Phonon and maxon instability in Bose–Einstein condensates with parity-time symmetric spin–orbit coupling

Parity-time ( PT ) symmetry has drawn great research interest in non-Hermitian physics. Recently, there is an emerging model of PT -symmetric spin–orbit coupling (SOC) which could be realized with spin-1/2 atomic Bose–Einstein condensates (BECs) when the two spin components are respectively subjected to momentum-dependent gain and loss (Qin et al 2022 New J. Phys. 24 063025). In this model, inter-atom interaction has no influence on the PT -symmetric plane waves. Thus, collective excitation playing the role of fingerprint of inter-atom interaction is investigated in this paper. For the phonon excitation, it is shown that the repulsive interaction between atoms in different spin states, which tends to drive the atoms to populate in only one spin component, can break the PT -symmetry, and leads to a phonon instability. Whereas, for the case of the phonon-maxon-roton collective excitation spectrum, the repulsive interaction between atoms in different spin states can lead to a maxon instability, which does not occur in Hermitian BEC systems (dipolar BECs or BECs with Raman induced SOC). Simulation of the time evolution of the plane wave solution against small noise shows that the maxon instability can result in the formation of a supersolid-like stripe pattern at the initial stage, but the non-Hermitian nature of the system finally destroys the pattern. The phase diagram of stability shows that the Rabi coupling and repulsive interaction between the atoms in the same spin state can stabilize the system.

Non-Hermitian parity-time (PT ) symmetry physics has experienced rapid developments in the last two decades, especially in the field of optics, where the balanced optical gain and loss can be modeled by a PT -symmetric Hamiltonian, and it is found that gain and loss can serve as new engineering tools and also lead to abundant new and unexpected features [34][35][36][37][38].These great developments inspired the interest in extending cold atom physics from Hermitian to non-Hermitian regime, with the prospect of exploring unidirectional transportation [39], non-Hermitian skin effect [40][41][42][43], soliton [44][45][46], double-well [47][48][49] and Hubbard model [50,51] in open environments, etc.In cold atom systems, controlled atomic loss can be realized by extracting atoms from the atomic gas via atom-light interaction [52]; while the atomic gain is proposed to be realized by injecting atoms into the atomic gas using an atom laser [53].In experiments, thanks to the resplendent controlling techniques, non-Hermitian cold atom systems have already been successfully realized in several different configurations, including spin-1/2 atomic gas [52,54], atomic gas in optical lattice [55] and synthetic momentum lattice [43,56]; furthermore, unique non-Hermitian phenomena such as PT -symmetry breaking [52] and chiral control of quantum states [54] have been demonstrated, and non-Hermitian many-body physics have also been examined [55].With the theoretical and experimental progress in non-Hermitian cold atom physics, collective excitations have begun to draw research interests recently, for example, in [57], the collective excitations in PT -symmetric Fermi superfluid are studied, and it is suggested that the topological critical phenomena offer a route toward perturbation-free quantum states; and in [58], it is shown that spontaneously broken PT -symmetry in collective excitation leads to enhanced collapse of BECs.
In the realization of a PT -symmetric BEC system, the atomic gain and loss are usually momentum-dependent, due to the momentum distribution of the atom laser [59,60] and the Doppler effect in atom-light interaction [61].It is proposed that when the two components of a spin-1/2 atomic BEC are respectively subjected to momentum-dependent balanced gain and loss, a PT -symmetric SOC can be artificially created [62].In this paper, we show that for such a PT -symmetric SOC BEC system, its collective excitation spectrum can exhibit either a pure phonon or a phonon-maxon-roton structure, and the interplay between inter-atom interaction and PT -symmetric SOC leads to abundant new instability physics.Even the repulsive interaction can result in a phonon instability, and the instability dynamics are not characterized by bright soliton formation or collapse of the condensate as in the Hermitian BEC systems; instead, the non-Hermitian nature of the system leads to an amplification of the matter wave density.In the case of the phonon-maxon-roton excitation spectrum, maxon instability, rather than the usual roton instability, could be induced.At the initial stage, the maxon instability leads to the formation of a supersolid-like stripe pattern, but after a long time of evolution, the non-Hermitian nature of the system destroys this stripe pattern.
The following contents of this paper are organized as follows.We first introduce the physical model and derive the Bogoliubov-de Gennes (BdG) equation which governs the collective excitations in section 2. In section 3, we numerically solve the BdG equation to obtain the collective excitation spectrum, and directly simulate the nonlinear Schrödinger equation to examine the instability dynamics.This section is split into four sub-sections.In section 3.1, we discuss the phonon instability of the zero-momentum states, and in section 3.2, we discuss the maxon instability of the plane wave states.In section 3.3, we present the phase diagram of stability.In section 3.4, we examined the phase transition between zero-momentum and plane wave states.Finally, the work is summarized in section 4.

