}$ symmetry alone remains topologically obstructed and is not represented in a simple site-to-site fashion. We also analyze the gap and the dispersion of single particle and single hole excitations above a strong coupling ground state at integer fillings, which we find to be dominated by the on-site and the nearest neighbor terms of a triangular lattice hopping model, with a minimum at the center of the moire Brillouin zone. Finally, we use the insight gained from this real space description to understand the dependence of the gap and the effective mass on the range of the screened Coulomb interaction.

cond-mat.str-el↗
arXiv2021

Cascades between light and heavy fermions in the normal state of magic angle twisted bilayer graphene

We present a framework for understanding the recently observed cascade transitions and the Landau level degeneracies at every integer filling of twisted bilayer graphene. The Coulomb interaction projected onto narrow bands causes the charged excitations at an integer filling to disperse, forming new bands. If the excitation moves the filling away from the charge neutrality point, then it has a band minimum at the moire Brillouin zone center with a small mass that compares well with the experiment; if towards the charge neutrality point, then it has a much larger mass and a higher degeneracy. At a non-zero density away from an integer filling the excitations interact. The system on the small mass side has a large bandwidth and forms a Fermi liquid. On the large mass side the bandwidth is narrow, the compressibility is negative and the Fermi liquid is likely unstable. This explains the observed sawtooth features in compressibility, the Landau fans pointing away from charge neutrality as well as their degeneracies. By providing a description of the charge itineracy in the normal state this framework sets the stage for superconductivity at lower temperatures.

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arXiv2020

Towards the hidden symmetry in Coulomb interacting twisted bilayer graphene: renormalization group approach

We develop a two stage renormalization group which connects the continuum Hamiltonian for twisted bilayer graphene at length scales shorter than the moire superlattice period to the Hamiltonian for the active narrow bands only which is valid at distances much longer than the moire period. In the first stage, the Coulomb interaction renormalizes the Fermi velocity and the interlayer tunnelings in such a way as to suppress the ratio of the same sublattice to opposite sublatice tunneling, hence approaching the so-called chiral limit. In the second stage, the interlayer tunneling is treated non-perturbatively. Via a progressive numerical elimination of remote bands the relative strength of the one-particle-like dispersion and the interactions within the active narrow band Hamiltonian is determined, thus quantifying the residual correlations and justifying the strong coupling approach in the final step. We also calculate exactly the exciton energy spectrum from the Coulomb interactions projected onto the renormalized narrow bands. The resulting softening of the collective modes marks the propinquity of the enlarged ("hidden") $U(4)\times U(4)$ symmetry in the magic angle twisted bilayer graphene.

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arXiv2020

A diagnosis of explicit symmetry breaking in the tight-binding constructions for symmetry-protected topological systems

It has been well established that for symmetry protected topological systems, the non-trivial topology prevents a real space representation using exponentially localized Wannier wavefunctions (WFs) in all directions, unless the protecting symmetry is sacrificed as an on-site transformation. This makes it challenging to determine the symmetry of various physical observables represented using such WFs. In this work, we propose a practical method for overcoming such challenges using the Kane-Mele model as a concrete example. We present a systematic procedure for diagnosing the symmetry of any observables, as well as a method for constructing symmetric operators up to arbitrary truncation accuracy.

cond-mat.str-el↗
arXiv2020

Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene

We use the DMRG to study the correlated electron states favored by the Coulomb interaction projected onto the narrow bands of twisted bilayer graphene within a spinless one-valley model. The Hilbert space of the narrow bands is constructed from a pair of hybrid Wannier states with opposite Chern numbers. Depending on the parameters in the BM model, the DMRG in this basis determines the ground state at one particle per unit cell to be either QAH state or a state with no Hall effect which is nearly a product state. Based on this form, we then apply the variational method to study their competition, thus identifying three states: the QAH, a gapless $C_2T$ symmetric nematic, and a gapped $C_2T$ symmetric stripe. All three states are nearly degenerate at the realistic parameters of the BM model. The single particle spectrum of the nematic contains either a quadratic node or two close Dirac nodes near $Γ$. Motivated by the Landau level degeneracy found in this state, we propose it to be the state observed at the charge neutrality point once spin and valley degeneracies are restored. The optimal period for the $C_2T$ stripe state is found to be $2$ unit cells. In addition, using the fact that the topological charge of the nodes in the $C_2T$ nematic phase is no longer described simply by their winding numbers once the translation symmetry is broken, but rather by certain elements of a non-Abelian group that was recently pointed out, we identify the mechanism of the gap opening within the $C_2T$ stripe state. Although the nodes at the Fermi energy are locally stable, they can be annihilated after braiding with other nodes connecting them to adjacent (folded) bands. Therefore, if the translation symmetry is broken, the gap at one particle per unit cell can open even if the system preserves the $C_2T$ and valley $U(1)$ symmetries, and the gap to remote bands remains open.

cond-mat.str-el↗
arXiv2020

Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening

The ability to control the strength of interaction is essential for studying quantum phenomena emerging from a system of correlated fermions. For example, the isotope effect illustrates the effect of electron-phonon coupling on superconductivity, providing an important experimental support for the BCS theory. In this work, we report a new device geometry where the magic-angle twisted bilayer graphene (tBLG) is placed in close proximity to a Bernal bilayer graphene (BLG) separated by a 3 nm thick barrier. Using charge screening from the Bernal bilayer, the strength of electron-electron Coulomb interaction within the twisted bilayer can be continuously tuned. Transport measurements show that tuning Coulomb screening has opposite effect on the insulating and superconducting states: as Coulomb interaction is weakened by screening, the insulating states become less robust, whereas the stability of superconductivity is enhanced. Out results demonstrate the ability to directly probe the role of Coulomb interaction in magic-angle twisted bilayer graphene. Most importantly, the effect of Coulomb screening points toward electron-phonon coupling as the dominant mechanism for Cooper pair formation, and therefore superconductivity, in magic-angle twisted bilayer graphene.

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