Abstract
Experimental advances in the fabrication and characterization of fewlayer materials stacked at a relative twist of small angle have recently shown the emergence of flat energy bands. As a consequence electron interactions become relevant, providing inroads into the physics of strongly correlated twodimensional systems. Here, we demonstrate by combining large scale ab initio simulations with numerically exact strong correlation approaches that an effective onedimensional system emerges upon stacking two twisted sheets of GeSe, in marked contrast to all moiré systems studied so far. This not only allows to study the necessarily collective nature of excitations in one dimension, but can also serve as a promising platform to scrutinize the crossover from two to one dimension in a controlled setup by varying the twist angle, which provides an intriguing benchmark with respect to theory. We thus establish twisted bilayer GeSe as an intriguing inroad into the strongly correlated physics of lowdimensional systems.
Introduction
Understanding emergent, strongly correlated quantum phenomena in complex manybody interacting and low dimensional materials is one of the main driving forces in modern condensed matter research. Strongly correlated systems are fascinating as they challenge our understanding of quantum mechanics fundamentally, but are also highly relevant to technological advances, such as the quest for room temperature superconductivity, ultradense and ultrafast memory solutions, as well as quantum computing platforms^{1}, to name a few. In this context, the study of lowdimensional systems has revealed a zoo of surprising insights into quantum collective behavior of manybody systems, some of which already find far reaching applications in everyday life, e.g., in computer memory (magnetism) and magnetic resonance imaging techniques (superconductivity).
Recently, twisted bilayer graphene^{2,3,4,5,6} and other van der Waals materials stacked atop each other at a twist^{7,8,9,10,11,12,13} have been proposed as material realizations of twodimensional correlated physics that afford an unprecedented level of control. Previous studies concentrate on fewlayer films featuring a \(6{0}^{\circ }\) or \(12{0}^{\circ }\) rotational symmetry stacked at a twist. By forming a large moiré supercell at small twist angles, a quasitwodimensional system with quenched and tunable kinetic energy scales emerges, thereby drastically enhancing the role of electronic interactions.
Surprisingly, we report here that if instead we consider layered systems stacked at a small twist angle for which the monolayers have a rectangular lattice with only mirror symmetry, an effectively onedimensional system with quenched kinetic energy scales (flat bands) emerges. This elevates the concept of moiré systems to include the broad and exciting realm of onedimensional quantum systems, which from a theory point of view is ideal to study quantum manybody effects, because powerful theoretical tools (such as bosonization, tensor network approaches and the Bethe ansatz^{14,15,16}) can be employed to obtain a nearly complete picture of its collective nature and effects of strong correlations. Remarkably, we find that varying the twist angle smoothly interpolates between an effectively onedimensional and a twodimensional system at low energies, permitting experimental studies of the dimensional crossover in a clean and controllable manner.
To illustrate this point we perform largescale ab initio based simulations of two sheets of GeSe stacked at a twist, where GeSe belongs to the family of 2D groupIV monochalcogenides^{17} and has a similar structure as phosphorene [see Fig. 1a, b]. 2D GeSe exhibits high air stability and thin GeSe films down to a monolayer have been studied extensively in experiments for their applications in phototransitors and nearinfrared photodetectors^{18,19,20,21,22,23,24}. 2D GeSe is also predicted to exhibit giant piezoelectricity^{25,26}, roomtemperature ferroelectricity^{27,28} and ferroelasticity^{29,30}, strong visiblelight absorbance^{31} and a large bulk photovoltaic effect^{32}. This renders GeSe an interesting choice as much prior expertize on the (untwisted) material exist and samples are experimentally available. Furthermore, recently an Eshelby twist has already been realized for GeSe, as well as in the structurally similar system GeS^{33}.
With extensive ab initio calculations, we explicitly demonstrate that a quasionedimensional system emerges for twisted bilayer GeSe at small twist angles, where the degree of “onedimensionality” increases with decreasing angle. Upon including interactions we show that this system is an effective realization of the socalled ionic Hubbard model. This model has attracted a lot of research attention in the past^{34,35,36,37,38,39,40}, because it features many interesting prototypical (correlated) phases of matter, including band insulators, Mott insulators, bond density waves and Luttinger liquids, and hosts Ising, as well as Kosterlitz–Thouless quantum phase transitions. As a consequence we find that in twisted GeSe all these different phases of matter can be accessed and their respective phase transitions can be studied in a controllable condensed matter setup. We explicitly outline the phase diagram including all the above mentioned phases of matter upon varying the filling (experimentally tunable by gating), as well as the ratio of kinetic and interaction energy scales (tunable by the twist angle) at temperatures accessible within current experimental limitations. Furthermore, twisted bilayer GeSe constitutes a unique system for the controlled study of the crossover between twodimensional and onedimensional physics via varying the twist angle using the experimental setup outlined in ref. ^{41}, which can be used to shed light on this interesting regime from an experimental viewpoint in the future. This condensed matter based benchmark system could complement results from more conventional quantum simulation platforms^{42,43,44} in the future in terms of scalability of system size and operation temperature. Twisted bilayer GeSe, as we demonstrate, is thus an ideal inroad into the strongly correlated nature of lowdimensional systems.
