The Quantum Hall ferroelectric helix in bilayer graphene

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Submitted on 7 May 2019

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Thierry Jolicoeur, Csaba Töke, Inti Sodemann

To cite this version:

Thierry Jolicoeur, Csaba Töke, Inti Sodemann. The Quantum Hall ferroelectric helix in bilayer

graphene. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American

Phys-ical Society, 2019, 99 (11), �10.1103/PhysRevB.99.115139�. �hal-02123104�

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Thierry Jolicoeur

1 _{, Csaba T˝oke}

2,3

_{, and Inti Sodemann}

4

1 _{Laboratoire de Physique Th´}

_{eorique et Mod`}

_{eles statistiques, CNRS,}

Universit´

e Paris-Sud, Universit´

e Paris-Saclay, 91405 Orsay, France

2 Department of Theoretical Physics, Budapest University of

Technology and Economics, Budafoki ´

ut 8, H-1111 Budapest, Hungary

3 BME “Momentum” Exotic Quantum Phases Research Group, 1111 Budapest, Budafoki ´

ut 8, Hungary and

4 Max-Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany

(Dated: December, 2018)

We re-examine the nature of the ground states of bilayer graphene at odd integer filling factors

within a simplified model of nearly degenerate n = 0 and n = 1 Landau levels. Previous

Hartree-Fock studies have found that ferroelectric states with orbital coherence can be stabilized by tuning

the orbital splitting between these levels. These studies indicated that, in addition to a uniform

ferroelectric state, a helical ferroelectric phase with spontaneously broken translational symmetry

is possible. By performing exact diagonalization on the torus, we argue that the system does not

have a uniform coherent state but instead transitions directly from the uniform incoherent state

into the ferroelectric helical phase. We argue that there is a realistic prospect to stabilize the helical

ferroelectric state in bilayer graphene by tuning the interlayer electric field in a model that includes

all single particle corrections to its zero energy eight-fold multiplet of Landau levels.

PACS numbers: 73.43.-f, 73.22.Pr, 73.20.-r

I. INTRODUCTION

The last two decades have witnessed a rapid and formidable increase in the richness of the quantum Hall physics

of monolayer and bilayer graphene

1–15

_{. In particular, bilayer graphene (BLG) possesses a unique zeroth Landau level}

manifold which features two nearly degenerate cyclotron orbital degrees of freedom with n = 0 and n = 1 character

16 _,

in addition to the two spins and two valleys making up a manifold with a total of eight levels. The interplay of

strong Coulomb interactions, that stabilize integer quantum Hall states via quantum Hall ferromagnetism

17 _{, and the}

single particle splitting terms, can lead to an intricate variety of coherent states in this system

18–23

_{. A notoriously}

interesting possibility, pointed out in Ref. 18, is the coherence between the states with different orbital character

n = 0/n = 1. This coherence breaks spontaneously the real space inversion symmetry, resulting in the formation of

a type of quantum Hall ferroelectric state, which is expected to have a linearly dispersing Goldstone mode (this form

of quantum Hall ferroelectric state is distinct from that proposed in Ref. 24 which results from spontaneous valley

polarization). Following these initial studies, it was later argued based on Hartree-Fock theory

21 _{, that an analogue}

of Dzyaloshinskii-Moriya interaction allowed by the breaking of inversion symmetry, can drive the softening of this

Goldstone mode at a finite wave-vector, leading to the formation of ferroelectric helical state. The phase diagram

as a function of the single-particle splitting of the n = 0/n = 1 orbitals obtained from Hartree-Fock is depicted in

Fig.1(b), which additionally features a Wigner crystal state.

To this date there are very few studies incorporating correlation effects beyond Hartree-Fock on this problem. One

exact diagonalization study

25 _{restricted itself to the case of zero single particle splitting between n = 0 and n = 1}

orbitals, and found that in such case the ground state is a trivial integer quantum Hall state adiabatically connected

to the fully polarized state into the n = 0 orbital with a full gap to all excitations. As we will see, however the

interesting ferroelectric states predicted by Hartree-Fock theory tend to occur at negative splitting when the n = 1

orbital is energetically favored, explaining why they were not observed in Ref. (25).

