Excitonic Order Parameter Research Articles

Antiferromagnitnyy eksitonnyy dielektrik

The effective the two-band Hamiltonian is obtained for iridium oxides with account for strong electron correlations (SEC) and the spin–orbit interaction. The intraatomic electron correlations in iridium ions induce the formation of Hubbard fermions (HF) filling the states in the valence band. Another consequence of SEC is associated with the emergence of the antiferromagnetic (AFM) exchange interaction between HF in accordance with the Anderson mechanism. As a result, a long-range antiferromagnetic order is established in the system, and in the conditions of band overlapping, the intersite Coulomb interaction induces a phase transition to the excitonic insulator (EI) state with a long-range AFM order. The system of integral self-consistent equations, the solution to which determines the excitonic order parameter components Δi, j(k), sublattice magnetization M, Hubbard fermion concentration nd, and chemical potential μ, is obtained using the atomic representation, the method of two-time temperature Green’s functions, and the Zwanzig–Mori projection technique. The symmetry classification of AFM EI phases is performed, and it is shown that in the nearest neighbor approximation, state Δi, j(k) with the s-type symmetry corresponds to the ground state, while the phases with the d- and p-symmetries are metastable.

Ginzburg-Landau action and polarization current in an excitonic insulator model of electronic ferroelectricity

In comparison to transport of spin polarization in ferromagnets, transport of electric polarization in ferroelectrics remains less explored. Taking an excitonic insulator model of electronic ferroelectricity as a prototypical example, we theoretically investigate the low-energy dynamics and transport of electric polarization by microscopically constructing the Ginzburg-Landau action. We show that, because of the scalar nature of the excitonic order parameter, only the longitudinal fluctuations are relevant to the transport of electric polarization. We also formulate the electric-polarization diffusion equation, in which the electric-polarization current is defined purely electronically without recourse to the lattice degrees of freedom.

Electron-phonon correlations inducing excitonic excitations in semimetal and semiconducting materials

The influence of phonons on the low-energy excitonic excitations at zero temperature in the extended Falicov-Kimball model has been investigated. In the framework of the unrestricted Hartree-Fock approximation, a set of self-consistent equations for the excitonic condensate order parameter and a lattice distortion is derived when both electron-phonon coupling and electron-hole Coulomb interaction are treated on an equal footing. The low-energy excitation properties of the excitonic condensate are addressed in signatures of the optical conductivity and the dynamical excitonic susceptibility function. The real part of the optical conductivity is evaluated by the Kubo linear response theory and the imaginary part of the dynamical excitonic susceptibility is found by adapting the random phase approximation. In the semimetal state, one always finds a sharp peak in the optical conductivity spectrum indicating the stability of the excitonic condensation in the BCS type if the correlation between electrons and phonons becomes significant. In contrast, the peak is smeared out on the semiconducting side indicating the stability of the BEC-type excitonic condensate. In this semiconducting side, the sharp peak signature appears and the system turns to the BCS-type excitonic condensation state by increasing the electron-phonon correlations. In either the semimetal or the semiconducting normal state, increasing the electron-phonon correlations always reinforces a low-energy sharp peak in the dynamical excitonic susceptibility spectrum, indicating the existence of the tightly bound excitonic excitations before the condensation state. Specifically, on the semiconducting side, the ``halo'' phase with the preformed excitons exiting outside of the BEC-excitonic condensation state has been specified. The halo phase becomes more recognizable by raising the electron-phonon correlations.

Excitonic properties in a double-layer graphene

This paper investigates theoretically the excitonic condensation state at zero temperature in a double-layer graphene structure. In the framework of the unrestricted Hartree–Fock approximation, the electron-hole system in the structure described in the two-band electronic model is analyzed and one finds a set of self-consistent equations determining the excitonic order parameter. The optical properties of the excitonic condensation state then are examined in the Kubo linear optical response theory. Our results indicate that in the case of sufficiently large Coulomb interaction, the BEC excitonic condensation state might occur at low electronic excitation density. By turning the external electric field, the superfluid state stabilizes in the BCS-type excitonic condensate. The optical conductivity spectrum also provides us more insight into the excitonic condensation states.

