Efficient simulation of moiré materials using the density matrix renormalization group

Abstract

We present an infinite density-matrix renormalization group (DMRG) study of an interacting continuum model of twisted bilayer graphene (tBLG) near the magic angle. Because of the long-range Coulomb interaction and the large number of orbital degrees of freedom, tBLG is difficult to study with standard DMRG techniques-even constructing and storing the Hamiltonian already poses a major challenge. To overcome these difficulties, we use a recently developed compression procedure to obtain a matrix product operator representation of the interacting tBLG Hamiltonian which we show is both efficient and accurate even when including the spin, valley, and orbital degrees of freedom. To benchmark our approach, we focus mainly on the spinless, single-valley version of the problem where, at half filling, we find that the ground state is a nematic semimetal. Remarkably, we find that the ground state is essentially a k-space Slater determinant, so that Hartree-Fock and DMRG give virtually identical results for this problem. Our results show that the effects of long-range interactions in magic angle graphene can be efficiently simulated with DMRG and open up a new route for numerically studying strong correlation physics in spinful, two-valley tBLG, and other moiré materials in future work.

Realization of topological Mott insulator in a twisted bilayer graphene lattice model

“…Our real-space interaction-only model therefore establishes the microscopic mechanism of the evolution between the insulating stripe and QAH phases. Our effective model and its unbiased numerical solution therefore revealed the essence of the physics in this particular regime, and is also consistent with other theoretical calculations 27 , 28 , 44 , 46 .…”

Strain-Induced Quantum Phase Transitions in Magic-Angle Graphene

“…Our DMRG considers both valley degrees of freedom, which is essential for correctly identifying the even-integer insulators. Similar to earlier works on singlevalley models [31,32], we find that DMRG and SCHF agree remarkably well. In particular, DMRG confirms the presence of K-IVC order at ν ¼ −2 in the absence of strain.…”

“…Further evidence for the importance of strain comes from symmetry considerations. In the absence of strain, models at even-integer filling show that, although the ground state has K-IVC order, there is a close competitor whose energy is only slightly higher: a nematic semimetal [22,28,[30][31][32]. As elucidated in Ref.…”

“…Following the methods developed in Refs. [31,32,48], here we use infinite DMRG to study H compactified onto a infinitely long cylinder of circumference L y moiré cells. SCHF finds that the ground state is perfectly spin polarized for ϵ ≲ 0.2%, so we accelerate our DMRG calculations by assuming full spin polarization of the narrow bands at ν ¼ −2 while keeping both valleys [42].…”

Domain wall competition in the Chern insulating regime of twisted bilayer graphene

“…[47], which however does not report results for inhomogeneous substrates. Also, recent advances [48,49] suggest that some of the results reported here could be within reach of tensor-network techniques, providing a complementary perspective.…”