In this paper we concentrate on phases with full spin and valley polarization and just keep the two orbital degrees

of freedom n = 0 and n = 1, and restrict to total filling factor ν = 1. Using the torus geometry we diagonalize

exactly the Hamiltonian for up to 14 electrons, which allows us to obtain information on ground states and some of

the low-lying excited states. We pay special attention to the role of a nontrivial particle-hole symmetry, identified

in Ref. (26), which maps states with filling ν onto states with 2

− ν in the two orbital manifold. Remarkably, this

symmetry acts non-trivially on the bare single particle-splitting between the n = 0 and n = 1 orbitals, and for a

Hamiltonian to be particle-hole invariant under this symmetry it must have a negative splitting favoring the n = 1

orbital.

We will show that the simple polarized incoherent phase observed in Ref.(25) can be captured by perturbation theory

and show that its excited states can be reproduced by time-dependent Hartree-Fock theory. We also find that for some

range of orbital splitting there is evidence for a phase with broken translation symmetry that we identify as the helical

ferroelectric phase seen in HF calculations. However, we find no evidence for a spatially uniform orbitally coherent

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Exact Diagonalization:

Incoherent

n=0 polarised

Ferroelectric

Helix

Incoherent

n=1 polarised

h

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c1

⇠ 0.1

h

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c2

⇠ 0.22

Incoherent

n=0 polarised

Ferroelectric

Helix

Incoherent

n=1 polarised

WC

Uniform

coherent

Uniform

h

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h

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a)

b)

Hartree-Fock:

FIG. 1: Summary of the phases observed in the orbitally mixed n = 0/1 as a function of the orbital single particle splitting h.

(a) The exact diagonalization results of the current study indicate the presence of only two phases: an incoherent adiabatically

connected to the maximally polarized states, and a ferroelectric helical state. The red star indicates the orbital bias at which

particle hole symmetry is achieved for which the system has equal population of the n = 0/1 orbitals. (b) The Hartree-Fock

studies find two other phases: the uniform orbitally coherent ferroelectric and a Wigner crystal (WC), see e.g. Ref .(21) for a

detailed estimate of the boundaries.

state, because we do not observe any translationally invariant phase with its characteristic Goldstone mode. We

depict the approximate phase diagram resulting from our study in Fig.1(a). We will also show that there is a realistic

prospect to realise the ferroelectric helical state in BLG. Recent experiments have achieved a detailed understanding

and a remarkable degree of control over the single particle splittings in the zeroth Landau level manifold of BLG

27–29

_.

Because the valley degree of freedom is locked to the layer index in the zeroth Landau level of BLG, this degree of

freedom can be easily controlled by applying an interlayer bias. The spin splitting on the other hand can be controlled

with in-plane fields. It is therefore, possible to achieve the conditions in which the system is valley and spin polarized

and the relevant active degrees of freedom are the n = 0 and n = 1 orbitals. The splitting between the n = 0 and

n = 1 levels is intimately related to hopping terms that break particle-hole invariance in bilayer graphene

30 _{. We will}

also show, however, that there is a way to experimentally control the splitting between the n = 0 and n = 1 orbitals

by tuning the interlayer bias to sufficiently large values.

Our paper is organized as follows, in section II we discuss the realistic band structure of BLG and show that the

level crossing between n = 0/n = 1 lies within a realistic range of parameters. In section III we explain the peculiar

particle-hole symmetry of the model obtained by assuming full spin and valley polarization. Section IV contains some

definitions of finite-size torus wavefunctions. Section V summarize the Hartree-Fock treatment. In section VI we give

results of exact diagonalization studies. Section VII is devoted to the incoherent phase and its excitations. Section

VIII contains our findings about the broken translation symmetry phase. Our conclusions are presented in section

IX.