Low-energy excitonic condensate excitations in semimetal and semiconducting materials

The low-energy excitation properties of the excitonic condensate in the semimetal and semiconducting materials are inspected in the extended Falicov-Kimball model. In the framework of the unrestricted Hartree-Fock approximation, we have released a set of self-consistent equations so the excitonic condensate order parameter and then the optical conductivity and the dynamical excitonic susceptibility function are evaluated. We find that the optical conductivity becomes significant and shows a sharp peak at a frequency that is twice as of the excitonic condensate order parameter once the system settles in the excitonic condensate. In the meanwhile, out of the excitonic condensation state, the dynamical excitonic susceptibility function in the semimetal state releases a dominant peak at low frequency if Coulomb interaction is sufficiently large. This character indicates the possibility of the resonance bound excitonic condensate even in high-temperature normal semimetals. The resonance is dramatically reinforced inside of the semiconducting state and that specifies the stability of the bound excitonic coherent state of the preformed excitons in the semiconducting materials.

Excitonic condensation and metal-semiconductor transition in AA bilayer graphene in an external magnetic field

In this paper, the effects of the external transverse magnetic field $B$ (perpendicular to the surface of the layers) on the electronic and excitonic properties are studied in the AA-stacked bilayer graphene (BLG). The effects of the Coulomb interactions and excitonic pairing have been taken into account and analyzed in detail within the bilayer Hubbard model. Both half-filling and partial filling regimes have been taken into account and the magnetic field dependence of a series of physical parameters was found. It is shown that the difference between the average electron concentrations in the layers vanishes at some critical value of magnetic field $B_{c}$ and the chemical potential is calculated numerically above and below that value. The role of the Coulomb interactions on the average carrier concentrations in the layers has been analyzed, and the excitonic order parameters have been calculated for different spin orientations. We found a possibility for the particle population inversion between the layers when varying the external magnetic field. The calculated electronic band structure in the AA-BLG shows the presence of metal-semiconductor transition, governed by the strength of the applied magnetic field or the interlayer interaction potential. We show that for high magnetic fields the band-gap is approaching the typical values of the gaps in the usual semiconductors. It is demonstrated that, at some parameter regimes, the AA-BLG behaves like a spin-valve device, by permitting the electron transport with only one spin direction. All calculations have been performed at the zero-temperature limit.

Excitonic insulators and Gross-Neveu models

We introduce a generalized Gross-Neveu (GN) model to describe the excitonic instabilities in two different systems: a small overlap semi-metal (SM) and a small gap semi-conductor (SMC), both in two (2d) and three-dimensions (3d). We identify the excitonic order parameter (EOP) and obtain the effective potential within the Large $N$ limit approach where the GN model can be exactly solved. We obtain the excitonic insulator (EI) phase diagrams as a function of temperature, chemical potential, overlap between bands and gaps of the system. We show that the EI may undergo first- or second-order thermal transitions depending on the regime whereupon this phase is approached. We also investigate the expected thermodynamic signatures for the specific heat above the fine-tuned excitonic quantum critical point (EQCP), in both 2d and 3d, in the SMC regime. We show that the EQCP is a different kind of critical point since although the EOP vanishes at the EQCP, there is always a finite gap in the SMC regime. We find that for high temperatures, the specific heat might exhibit a scaling behavior in the form $C_V/T \propto T^{(d-z)/z}$, where $d$ is the dimension of the system and $z$ is the dynamical critical exponent. The very low temperature behavior has a dominant exponential thermally activated term due to the presence of a gap that does not vanish at the excitonic transition.

Photoinduced dynamics of a quasicrystalline excitonic insulator

We study the photoinduced dynamics of the excitonic insulator in the two-orbital Hubbard model on the Penrose tiling by means of the time-dependent real-space mean-field approximation. We show that, with a single-cycle electric-field pulse, the bulk (spatially averaged) excitonic order parameter decreases in the Bardeen-Cooper-Schrieffer (BCS) regime, while it increases in the Bose-Einstein condensate (BEC) regime. This behavior is in common with the excitonic insulator in the square lattice reported previously. To explore the dynamics peculiar to the Penrose tiling, we examine the coordination number dependence of observables and analyze the perpendicular space. In the BEC regime, characteristic oscillations of the electron number at each site are induced by the pulse, which are not observed in normal crystals with single coordination number. On the other hand, the dynamics in the BCS regime is characterized by drastic change in the spatial pattern of the excitonic order parameter. Our results imply that potentially interesting nonequilibrium physics is caused by rich local structural patterns involved in quasicrystals.