Results
Density functional characterization
We start by discussing the ab initio band structure results for twisted GeSe. In Fig. 1 we show the density functional theory (DFT) characterization of two sheets of stacked GeSe at a twist (see Methods section). The atomic structure of a single sheet of GeSe resides in a rectangular lattice [panels (a) and (b)]. Starting from a perfectly aligned AAstacking bilayer, different moiré patterns are formed when the top (or the bottom) layer is twisted with angle \(\phi\) ranging from \({0}^{\circ }\) to \(18{0}^{\circ }\) with respect to the other layer.
The systems with twist angles \(\phi\) and \(18{0}^{\circ }\)\(\phi\), which we refer to as configurations A and B, respectively, share supercells of the same size. The supercell for system B is shown in Fig. 1c, and the corresponding supercell for system A can be found in the method section. We will focus on configuration B in this work and we refer to the twist angle of \(18{0}^{\circ }\)\(\phi\) as \(\phi\) for simplicity. Similar to the results reported for hexagonal or triangular lattice systems^{2,3,4,5,6,8,9,45} we find the emergence of flat bands (which as in the case of twisted BoronNitride^{8} does not rely on tuning to magic angles) at the edges of the conduction and valence bands at small twist angles. However, in marked contrast to these other systems surprisingly some of the low energy bands disperse only along one direction in real space. This is most obvious for bands at the bottom of the conduction bands [see panel (d) and (e)], which are only dispersive along the \(\Gamma\)X (or YS) direction and dispersionless along the perpendicular \(\Gamma\)Y (or XS) direction. We carefully checked these results against varying the functionals used in our DFT calculations, which give slightly different relaxed atomic geometry. To this end we compare results obtained within the local density approximation (LDA) to those obtained employing a generalized gradient approximation (GGA) with van der Waals corrections. We find very consistent behavior upon varying the choice of functionals (see Methods section). Remarkably, we find that the moiré system at small angle shows a quasionedimensional chainlike staggered charge distribution in real space [see panel (g)] for states in the flat bands, with pairs of wires in the unit cell, each of which displays an alternating sequence of large and small charge puddles. To capture this behavior, we fit the lowenergy moiré bands using an anisotropic tightbinding model with a staggered onsite potential (see Methods section). Panel (f) summarizes results for such fits obtained within a LDA and GGA. We find (robust to changing the functionals used in DFT) that the ratio between intrawire and interwire couplings decreases with decreasing twist angle, which tunes the system continuously to the onedimensional limit.
If we neglect the coupling between the onedimensional wires at small twist angle, then a simple model that accurately describes the dispersion and charge modulation along the wire is given by a Hamiltonian with nearestneighbor hopping \(t\) and featuring a staggered onsite potential \({\epsilon }_{0}\)
with \({n}_{i,\sigma }={c}_{i,\sigma }^{\dagger }{c}_{i,\sigma }\) the occupancy at site \(i\). The corresponding dispersion has two branches \({E}_{k}^{\pm }={\pm} \!\sqrt{4{t}^{2}{\cos }^{2}(k)+{\epsilon }_{0}^{2}}\).
Starting from the LDA DFT results at a twist angle of \(\phi =6.6{1}^{\circ }\), \({\epsilon }_{0}=0.001337\)eV can be read off by the gap magnitude at the zone edge, and an optimal fit of the single remaining parameter \(t={t}_{\parallel }\) (see Methods section) is shown in Fig. 2a. At this angle we find \({\epsilon }_{0}/t\approx 1.3\), placing the system in the interesting regime where kinetic energy terms and staggering potential compete in their order of magnitude. More details about the fit, as well as the crossover from two dimensions to one can be found in the Methods section.