Model
We consider a one-dimensional pseudo-spin-1/2 atomic BEC system.As shown in [62], when the two spin components experience momentum-dependent balanced gain and loss respectively, an imaginary SOC can be artificially created, and in momentum space, the imaginary SOC Hamiltonian reads Here, h is the Planck constant, k x is the wave vector operator (i.e.hk x is the momentum operator), m is the mass of the BEC atom, γ is the rate of the balanced gain and loss, Ω is the Rabi coupling strength between the two pseudo-spin states, Θ (k x ) is a dimensionless real function describing the momentum-dependence profile of the balanced gain and loss.This Hamiltonian is PT -symmetric, in the sense that P represents the exchanging of the two spin components, and T performs the complex conjugation [34].In this paper, we assume that Θ (k x ) takes a Lorentz form with σ being the spectrum width of the balanced gain and loss.In experiments, the loss can be realized by exciting the atoms to excited states with a laser [52], and the following photon recoil will eject them out from the condensate, due to Doppler shift, the atomic loss realized in such a way has a momentum-dependence [61]; while the momentum-dependent gain would be realized by injecting atoms into the condensate using an atom laser with appropriate momentum distribution [59,60].Owing to the remarkable controlling techniques in cold atoms, one would expect that the imaginary SOC parameters are controllable.
In the atomic BEC system, atoms also interact with each other via s-wave collision.In coordinate space, the interaction Hamiltonian is given by where T is the wave function of the condensate, g ↑↑ and g ↓↓ represent the interaction strength between atoms with the same spin state (intra-spin interaction), while g ↑↓ and g ↓↑ represent the interaction strength between atoms with opposite spins (inter-spin interaction).These interaction parameters are related to the transverse trapping potential and s-wave scattering length, [63], with i, j =↑, ↓ and ω ⊥ being the transverse trapping frequency.Therefore, they can be tuned by adjusting the transverse trap, or by applying the Feshbach resonance technique [64].We assume these interaction parameters fulfill relations g ↑↑ = g ↓↓ = g 0 and g ↑↓ = g ↓↑ = g 1 , such that the interaction does not explicitly break the PT -symmetry.In the case of g 0 < 0 and g 1 < 0, the attractive interaction usually leads to the formation of spatially localized matter wave solitons [1], which would be interesting, but is not the subject of the present work.In this paper, we devote our efforts to study the spatially homogeneous matter waves, therefore repulsive interaction with g 0 > 0 and g 1 > 0 are assumed in the following analysis , and for succinctness of the formulas, we also apply the natural unit system h = m = 1.Combining the single-particle imaginary SOC Hamiltonian (1) and the interaction Hamiltonian (3), the dynamics of the system are governed by the nonlinear Schrödinger equation Here, the spatial Fourier transform F and the related inverse Fourier transform F −1 are introduced to link the coordinate and momentum spaces, since H 0 is expressed in momentum space, while H n and Ψ are expressed in coordinate space.

Nonlinear waves and dispersion relations
We investigate the case in which the BEC wave function takes a plane wave form, i.e.
where A ↑,↓ are amplitudes of the spin-↑ and spin-↓ components, k 0 is the momentum (wave vector) of the plane wave, and µ is the chemical potential.Inserting equation ( 5) into the nonlinear Schrödinger equation ( 4), and applying the normalization constrain , we obtain the following stationary coupled nonlinear equations with , the nonlinear terms in equations ( 6) and (7) disappear, hence the solutions are the same as those in the linear case, but the chemical potential is shifted by a value of g.Therefore, adopting the linear case results in [62], when γ 2 Θ 2 (k 0 ) ⩽ Ω 2 , the nonlinear system also admits PT -symmetric solutions and the corresponding nonlinear dispersion relation curve also splits into two branches with pure real values Depending on the specific values of the parameters, the dispersion relation curve can show very different manners.
, both the upper and lower branch have only one minimum at k 0 = 0, see figure 1(a).And in this case, the ground state is a zero-momentum state.When Ω > γ, σ < σ c , the upper branch still has only one minimum at k 0 = 0; however, for the lower branch, k 0 = 0 becomes a local maximum point, and two minima emerge at its two sides, k 0 = ±k m with k m being the solutions of dµ − /dk 0 = 0.This is to say that the lower branch exhibits a double-valleys structure in this case, see figure 1(b).The ground state is degenerate because it can fall in either of the two valleys.Since the bottoms of the two valleys have non-zero momentum ±k m , the ground state is a plane wave state.When Ω < γ, the chemical potential becomes complex, that is the PT -symmetry is spontaneously broken, and the matter waves will undergo spontaneous decay or amplification [62].