Theoretical study of the dynamics of the excitonic insulator order parameter of the Falicov–Kimball model excited by ultrashort laser pulse

The non-equilibrium dynamics of the excitonic insulator order parameter in the excitonic insulator phase of the Falicov–Kimball model excited by ultrashort laser pulse is studied. The Keldysh–Schwinger functional integral method with the saddle point approximation is employed. The numerical solution for the order parameter obtained from the saddle point equation is found to exhibit oscillatory behaviors, whose frequencies are consistent with the energy gap of the excitonic insulator. The fluctuations of the order parameter around the saddle point are identified with the collective modes in non-equilibrium states, and their Green’s functions are explicitly found in terms of the saddle point solution. Also, the interaction vertices responsible for the decay of the non-equilibrium order parameter is found, and the lowest order contribution for the decay process is expressed in terms of the vertices and the Green’s functions of the non-equilibrium collective modes. In the presence of a particular electron–phonon interaction, a hybrid mode of the excitonic order parameter and the phonon can be naturally defined in functional integral and its saddle point equation can be derived. It is shown that the saddle point solution with electron–phonon interaction is consistent with the experimental data and the existing theoretical result.

Third-harmonic generation in excitonic insulators

We study third-harmonic generation (THG) in an excitonic insulator (EI) described in a two-band correlated electron model. Employing the perturbative expansion with respect to the external electric field, we derive the THG susceptibility taking into account the collective dynamics of the excitonic order parameter. In the inversion-symmetric EI, the collective order parameter motion is activated at second order of the external field and its effects arise in THG. We find three peaks in the THG susceptibility at energies $\hbar \Omega = \Delta_g/3$, $\Delta_g/2$, and $\Delta_g$, where $\Delta_g$ is the band gap. While the THG response at $\Delta_g/3$ is caused by bare three-photon excitation of the independent particle across the band gap, the latter two peaks involve the motion of the order parameter activated at second order. The resulting resonant peaks are prominent in particular in the BCS regime but they become less significant in the BEC regime. We demonstrate that the resonant peak originated by the collective excitation is observable in the temperature profile of the THG intensity. Our study suggests that the THG measurement should be promising for detecting the excitonic collective nature of materials.

Second-Order Josephson Effect in Excitonic Insulators.

We show that in electron-hole bilayers with excitonic orders arising from conduction and valence bands formed by atomic orbitals that have different parities, nonzero interlayer tunneling leads to a second-order Josephson effect. This means the interlayer electrical current is related to the phase of the excitonic order parameter as J=J_{c}sin2θ instead of J=J_{c}sinθ and that the system has two degenerate ground states at θ=0,π that can be switched by an interlayer voltage pulse. When generalized to a three dimensional stack of alternating electron-hole planes or a two dimensional stack of chains, the ac Josephson effect implies that electric field pulses perpendicular to the layers and chains can steer the order parameter phase between the two degenerate ground states, making these devices ultrafast memories. The order parameter steering also applies to the excitonic insulator candidate Ta_{2}NiSe_{5}.

Mass Imbalance Effects in the Excitonic Condensation of the Extended Falicov–Kimball Model

The ground‐state properties of the excitonic condensation state in the effects of the mass imbalance are discussed in the framework of the spinless extended Falicov–Kimball model, in which the hopping of the f electrons is involved. Using the unrestricted Hartree–Fock approximation, a set of self‐consistent equations are obtained so the excitonic condensate order parameter is determined. In this sense, the real part of the optical conductivity is analytically calculated based on the Kubo formula. Phase diagram releases a complex phase structure of the excitonic condensation state with a strong depression of its Bardeen–Cooper–Schrieffer (BCS)–Bose–Einstein condensation (BEC) crossover in the influence of the mass imbalance. The impression of the mass imbalance affecting the excitonic condensate is also addressed in the signature of the optical conductivity. The finding delivers the natural scenario of the excitonic condensation state applied for a wide range of materials, from transition‐metal dichalcogenides to double‐layer systems such as a semiconductor double quantum well, double‐bilayer graphene, or double‐layer graphene.

