Modulated phases of graphene quantum Hall polariton fluids

There is a growing experimental interest in coupling cavity photons to the cyclotron resonance excitations of electron liquids in high-mobility semiconductor quantum wells or graphene sheets. These media offer unique platforms to carry out fundamental studies of exciton-polariton condensation and cavity quantum electrodynamics in a regime, in which electron–electron interactions are expected to play a pivotal role. Here, focusing on graphene, we present a theoretical study of the impact of electron–electron interactions on a quantum Hall polariton fluid, that is a fluid of magneto-excitons resonantly coupled to cavity photons. We show that electron–electron interactions are responsible for an instability of graphene integer quantum Hall polariton fluids towards a modulated phase. We demonstrate that this phase can be detected by measuring the collective excitation spectra, which is often at a characteristic wave vector of the order of the inverse magnetic length.

where v D ≈ 10 6 m/s is the Dirac velocity. Here σ = (σ x , σ y ) is a 2D vector of Pauli matrices acting on sublattice degrees-of-freedom and p = −i ∇ r is the 2D momentum measured from one of the two corners (valleys) of the We work in the Landau gauge A 0 = −Byx. In this gauge the canonical momentum along thex direction, p x , coincides with the magnetic translation operator [3] along the same direction and it commutes with the Hamiltonian H 0 . Thus, the eigenvalues of p x are good quantum numbers. A complete set of eigenfunctions of the Hamiltonian H 0 in Eq. (2) is provided by the two component pseudospinors [4] r|λ, n, k = e ikx √ 2L where λ = + (−) denotes conduction (valence) band levels, n ∈ N is the Landau level (LL) index, and k is the eigenvalue of the magnetic translation operator in thex direction. In Eq. (3) w ±,n = 1 ± δ n,0 guarantees that the pseudospinor corresponding to the n = 0 LL has nonzero weight only on one sublattice. Furthermore, φ n (y) with n = 0, 1, 2, . . The spectrum of the Hamiltonian (2) has the well-known form [4] ε λ,n = λ ω c √ n .
Each LL has a macroscopic degeneracy is the spin-valley degeneracy and S = L 2 is the sample area.
The fully microscopic matter Hamiltonian is written as Here, H 0 is the second-quantized version of Eq. (2), where c † λ,n,k,ξ (c λ,n,k,ξ ) is a fermionic creation (annihilation) operator for an electron with band index λ, LL quantum number n, and eigenvalue of the magnetic translation operator along thex direction equal to k. The collective index ξ refers to the valley (K, K ) index and spin-projection along theẑ direction. The second term in Eq. (6), H ee , represents Coloumb interactions.
This term can be written as where n q is the Fourier transform of the electronic density operator and v q is the 2D Fourier transform of the Coulomb potential v q = 2πe 2 q .
Here κ r is the cavity dielectric constant. The real-space eld operators ψ † (r) and ψ(r) have been introduced in the main text.
We consider the integer quantum Hall regime in which a given number of LLs are fully occupied and the remaining ones are empty. Since the MDF Hamiltonian is particle-hole symmetric, we can consider, without loss of generality, the situation in which graphene is n-doped and the Fermi energy lies in the conduction band (λ = +). We denote by n = M the highest occupied LL, and the lowest empty LL is n = M + 1 We are interested in the case in which cavity photons with energy ω are nearly resonant with the energy dierence between the two conduction-band LLs n = M, M +1. In this limit, the fermionic Hilbert space can be reduced to the resonant doublet. From here on, we denote byc λ,n,k,ξ andc † λ,n,k,ξ operators for states which do not belong to the resonant doublet M, M + 1. We will keep using c λ,n,k,ξ and c † λ,n,k,ξ only for states which belong to the resonant doublet. We can rewrite the full microscopic Hamiltonian in Eq. (6) as the sum of three terms: The rst term H d contains only fermionic eld operators related to the resonant doublet M, M +1. The second term, H m , contains only eld operators of the typec λ,n,k,ξ ,c † λ,n,k,ξ : these degrees of freedom play the role of a medium for the resonant doublet. The third term, H dm , describes coupling between the resonant doublet and the medium degrees of freedom.
To obtain an eective matter Hamiltonian, we start from the fully microscopic Hamiltonian H mat and we treat in an exact fashion all the terms that involve eld operators (c λ,n,k,ξ and c † λ,n,k,ξ ) acting only on the doublet n = M, M + 1 in conduction band λ = +. All the other terms (containing c λ,n,k,ξ andc † λ,n,k,ξ ) are treated within the Hartree-Fock approximation [3].
The medium Hamiltonian is discarded. In the coupling term H dm , we only keep terms that separately conserve the number of particles in the M, M + 1 doublet and in the medium. These are terms of the formc † c † cc. Terms of the formc †c † cc are discarded. We therefore replacẽ After straightforward algebraic manipulations we reach the nal result for the eective matter Hamiltonian, which is best expressed in a pseudospin representation in which pseudospin up (down) corresponds to the M + 1 (M ) LL. To this end, we dene the following pseudospin operators: where m = 0, . . . , 3 and [τ m ] nn labels the matrix elements of a four-vector τ m of 2 × 2 matrices acting on the M, M + 1 doublet, specically τ 0 represents the 2 × 2 identity matrix and τ 1 , τ 2 , τ 3 represent the ordinary 2 × 2 Pauli matrices.
The nal eective matter Hamiltonian reads as following: Here where the summation over λ, n runs only values such that λ,n < +1,M .
The last term in Eq. This is a 4 × 4 Hermitian matrix, which can be decomposed into its real and imaginary parts: It is possible to show that every quantity V mm (q) is either purely real or purely imaginary: V mm (q) is purely imaginary if m = 1 or 2 (m = 0 or 3) and simultaneously m = 0 or 3 (m = 1 or 2); for any other value of m, m V mm (q) is purely real.

