Polariton Anomalous Hall Effect in TMDs - Sciencesconf.org€¦ · PRL 101, 106401 (2008) ⌦⌧ ex(q)= 1 4 X q0 |(q0)|2 X =±1 ⌦⌧ c (q 0 + q/2) Photon: Two polarizations. A⌫

Polariton Anomalous Hall Effect

in TMDs

Luis Martin-Moreno

Chiral modes in optics and electronics of 2D systems

Aussois,France,27/11/2018

In collaboration withF. Guinea, L.Chirolli,A. Gutierrez-Rubio (IMDEA, Madrid),F. J. Garcia-Vidal (IFIMAC, Madrid)

Outline

Excitons Polaritons have a Berry Curvaturewhich can originate from:

(i)Exciton BC(ii)Photon BC(iii)Coupling between them

In this talk:

- Evaluation of the relative contributions- Estimation of Lateral shifts upon Polariton motion

Model System:

Excitons in TMDs in a planar metallic cavity

ARTICLE

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1882

NATURE COMMUNICATIONS | 3:887 | DOI: 10.1038/ncomms1882 | www.nature.com/naturecommunications

© 2012 Macmillan Publishers Limited. All rights reserved.

A non-equilibrium charge carrier imbalance between valleys is the key to creating valleytronic devices1–9. The principal mechanism invoked in this context is circularly polarized

optical excitation3,4 . In this approach, the two valleys absorb left- and right-handed photons differently, a phenomenon referred to as circu-lar dichroism (CD). An essential condition for valley-selective CD in a honeycomb lattice is the absence of a centre of inversion. In the case of graphene, it was suggested that by interacting graphene with a sub-strate such that the centre of inversion can be obliterated, whereupon a gap opens up in each valley10,11. This strategy, however, is quite challenging experimentally. And even in its eventual realization, there can only be a gentle perturbation to graphene via the weak covalent coupling at large van der Waals separations from the substrate11.

Since its first isolation12,13, monolayer molybdenum disulphide (MoS2) has attracted immense attention. Many measurements have been performed to characterize the optical and transport proper-ties of this material12–16. In monolayer MoS2, two layers of sulphur atoms in a two-dimensional hexagonal lattice are stacked over each other in an eclipsed fashion. Each Mo sits in the centre of a trigonal prismatic cage formed by six sulphur atoms (Fig. 1a). Quite remark-ably in the context of current discussion, the natural stable struc-ture of free-standing monolayer MoS2 is a honeycomb lattice with inequivalent bipartite colouring, breaking the inversion symmetry (Fig. 1b).

While bulk MoS2 has an indirect bandgap, interestingly, when thinned to the monolayer limit, the material acquires direct band-gaps located exactly at the corners of the Brillouin zone. Indeed, with its 1.8 eV direct bandgap and unique two-dimensional structure embracing the high-symmetry valleys, monolayer MoS2 has much to offer in the exploration of novel electronic and optoelectronic devices and the associated physics (as compared with boron nitride, which has a bandgap of about 6.0 eV in the ultraviolet regime). It is quite natural to question whether it is possible to achieve valley-selective CD in this semiconducting atomic membrane, which will endow electrons in this material the valley degree of freedom, in addition to charge and spin that have been routinely explored in conventional device physics. We show here using ab initio numeri-cal simulation and experimental micro-photoluminescence that monolayer MoS2 possesses near-perfect valley-selective CD, a prop-erty rooted in its bulk symmetry and very much conducive to opto-electronic valley polarization needed for valleytronics.

ResultsDegree of circular polarization. The key quantity to assess is, therefore, the k-resolved degree of optical polarization, H(k),

between the top of the valence bands and the bottom of the conduction bands4

I X( , ) = | ( ) | | ( ) || ( ) | | ( ) |

.2 2

2 2k k kk kcv

cv cv

cv cv( (( (

This quantity is the difference between the absorption of left- and right-handed lights ( o ), normalized by total absorption, at each k-point and evaluated between the top of the valence bands (v) and the bottom of conduction bands (c). The band structure of monolayer MoS2 is shown in Fig. 2a. The dependence on the transition energy, =X F Fcv c v( ) = ( ) ( )k k k , is implicit through k. Here, the transition matrix element of circular polarization is (o ocv

xcv

ycvP iP( ) =1 2[ ( ) ( )]k k k/ . The interband matrix elements,

P pcvc v( ) = | |k k k ®Z Z , are evaluated using the density functional

perturbation theory, within the local-density approximation17 , as implemented in VASP18 . Spin-orbit coupling is not included in our calculations19,20. Briefly, a planewave basis set is employed at a

(1)(1)

Mo S

Figure 1 | The crystal structure of monolayer MoS2. (a) Coordination environment of Mo (blue sphere) in the structure. Sulphur is shown as golden spheres. (b) A top view of the monolayer MoS2 lattice, emphasizing the connection to a honeycomb lattice. In our calculations, we used an optimized structure at the level of local density approximation in density functional theory. The shaded region bounded by dashed lines corresponds to one primitive cell. The unit cell parameter is a = 3.12 Å, and the vertical separation between sulphur layers is 3.11 Å.

K+

K_

–1

–0.8

–0.4

–0.6

–0.2

0

0.2

0.4

0.6

0.8

1

1.8 eV

+s+s

+s

– l

– l – l

Figure 2 | Valley-selective CD of monolayer MoS2. (a) Top valence band (blue) and bottom conduction band (pink). The centre hexagon is the Brillouin zone colour-coded by the degree of circular polarization, H(k), as defined in the text. The vector connecting K + and K − is perpendicular to Mo-S bond in the crystal structure in Fig. 1b. (b) Schematic of phase winding on the MoS2 lattice that gives rise to the chiral optical selectivity. Left panel: the contribution to phase winding from the Bloch lattice phase, where T = o 1 is the valley index, and s = 1,2 corresponding to the S and Mo sites (isospin index). Right panel: the phase winding under a threefold rotation. The green axes indicate the rotation of local atomic coordinates that leads to the azimuth dissynchronization.