Direct observation of excitonic instability in Ta2NiSe5

Coulomb attraction between electrons and holes in a narrow-gap semiconductor or a semimetal is predicted to lead to an elusive phase of matter dubbed excitonic insulator. However, direct observation of such electronic instability remains extremely rare. Here, we report the observation of incipient divergence in the static excitonic susceptibility of the candidate material Ta2NiSe5 using Raman spectroscopy. Critical fluctuations of the excitonic order parameter give rise to quasi-elastic scattering of B2g symmetry, whose intensity grows inversely with temperature toward the Weiss temperature of TW ≈ 241 K, which is arrested by a structural phase transition driven by an acoustic phonon of the same symmetry at TC = 325 K. Concurrently, a B2g optical phonon becomes heavily damped to the extent that its trace is almost invisible around TC, which manifests a strong electron-phonon coupling that has obscured the identification of the low-temperature phase as an excitonic insulator for more than a decade. Our results unambiguously reveal the electronic origin of the phase transition.

Signatures of optical conductivity in double-layer graphene excitonic condensate

Signatures of the excitonic condensation state and its optical properties in biased double-layer graphene (DLG) are discussed. In the framework of the generic bilayer graphene model involving a Coulomb interaction between electrons in one layer and holes in the opposite layer, we have derived a set of self-consistent equations determining the excitonic condensate order parameter by use of the projector-based renormalization method. Based on the Kubo theory of the linear optical response, a formula of the optical conductivity is also obtained with full contributions of the quantum fluctuations. Our results elucidate a possibility of the excitonic condensation formation in the biased system once the dielectric thickness is sufficiently small. As a function of the external electric field, the excitonic condensation dome suppresses rapidly as enlarging the dielectric thickness. The impression of the dielectric thickness and external electric field affecting the excitonic condensate is also addressed in the signature of the optical conductivity. Our findings are worthwhile to understand the formation and stability of the excitonic condensation states not only in DLG but also in other double-layer systems or in semimetal-semiconductor transition materials.

Collective excitations of the U (1)-symmetric exciton insulator in a cavity

We investigate the equilibrium state and the collective modes of an excitonic insulator (EI) in a Fabry-P\'erot cavity. In an EI, two bands of a semiconductor or semimetal spontaneously hybridize due to the Coulomb interaction between electrons and holes, leading to the opening of a gap. The coupling to the electromagnetic field reduces the symmetry of the system with respect to phase rotations of the excitonic order parameter from $U(1)$ to $Z_2$. While the reduction to a discrete symmetry would in general lead to a gapped phase mode and enhance the stability of the ordered phase, the coupling to the cavity leaves the mean-field ground state unaffected. Its energy remains invariant under $U(1)$ phase rotations, in spite of the lower $Z_2$ symmetry imposed by the cavity. In dipolar gauge, this can be traced back to the balancing of the linear light-matter coupling and the dipolar self-interaction at zero frequency. At nonzero frequency, however, the collective excitations do reflect the lower $Z_2$ symmetry, which shows that fluctuations beyond mean-field could play a crucial role in finding the true phase at finite temperature.

Comparing the generalized Kadanoff-Baym ansatz with the full Kadanoff-Baym equations for an excitonic insulator out of equilibrium

We investigate out-of-equilibrium dynamics in an excitonic insulator (EI) with a finite momentum pairing perturbed by a laser-pulse excitation and a sudden coupling to fermionic baths. The transient dynamics of the excitonic order parameter is resolved using the full nonequilibrium Green's function approach and the generalized Kadanoff-Baym ansatz (GKBA) within the second-Born approximation. The comparison between the two approaches after a laser pulse excitation shows a good agreement in the weak and the intermediate photo-doping regime. In contrast, the laser-pulse dynamics resolved by the GKBA does not show a complete melting of the excitonic order after a strong excitation. Instead we observe persistent oscillations of the excitonic order parameter with a predominant frequency given by the renormalized equilibrium bandgap. This anomalous behavior can be overcome within the GKBA formalism by coupling to an external bath, which leads to a transition of the EI system towards the normal state. We analyze the long-time evolution of the system and distinguish decay timescales related to dephasing and thermalization.

