Real space representation of topological system twisted bilayer graphene as an example Jiawei Zang1Jie Wang2Antoine Georges2 3 4 5Jennifer Cano2 6and Andrew J. Millis1 2

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Real space representation of topological system: twisted bilayer graphene as an

example

Jiawei Zang,1, ∗Jie Wang,2Antoine Georges,2, 3, 4, 5 Jennifer Cano,2, 6 and Andrew J. Millis1, 2

1Department of Physics, Columbia University, 538 W 120th Street, New York, New York 10027, USA

2Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

3Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France

4CPHT, CNRS, ´

Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France

5DQMP, Universit´e de Gen`eve, 24 Quai Ernest Ansermet, CH-1211 Gen`eve, Switzerland

6Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA

We construct a Wannier basis for twisted bilayer graphene that is projected only from the Bloch

functions of the twisted bilayer ﬂat bands. The C3and C2Tsymmetries act locally on the Wannier

functions while the Wannier function charge density is strongly peaked at the triangular sites and

becomes fully sublattice-polarized in the chiral limit. The Wannier functions have a power-law tail,

due to the topological obstruction, but most of the charge density is concentrated within one unit cell

so that the on-site local Coulomb interaction is much larger than the further neighbor interactions

and in general the Hamiltonian parameters may be accurately estimated from a modest number

of Wannier functions. One exception is the momentum space components of the single-particle

Hamiltonian, where because of the topological obstruction convergence is non-uniform across the

Brillouin zone. We observe, however, that mixed position and momentum space representations

may be used to avoid this diﬃculty in the context of quantum embedding methods. Our work

provides a new route to study systems with topological obstructions and paves the way for the

future investigation of correlated states in twisted bilayer graphene, including studies of non-integer

ﬁllings and temperature dependence.

Introduction— The interplay of correlation physics and

band topology is of great current interest [1] and calls for

further development of theoretical methodologies. One

important issue is the choice of basis. If a theory is solved

exactly, the physical content is independent of the basis

choice. However quantum many-body systems cannot in

general be solved exactly and both the quality and the

physical content of any approximate solution are aﬀected

by the choice of basis. In studying the physics of strongly

interacting electrons in periodic potentials, spatially dis-

cretized representations such as Kanamori-Hubbard type

models [2] have proven very useful because correlation

physics typically involves strong local quantum and ther-

mal ﬂuctuations leading to local moment formation [3],

Mott transitions [4], orbital ﬂuctuations, low tempera-

ture entropy release [5–7], and related phenomena.

Localized representations may be built from the Wan-

nier functions derived [8] from Bloch functions provided

by a band theory calculation. However, for topologically

nontrivial bands the standard Wannierization procedure

encounters diﬃculties [9–12]. In these situations, Wan-

nier functions that transform locally under the relevant

point symmetries are power-law rather than exponen-

tially localized and while integrals such as those needed

for determining Hamiltonian parameters exist, the inte-

gral for the mean square position uncertainty hR2iis di-

vergent so the standard Maximally Localized Wannier

construction [8] cannot be directly applied.

Perhaps more importantly, in any Wannier descrip-

tion, the one-electron properties are described by a tight

∗jz3122@columbia.edu

binding model, and one must generically include an in-

ﬁnite number of hopping parameters to exactly repro-

duce the band structure. For the usual exponentially lo-

calized Wannier functions the one electron Hamiltonian

converges to the exact dispersion exponentially rapidly

as more hoppings are included, and the convergence is

uniform in momentum space. In the topologically ob-

structed case, the convergence is power-law in the num-

ber of included orbitals and, more seriously, the conver-

gence of the single particle (hopping) parameters of the

Hamiltonian is non-uniform in momentum space as will

be described in detail below. The non-uniform conver-

gence is related to constraints on the implementation of

point symmetries in momentum and position space repre-

sentations [13]. These diﬃculties have impeded the use of

Wannier representations in studying correlation physics

in topological bands.

In this paper we use the example of twisted bilayer

graphene (TBG) [14–17] to show that even for topolog-

ical bands useful Wannier representations may be con-

structed in which the Wannier functions faithfully repre-

sent important properties of the system and point sym-

metries act locally. Further, we observe that the real-

space representation of the single-particle part of the

Hamiltonian is not needed for many practical many-body

calculations. As we demonstrate explicitly using the ex-

ample of dynamical mean ﬁeld theory, a mixed posi-

tion/ momentum space representation can be employed,

in which the kinetic energy is expressed in the momentum

space basis of non-interacting eigenstates, so that all the

topological features are exact and well preserved, while

the interaction part may be expressed in position space

and inherit convenient locality and symmetry properties

arXiv:2210.11573v2 [cond-mat.mes-hall] 24 Oct 2022

from the Wannier functions [18–20].

Speciﬁcally, in this paper we explicitly construct a

Wannier basis for twisted bilayer graphene involving two

triangular site-centered Wannier functions per unit cell

derived from the two ﬂatbands per spin per valley and

show that these provide a physically intuitive and math-

ematically convenient real space picture. The two crucial

point symmetries C2Tand C3act locally on the Wan-

nier functions we construct, and within a unit cell their

charge density proﬁle corresponds closely to that found in

scanning probe experiments [21]. Although the Wannier

functions have a power-law decay arising from the topo-

logical obstruction, we ﬁnd that they are in practice very

localized, leading to a computationally convenient repre-

sentation of the interactions in which the on-site terms

are much bigger than the ﬁrst or farther neighbor terms.