- Direct bandgap 1.8 eV (in bulk, indirect gap of 1.3 eV)- Strong light absorption and electroluminescence - Strong exciton with large binding energy (stable at Room Temp.)

Semiconductors: Transition Metal Dicalchogenides

Single Layer

While Excitons in 3D Materials (GaAs): ~ 100 nm, ~ 5 meV…

In TMDs: 2D character and weak dielectric screening:enhanced Coulomb interactions

Much smaller sizes ~ 1 nm

Effects of Diracness: non-hydrogenic

Srivastava et al., PRL 115, 166802 (2015) J. Zhou et al., PRL 120, 077401 (2018)

Splendiani et al., Nano Lett. 10, 1271 (2010) Berkelbach, PRB 88, 045318 (2013)

excitons dominate the optical and optoelectronic respond in TMDCs

Larger binding energy ~ 10 kBTroom

Excitons in 2D TMDs

circular dichroism

carrier’s Berry curvature

Valley Hall and Exciton Anomalous Hall Effects in TMDCs

Exciton Hall effect

electron-hole move in same directions

M. Onga et al. Nat. Materials 16, 1193 (2017)

Valley Hall effect

electrons and holes move in opposite directions

K. Mak et al. Science 344, 1489 (2014)

S. Wang et al. Nano Lett. 16 (7), 4368 Z. Sun et al. Nat. Photonics 11, 491 (2017)

Monolayer in photonic cavity: Exciton-Polaritons

Rabi splittings were limited by the absorption features of thematerials with a maximum reported splitting of 86 meV at roomtemperature.29 Furthermore, in MoX2 monolayers, spin−orbitcoupling (SOC) induces the splitting of the excitonic transitionby ca. 150 meV such that both the so-called A and B excitontransitions (see Figure 1b) can simultaneously interact withcavity modes complicating the studies of such systems.29 The

WS2 monolayer has the advantage that it presents a muchsharper isolated absorption band as can be seen in Figure 1b. Inaddition it displays an intense photoluminescence (PL) peak at2.016 eV (Figure 1c). Hence WS2 constitutes a natural choicefor light−matter strong coupling.In this letter we demonstrate that, by coupling WS2

monolayers to metallic resonators, the magnitude and visibility

Figure 1. (a) Structure of a WS2 monolayer showing, from left to right, out-of-plane view, in-plane view, and the unit cell. (b) Transmittance ofmonolayers of WS2 (black solid curve) and MoS2 (blue solid curve) on quartz; the inset shows an optical micrograph of the WS2 flake in which theblue region is a monolayer and the bright region is a multilayer. (c) Emission spectrum taken from the monolayer region of WS2 under continuouswave (cw) 532 nm (2.330 eV) excitation. The inset displays the fluorescence image of the WS2 flake in which bright emission is observed only fromthe monolayer region under cw 470 ± 10 nm (2.64 ± 0.06 eV) excitation.

Figure 2. (a) Energy diagram of the strong coupling of the direct band gap transition ℏωe between the valence band (VB) and conductive band(CB) of monolayer WS2 to the first optical mode ℏωp of a cavity. The coherent coupling forms two polaritonic states P+ and P− separated by theRabi energy ℏΩR. (b) Schematic of the Ag Fabry−Perot (FP) cavity with WS2 monolayer placed at the center (the optical field maximum). (c)Schematic of the plasmonic hole array with WS2 monolayer physisorbed on top. Five nanometers of PMMA separates the monolayer from the metalsurface. (d) Schematic of the optical setup for angle resolved spectroscopy. White light source or cw 533 nm laser beam is selected by the flippedmirror M1 to pass through a beam splitter (BS) and a 40× objective lens (O1, N.A. = 0.6). The reflected beam or PL from the sample, collected bythe same O1, passes through a polarization analyzer (P), tube lens (L1), spatial filtering iris, imaging lens (formed by L2 and L3), Fourier lens (L3),and spectrometer coupled CCD. (e) Optical micrograph of a WS2 monolayer physisorbed on the plasmonic hole array; the monolayer region isdelineated by the red PL image of the same WS2 monolayer in (f).

Nano Letters Letter

DOI: 10.1021/acs.nanolett.6b01475Nano Lett. 2016, 16, 4368−4374

4369

W. Yao and Q. Niu, PRL 101, 106401 (2008)

ex(q) =

1

4

X

q0

|(q0)|2X

=±1

c (q

0 + q/2)

Photon: Two polarizations.

Aph(k) = (cos 1)e

ph(k) = kz/k

3

J. Segert, PRA 36, 10 (1987)

Polariton: the winding of exciton-photon coupling contributes

M. Onoda, et al., PRL 36, 083901 (2004)

Berry Curvature in Exciton Polaritons.

Exciton: From the Berry curvature of electrons: c (q)

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H =

X

q

!ph,qa†qaq + !ex,qb

†qbq + (gqe

imb†qaq +H.c.)

Karzig et al PRX 5, 031001 (2015)

signature of this nontrivial topology is a bulk gap in theexcitation spectrum with chiral edge modes as the only in-gap states (see Fig. 1). These edge modes provide a newrealization of a controllable one-waywaveguide for photons[3,4,19–23]. More conceptually, our proposal allows torealize topological photons at optical frequencies and, tothe best of our knowledge, constitutes the first example of atopological hybrid state treating light and matter degrees offreedom on the same footing. This is particularly interestingsince finite interactions in the excitonic component open theperspective of interacting topological states of polaritons.