Edge States of an Excitonic Insulator with the Spin—Orbit Interaction

The effect of the spin-orbit and Coulomb interactions on the symmetry and topological properties of the s, p, d, and s + d phases of an excitonic insulator has been considered by example of the inverted two-band structure of the HgTe quantum well. It has been found that only the p phase has a nontrivial topology, but it is metastable. It has been shown that the exchange interaction between fermions induces the ground state with the s + d symmetry of the excitonic order parameter. This phase of the excitonic insulator has zero Chern number; nevertheless, edge states occur in the case of open boundaries.

Onsite density matrix description of the extended Falicov-Kimball model at finite temperatures

We propose a single-site mean-field description, an analogue of Weiss mean-field theory, suitable for narrow-band systems with correlation-induced hybridisation at finite temperatures. Presently this approach, based on the notion of a fluctuating on-site density matrix (OSDM), is developed for the case of extended Falicov-Kimball model (EFKM). In an EFKM, an excitonic insulator phase can be stabilised at zero temperature. With increasing temperature, the excitonic order parameter (interaction-induced hybridisation on-site, characterised by the absolute value and phase) eventually becomes disordered, which involves fluctuations of both its phase and (at higher T) its absolute value. In order to build an adequate finite-temperature description, it is important to clarify the nature of degrees of freedom associated with the phase and absolute value of the induced hybridisation, and correctly account for the corresponding phase-space volume. We show that the OSDM-based treatment of the local fluctuations indeed provides an intuitive and concise description (including the phase-space integration measure). This allows to describe both the lower-temperature regime where phase fluctuations destroy the long-range order, and the higher temperature crossover corresponding to a decrease of the absolute value of hybridisation. In spite of the rapid progress in the studies of excitonic insulators, a unified picture of this kind has not been available to date. We briefly discuss recent experiments on ${\rm Ta_2 Ni Se_5 }$ and also address the amplitude mode of collective excitations in relation to the measurements reported for 1T--${\rm TiSe_2}$. Both the overall scenario and the theoretical framework are also expected to be relevant in other contexts, including the Kondo lattice model.

Many-body effects in twisted bilayer graphene at low twist angles

We study the zero-temperature many-body properties of twisted bilayer graphene with a twist angle equal to the so-called `first magic angle'. The system low-energy single-electron spectrum consists of four (eight, if spin label is accounted) weakly-dispersing partially degenerate bands, each band accommodating one electron per Moir{\'{e}} cell per spin projection. This weak dispersion makes electrons particularly susceptible to the effects of interactions. Introducing several excitonic order parameters with spin-density-wave-like structure, we demonstrate that (i)~the band degeneracy is partially lifted by the interaction, and (ii)~the details of the low-energy spectrum becomes doping-dependent. For example, at or near the undoped state, interactions separate the eight bands into two quartets (one quartet is almost filled, the other is almost empty), while for two electrons per Moir\'{e} cell, the quartets are pulled apart, and doublets emerge. When the doping is equal to one or three electrons per cell, the doublets split into singlets. Hole doping produces similar effects. As a result, electronic properties (e.g., the density of states at the Fermi energy) demonstrate oscillating dependence on the doping concentration. This allows us to reproduce qualitatively the behavior of the conductance observed recently in experiments [Cao et al., Nature {\bf 556}, 80 (2018)]. Near half-filling, the electronic spectrum loses hexagonal symmetry indicating the appearance of a many-body nematic state.

Spin texture and spin current in excitonic phases of the two-band Hubbard model

Using the mean-field approximation, we study the $k$-space spin textures and local spin currents emerged in the spin-triplet excitonic insulator states of the two-band Hubbard model defined on the square and triangular lattices. We assume a noninteracting band structure with a direct band gap and introduce $s$-, $p$-, $d$-, and $f$-type cross-hopping integrals, i.e., the hopping of electrons between different orbitals on adjacent sites with four different symmetries. First, we calculate the ground-state phase diagrams in the parameter space of the band filling and interaction strengths, whereby we present the filling dependence of the amplitude and phase of the excitonic order parameters. Then, we demonstrate that the spin textures (or asymmetric band structures) are emerged in the Fermi surfaces by the excitonic symmetry breaking when particular phases of the order parameter are stabilized. Moreover, in case of the $p$-type cross-hopping integrals, we find that the local spin current can be induced spontaneously in the system, which does not contradict the Bloch theorem for the absence of the global spin current. The proofs of the absence of the global spin current and the possible presence of the local spin currents are given on the basis of the Bloch theorem and symmetry arguments.