Interplay between photon condensation and electron-electron interactions in molecular systems

The gist: The global behavior of the system may also undergo a first order transition because of the action of the electron-electron interaction, except for half-filling and single electron cases, which are necessarily second order phase transitions

Model and Setup

The study investigates a minimal molecular model consisting of square planar plaquettes hosting multiple electrons, whose dynamics is governed by a tight-binding Hamiltonian supplemented by on-site Hubbard repulsion. This system is coupled to a spatially nonuniform cavity mode to analyze the emergence of photon condensation originating from the paramagnetic Van Vleck mechanism. The system is modeled with square plaquettes having ns = 4 sites, described by a tight-binding Hamiltonian that includes only nearest-neighbor hopping processes and no spin-flip term. The kinetic Hamiltonian of the system of N molecules is then Tˆ0 = X N k=1 Tˆk, (1a) where the operator ˆc† jzk (ˆcjszk) creates (destroys) an electron with spin z-projection sz at the site j of the kth molecule. The total system Hamiltonian is written as Hˆe = Tˆ0 + Uˆ = X N k=1 Hˆk, (4) where Hˆk = Tˆk + Uˆk represents the single-molecule Hamiltonian of the k-th molecule.

Electronic States and Interactions

For a generic single molecule Hamiltonian Ĥ k one has to work in a many-body basis. When multiple electrons are introduced, their confinement in a reduced space can make the role of the electron-electron interactions significantly important. The Coulomb repulsion is modeled through an on-site Hubbard term Uˆ = X N k=1 Uˆk, (3a) where U > 0. The total system Hamiltonian is written as Hˆe = Tˆ0 + Uˆ = X N k=1 H k, (4) where H k = T k + U k represents the single-molecule Hamiltonian of the k-th molecule. In this context, we model the Coulomb repulsion through an on-site Hubbard term [63]. The total number of electrons Ne exceeds half-filling, Ne > ns, one can use particle-hole symmetry to map these situations onto cases with an electron number of 2ns − Ne.

Photon Condensation and Phase Transitions

Photon condensation occurs when the photonic order parameter α = ⟨ϕα aˆ ϕα⟩ / √N reaches a nonzero value in the thermodynamic limit. The global behavior is analyzed for different electronic filling factors, and we find that, except for the special cases of half-filling and single electron, where the transition, if it occurs, is necessarily a second order phase transition, the global system may also undergo a first order transition because of the action of the electron-electron interaction. The polaritonic excitation energies are analyzed to provide clear spectroscopic signatures of the magnetostatic instability and of its order.

Role of Electron-Electron Repulsion

The electron–electron interaction strength acts as an additional control parameter in addition to the light–matter coupling. Depending on the chosen filling factor, the electron–electron interaction can either renormalize the critical coupling at which photon condensation emerges as a second order phase transition, or alter the character of the transition so that it becomes a first order phase transition. The analysis shows that for Ne = 2 electrons per molecule, when U = t, there are two particular hopping parameters τ1 −t, such that for any value of τ where τ1 ≤ τ ≤ τ2 the ground state is the degenerate triplet of energy ϵ0 +ϵ1. In contrast, outside this specified range, the ground state is a non-degenerate singlet, resides in the spin sector Sz = 0, and its energy value depends on U.

Polaritonic Signatures

Polaritons are novel hybrid light-matter states arising as collective excitations above the mean-field ground state. The polaritonic Hamiltonian is derived by expanding the full Hamiltonian up to the second order with respect to the photonic fluctuations δaˆ and δaˆ†. The resulting 2s × 2s matrix Ξ, written in the basis (δa, δ ˆ aˆ†, ˆbl, ˆb†l), is explicitly expressed as Eq. (B19). The eigenvalues of this Hopfield matrix Ξ correspond to the polaritonic excitation energies. Polaritonic modes display either a softening or a step-like change, depending on whether the transition is continuous or discontinuous.

Conclusion

In summary, when the electron occupation number is Ne = 2, the repulsive interaction between electrons can significantly alter the character of photon condensation, provided the microscopic parameters lie within a certain range. This discussion can also be extended to the situation with Ne = 3 electrons per molecule, in which photon condensation can emerge either as a first order or as a second order phase transition. For Ne = 4 (half-filling), photon condensation always occurs as a second order phase transition, and the effect of the electron–electron repulsion consists only of a renormalization of the parameters at which the transition occurs. The study concludes that when Ne = 4, photon condensation is suppressed when the electron-electron repulsive interaction exceeds approximately Uc ≈ 1.4t at τ = t.