II. TOPOLARITONS

Polaritons are superpositions of photons and excitonsthat can be described by a Hamiltonian of the form

H ¼X

q

½ωCq a

†qaq þ ωX

q b†qbq þ ðgqb†qaq þ H:c:Þ&; ð1Þ

where the operators a†q and b†q create photons and excitonswith momentum q, respectively. We assume that theexcitons and photons are both confined to two dimensions;i.e., q ¼ ðqx; qyÞ. For the excitons, this can be achievedusing a quantum well, while photons can be trapped usinga microcavity or waveguide. The exciton dispersion ωX

q ¼q2=2mX þ ωX

0 (setting ℏ ¼ 1) describes its center-of-massmotion, while the energy gap ωX

0 for creating an excitonis given by the difference between the bare particle-holeexcitation gap and the exciton binding energy. We denotethe dispersion of the cavity photon by ωC

q .The crucial ingredient for generating topolaritons is the

exciton-photon coupling gq, which describes the creation ofan exciton by photon absorption and vice versa. Here, werequire gq to wind according to

gq ¼ gqeimθq ; ð2Þ

where gq is the amplitude of the exciton-photon interaction(or Rabi frequency), m is a nonzero integer, and θq denotesthe polar angle of q.To reveal the nontrivial topology, we diagonalize the

Hamiltonian Eq. (1) in terms of polariton operators

P'q ¼ eP'q ⋅

! aq

bq

"; ð3Þ

with spectrum ωP'q ¼ 1

2ðωCq þωX

q Þ ' 12 ½ðω

Cq −ωX

q Þ2þ4g2q&1=2,where the ' sign refers to the upper and lower polaritonband, respectively (see Fig. 2). The vectors eP'q describe therelative strength between the photon and exciton compo-nents of the polariton wave function and can be interpretedas a spinor. The nontrivial form of the coupling Eq. (2)leads to a winding of this spinor of the form

eP'q ¼ 1ffiffiffi2

p ' e−imθq

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 ' βq

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ∓ βq

p

!

ð4Þ

where βq ¼ ðωCq − ωX

q Þ=½ðωCq − ωX

q Þ2 þ 4g2q&1=2. The spinorof the lower polariton band “flips” from photonic toexcitonic (vice versa for the upper polariton) far awayfrom the resonance as described by the limits βq¼0 ¼ −1and βq→∞ ¼ 1. Combined with the winding eimθq , this flipleads to a full wrapping of the unit sphere by the spinor eP'q ,thereby leading to a nontrivial topology in full analogy tofermionic topological systems. This can be confirmed bycalculating the Chern number of the upper and lowerpolariton bands. Indeed, we find C' ¼ ' m. Note thatthe mechanism leading to nontrivial topology is distinctlydifferent from the original proposal by Haldane and Raghu[1,2]. Here, the nontrival Berry curvature arises directlyfrom the (winding) hybridization between two ordinaryexciton and photon bands, not from the gapping out ofsymmetry-protected band touchings (e.g., Dirac cones) inthe photonic spectrum [24].A consequence of the nontrivial Chern number is the

presence of chiral polaritonic edge modes. In our setting,edges are defined by the confinement of the excitons andphotons. While excitons are naturally restricted to thequantum well, photons can be confined, e.g., by usingreflecting mirrors or a suitable photonic band gap at theedges of the system. Another possibility would be toconfine the photons in a dielectric slab waveguide endingat the system edges.Besides the existence of bands with nontrivial Chern

numbers, the stability of the edge modes also requires aglobal energy gap (i.e., present for all momenta) betweenthe upper and lower polariton bands. Above, we assumethat the bare exciton and photon dispersions are far(negatively) detuned at q ¼ 0. In that case, the energy ofthe lower polariton branch essentially coincides with thebare exciton dispersion for low momenta. Since the lowerpolariton branch approaches the bare exciton dispersion forlarge q, it thus appears impossible to open a gap for a

FIG. 2. Schematic polariton spectrum ωP'q . The upper and

lower polariton bands are drawn in red and blue, while theuncoupled bands (gq ¼ 0) are shown as thin black lines forcomparison. Because of the winding structure of the coupling, thetwo polariton bands acquire a nontrivial topology with finiteChern numbers C' ¼ ' 1 in the case of m ¼ 1 [see Eq. (2)]. Theassociated edge modes are indicated by a dashed purple line.

TOPOLOGICAL POLARITONS PHYS. REV. X 5, 031001 (2015)

031001-2

Valley Hall and Exciton Hall effect in TMDCs

Valley Hall effect

electrons and holes move in opposite directions

Exciton Hall effect

electron-hole move in same directions

M. Onga et al. Nat. Materials 16, 1193 (2017)

circular dichroism

K. Mak et al. Science 344, 1489 (2014)

carrier’s Berry curvature

Which contribution dominates?

What is the Berry Curvature of an Exciton polariton?

(i) We need the exciton-photon hamiltonian

Model for electrons

K K' K K'Model for electrons

K K' K K'

tan =vk

, tan =kykx

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H = h0 +h · <latexit sha1_base64="Syvz726Rc4tL7KIdZrwnXkhdppg=">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</latexit><latexit sha1_base64="Syvz726Rc4tL7KIdZrwnXkhdppg=">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</latexit><latexit sha1_base64="Syvz726Rc4tL7KIdZrwnXkhdppg=">AAACl3icbVHbihNBEO2Mt3W8bFafxJfGIK64LDMiKIK4oLh58GEXzW4gE0JNTyVp0pehu2bZMOQT/Bpf9UP8G3uSCJusBQ2nz6mq7jqVl0p6SpI/rejGzVu37+zcje/df/Bwt7336MzbygnsCaus6+fgUUmDPZKksF86BJ0rPM9nnxr9/AKdl9Z8p3mJQw0TI8dSAAVq1H7R5R/4dJTwVzy7QFFPFzwThaXVLfNyomERx/Go3UkOk2Xw6yBdgw5bx8lor9XPCisqjYaEAu8HaVLSsAZHUihcxFnlsQQxgwkOAjSg0Q/r5UQL/jwwBR9bF44hvmSvVtSgvZ/rPGRqoKnf1hryf9qgovG7YS1NWREasXpoXClOljf28EI6FKTmAYBwMvyViyk4EBRM3Hhl2btEsTFJfVkZKWyBW6yiS3LByMwjaZCmmao+lkrxb2D8Pz40bIT9z3IiyR98DTsyB8cOcfbyanLYRbrt/HVw9vowDfj0Tefo/XorO+wpe8b2WcresiPWZSesxwT7wX6yX+x39CT6GH2JuqvUqLWuecw2Ijr9C0bZy/4=</latexit><latexit sha1_base64="Syvz726Rc4tL7KIdZrwnXkhdppg=">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</latexit>

h0 = ∆/2,h = (τ v kx, v ky,∆) = h(sin θ cosΨ, sin θ sinΨ, cos θ)

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ARTICLE

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1882

NATURE COMMUNICATIONS | 3:887 | DOI: 10.1038/ncomms1882 | www.nature.com/naturecommunications

© 2012 Macmillan Publishers Limited. All rights reserved.