Correlation effects
Next, we model electron interaction effects. At this point we have no definite way to pinpoint the range of the interactions and rather adopt the vantage point that screening will promote rather short ranged interactions. To this end we include an onsite repulsion with
as the dominant contribution. The interactions are written in a particlehole symmetric way for convenience which amounts to an overall shift in chemical potential. This model is known in the literature as the ionic Hubbard model; a paradigmatic model to study the transition from band insulators (BI) to Mott insulators (MI) as the interactions are increased and was investigated extensively at halffilling^{34,35,36,37,38,39,40}. It is now well understood that this transition occurs via an intermediate bond order wave state (BOW), in which interaction induced spontaneous dimerization leads to alternating strong and weak bonds. The transition from BI to BOW is of the Ising, second order type, while the second transition from the BOW to the MI state is of the KosterlitzThouless (KT) type^{34,35,36,37,38}. Twisted GeSe thus provides an inroad into this highly intriguing physics and can, depending on the parameters, potentially realize all of these different phases. So far we have used a (zero temperature) ab initio analysis of the band structure. For experiments, however, an important question is whether and how the emergent, correlated phases manifest at finite but still low temperature. This can be simulated efficiently for any chemical potential \(\mu\), as well as \(U\) and \({\epsilon }_{0}\) using density matrix renormalization group (DMRG) (see Methods section) taking the ab initio band structure as an input (at higher temperature the band structure itself might be affected but this regime is not the one we focus on in this work).
Much is known about the phases in the ionic Hubbard model and how to characterize them^{34,35,36,37,38}, particularly at halffilling. We summarize how to distinguish these phases in Table 1, where we characterize the four different phases band insulator, bond ordered wave, Mott insulator and Luttinger liquid by whether they display a charge gap, spin gap and a staggered bond dimer order. A checkmark signals that the phase displays a nonzero value of the gap or order, while a cross denotes the absence thereof. By calculating the static susceptibility to magnetization \({\chi }^{M}\), charging \({\chi }^{C}\) and bond ordering \({\chi }^{BOW}\) upon including a small seed perturbation in magnetic field, onsite potential or bond dimerization, respectively, we determine the spin and charge gaps, as well as the bond ordering tendencies (see Supplementary Note 1). For the smaller angle of \(\phi =6.6{1}^{\circ }\), we show \({\chi }^{M}\) and \({\chi }^{BOW}\) given a small seed \(s/t=1{0}^{2}\) in Fig. 2b, c. By calculating the static susceptibilities in this fashion and varying \(U/t\), as well as \(\mu /t\) (corresponding experimentally to controlling the angle, as well as back gate) we can map out the phase diagram by using Table 1. Panel (d) of Fig. 2 shows the full phase diagram we obtain this way. The BOW state occupies only a tiny fraction of the phase diagram and most likely requires fine tuning to be seen in experiments, especially at finite temperature.
The different phases of matter manifest prominently in transport experiments with the insulating gap scaling either with \({\epsilon }_{0}\) or \(U\) in the BI and MI case, respectively, while showing characteristic powerlaw suppression in temperature in the LL regime. Scanning tunneling microscopy (STM) will reveal either a charge gap (BI and MI) with different temperature scaling or a powerlaw suppression of the density of states in the LL case. Both transport and STM have recently been successfully put forward in the twisted van der Waals material’s context^{2,3,4,5,6}. Furthermore, specific heat and spinspin correlation functions can be monitored to distinguish between these phases. In panel (e) and (f) of Fig. 2 we show the specific heat \(c=\partial E/\partial T\), as well as the spinspin correlation \({C}_{S}(x)\) at half filling for two values of \(U/t=0\) (BI) and \(U/t=8\) (MI). The specific heat in (e) at large inverse temperature \(1/T\) is exponentially suppressed in the BI case while for a MI we find a linear behavior which is one of the hallmarks of the emergent gapless spinexcitations. We find that at \(1K\) the system starts to show clear MI behavior (specific heat \(c\) turns linear) for \(U/t=8\). Panel (f) depicts the real space spinspin correlation function. The BI phase is characterized by an exponential suppression of these correlation functions, while one of the hallmarks of the MI state are long range algebraic correlations \({C}_{S} \sim {x}^{1}\) at \(T=1/8K\), at least for small enough distances compared to \(1/T\) (after which correlations fall off exponentially). We complement this by studying the chargecharge correlation function \({C}_{C}\) obtained for finite doping \(\mu /t=3\) shown in panel (g). The longranged powerlaw decay (dashed line) in the correlation functions falls of as approximately \({C}_{S} \sim {x}^{1.9}\) which indicates a weakly correlated LL state. Importantly, the temperatures for which all of these predictions can be measured are on the Kelvin scale and thus within experimental reach.