II. BAND STRUCTURE OF BILAYER GRAPHENE

The four band model containing all leading corrections to bilayer graphene’s Hamiltonian for a single spin and

valley in the presence of a magnetic field has the form

30 _:

H

4b

= ω

0 





u

2ω

0 a

†

₋

γ

4 γ

0 a

†

₋

γ

3 γ

0 a

a

∆

_2ω

0 +u

₀

γ

1 ω

0 −

γ

4 γ

0 a

†

−

γ

4 γ

0 a

γ

1 ω

0 ∆

0 _−u

2ω

0 a

†

−

γ

3 γ

0 a

†

−

γ

4 γ

0 a

a

−

u

2ω

0 





.

(1)

Here ω

0 =

√

2v

0 /`

≈ 30.6meVpB[T], with the magnetic length ` = p~c/eB, γ

0 ≈ 2.61eV, γ

1 ≈ 361meV, γ

3 ≈

283meV, γ

4 ≈ 138meV, ∆

0 ≈ 15meV, and u is the energy difference between top and bottom layers controlled by a

perpendicular electric field. This Hamiltonian has been successfully employed in detailed modeling of Landau levels

for integer

27 _{and fractional quantum Hall states recently}

28 _.

In this Hamiltonian the terms ∆

0 and γ

4 break particle-hole symmetry and induce a splitting between the n = 0

and n = 1 LLs at zero interlayer bias u which favors n = 0. This renders the ground state trivially polarized into

n = 0 at u = 0, and therefore under normal conditions one would not expect the physics that we will describe in this

paper to appear. However, as illustrated in Fig. 2, the interlayer bias can be used to control the splitting between

(4)

20

40

60

80

100

120 -0.4

-0.2

0.2

0.4

0.6 u[meV]

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E/!

0

<latexit sha1_base64="SCzYSFaBtmKWLkSzICr8p70TL78=">AAAB8XicbZDLSgMxFIbP1Futt6pLN8EiuKozIuiyKILLCvaC7VAyaaYNzWVIMkIZ+hZuXCji1rdx59uYtrPQ1h8CH/85h5zzRwlnxvr+t1dYWV1b3yhulra2d3b3yvsHTaNSTWiDKK50O8KGciZpwzLLaTvRFIuI01Y0upnWW09UG6bkgx0nNBR4IFnMCLbOerw96ypBB7jn98oVv+rPhJYhyKECueq98le3r0gqqLSEY2M6gZ/YMMPaMsLppNRNDU0wGeEB7TiUWFATZrONJ+jEOX0UK+2etGjm/p7IsDBmLCLXKbAdmsXa1Pyv1kltfBVmTCappZLMP4pTjqxC0/NRn2lKLB87wEQztysiQ6wxsS6kkgshWDx5GZrn1cDx/UWldp3HUYQjOIZTCOASanAHdWgAAQnP8ApvnvFevHfvY95a8PKZQ/gj7/MHta+QRg==</latexit><latexit sha1_base64="SCzYSFaBtmKWLkSzICr8p70TL78=">AAAB8XicbZDLSgMxFIbP1Futt6pLN8EiuKozIuiyKILLCvaC7VAyaaYNzWVIMkIZ+hZuXCji1rdx59uYtrPQ1h8CH/85h5zzRwlnxvr+t1dYWV1b3yhulra2d3b3yvsHTaNSTWiDKK50O8KGciZpwzLLaTvRFIuI01Y0upnWW09UG6bkgx0nNBR4IFnMCLbOerw96ypBB7jn98oVv+rPhJYhyKECueq98le3r0gqqLSEY2M6gZ/YMMPaMsLppNRNDU0wGeEB7TiUWFATZrONJ+jEOX0UK+2etGjm/p7IsDBmLCLXKbAdmsXa1Pyv1kltfBVmTCappZLMP4pTjqxC0/NRn2lKLB87wEQztysiQ6wxsS6kkgshWDx5GZrn1cDx/UWldp3HUYQjOIZTCOASanAHdWgAAQnP8ApvnvFevHfvY95a8PKZQ/gj7/MHta+QRg==</latexit><latexit sha1_base64="SCzYSFaBtmKWLkSzICr8p70TL78=">AAAB8XicbZDLSgMxFIbP1Futt6pLN8EiuKozIuiyKILLCvaC7VAyaaYNzWVIMkIZ+hZuXCji1rdx59uYtrPQ1h8CH/85h5zzRwlnxvr+t1dYWV1b3yhulra2d3b3yvsHTaNSTWiDKK50O8KGciZpwzLLaTvRFIuI01Y0upnWW09UG6bkgx0nNBR4IFnMCLbOerw96ypBB7jn98oVv+rPhJYhyKECueq98le3r0gqqLSEY2M6gZ/YMMPaMsLppNRNDU0wGeEB7TiUWFATZrONJ+jEOX0UK+2etGjm/p7IsDBmLCLXKbAdmsXa1Pyv1kltfBVmTCappZLMP4pTjqxC0/NRn2lKLB87wEQztysiQ6wxsS6kkgshWDx5GZrn1cDx/UWldp3HUYQjOIZTCOASanAHdWgAAQnP8ApvnvFevHfvY95a8PKZQ/gj7/MHta+QRg==</latexit><latexit sha1_base64="SCzYSFaBtmKWLkSzICr8p70TL78=">AAAB8XicbZDLSgMxFIbP1Futt6pLN8EiuKozIuiyKILLCvaC7VAyaaYNzWVIMkIZ+hZuXCji1rdx59uYtrPQ1h8CH/85h5zzRwlnxvr+t1dYWV1b3yhulra2d3b3yvsHTaNSTWiDKK50O8KGciZpwzLLaTvRFIuI01Y0upnWW09UG6bkgx0nNBR4IFnMCLbOerw96ypBB7jn98oVv+rPhJYhyKECueq98le3r0gqqLSEY2M6gZ/YMMPaMsLppNRNDU0wGeEB7TiUWFATZrONJ+jEOX0UK+2etGjm/p7IsDBmLCLXKbAdmsXa1Pyv1kltfBVmTCappZLMP4pTjqxC0/NRn2lKLB87wEQztysiQ6wxsS6kkgshWDx5GZrn1cDx/UWldp3HUYQjOIZTCOASanAHdWgAAQnP8ApvnvFevHfvY95a8PKZQ/gj7/MHta+QRg==</latexit>