A non-equilibrium charge carrier imbalance between valleys is the key to creating valleytronic devices1–9. The principal mechanism invoked in this context is circularly polarized

optical excitation3,4 . In this approach, the two valleys absorb left- and right-handed photons differently, a phenomenon referred to as circu-lar dichroism (CD). An essential condition for valley-selective CD in a honeycomb lattice is the absence of a centre of inversion. In the case of graphene, it was suggested that by interacting graphene with a sub-strate such that the centre of inversion can be obliterated, whereupon a gap opens up in each valley10,11. This strategy, however, is quite challenging experimentally. And even in its eventual realization, there can only be a gentle perturbation to graphene via the weak covalent coupling at large van der Waals separations from the substrate11.

Since its first isolation12,13, monolayer molybdenum disulphide (MoS2) has attracted immense attention. Many measurements have been performed to characterize the optical and transport proper-ties of this material12–16. In monolayer MoS2, two layers of sulphur atoms in a two-dimensional hexagonal lattice are stacked over each other in an eclipsed fashion. Each Mo sits in the centre of a trigonal prismatic cage formed by six sulphur atoms (Fig. 1a). Quite remark-ably in the context of current discussion, the natural stable struc-ture of free-standing monolayer MoS2 is a honeycomb lattice with inequivalent bipartite colouring, breaking the inversion symmetry (Fig. 1b).

While bulk MoS2 has an indirect bandgap, interestingly, when thinned to the monolayer limit, the material acquires direct band-gaps located exactly at the corners of the Brillouin zone. Indeed, with its 1.8 eV direct bandgap and unique two-dimensional structure embracing the high-symmetry valleys, monolayer MoS2 has much to offer in the exploration of novel electronic and optoelectronic devices and the associated physics (as compared with boron nitride, which has a bandgap of about 6.0 eV in the ultraviolet regime). It is quite natural to question whether it is possible to achieve valley-selective CD in this semiconducting atomic membrane, which will endow electrons in this material the valley degree of freedom, in addition to charge and spin that have been routinely explored in conventional device physics. We show here using ab initio numeri-cal simulation and experimental micro-photoluminescence that monolayer MoS2 possesses near-perfect valley-selective CD, a prop-erty rooted in its bulk symmetry and very much conducive to opto-electronic valley polarization needed for valleytronics.

ResultsDegree of circular polarization. The key quantity to assess is, therefore, the k-resolved degree of optical polarization, H(k),

between the top of the valence bands and the bottom of the conduction bands4

I X( , ) = | ( ) | | ( ) || ( ) | | ( ) |

.2 2

2 2k k kk kcv

cv cv

cv cv( (( (

This quantity is the difference between the absorption of left- and right-handed lights ( o ), normalized by total absorption, at each k-point and evaluated between the top of the valence bands (v) and the bottom of conduction bands (c). The band structure of monolayer MoS2 is shown in Fig. 2a. The dependence on the transition energy, =X F Fcv c v( ) = ( ) ( )k k k , is implicit through k. Here, the transition matrix element of circular polarization is (o ocv

xcv

ycvP iP( ) =1 2[ ( ) ( )]k k k/ . The interband matrix elements,

P pcvc v( ) = | |k k k ®Z Z , are evaluated using the density functional

perturbation theory, within the local-density approximation17 , as implemented in VASP18 . Spin-orbit coupling is not included in our calculations19,20. Briefly, a planewave basis set is employed at a

(1)(1)

Mo S

Figure 1 | The crystal structure of monolayer MoS2. (a) Coordination environment of Mo (blue sphere) in the structure. Sulphur is shown as golden spheres. (b) A top view of the monolayer MoS2 lattice, emphasizing the connection to a honeycomb lattice. In our calculations, we used an optimized structure at the level of local density approximation in density functional theory. The shaded region bounded by dashed lines corresponds to one primitive cell. The unit cell parameter is a = 3.12 Å, and the vertical separation between sulphur layers is 3.11 Å.

K+

K_

–1

–0.8

–0.4

–0.6

–0.2

0

0.2

0.4

0.6

0.8

1

1.8 eV

+s+s+s

– l

– l – l

Figure 2 | Valley-selective CD of monolayer MoS2. (a) Top valence band (blue) and bottom conduction band (pink). The centre hexagon is the Brillouin zone colour-coded by the degree of circular polarization, H(k), as defined in the text. The vector connecting K + and K − is perpendicular to Mo-S bond in the crystal structure in Fig. 1b. (b) Schematic of phase winding on the MoS2 lattice that gives rise to the chiral optical selectivity. Left panel: the contribution to phase winding from the Bloch lattice phase, where T = o 1 is the valley index, and s = 1,2 corresponding to the S and Mo sites (isospin index). Right panel: the phase winding under a threefold rotation. The green axes indicate the rotation of local atomic coordinates that leads to the azimuth dissynchronization.