How it works

The system is modeled by a tight-binding Hamiltonian that accounts for hopping within the planar structure, together with an on-site electron-electron interaction potential. The kinetic Hamiltonian of the system of N molecules is then Tˆ0 = X N k=1 Tˆk, (1a) where the operator ˆc† jzk (ˆcjszk) creates (destroys) an electron with spin z-projection sz at the site j of the kth molecule. The total system Hamiltonian is written as Hˆe = Tˆ0 + Uˆ = X N k=1 H k, (4) where H k = T k + U k represents the single-molecule Hamiltonian of the k-th molecule.

The analysis involves coupling this system to a spatially nonuniform cavity mode, which leads to the emergence of a magnetostatic instability, namely photon condensation originating from the paramagnetic Van Vleck mechanism. The gauge invariance principle forbids photon condensation when the vector potential A(r) is spatially uniform, but this limitation can be overcome by considering finite momentum exchanged between the photonic mode and the material system. This implies that A(r) acquires a spatial dependence such that it represents a nonconservative field, i.e. ∇ × A(r) ≠ 0.

Mean-Field Theory of Photon Condensation

In the thermodynamic limit N ≫ 1, the ground state of the complete Hamiltonian is approximated as a direct product F⟩ ⊗ ϕα⟩. Photon condensation occurs when the photonic order parameter α = ⟨ϕα aˆ ϕα⟩ / √N reaches a nonzero value in the thermodynamic limit. The mean-field molecular Hamiltonian HˆMF(α) exclusively involves the electronic degrees of freedom. The global behavior is analyzed for different electronic filling factors, and we find that, except for the special cases of half-filling and a single electron, where the transition, if it occurs, is necessarily of the second order

Polaritons

Polaritons are novel hybrid light-matter states arising as collective excitations above the mean-field ground state. The polaritonic Hamiltonian is quadratic in the bosonic fields by diagonalizing the corresponding Hopfield matrix, one directly obtains the associated spectrum [91]. The resulting 2s × 2s matrix Ξ, written in the basis (δa, δ ˆ aˆ†, ˆbl, ˆb†l), is explicitly expressed as Eq. (B19). The eigenvalues of this Hopfield matrix Ξ correspond to the polaritonic excitation energies.

Conclusion

In this work, we investigated how electron–electron repulsion affects both the onset and the nature of the photon condensation phase transition in a system composed of many coplanar molecules coupled to a single quantized magnetic mode. For multiple electrons per molecule, our results demonstrate that photon condensation can occur even in the presence of electron–electron repulsion, which can substantially reshape the phase diagram compared to the non-interacting case. In particular, depending on the number of itinerant electrons and the microscopic parameters that enter the molecular Hamiltonian, photon condensation can manifest as either a first or a second order phase transition. Furthermore, we have demonstrated that the polaritonic spectrum can be used to probe the character of the phase transition. Possible future developments may include the investigation of reflectivity and transmission spectra in the vicinity of the phase transition [92].

1. Improved AI systems can accurately predict whether photon condensation will manifest as a first-order or second-order phase transition by analyzing parameters like electron filling factor and electron-electron interaction strength, as the paper states that the global system may also undergo a first order transition because of the action of the electron-electron interaction.

2. AI systems can determine if photon condensation is suppressed by strong repulsion in certain regimes, specifically noting that photon condensation is suppressed when the electron-electron repulsive interaction exceeds approximately Uc ≈ 1.4t at τ = t for half-filling, which is reminiscent of the Hubbard model's Mott transition.

3. AI systems can identify critical microscopic parameters that govern phase transitions by analyzing spectroscopic signatures, such as polaritonic excitation energies, to distinguish between continuous and discontinuous transitions in the polaritonic spectrum.

4. AI systems can predict the nature of the transition based on the magnetic response of the matter subsystem, specifically by examining whether it crosses over from a Van Vleck paramagnetic regime to a diamagnetic one when considering the resulting modifications of the magnetic response of the matter subsystem.

5. AI systems can model and analyze polaritonic excitations to characterize phase transitions by observing that polaritonic modes display either a softening or a step-like change, depending on whether the transition is continuous or discontinuous.