N = 0

<latexit sha1_base64="eeh467HImJcrSpBKu6GaIKH938k=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F25vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBlnmNUw==</latexit><latexit sha1_base64="eeh467HImJcrSpBKu6GaIKH938k=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F25vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBlnmNUw==</latexit><latexit sha1_base64="eeh467HImJcrSpBKu6GaIKH938k=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F25vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBlnmNUw==</latexit><latexit sha1_base64="eeh467HImJcrSpBKu6GaIKH938k=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F25vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBlnmNUw==</latexit>

N = 1

<latexit sha1_base64="1br6hnYpKkg4W90JG/0e2yTSXOE=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F15vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBl/2NVA==</latexit><latexit sha1_base64="1br6hnYpKkg4W90JG/0e2yTSXOE=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F15vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBl/2NVA==</latexit><latexit sha1_base64="1br6hnYpKkg4W90JG/0e2yTSXOE=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F15vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBl/2NVA==</latexit><latexit sha1_base64="1br6hnYpKkg4W90JG/0e2yTSXOE=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIoVB1/12Ciura+sbxc3S1vbO7l55/6Bp4lQz3mCxjHU7oIZLoXgDBUreTjSnUSB5KxjdTOutJ66NiNUjjhPuR3SgRCgYRWs93F15vXLFrbozkWXwcqhArnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHYuKRtz42WzVCTmxTp+EsbZPIZm5vycyGhkzjgLbGVEcmsXa1Pyv1kkxvPQzoZIUuWLzj8JUEozJ9G7SF5ozlGMLlGlhdyVsSDVlaNMp2RC8xZOXoXlW9Szfn1dq13kcRTiCYzgFDy6gBrdQhwYwGMAzvMKbI50X5935mLcWnHzmEP7I+fwBl/2NVA==</latexit>