Spinor

( )Valley

Electron Hamiltonian

For each valley:

Wannier excitons

electron

hole

binding energy

Prada, Elsa, et al. PRB 91, 245421 (2015)

quadraticdispersion

CM

O

Center of mass: plane wave

Relative motion:variational wavefunction

Wannier s-wave Excitons: Variational Wavefunction

Wannier excitons

electron

hole

binding energy


quadraticdispersion

CM

O




Wannier excitons

electron

hole

binding energy


quadraticdispersion

CM

O




Photons in a cavityPhotons in a cavity

Perfect metal:

Metal

Metal

MoS2 layer

+ + + + + + + + +

LzCavity widthencoded here


Perfect metal:

Metal

Metal

MoS2 layer

+ + + + + + + + +


fQ =Q

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Perfect metal:

Metal

Metal

MoS2 layer

+ + + + + + + + +


fQ =Q

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From electron-phonon to exciton-photon coupling

1. Electron-Photon coupling

(i) Expand Hamiltonian close to minima H(k) H0 + kiHi1

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sha1_base64="4mbIK9g+5W1+uIlRErUBZPeKG3w=">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</latexit><latexit sha1_base64="4mbIK9g+5W1+uIlRErUBZPeKG3w=">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</latexit><latexit sha1_base64="4mbIK9g+5W1+uIlRErUBZPeKG3w=">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</latexit>

(ii) Minimal substitution p p eA<latexit sha1_base64="TR9W8CA/1bHNY2vnJlpnMcAa6/M=">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</latexit><latexit sha1_base64="TR9W8CA/1bHNY2vnJlpnMcAa6/M=">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</latexit><latexit sha1_base64="TR9W8CA/1bHNY2vnJlpnMcAa6/M=">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</latexit><latexit sha1_base64="TR9W8CA/1bHNY2vnJlpnMcAa6/M=">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</latexit>

H(k) H(k) ev Ai Hi

1<latexit sha1_base64="1aVt2vqLUxTjOIPchG2TF7Ic98A=">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</latexit><latexit sha1_base64="1aVt2vqLUxTjOIPchG2TF7Ic98A=">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</latexit><latexit sha1_base64="1aVt2vqLUxTjOIPchG2TF7Ic98A=">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</latexit><latexit sha1_base64="1aVt2vqLUxTjOIPchG2TF7Ic98A=">AAACrHicbVFNa9tAEF2rX2n65bTHXpaaggOukUKhIfSQ0kJ86CGlcWKwVDFaj+zFq5XYHaUxQn+mv6bX9tZ/05XiQux0YNnHezOzO2+SQklLvv+n4925e+/+g52Hu48eP3n6rLv3/NzmpRE4FrnKzSQBi0pqHJMkhZPCIGSJwotk+bHRLy7RWJnrM1oVGGUw1zKVAshRcff9qL/c56GR8wWBMfl33hJveJgaEBXyy7oKFwmYmocDzj/EsrlHcfCtknXc7flDvw1+GwRr0GPrOI33OpNwlosyQ01CgbXTwC8oqsCQFArr3bC0WIBYwhynDmrI0EZVO2bNXztmxtPcuKOJt+zNigoya1dZ4jIzoIXd1hryf9q0pPQwqqQuSkItrh9KS8Up541nfCYNClIrB0AY6f7KxQKcP+Sc3Xil7V2g2Jikuiq1FPkMt1hFV2TAkRYpA6mbqaoTqRT/Ctr+413DRuh/knNJdvDZLU4PTgzicv9msttFsO38bXB+MAwc/vK2d3y03soOe8lesT4L2Dt2zEbslI2ZYD/YT/aL/faG3pk39aLrVK+zrnnBNsJL/wISjNPS</latexit>

Hx1 = x, Hy

1 = y<latexit sha1_base64="544Qrztlfa8fpi7Ee5rqEIKWNMA=">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</latexit><latexit sha1_base64="544Qrztlfa8fpi7Ee5rqEIKWNMA=">AAACpXicbVHdShtBFJ6stlX7F/XSm6Gh1EKQXWlRCoJgQS8UlJoYyKbL2clJHDIzu8ycLVkWn8Sn6W37BH2bzsYUTOwZBj6+7/yfNFfSURj+aQQrq8+ev1hb33j56vWbt83Nra7LCiuwIzKV2V4KDpU02CFJCnu5RdCpwpt0clLrNz/QOpmZaypzHGgYGzmSAshTSfPzWRJ9n/IjzmOCgsdOjjUk0zaP2/XjtVwuy2XSbIV74cz4UxDNQYvN7TLZbPTiYSYKjYaEAuf6UZjToAJLUii824gLhzmICYyx76EBjW5Qzea74+89M+SjzPpviM/YxxEVaOdKnXpPDXTrlrWa/J/WL2h0OKikyQtCIx4KjQrFKeP1svhQWhSkSg9AWOl75eIWLAjyK12oMsudo1iYpJoWRopsiEusoilZ8KRD0iBNPVV1KpXi38C4f7xPWAu7X+VYkmuf+4uZ9qlFnHx87OxvES1v/ino7u9FHl99ah1/mV9lje2wd2yXReyAHbMzdsk6TLB79pP9Yr+DD8FFcB10H1yDxjxmmy1YkPwFncrQtg==</latexit><latexit sha1_base64="544Qrztlfa8fpi7Ee5rqEIKWNMA=">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</latexit><latexit sha1_base64="544Qrztlfa8fpi7Ee5rqEIKWNMA=">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</latexit>

WEM = ev Ai Hi

1<latexit sha1_base64="DD/rDJc46pfBAlwV9/mcpTfIlho=">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</latexit><latexit sha1_base64="DD/rDJc46pfBAlwV9/mcpTfIlho=">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</latexit><latexit sha1_base64="DD/rDJc46pfBAlwV9/mcpTfIlho=">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</latexit><latexit sha1_base64="DD/rDJc46pfBAlwV9/mcpTfIlho=">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</latexit>

with

2. Exciton-Photon coupling

(i) Project WEM onto

Electron-photon coupling

Project onto subspace spanned by

P. San-Jose, et al., PRX 6.3, 031046 (2016)

and







( )(ii) Expand Electron Wavefunction to O(Q2)



Electronic wavefunction to second order in Q:




Electronic wavefunction to second order in Q:


