Quantum tricriticality in a generalized quantum Rabi system

Quantum tricriticality, a unique form of high-order criticality, is expected to exhibit fascinating features including unconventional critical exponents and universal scaling laws. However, a quantum tricritical point (QTCP) is much harder to access, and the corresponding phenomena at tricriticality have rarely been investigated. In this study, we explore a tricritical quantum Rabi model, which incorporates a non-trivial parameter to adjust the coupling ratio between a cavity and a three-level atom. The QTCP emerges at the intersection of first- and second-order superradiant phase transitions according to Landau theory. By using finite-frequency scaling analysis on quantum fluctuations and the average photon number, universal critical exponents differentiate the QTCP from the second-order critical point. Our results indicate that the phase transition at the tricritical point goes beyond the conventional second-order phase transition. Our work explores an interesting direction in the generalization of the well-known Rabi model for the study of higher-order critical points due to its high control and tunability.


Introduction
Quantum phase transition (QPT) is a central issue in the study of many-body quantum phenomena at zero temperature [1].Characterizing universal phase transition phenomena and identifying critical exponents are essential in understanding phase transitions.Quantum critical points are often observed as a divergence point of an order parameter in continuous phase transitions by adjusting external physical parameters such as magnetic fields [2,3].In contrast to conventional critical points, a quantum tricritical point (QTCP) arises where a continuous phase transition changes into a discontinuous one.QTCPs were originally found in He 3 -He 4 mixtures in finite temperature phase diagrams, which were characterized by the Landau theory of phase transitions [4].Tricriticality is difficult to access in real materials, but can be found, for example, in itinerant ferromagnets [5] and metallic magnets [6][7][8][9].Several experimental and theoretical works indicate unconventional quantum criticalities resulting from quantum tricriticalities in many-body systems [9][10][11][12][13][14] .
QPTs in light-matter interaction systems have been extensively studied in recent years, leading to the discovery of exotic quantum phases in quantum many-body systems [15][16][17].A well-known quantum phenomenon is the superradiant phase transition, which occurs when a collection of two-level atoms undergoes spontaneous emission [18][19][20][21], This phenomenon has been observed in Bose-Einstein condensate gas experiments [22] and degenerate Fermi gas experiments [23].The quantum Rabi model, consisting of a two-level system and a bosonic field mode, also exhibits a superradiant phase transition in an infinite frequency ratio limit analogous to a thermodynamic limit [24][25][26][27][28].This has been achieved in quantum simulations [29,30].Significant efforts have been dedicated to exploring the existence of QPTs in few-body systems in finite Jaynes-Cummings lattice systems [31], anisotropic quantum Rabi and Rabi-star model [32][33][34], and quantum Rabi ring with an artifical field [35][36][37].The QPTs in a few-body system offer an avenue for investigating nontrivial criticality and exotic phases due to the high degree of tunability.
In this study, we explore a generalization of the well-known Rabi model, aiming to find profound high-order criticality.We introduce a tricritical quantum Rabi model that incorporates a tunable parameter that gauges the ratio between the coupling strengths of the cavity and a three-level atom.By using Landau A three-level atom is coupled to a single-mode cavity.Three levels of the atom are coupled by cavity-assisted Raman transitions, for which the atomic transition ratio γ is tuned from the side by two driving lasers in red and yellow lines.(b) Average value of the atom energy ⟨h⟩ in the γ − λ plane for the phase transitions from the NP (λ < λc) to the second-order SR and first-order SR phases (λ > λc), respectively.The white solid line is a second-order critical line while the blue dashed line is a first-order critical line, respectively.The QTCP is marked by a red dot.In all our calculations, we set ω = 1 as the units for frequency.theory, we expand the ground-state energy in terms of an order parameter, revealing superradiant phase transitions of the first and second orders.Notably, a QTCP emerges at the boundary between critical lines for first-and second-order quantum phase transitions.At the QTCP, the scaling exponents of quantum fluctuations and the mean photon number are different from the exponents at the second-order critical point.Our findings indicate that the QTCP belongs to a distinct universality class with a unique universal critical exponent, which differs from that of the conventional quantum Rabi model.