BdG equation for the collective excitation
Although the inter-atom interaction has no influence on the PT -symmetric plane wave solutions, its effect would appear in the collective excitations of the system.Moreover, in cold atom physics, the collective excitations can be experimentally investigated by the Bragg spectroscopy technique [65,66], thus in this paper we will examine the collective excitations of the system in detail.
Using Bogoliubov transformation [2], the perturbation is further written as where u ↑,↓ , v ↑,↓ , q, ω are respectively the amplitude, quasi-momentum, and frequency of the perturbation.Substituting equations ( 9) and ( 10) into the nonlinear Schrödinger equation ( 4), and only keeping the terms linear in perturbation, we obtain the following BdG equation [21,[67][68][69][70]] with where If the eigen frequency of the BdG equation comes out to be a complex number [Im(ω) ̸ = 0], according to equation (10), the initial small perturbations will undergo an exponential amplification during the afterward evolution, that is to say, the ground state is dynamically unstable.Otherwise, if ω is purely real [Im(ω) = 0], the ground state would be stable against small initial perturbations.
Besides the dynamical instability, the energetic instability is another instability mechanism that is widely considered in nonlinear physics [71][72][73].It states that if the eigen values of diag{1, 1, −1, −1} • H BdG are negative, the perturbations will lower the ground state energy, and the system is energetically unstable.Here, for the system considered in this paper, as has been shown in section 2.1, due to the specific PT -symmetry, the nonlinear interaction only shifts the energy by a constant value, therefore the system would be always energetically stable.Collective excitation spectra for the zero-momentum ground states.The real (a1), (b1) and imaginary (a2), (b2) parts of the collective excitation frequency ω are plotted as a function of quasi-momentum q.In panels (a1), (a2), the interaction parameters are g0 = 1.0 and g1 = 0.3, while in panels (b1), (b2) the interaction strength parameters are g0 = 1.0 and g1 = 1.5.In all these panels, the PT -symmetric SOC parameters are Ω = 1.1, γ = 1.0 and σ = 2.5, which are the same as those in figure 1(a).Figure 3. Phonon speed for zero-momentum ground states as a function of inter-spin interaction strength g1.When g1 < 1.28, the corresponding collective excitation spectrum is purely real, and the phonons have a non-zero speed, vp > 0, the ground state is stable.Otherwise, the phonon speed drops to zero, and an imaginary part emerges in the collective excitation spectrum, the ground state becomes unstable.Other parameters are Ω = 1.1, γ = 1.0, σ = 2.5 and g0 = 1.0, which are the same as those in figure 2.

Numerical results
In this section, we first solve the BdG equation (11) to obtain the collective excitation spectrum, which can give an insight into the stability of the BEC matter waves.The corresponding stability dynamics of the system are examined by directly simulating the nonlinear Schrödinger equation (4).