The rest of this paper is organized as follows. We

ﬁrst summarize relevant aspects of the physics of twisted

bilayer graphene and then review the continuum model

that is believed to describe the low energy physics and

summarize the symmetry of the eigenstates. Next we

construct the Wannier functions and present a detailed

analysis of their symmetry properties and spatial struc-

ture. Then we calculate the interaction and show how to

use a mixed real and momentum space representation to

study correlated states. Finally we present a summary

and broader outlook.

FIG. 1. (a) Representation of hexagonal moir´e lattice with tri-

angular (hexagon-centered) AA sites and hexagonal (AB/BA)

sites indicated. Solid lines show the hexagonal lattice with

vertices at the AB/BA sites and dashed lines show the tri-

angular lattice with vertices at the AA sites. (b) Sketch

of AB/BA site-centered Wannier functions. Blue and or-

ange dots show the hexagonal AB and BA sites respectively;

shaded blue and orange ellipses show the charge densities of

the corresponding two Wannier functions and demonstrate

the spatially extended “ﬁdget spinner” shape required for the

charge density of the Wannier functions to correspond to the

physical charge density. (c) Sketch of the two Wannier func-

tions centered at the triangular AA site. At the coarse grained

level shown here the charge densities of the two orbitals are

the same.

Twisted bilayer graphene— Twisted bilayer graphene

(TBG) is a system comprised of two graphene sheets

stacked one on top of the other at a small twist an-

gle. At commensurate twist angles one ﬁnds a moir´e

pattern characterized by a hexagonal unit cell as shown

in panel (a) of Fig. 1. The hexagonal unit cell may be

very large compared to the lattice constant of the under-

lying graphene. At the experimentally interesting carrier

concentrations and appropriate small twist angles, cal-

culations [16,17] indicate that there are eight relevant

bands, two per spin per valley. The absence of spin-orbit

coupling and the exponential suppression of intervalley

mixing in large unit cells means that the spin and valley

quantum numbers may be thought of as internal quan-

tum numbers attached to two bands of electrons that

may be described as eigenstates of a ‘continuum’ model

[17] involving a Dirac dispersion in each plane and a spa-

tial periodicity deﬁned by the interplane coupling. These

two bands are often referred to as ‘ﬂatbands’ because for

appropriate twist angles they are well separated from the

other bands and their dispersion can be very weak rela-

tive to the dispersion of other bands. The ﬂatbands of

TBG are of intense current interest for the wide variety

of novel phases and correlated electron physics they host

[14,15].

Density functional theory [16] and scanning probe ex-

periments [21] conﬁrm that for almost all momenta in the

moir´e Brillouin zone the charge density of the ﬂat band

electrons is concentrated at the triangular (hexagon-

center) AA sites, suggesting that the real-space descrip-

tion should be based on two Wannier orbitals (each with

spin and valley quantum numbers) that are centered at

the triangular sites, transform appropriately under the

point symmetries and are constructed from the ﬂat band

states.

However, it is known that the ﬂat bands in TBG ex-

hibit a topological obstruction that prohibits the con-

struction of exponentially-localized Wannier functions on

which the relevant symmetries act locally [9,10]. Several

strategies have been proposed to resolve the problem.

One of them is to avoid the problem and work entirely in

momentum space [22]. This approach is suitable for con-

structing ground states and for Hartree-Fock based stud-

ies, but is not convenient for several many-body methods

beyond Hartree-Fock. Building Wannier functions from

a large number of bands [23–25] removes the topological

obstruction at the cost of greatly widening the energy

range that must be considered and complicating the the-

oretical model.

The topological obstruction may also be avoided by

building exponentially-localized Wannier functions cen-

tered at the hexagonal AB and BA sites [13,26,27], as

shown in Fig. 1(b). The C2Tsymmetry of TBG acts

non-locally on these states. Additionally, the need to

capture the triangular site centered charge densities us-

ing hexagon-center Wannier functions leads to a spatially

extended “ﬁdget spinner” shape [26,27] that leads to

highly non-local real space interactions [28]. Alternative

representations of the local interaction physics are there-

fore desirable.

Continuum model and its symmetries.— The contin-

uum Hamiltonian Hη(r) for one valley η=±is a 4 ×4

matrix with the basis (A1, B1, A2, B2), corresponding to

the two layers 1, 2 and two sublattices A, B of monolayer

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Realspacerepresentationoftopologicalsystem:twistedbilayergrapheneasanexampleJiaweiZang,1,JieWang,2AntoineGeorges,2,3,4,5JenniferCano,2,6andAndrewJ.Millis1,21DepartmentofPhysics,ColumbiaUniversity,538W120thStreet,NewYork,NewYork10027,USA2CenterforComputationalQuantumPhysics,FlatironInstitute,1625thA...

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Real space representation of topological system twisted bilayer graphene as an example Jiawei Zang1Jie Wang2Antoine Georges2 3 4 5Jennifer Cano2 6and Andrew J. Millis1 2.pdf

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Real space representation of topological system twisted bilayer graphene as an example Jiawei Zang1Jie Wang2Antoine Georges2 3 4 5Jennifer Cano2 6and Andrew J. Millis1 2

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