= TE, TM<latexit sha1_base64="Zj7mDyQ2OQAf1pDtewbMBbfR2yo=">AAACnXicdVHdShtBFJ5sbWu1P7Fe9qJDQ8FCKjtSaiwIQhW90KLUmEA2hNnJ2WTIzOwyc7YYln2KPk1v61P0bZyNUYzYAwPffN/5P3GmpMMw/FcLniw9ffZ8+cXK6stXr9/U195euDS3AtoiVantxtyBkgbaKFFBN7PAdaygE0++V3rnF1gnU3OO0wz6mo+MTKTg6KlB/XNkcrpLIwUJRkWkOY6tLs4Pyia9+5yUkZWjMUbloN4IN8MwZIzRCrDtr6EHOzutLdairJK8NcjcTgdrtW40TEWuwaBQ3LkeCzPsF9yiFArKlSh3kHEx4SPoeWi4BtcvZnOV9KNnhjRJrX8G6Yy9H1Fw7dxUx96z6tU91CryMa2XY9LqF9JkOYIRN4WSXFFMabUkOpQWBKqpB1xY6XulYswtF+hXuVBlljsDsTBJcZkbKdIhPGAVXqLlnnSAmktTTVUcSqXoT27cLe8TVsLGvhxJdM1jfynTPLQAk0/3nf0tbhdO/w8utjaZx2dfGnvf5ldZJu/IB7JBGNkme+SInJI2EeQ3+UP+kqvgfXAQHAc/blyD2jxmnSxY0LkGoCjRGQ==</latexit><latexit sha1_base64="Zj7mDyQ2OQAf1pDtewbMBbfR2yo=">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</latexit><latexit sha1_base64="Zj7mDyQ2OQAf1pDtewbMBbfR2yo=">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</latexit><latexit sha1_base64="Zj7mDyQ2OQAf1pDtewbMBbfR2yo=">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</latexit>

= ±1<latexit sha1_base64="uA4HQz4LS7In12Ldz4dZH1o1LvE=">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</latexit><latexit sha1_base64="uA4HQz4LS7In12Ldz4dZH1o1LvE=">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</latexit><latexit sha1_base64="uA4HQz4LS7In12Ldz4dZH1o1LvE=">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</latexit><latexit sha1_base64="uA4HQz4LS7In12Ldz4dZH1o1LvE=">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</latexit>

Exciton-photon coupling

Changing polarization basis onto the circular polarization basis set(with respect to , not to !!):

a+ =1√2[aTM − ı aTE] e−ıφk

a− =1√2[aTM + ı aTE] eıφk

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Electron-photon couplingWinding phases

Dependenceon Lz

+O(Q2

(∆/v)2)

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γ =eκ∆√

π!ωkϵ0LzF0(κ)

<latexit sha1_base64="9IqeavRuOYA9QKI8bmfVxEUEIqU=">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</latexit><latexit sha1_base64="EyeFMgLeRQ4sQS0IHfQaM3EbVZU=">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</latexit><latexit sha1_base64="EyeFMgLeRQ4sQS0IHfQaM3EbVZU=">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</latexit><latexit sha1_base64="jd/dV4yVuFhxVu0cbWFq8s67Snc=">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</latexit>

F0(κ) =1κ

!1κ+

1κ+ 1

"

<latexit sha1_base64="b3d2kwWt2ykbhLJk70g5VWowmZg=">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</latexit><latexit sha1_base64="6NNChXQjSTTS3979IEYq8++AnYg=">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</latexit><latexit sha1_base64="6NNChXQjSTTS3979IEYq8++AnYg=">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</latexit><latexit sha1_base64="PxZZrXkWhAkUCtfNwC5qIL0TzJI=">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</latexit>

For MoS2

κ =aex∆v

≈ 3.7 forMoS2<latexit sha1_base64="UB3qHZ8lyynjbOTyVLJOopHKANE=">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</latexit><latexit sha1_base64="5Br8njlZlFSAToQMR9T8DjjolmE=">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</latexit><latexit sha1_base64="5Br8njlZlFSAToQMR9T8DjjolmE=">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</latexit><latexit sha1_base64="fMWVGcr1cIfo7NKovGYnQqXr6SM=">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</latexit>

F0 ≈ 0.13<latexit sha1_base64="1relKA2rzywecoPh0ZPmeCH3iNY=">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</latexit><latexit sha1_base64="UgayIZkgTdvV1hZaJQD5an7ZWog=">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</latexit><latexit sha1_base64="UgayIZkgTdvV1hZaJQD5an7ZWog=">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</latexit><latexit sha1_base64="OzH6xYqWy2fQ3OA9NvC21Jnr0uo=">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</latexit>

hTE,τph−ex(Q, kz) = γ exp−ıτφQ

hTM,τph−ex(Q, kz) = ı

kzkγ exp−ıτφQ

<latexit sha1_base64="MmahCxaCac0FSAhjk0dtoaqGnYI=">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</latexit><latexit sha1_base64="WFH1RLFt7m9H6/HAez6+Ya33oDE=">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</latexit><latexit sha1_base64="WFH1RLFt7m9H6/HAez6+Ya33oDE=">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</latexit><latexit sha1_base64="DcUH5yNho6+XJkkuZUG6hP5Nytg=">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</latexit>

winding phase

H =

2

66664

!ex i0 0 i0q2

4k2ze2i

i0 !ph i0q2

4k2ze2i 0

0 i0q2

4k2ze2i

!ex i0

i0q2

4k2ze2i 0 i0 !ph

3

77775small q expansion

= 1 = 1 = 1 = 1

Polariton Hamiltonian in circular polarization basis.

fine splitting 5 meV

compatible with photon linewidth

S. Dufferwiel et al

Nat. Comm. 6, 8579 (2015)