Supplementary Note 2
We consider a graphene sheet coupled to the electromagnetic eld in a cavity.
Among all the processes described by H int , we take into account only resonant terms which describe the photon-induced electronic transitions between LLs M and M + 1 in conduction band. This approximation is called rotating wave approximation (RWA). In the RWA, the light-matter interaction Hamiltonian becomes where a q,L = (a q,x − ia q,y )/ √ 2 [a † q,L = (a † q,x + ia † q,y )/ where and γ ≡ |α| 2 + |β| 2 . (32) We note that the commutator between the operator d † and its hermitian conjugate d is given by: Calculating the expectation value of [d, d † ] on the variational state introduced in Eq. (3) of the main text and taking the low-density limit θ k 1, we nd: As in Supplementary Note 1, the sum over λ, n runs only over indices such that λ,n < +,M . We note that a ee involves only the two resonant LLs M and M + 1. Supplementary Figure 1 shows the quantity a ee (in units of α ee ω c ) as a function of the LL index M in the interval 1 ≤ M ≤ 15. Here α ee = e 2 /(κ r v D ) ≈ 2.2/κ r .
The symmetric and antisymmetric pseudospin-pseudospin interactions can be decomposed as following: The direct contributions are given by: The exchange contributions are given by: where N φ has been introduced in the main text.
Moreover, we nd that Supplementary Note 5 The correction ∆ ee in Eq. (35) to the cyclotron transition energy is related to the renormalization [6,7,8] of the Dirac velocity v D due to exchange interactions, which occurs also in the absence of a magnetic eld. It is wellknown [9,10,11] that ∆ ee is logarithmically divergent, i.e.
where n max is a cut-o dened below and the constant C M depends on the highest-occupied LL with index M . For example, C 0 −1.017 and C 1 −2.510. The Dirac model applies over a large but nite energy region, so we dene a high-energy cut-o W in valence band. At any given magnetic eld B, the integer n max represents the number of LLs in the valence band with energy larger than W , i.e. where Shizuya [11] in writing the correction to cyclotron transition energy in terms of the renormalized Dirac velocity: The quantity v D is the bare Dirac velocity and δv D = α QED c/(8κ r )[ln(B W /B) + C M ] is the correction to the Dirac velocity, where α QED = e 2 /( c) 1/137 is the QED ne-structure constant. This allows us to x B W = 450 Tesla to make sure that v D matches the value v D = 1.12 × 10 6 m/s measured [12] from the intraband 0 → 1 cyclotron transition energy at B = 18 Tesla and in a sample with κ r = 5. We have xed the bare Dirac velocity to v D = c/300.

Supplementary Note 6
In this Supplementary Note we discuss the relative role of symmetric and antisymmetric pseudospin-pseudospin interactions in determining the precise form of the phase diagrams shown in Figure 2 of the main text.
We start by setting to zero all the symmetric interactions in the energy functional introduced in Eq. (7) of the main text: J = 0. In this case, we nd that the unstable regions (grey-shaded regions in Figure 2 of the main text) considerable expand. This is illustrated for the case κ r = 15 and M = 1 in Supplementary Figure 3.

Supplementary Note 7
In this Supplementary Note we explain more in detail the approach we have followed to nd the elementary excitations of the polariton uid.
In the spin-chain language introduced in the main text, the state |ψ of the homogeneous uid phase represents a collinear ferromagnet in which the expectation value of the spin operator S(q) ≡ [S 1 (q), S 2 (q), S 3 (q)] T is non-zero only at q = 0 and oriented along the direction forms an orthonormal set, which will be used below to construct low-energy collective excitations above the ground state |ψ .
In the time-dependent Hartree-Fock (or generalized random phase) approximation [13], the spin wave spectrum can be calculated by making use ξξ + (q) and keeping terms up to linear order in α qν = a q,ν and r qξ = ρ ξξ − (q) , we nd a homogeneous system of rst-order linear dierential equations. Eigenmodes are found by replacing i ∂ t → Ω. We obtain an eigenvalue problem, whose solution gives the spin wave spectrum. We nd six independent collective modes, four