Phonon instability of the zero-momentum waves
In figure 2, we show the collective excitation spectra (ω-q relationship) for the zero-momentum ground state, corresponding to the bottom of the single-valley nonlinear dispersion relationship in figure 1(a).In figures 2(a1) and (a2), where the interaction strength parameters are g 0 = 1.0 and g 1 = 0.3, the collective excitation spectrum has purely real values, which indicates that the corresponding ground state is stable against initial small perturbations, and moreover, the low energy excitation frequency ω is approximately linearly proportional to the small quasi-momentum q, showing a phonon feature of the excitation.The phonon speed v p can be determined by the slope of the spectrum at q = 0. Increasing g 1 , we found that the phonon speed undergoes a sharp drop near a critical value of 1.28, see figure 3. Exceeding this critical value, further increasing g 1 will lead to the emergence of an imaginary part in the phonon regime of the collective excitation spectrum, which indicates a phonon instability of the matter waves.Such an example of complex collective excitation spectrum is shown in figures 2(b1) and (b2) for parameter g 1 = 1.5.Considering that we are now dealing with repulsive interaction, the phonon instability found here is in sharp contrast with the conventional Hermitian BEC systems where phonon instability usually results from the attractive interaction [1,2].
To confirm the phonon instability induced by repulsive interaction, we examine the dynamical evolution of the system by directly simulating the nonlinear Schrödinger equation ( 4) using the operator splitting technique [74].As shown in section 2, the Hamiltonian naturally splits into two parts, H 0 and H n , we evolve In panel (b1), the color scale is cut off at the value of 12, therefore the yellow colored region represents a very large value of the matter wave density.Panels (a2), (b2): time evolution of the total atom number (green dashed line), and the maximum value of matter wave density (violet solid line).N0 is the initial total atom number.Note that in panel (b2), the vertical axes are scaled by a factor of 10 6 .The inset panels show the time evolution of the ratio between spin-↓ and spin-↑ atom numbers, N ↓ /N ↑ .In panels (a1), (a2), the interaction strength parameters are g0 = 1.0, g1 = 0.3, the time evolution is stable.In panels (b1), (b2), the interaction strength parameters are g0 = 1.0, g1 = 1.5, and the time evolution is unstable.The PT -symmetric SOC parameters Ω = 1.1, γ = 1.0, and σ = 2.5 are used in all these panels.the wave function with the formula Here, the H n evolution step can be calculated directly in coordinate space, and the H 0 evolution step is calculated in momentum space where it can be easily diagonalized.The coordinate space and momentum space are linked by the forward and inverse fast Fourier transformation.We note that due to the non-Hermitian feature of the system, in the simulation a very small ∆t should be used, otherwise, the simulation would be numerically unstable.The initial wave function is the exact ground state solution plus a 2% random perturbation, i.e. ψ ↑,↓ (x, t = 0) = A ↑,↓ e ik0x [1 + 0.02ξ (x)] with ξ (x) being random numbers uniformly distributed in the range of (−1, 1).And, a periodical boundary condition is applied.
As shown in figure 4, the dynamical evolution result obtained with interaction parameters g 0 = 1.0 and g 1 = 0.3 is stable against the small perturbations, while the evolution for g 0 = 1.0 and g 1 = 1.5 is unstable, in agreement with the analysis of the collective excitation spectrum.In the conventional Hermitian BEC systems with attractive interaction, the phonon instability leads to the formation of bright solitons in the one-dimensional case (or collapse of the condensate in the two and three-dimensional cases) [1][2][3][4][5][6][7][8][9][10].However, in our non-Hermitian SOC BEC system, the phonon instability breaks the PT -symmetry, i.e. breaks the balance between gain and loss, and leads to the dominant role of the amplifying spin component, see figure 4(b2) and the inset.