B = 25T

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FIG. 2: Landau levels for a single spin-valley in BLG as a function of interlayer bias. The bias can tune a level crossing between

n = 0 and n = 1. There is a sizable energy separation from higher Landau levels throughout justifying the projection into the

two n = 0 and n = 1 LLs.

these levels and successfully induce a level crossing making the n = 1 lower in energy as we desire. The value required

to achieve this crossing is independent of the magnetic field and, neglecting the trigonal warping term (γ

3 → 0), can

be estimated to be :

u

0 =

2γ

4 γ

1 γ

0 + ∆

0 1 +

γ

4 γ

0

2 !

≈ 53.2meV.

(2)

which should be within experimental reach. In Fig. 3 we show this value as green dots (including the trigonal warping

term). At this value one expects the boundary between the state fully polarized into n = 0 and the coherent states.

The orange dots are the required values in order to overcome the splitting produced by the exchange interactions with

the vacuum (analogous to the well-known Lamb shift in atomic physics)

26 _{for bare Coulomb interactions so that the}

system has approximate particle-hole symmetry. The blue dots depict the expected Hartree-Fock boundary between

coherent and fully n = 1 polarized states

21 _{. Taking bare Coulomb interactions, these two boundaries correspond to}

the single particle energy splitting given respectively by:

E

1 − E

0 =

Lamb

=

e

2 `

1

16 r π

2 ,

E

1 − E

0 =

orb.pol.

=

e

2 `

1

8 r π

2 .

Finally, in Fig. 4 we illustrate the overlaps of the LL levels in the full four-band model with the idealized two-band

model as a function of the interlayer bias and field. We see an excellent overlap for n = 0 and a good overlap for

n = 1. The latter is known to decrease as the magnetic field increases

28 _{, but as we see there is no substantial change}

as a function of interlayer bias.

III. PARTICLE-HOLE SYMMETRY BETWEEN n = 0 AND n = 1 LANDAU LEVELS

Consider the two-band model of bilayer graphene :

H

K

=

0 −(Π

x

− iΠ

y

)

2 /2m

∗

−(Π

x

+ iΠ

y

)

2 /2m

∗

0 (3)

where m

∗

_{is an effective mass and Π}

x,y

= p

x,y

+ eA

x,y

/c. This is the Hamiltonian concerning valley K and spin is

polarized. The single-particle spectrum has then two zero-energy states :

(5)

15

20

25

30

35

40

20

40

60

80

100 u[meV]

B[T]

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N = 0 polarized

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N = 1 polarized

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N = 0 + 1 coherence

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FIG. 3: Expected schematic phase diagram as a function of interlayer bias u and perpendicular magnetic field. Identification of

phases come Hartree-Fock calculations. The green dots are where the bare single particle energies of n = 0 and n = 1 become

degenerate. The yellow dots lie where approximate particle symmetry is expected in the presence of Coulomb interactions and

the blue dots are the expected location at which the system polarizes into n = 1. We assume that the bilayer graphene sample

is on top of a hexagonal boron nitride substrate with screening constant ≈ 6.6 as in ref.(28).

u[meV]

0

20

40

60

80

100

0.75

0.80

0.85

0.90

0.95

1.00 |h

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0 | i|

2 B = 25T

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N = 0

N = 1

|h

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0 | i|

2 N = 0

N = 1

15

20

25

30

35

40

0.70

0.75

0.80

0.85

0.90

0.95

1.00 u = 53meV

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B[T]

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FIG. 4: Overlaps between the ideal two-band model Landau level wavefunctions and the four-band model wavefunctions as a

function of interlayer bias and magnetic field.

and the remainder of the spectrum depends upon the magnetic field :

E

nK

= sign(n)ω

c

p|n|(|n| − 1), |n| ≥ 2,

(5)

where n is an integer positive or negative and ω

c

= eB/m

∗

. The corresponding eigenstates are given by :

Φ

n

=

φ

₀

n

,

n = 0, 1

(6)