At strong coupling

q/kz ≈ 0.3<latexit sha1_base64="uh36r14G1yE1xTXqm+3/uVkbg3U=">AAACgXicbVFdaxNBFL3ZWq31o4m+6ctgEVoocbc+KPoSUKgPPlQ0bSAJ4e7kJh0yOzPO3NXEpT/FV/1N/htnkwpN6oULh3Pu982dVoHT9E8j2bq1ffvOzt3de/cfPNxrth6dBVt6SV1ptfW9HANpZajLijX1nCcsck3n+exdrZ9/Ix+UNV944WhY4NSoiZLIkRo1W19fzEY/xACd83Yu0vbLUXM/badLEzdBdgX2O09guwMAp6NWozcYW1kWZFhqDKGfpY6HFXpWUtPl7qAM5FDOcEr9CA0WFIbVcvZL8TwyYzGxPrphsWSvZ1RYhLAo8hhZIF+ETa0m/6f1S568HlbKuJLJyFWjSakFW1EfQoyVJ8l6EQFKr+KsQl6gR8nxXGtdlrUdybVNqnlplLRj2mA1z9ljJANxgcrUW1UnSmvxGU34x8eCtXDwXk0Vh6OP8Rvm6MQTzQ6vB8dfZJuXvwnOjttZxJ/iU97AynbgKTyDA8jgFXTgA5xCFyR8h5/wC34nW8lhkibHq9CkcZXzGNYsefsXfrvE5A==</latexit><latexit sha1_base64="yefZD4i1g/WZrQRyPE9Ei35/VIE=">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</latexit><latexit sha1_base64="yefZD4i1g/WZrQRyPE9Ei35/VIE=">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</latexit><latexit sha1_base64="sZf7MzwyB3g0jNfLmWljgYaL2jY=">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</latexit>

Lz = 0.35µm<latexit sha1_base64="+O9zpdohknwiKyuvPZHzNFrRvBA=">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</latexit><latexit sha1_base64="+O9zpdohknwiKyuvPZHzNFrRvBA=">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</latexit><latexit sha1_base64="+O9zpdohknwiKyuvPZHzNFrRvBA=">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</latexit><latexit sha1_base64="+O9zpdohknwiKyuvPZHzNFrRvBA=">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</latexit>

Composite-particle state:

Berry Curvature of composite particles

A. Gutierrez et al., Phys. Rev. Lett. 121, 137402 (2018)

BRIEF ARTICLE

THE AUTHOR

(1) ~An = ı hn|r~q |ni

(2) |ni =X

i

in |eii

(3) ~Aintjk = ı hej |r~q |eki

(4) ~An = ıX

j

jn r~q

jn +

X

j,k

jn~Aintjk

kn

1

elementary constituent

encode the coupling

BRIEF ARTICLE

THE AUTHOR


(2) |ni =X

i

in |eii


(4) ~An = ıX

j

jn r~q

jn +

X

j,k

jn~Aintjk

kn

1

BRIEF ARTICLE

THE AUTHOR


(2) |ni =X

i

in |eii


(4) ~An = ıX

j

jn r~q

jn +

X

j,k

jn~Aintjk

kn

1

Defining the “intrinsic” Berry connection

BRIEF ARTICLE

THE AUTHOR


(2) |ni =X

i

in |eii


(4) ~An = ıX

j

jn r~q

jn +

X

j,k

jn~Aintjk

kn

1

extrinsic intrinsic

Berry connection of composite particle state:

BRIEF ARTICLE

THE AUTHOR


(2) |ni =X

i

in |eii


(4) ~An = ıX

j

jn r~q

jn +

X

j,k

jn~Aintjk

kn

1

Berry Curvature:

BRIEF ARTICLE

THE AUTHOR


(2) |ni =X

i

in |eii


(4) ~An = ıX

j

jn r~q

jn +

X

j,k

jn~Aintjk

kn

(5) ~n = ~r~q ~An

1

BRIEF ARTICLE

THE AUTHOR


(2) |ni =X

i

in |eii


(4) ~An = ıX

j

jn r~q

jn

| z

+X

j,k

jn~Aintjk

kn

| z

(5) ~n = ~r~q ~An

1

Effective Description of Polaritons.

A. Gutierrez et al., Phys. Rev. Lett. 121, 137402 (2018)see also O. Bleu et al., PRL 121, 020401 (2018)

and the poster in this conference!

He↵ = 0(q) +

↵q

2e2i

↵q2e2i

Two Effective 2x2 Hamiltonians (for LPs and for UPs) analogous to gapped bilayer graphene ⇒ analytical results

Schrieffer-Wolf transformation

/ Vz exciton Zeeman( and ! depends on γ)

need to break time-reversal symmetry

n = 0 opposite valley/polarization cancel

apply magnetic field

Due to the “wrong” valley-polarization coupling:

structure of the BC. A nonzero Berry curvature arises iftime-reversal symmetry is broken, and a magnetic fieldinduces a Zeeman coupling [19,27]. Alternatively, a polar-iton Hall current will exist when the initial state is given bya finite population of chiral excitations, which can becreated using circularly polarized light.In the presence of a Zeeman coupling Vz, the BC is

smeared around small momenta. A simple effectiveHamiltonian for either the UP (s ¼ 1) or the LP (s ¼ −1)branches can be obtained by treating γ and Γ as perturbativeparameters and considering Vz ≪ jEph − Eexj. At smallmomenta,

Hseff ¼ fsðqÞ þ

! ΔsðqÞ αq2e2iϕ

α%q2e−2iϕ −ΔsðqÞ

"; ð11Þ

with α¼−4γΓ=ðEex−EphÞ, Γ ¼ Γ=q2, and fs and Δs ∝Vzgiven in Supplemental Material [32]. The correspondencebetween the polaritonic branches and gapped (gapless forVz ¼ 0) bilayer graphene [52] becomes manifest here,providing us with a qualitative understanding of the EBC.The BC then reads