Next, we highlight the signatures accessible via STM. We compute the density of states \(\rho\) at the even lattice sites \(i\) by simulating the real time dynamics of \(\langle {c}_{i,\uparrow }{c}_{i,\uparrow }^{\dagger }(t)\rangle\) and taking the Fourier transform. Via the dissipation fluctuation theorem the local density of states can be obtained from this by dividing out the Fermidistribution \(f(\omega )\) (see Methods for details). The results are summarized in Fig. 3 for temperatures in the Kelvin regime. At small \(U\) we find that the single particle gap scales with \({\sim} {\epsilon }_{0}\), while the Mott insulating gap scales as \({\sim} U\). Overall the behavior of the gap first decreases (with a minimum close to the BOW phase) and then increases as \(U\) is increased. The spectral features of the density of states can be used to clearly distinguish experimentally which phases are realized in the system.
Discussion
We have established that twisted bilayer GeSe is an exciting platform to study strongly correlated onedimensional physics and the crossover from one to two dimensions in a highly tunable manner. We find that upon marrying ab initio materials characterization and strong correlations a onedimensional ionic Hubbard Model arises, which shows many prototypical features and phases of strongly correlated onedimensional systems. These can be probed by experiments on twisted bilayer GeSe in accessible temperature regime, albeit on much enlarged moiré length scales. In twisted bilayer GeSe at small twist angles the spinorbit splitting for the effectively onedimensional system is negligible. Future research should address the questions whether in other moiré systems a stronger spinorbit coupling can be realized. If so this would provide a highly controllable platform to realize Majorana edge state in these effective wires, by coupling the system to a conventional \(s\)wave superconducting substrate.
In the final stages of writing this article ref. ^{46} appeared, which supports the message of this paper.
Methods
Details about the DFT treatment
We employed the Vienna Ab initio simulation package (VASP) to perform the ground state DFT calculations^{47}. The basis was chosen to be plane waves with an energy cutoff of 450 eV and the pseudo potentials are generated using the projector augmented wave method (PAW)^{48}. The exchangecorrelation functions are treated in the local density approximation (LDA)^{49}. We complement our calculations by also considering the exchangecorrelation functionals treated in the generalized gradient approximation (GGA)^{50} and find results consistent with LDA. A 1 × 1 × 1 momentum grid is used for the ground state and relaxation calculations. The experimental lattice constants for bulk GeSe (a = 4.38 \({\AA}\), b = 3.82 \({\AA}\)) are employed for the construction of the supercell of twisted bilayer GeSe. In order to satisfy the commensurate condition, the a lattice constant is slightly expanded by 0.68%. As periodic boundary condition are applied, a vacuum region larger than 15 \({\AA}\) is added in the zdirection perpendicular to the layers to avoid artificial interaction between the periodic slabs. We relax all the atoms in order to avoid artificial effects as known from unrelaxed structures for other moiré systems^{51,52,53}. Throughout the relaxation, all the atoms are relaxed until the force on each atom converges to values smaller than 0.01 eV/\({\AA}\). In the GGA calculations, van der Waals corrections are applied using the DFTD3 method of Grimme^{54}. To visualize the charge density distributions of the lowenergy states of twisted bilayer GeSe we employ the VESTA code^{55}. There exist two inequivalent configurations called A and B in the main text, which are illustrated and characterized in Fig. 4. The supercell of twisted bilayer GeSe with twist angles at 10.99\({}^{\circ }\), 8.26\({}^{\circ }\) and 6.61\({}^{\circ }\) contain 872, 1544 and 2408 atoms, respectively.
Details about the fitted band structure and 1D2D crossover
We use a simple tight binding model to describe the dispersion at all angles calculated within DFT. We consider a nextnearestneighbor lattice model on a rectangular grid with a 2 by 2 sites unit cell:
with \({n}_{i,}={c}_{i}^{\dagger }{c}_{i}\) the occupancy at site \(i=({i}_{x},{i}_{y})\). We fit the dispersion varying the nearestneighbor hopping amplitudes along the \(x\) direction (\({t}_{\perp }\)), along the \(y\) direction (\({t}_{\parallel }\)), the nextnearest hopping along the diagonal (\({t}_{D}\)), as well as the onsite potentials \({\epsilon }_{i}\). We consider a 2 by 2 unit cell so \({\epsilon }_{i}\) can take 4 different values \(({\epsilon }_{0,0},{\epsilon }_{1,0},{\epsilon }_{0,1},{\epsilon }_{1,1})\)
Fitting the bands for three different twist angles \(\phi =10.9{9}^{\circ }\), \(\phi =8.2{6}^{\circ }\), and \(\phi =6.6{1}^{\circ }\) yields the values reported in Table 2. Clearly, as one approaches smaller twist angles the onedimensional character of the system emerges and the residual chainchain coupling along the x direction \({t}_{\perp }\) and \({t}_{D}\) becomes negligible. This is further illustrated in Fig. 5 where we show the ab initio characterization of the dispersion for the same angles, as well as the corresponding fits. The bands show more appreciable residual dispersion along the \(XS\) direction at larger angle, signaling the crossover from 1D to 2D as the angle is increased. Therefore the effective dimensionality of the system can be tuned by the twist angle and twisted bilayer GeSe provides a tunable platform to study the 2D to 1D crossover.