Maxon instability of the plane waves
Figure 5 shows the collective excitation spectra for the plane wave ground state, corresponding to the bottom of double-valleys nonlinear dispersion relationship in figure 1(b).For the interaction parameters g 0 = 1.0 and g 1 = 0.3, as shown in figures 5(a1) and (a2), the excitation spectrum is purely real, and has a phonon-maxon-roton-like structure [11][12][13][14][15][16][17][18][19][20][21][22][23][24].Around q = 0, it shows a phonon-like linear dependence of ω on q, and the spectrum is anisotropic, the positive and negative slope of the spectrum is different, which means the positive and negative direction propagating phonons will have different speeds v p,+ and v p,− .At a finite value of q, there appears a roton minimum.Between the phonon and roton regimes, there is a maxon peak, and an energy gap ∆ m opens here.When we increase g 1 , it is found that the width of this maxon energy gap is reduced, and at a critical value it closes ∆ m = 0, see figure 6(a).Further increasing g 1 , we found that an imaginary part of the excitation frequency ω emerges in the two maxon regimes, which signifies a maxon instability, as shown in figures 5(b1) and (b2) for g 1 = 0.9.For the phonons, we found that their speed undergoes a monotonous increase when g 1 increases, see figure 6(b).In the dipolar BECs [11][12][13][14][15][16][17][18], and conventional Hermitian SOC BEC systems realized by the Raman coupling scheme [19][20][21][22][23][24][25], the collective excitation spectrum can also exhibit a phonon-maxon-roton structure, but in these systems, the instability appears in the roton regime, rather than the maxon regime.
The stability analysis results are numerically tested by solving the nonlinear Schrödinger equation, and the numerical results are shown in figure 7.For parameters g 0 = 1.0 and g 1 = 0.3, figures 7(a1) and (a2) show a stable time evolution of the ground state plane wave solution.For parameters g 0 = 1.0 and g 1 = 0.9, figures 7(b1) and (b2) show that in the beginning evolution stage (t < 20), a supersolid-like stripe pattern is Collective excitation spectra for the plane wave ground states.The real (a1), and imaginary (a2), (b2) parts of the collective excitation frequency ω are plotted as a function of quasi-momentum q.In panels (a1), (a2), the interaction parameters are g0 = 1.0 and g1 = 0.3, while in panels (b1), (b2) the interaction strength parameters are g0 = 1.0 and g1 = 0.9.In all these panels, the PT -symmetric SOC parameters are Ω = 1.1, γ = 1.0, σ = 0.5, which are the same as those in figure 1(b).formed, due to the interference between matter waves in the two maxon instability regimes with opposite momentum (figure 5(b2)); however, with the increasing of time, this stripe pattern is finally destroyed, as the result that inter-atom interaction leads to the breaking of PT -symmetry and consequently the matter wave density is amplified.In the yellow region, γ > Ω, the PT -symmetry is spontaneously broken, see section 2.1.In the gradient color (from violet to yellow) region, max {|Im (ω)|} > 0, the ground state is dynamically unstable.In the two black regions, max {|Im (ω)|} = 0, the ground state is stable.The white dashed line represents σc = σ, below this line σc < σ, while above this line σc > σ, therefore the stable zero-momentum ground states are located in the 'Stable-1' region, while the stable plane wave ground states are located in the 'Stable-2' region.Other parameters used to plot this figure are σ = 1.0, g0 = 1.0, and g1 = 3.0.

Phase diagram of stability
In this part, we study the stable and unstable regions of the ground state matter waves in the parametric space.Firstly, we examine this problem in the parameter space spanned by interaction parameters g 0 and g 1 .The results are shown in figure 8, where the maximum value of |Im (ω)| is plotted with colormap.In the black regions, we have max {|Im (ω)|} = 0, this means that the ground state is stable.While, in the gradient color (from violet to yellow) regions, max {|Im (ω)|} > 0, the ground state is unstable.Increasing the intra-spin interaction strength g 0 tends to stabilize the ground state waves; while increasing the inter-spin interaction strength g 1 is prone to destroy the stability.Physically, the intra-spin repulsive interaction tends to drive the atoms to equally populate at both the two spin states, such that the gain and loss can be well balanced with each other, and the system is stable; however, the inter-spin repulsive interaction tends to drive the atoms to populate only one of the two spin states, leading to the breaking of the PT -symmetry, therefore the gain and loss can no longer be balanced with each other.
We also examined the influence of PT -symmetric SOC parameters on the stability.In figure 9, a phase diagram of stability is plotted in the Ω-γ parameter space.In the γ > Ω yellow region, as discussed in section 2.1 and [62], the PT -symmetry is spontaneously broken.The γ < Ω region is split into two parts by the dashed line of σ = σ c , below this line the ground state is a zero-momentum one, while above this line the ground state is a plane wave state.The stability of the ground state is also determined by max {|Im (ω)|}: in the max {|Im (ω)|} = 0 black region, it is stable; while in the max {|Im (ω)|} > 0 gadient color (from violet to yellow) region, it is unstable.We see that the increase of γ tends to destroy the stability, in contrast, the increase of Ω will help to stabilize the system.This can also be understood by the fact that Rabi coupling Ω tends to drive the atoms equally populate the two spin states, while γ tends to make a difference.

Zero-momentum to plane wave phase transition
From the above discussion, we see that the ground state of the system would undergo a zero-momentum to plane wave phase transition when σ passes through the critical point σ c .In the two different phases, the collective spectra have distinct features, thus collective excitation would possibly serve as a probe for this phase transition.In figure 10, we studied the phonon speed during the phase transition.It is seen that in the zero-momentum phase, the positive and negative direction propagating phonons have the same speed; while in the plane wave phase, they propagate at different speeds.The bifurcation happens at the phase transition point σ = σ c .