Φ

n

=

1 √

2 φ

|n|

φ

|n|−2

,

n

_{≤ −2, Φ}

n

=

1 √

2 −φ

|n|

φ

|n|−2

,

n

_{≥ +2,}

(7)

(6)

where φ

n

are standard cyclotron eigenstates for particles with parabolic dispersion relation. We now consider two-body

interactions written in second quantization :

V =

1 ₂

X

1234

V

1234

c

†

n

1 ,j

1 c

†

n

2 ,j

2 c

n

3 ,j

3 c

n

4 ,j

4 (8)

with Landau level indices n

i

= 0, 1 and guiding center coordinates j

i

and the big indices

{1234} are a short-hand for

the combined indices. We now use the particle-hole conjugation operator

_{C which is defined by :}

C

†

_c

†

nj

C = c

nj

.

(9)

When acting on Eq.(8) we find :

C

†

VC = V +

X

12M

c

†

1 c

2 (V

1M 2M

+ V

M 2M 1

− V

2M M 1

− V

M 21M

) + const

(10)

The particle-hole transformation generates an additional one-body term which can written as :

∆

0 X

m

c

†

0m

c

0m

+ ∆

1 X

m

c

†

1m

c

1m

,

(11)

The constants ∆

0,1

are geometry-dependent and they are distinct ∆

0 6= ∆

1 . So even if we start with exactly degenerate

n = 0 and n = 1 LLs the particle-hole symmetry leads to a nontrivial splitting between them. If we now focus on the

state with total filling factor ν = 1 then the particle-symmetry above implies that the full spectrum of the Hamiltonian

Eq.(8) should be the same after addition of the one-body splitting Eq.(11).

IV. TORUS DEFINITIONS

We consider a torus that is obtained by periodic boundary conditions applied to a rectangular domain spanned by

vectors L

1 = L

x

ˆ

and L

2 = L

y

ˆ

y. The aspect ratio is defined as :

AR =

L

x

L

y

.

(12)

Periodic boundary conditions are imposed by magnetic translations

31 _{t(L) = exp}

i

~

L

· Π − i

ˆ

z·(L×r)

`

2 . Using the

Landau gauge A = Bxˆ

y

for in this study, these act as t(L)ψ(r) = exp(

iyˆ

_`

x·L

2 )ψ(r + L). The boundary conditions read

t(L

1,2

)ψ(r) = ψ(r). The two conditions are compatible only if the rectangle is pierced by an integral number of flux

quanta :

N

φ

=

|L

1 × L

2 |

2π`

2 =

L

x

L

y

2π`

2 .

(13)

The n = 0 states can be written as

32 :

η

0m

(z) =

1 p`L

2 √

π

ϑ

m/N

₀

φ

iπN

φ

z

L

2 N

φ

i

e

−

2`2

x2

,

(14)

where m = 0, 1, . . . , (N

φ

− 1), z = x + iy, and we have used Jacobi elliptic functions with characteristics

33 :

ϑ

a

_b

(z

|τ) =

∞

X

n=−∞

e

iπτ (n+a)

2 +2i(n+a)(z+bπ)

.

(15)

Higher Landau orbitals are obtained using the LL raising operator, which in our gauge takes the form : ˆ

a

†

₌

`

√

2 i∂

x

+ ∂

y

−

ix

`

2 =

`

√

2 2i∂

z

−

ix

`

2 . Thus we have :

η

n,j

(z) =

(ˆ

a

†

₎

n

√

n!

η

n0

(z) =

(16)

=

1 (L

y

√

π)

1/2

X

k

H

n

x

`

−

kL

x

`

−

2πj`

L

y

exp

_−i

kL

x

`

2 +

2πj

L

y

₋

1 2`

2 x

_{− kL}

x

−

2πj`

2 L

y

2 !

,

where H

n

is a a Hermite polynomial. We note that η

nm

(z) is normalized for the principal domain of the torus :

Z

L

y

0 dy

Z

L

x

0 dxη

∗

n

0 _m

0 (x + iy)η

nm

(x + iy) = δ

nn

0 δ

_mm

0 .

(17)