For the smallest twist angle of \(\phi =6.6{1}^{\circ }\), which we concentrate on in the main text when discussing correlation effects, the dispersion along \(x\) is negligible and we can set \({t}_{\perp }\approx 0\), \({t}_{D}\approx 0\), as well as label \(t={t}_{\parallel }\). Subtracting of the trivial mean potential shift of \({\epsilon }_{{\rm{s}}hift}=({\epsilon }_{x,0}+{\epsilon }_{x,1})/2\) and defining \({\epsilon }_{0}=({\epsilon }_{x,0}{\epsilon }_{x,1})/2\) we recover Eq. (1) of the main text (as well as reinstating the spin degree of freedom).
Treating electron correlations
We treat correlations in a numerically exact tensor network based approach formulated in matrix product states^{16}. We exploit the twosite translation invariance of the infinite system and set up the tensor network algorithm directly for the infinite dimensional limit. To treat finite temperature we use the purification scheme described in part 7 of ref. ^{16} and rewrite the unity operator, corresponding to an infinite temperature density matrix \(\rho \sim 1\) in terms of a wavefunction in combined physical and auxiliary Hilbert space. Subsequently we “cool” the density matrix to temperature \(T=1/\beta\), where \(\rho \sim {e}^{\beta H}\), by applying an imaginary time evolution algorithm. We converge the bond dimension such that numerically exact results are obtained and perform a fourth order TrotterSuzuki decomposition with small enough steps in imaginary time \(\Delta \beta =0.01\), such that the decomposition does not yield an appreciable approximation. A fourth order decomposition is chosen for numerical convenience allowing for larger time steps then a second order scheme reducing the overall numerical resources needed. In the Supplementary Note 2 the convergence of all numerical parameters is benchmarked explicitly in the noninteracting limit.
Calculating the density of states
To calculate the density of states we use a simulation in real time (and at finite temperature) to obtain the
For this we use the ideas put forward in ref. ^{56}. This is essential to reach long enough times, such that a meaningful Fourier transform can be taken with a Hanning type window function, compare Fig. 6a. The maximum time reached by the simulation thus limits the frequency resolution and introduces natural broadening in the Fourier transform. This procedure is employed for the Data shown in Fig. 3a, c where the \(U/t\) is either large or small both cases in which the entanglement growth is quite moderate. For the data shown Fig. 3b which is \(U/t=4\) the entanglement growth is much more severe and even after employing the ideas of ref. ^{56}, the time scales are limited. To this end we utilize a linear prediction algorithm to extend the time scales, see Fig. 6b.
Data availability
All data generated and analyzed during this study are available from the corresponding author upon reasonable request.
Code availability
All the custom codes used in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the European Research Council (ERC2015AdG694097) and Grupos Consolidados (IT57813). The Flatiron Institute is a division of the Simons Foundation. L.X. acknowledges the European Unions Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No. 709382 (MODHET). M.C. is supported by the Flatiron Institute, a division of the Simons Foundation. D.M.K. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence StrategyCluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1390534769. Gefördert durch die Deutsche Forschungsgemeinschaft(DFG) im Rahmen der Exzellenzstrategie des Bundes und der LänderExzellenzcluster Materie und Licht für Quanteninformation (ML4Q) EXC 2004/1390534769. D.M.K. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under RTG 1995. We acknowledge support by the Max Planck InstituteNew York City Center for NonEquilibrium Quantum Phenomena. D.M.R.G. calculations were performed with computing resources granted by RWTH Aachen University under projects prep0010. We acknowledge computing resources from Columbia University’s Shared Research Computing Facility project, which is supported by NIH Research Facility Improvement Grant 1G20RR03089301, and associated funds from the New York State Empire State Development, Division of Science Technology and Innovation (NYSTAR) Contract C090171, both awarded April 15, 2010.
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L.X. provided DFT results, D.M.K. performed the DMRG calculations. D.M.K., L.X., M.C., and A.R. interpreted the results and participated in writing the paper.
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Kennes, D.M., Xian, L., Claassen, M. et al. Onedimensional flat bands in twisted bilayer germanium selenide. Nat Commun 11, 1124 (2020). https://doi.org/10.1038/s41467020149470
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