Summary
In the BEC system with PT -symmetric SOC, inter-atom interaction has no influence on the PT -symmetric plane wave solutions, therefore collective excitations and stability of the plane waves become a fingerprint to explore the effect of inter-atom interaction.The collective excitation spectrum of the system has either a pure phonon or a phonon-maxon-roton structure.These spectrum structures are analogous to those in BECs with conventional Hermitian SOC; however, the corresponding stability features are quite different.The repulsive inter-spin interaction tends to drive atoms to populate only one spin component, thus increasing this interaction can lead to spontaneous breaking of the PT -symmetry, and consequently, a phonon instability of the system is induced.In contrast, in the Hermitian BEC systems, the phonon instability is usually induced by an attractive interaction.In the case of the phonon-maxon-roton collective excitation spectrum, the repulsive inter-spin interaction can lead to a maxon instability, which does not occur in dipolar BECs or BECs with Raman induced SOC.Simulation of the time evolution of the plane wave solutions against small noise shows that the maxon instability can first generate a supersolid-like stripe pattern, but because of the non-Hermitian nature of the system, the stripe pattern can not sustain for a long time.The phase diagram of stability shows that the Rabi coupling and interaction between the atoms in the same spin state can stabilize the system.

Figure 4 .
Figure 4. Time evolution of the zero-momentum ground states.Panels (a1), (b1): colormap plot of the matter wave density ρ(x, t) = ψ ↑ (x, t) 2 + ψ ↓ (x, t) 2 .In panel (b1), the color scale is cut off at the value of 12, therefore the yellow colored region represents a very large value of the matter wave density.Panels (a2), (b2): time evolution of the total atom number (green dashed line), and the maximum value of matter wave density (violet solid line).N0 is the initial total atom number.Note that in panel (b2), the vertical axes are scaled by a factor of 10 6 .The inset panels show the time evolution of the ratio between spin-↓ and spin-↑ atom numbers, N ↓ /N ↑ .In panels (a1), (a2), the interaction strength parameters are g0 = 1.0, g1 = 0.3, the time evolution is stable.In panels (b1), (b2), the interaction strength parameters are g0 = 1.0, g1 = 1.5, and the time evolution is unstable.The PT -symmetric SOC parameters Ω = 1.1, γ = 1.0, and σ = 2.5 are used in all these panels.

Figure 6 .
Figure 6.Maxon gap (a) and phonon speeds (b) for the plane wave ground states as a function of inter-spin interaction strength g1.Panel (a): when g1 < 0.43, the corresponding collective excitation spectrum is purely real and opens a gap at the maxon regime, ∆m > 0, the ground state is stable; and when g1 > 0.43, this gap closes, ∆m = 0, and an imaginary part emerges in the collective excitation spectrum, the ground state becomes unstable.Panel (b): speeds v p,± of the phonons propagating in positive and negative directions increase monotonously with g1.Other parameters are Ω = 1.1, γ = 1.0, σ = 0.5 and g0 = 1.0, which are the same as those in figure 5.

Figure 9 .
Figure 9. Phase diagram of stability in Ω-γ space.In the yellow region, γ > Ω, the PT -symmetry is spontaneously broken, see section 2.1.In the gradient color (from violet to yellow) region, max {|Im (ω)|} > 0, the ground state is dynamically unstable.In the two black regions, max {|Im (ω)|} = 0, the ground state is stable.The white dashed line represents σc = σ, below this line σc < σ, while above this line σc > σ, therefore the stable zero-momentum ground states are located in the 'Stable-1' region, while the stable plane wave ground states are located in the 'Stable-2' region.Other parameters used to plot this figure are σ = 1.0, g0 = 1.0, and g1 = 3.0.

Figure 10 .
Figure 10.Phonon speed during a plane wave to zero-momentum ground state phase transition.In the zero-momentum regime, the speed of the positive and negative direction propagating phonons are the same; while in the plane wave regime, they take different values v p,+ and v p,− .The bifurcation happens at the phase transition point σ = σc = 2.09.Other parameters used to plot this figure are γ = 1.0,Ω = 1.1, g0 = 1.0, and g1 = 0.3.