Ωs;λpolðqÞ ¼

2λΔs

αq2

½q2 þ ðΔs=αÞ2'3=2: ð12Þ

The BC for a generic parameter regime and Vz ¼ γ=5 isshown in Fig. 2 for one UP and one LP branch, the BC in thetwo other bands having the opposite sign. A breakdown ofthe different terms in Eq. (8) is presented therein. We candistinguish two different regions within the strong-couplingregimewhere either the intrinsic or the extrinsic contributionstoΩ dominate. Forq → 0, the EBCgoes to zero, whereas theIBC remains finite (see the inset in Fig. 2). In that range ofmomenta, the polariton behaves as merely inheriting the BCof its constituents.Remarkably, it turns out that the BC of excitons is 6

orders of magnitude smaller than that of photons and,therefore, can be neglected. This fact allows the interpre-tation of the dotted curves in Fig. 2 in simple terms: Agreater photonic component in the polaritonic eigenstateyields a higher value of the IBC. This behavior combineswith the decay ofΩph with q to make the IBC peak at q ¼ 0for LPs and at q > 0 for UPs.As for the dominance of the EBC, it happens near the

crossing between the bare exciton and the photon bands. Itbecomes an order of magnitude greater than its counterpartfor UPs and around twice as much as the IBC for the LPs.There is a significant difference in the BC between the UPand LP branches: The former reaches higher absolutevalues and peaks at larger momenta. Once more, thesefeatures can be understood in terms of the competitionbetween two factors. First, the proximity in energy betweena pair of either UP or LP branches is expected to increasethe value of the EBC. Notice that UP dispersion relationscome closer to each other with q, whereas the opposite

happens with LPs’ see Fig. 2(b). Second, the EBC isexpected to peak at momenta where the coupling is thegreatest. As a result of this balance, the location of the EBCpeaks in Fig. 2, which shift away from qsc, can be explained.We emphasized that photons are the only significant sourceof BC for polaritons either via the IBC or by means of thestrong coupling with matter.Polariton anomalous Hall effect.—The BC manifests in

a Hall current perpendicular to an applied in-plane force.On a general basis, assuming a polaritonic wave packet inthe s branch and the λ split band, centered at rc ¼ ðxc; ycÞand qc ¼ ðqc;x; qc;yÞ and with energy Es;λ

polðrc;qcÞ, we candescribe the evolution of its coordinates by semiclassicalequations of motion that include the BC through ananomalous velocity [53,54]:

_rc ¼∂Es;λ

pol

∂qc− _qc ×Ωs;λ

polðqcÞ; _qc ¼ −∂Es;λ

pol

∂rc : ð13Þ

Because of the anomalous term, a polariton anomalous Halleffect naturally arises when hybrid modes are accelerated.An anomalous exciton current was observed in Ref. [12],where the nonzero _qc was provided by a thermal gradientapplied to the sample. The same scheme can be applied topolaritons in the strong-coupling regime. Further alterna-tives to achieve wave packet accelerations, like a gradient inthe cavity thickness [55], can be also borne in mindregarding experimental realizations.

(b)

(c)

(a)

FIG. 2. (a) Schematics of the Hall drift experienced by LP andUP upon application of a thermal gradient. (b) Zeeman splitpolariton spectrum. (c) Extrinsic (dashed line), intrinsic (dottedline), and total (solid line) BC for the UP of highest (red) and theLP of lowest (blue) energy. Dashed black lines are a guide to theeye, with qsc such that EexðqscÞ ¼ EphðqscÞ. The lower insetenlarges the region of small momenta that appears squared ingreen. The model of Eq. (4) and the Berry connection given byEq. (8) were used in this section.

PHYSICAL REVIEW LETTERS 121, 137402 (2018)

137402-4

Composite Berry Curvature

Vz = γ/5<latexit sha1_base64="4nxaoTOd53gl/JUSXOt8p//E84Q=">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</latexit><latexit sha1_base64="81AS91hdV6DnPVqcCLXixQxM1Ug=">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</latexit><latexit sha1_base64="81AS91hdV6DnPVqcCLXixQxM1Ug=">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</latexit><latexit sha1_base64="PQgbBjtiAJdl1+y3ehPAucuo8Z4=">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</latexit>

INTRINSIC (dotted) dominates small qEXTRINSIC (dashed)

dominates strong coupling



Hseff ¼ fsðqÞ þ



"; ð11Þ


Ωs;λpolðqÞ ¼

2λΔs

αq2

½q2 þ ðΔs=αÞ2'3=2: ð12Þ







_rc ¼∂Es;λ

pol



pol

∂rc : ð13Þ


(b)

(c)

(a)



137402-4



Hseff ¼ fsðqÞ þ



"; ð11Þ


Ωs;λpolðqÞ ¼

2λΔs

αq2

½q2 þ ðΔs=αÞ2'3=2: ð12Þ







_rc ¼∂Es;λ

pol



pol

∂rc : ð13Þ


(b)

(c)

(a)



137402-4

- The Exciton Berry curvature is negligible

Ωs,λext =

2λ∆s

α

q2!q2 + (∆s/α)2]3/2

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- “Coupling” Berry curvature

rc =@Epol

@qc qpol(qc) qc = @Epol

@rc

quasi-classical equation of motion for wavepacket

Exciton-Polariton Anomalous Hall Effect

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Hseff ¼ fsðqÞ þ



"; ð11Þ


Ωs;λpolðqÞ ¼

2λΔs

αq2

½q2 þ ðΔs=αÞ2'3=2: ð12Þ







_rc ¼∂Es;λ

pol



pol

∂rc : ð13Þ


(b)

(c)

(a)



137402-4

• Cavity-modified selection rules at finite momentum.

• Berry curvature of composite particles.

• Polariton fine splitting.

Summary

H =

2

66664

!ex i0 0 i0q2

4k2ze2i

i0 !ph i0q2

4k2ze2i 0

0 i0q2

4k2ze2i

!ex i0

i0q2

4k2ze2i 0 i0 !ph

3

77775

= 1 = 1 = 1 = 1

• Exciton Polaritons have a Berry curvature arising from the

photon and the coupling ones.

A. Gutierrez et al., Phys. Rev. Lett. 121, 137402 (2018)

Documents

Polariton Anomalous Hall Effect in TMDs - Sciencesconf.org€¦ · PRL 101, 106401 (2008) ⌦⌧ ex(q)= 1 4 X q0 |(q0)|2 X =±1 ⌦⌧ c (q 0 + q/2) Photon: Two polarizations. A⌫