Tricritical quantum Rabi model
We consider a tricritical quantum Rabi system, which describes a three-level atom uniformly coupled to a single-mode cavity.The Hamiltonian of this system is a generalization of the well-known quantum Rabi model and reads where a(a † ) denotes the photon annihilation (creation) operator of the single-model cavity with the frequency ω. g is the atom-cavity coupling strength, Ω characterizes the atom energy splitting.The dipole operator d of the atom and the single-atom Hamiltonian h are defined as The dipole operator d incorporates a nontrivial parameter γ that tunes the strength ratio of the atomic transitions between |1⟩ ↔ |0⟩ and |0⟩ ↔ |−1⟩, for which |ε i ⟩ (ε i = 0, ±1) is the eigenstates of h of the three-level atom.γ plays a crucial influence on the effective coupling strength between the cavity and the atom.For an experimental realization in figure 1(a), three hyperfine levels could be the ones on the F = 1 ground state of 87 Rb.Three levels are coupled through the cavity and laser fields, which can be realized by cavity-assisted Raman transitions [38].The three-level atom interacts with two coupling lasers with different frequencies, which control the atomic transition ratio γ.For convenience, we set a scaled dimensionless coupling strength as λ = g/ √ Ωω.For a weak atom-cavity coupling λ, the excitation tends to zero, which corresponds to the normal phase (NP).As λ increases to a critical value λ c , the photon population becomes macroscopic, and the system enters a superradiant (SR) phase.SR phase transitions occur in the infinite frequency limit denoting η = Ω/ω → ∞, which is analogous to the infinite limit in the quantum Rabi model [25,26].Such phase transition is possible to be simulated in trapped-ion systems with the large detuning Ω ≫ ω, which can be realized by tuning the laser beam close to the blue and the red motional sidebands [30].

Superradiant phases and a tricritical point
In the superrdiant phases, the excitation is proportional to η due to the macroscopic population.Then we follow a mean-field approach by shifting the bosonic operator with respected to their mean value, a → a + β with β = ⟨a⟩ ∝ √ η.The Hamiltonian in equation ( 1) can be written as where h and d are the single atom operators given in equation ( 2).The ground-state energy is determined by a non-zero value of β minimizing the energy of the term H 0 .Figure 1(b) shows the average value ⟨h⟩ depending on γ and λ, which is obtained by the numerical variational method.In the regime of NP with coupling strength λ < λ c , ⟨h⟩ equals −1 corresponding to the atom in the lowest state.As λ exceeds the critical value λ c , ⟨h⟩ smoothly grows with the increasing λ, revealing a second-order SR phase transition.On the other hand, when γ is below a critical value γ TCP , ⟨h⟩ increases sharply at the critical value λ c .The discountinuous jump characterizes a first-order phase transition.Moreover, the critical lines of the first-and second-order phase transitions intersect at a QTCP (γ TCP , λ TCP ) in a red dot.We analyze the emergence of the first-and second-order phase transitions in the following using the Landau theory approach.
Using the rescaled order parameter α = 2λβ/ √ η equation ( 3) can be reduced to A Landau theory of phase transition is the basis theoretical framework to describe phase transitions involving symmetry breaking.For the phase transition breaking a Z 2 symmetry of the Hamiltonian, the mean-field ground-state energy E MF is an even function of the order parameter α.E MF can be expanded as a Taylor series in terms of α 2 around α = 0: The coefficients c k are obtained using the perturbation theory by treating h as an unperturbed Hamiltonian and αd term in equation ( 3) as the perturbation.Considering Landau theory, we perform the sixth-order perturbation energy by keeping the expansion up to order α 6 as where the coefficients )/4 are given in the appendix.According to the first derivatives of E MF with respect to α, dE MF /dα = 0, we obtain the minimum values and α = 0.The mean-field energy E MF has one global minimum at α = 0 if c 1 > 0 and c 2 2 < 3c 1 c 3 , and has two global minimums at α ± if c 1 < 0, c 2  2 > 3c 1 c 3 and c 2 ⩾ 0. The corresponding second-order critical boundary, where two global minimums merge into one, is given by c 1 = 0 and c 2 ⩾ 0.Moreover, E MF has three minimum at α = 0 and α ± if c 1 > 0, c 2 < 0 and c 2  2 > 3c 1 c 3 , which indicates a first-order phase transition.So the tricritical point, where three minimums merge into two when c 1 = c 2 = 0 [4,39].It is the intersection of the first-and second-order phase transitions, so-called the QTCP.Therefore, the second-order critical boundary is obtained when c 1 = 0 It fits well with the critical line of the second-order phase transition in figure 1(b) in the white solid line.The tricritical point is determined when c 1 = 0 and c 2 = 0 The location of the QTCP is marked with a red dot in figure 1(b).When γ ⩾ γ TCP , the ground-state energy E MF has two global minima at α ± , signaling the second-order phase transition.When γ < γ TCP , E MF has three local minima at α ± and α = 0.As the phase transition is crossed, the global minimum changes from α = 0 to α ± .The discontinuous jump of the energy in the global minimum location indicates a first-order phase transition in the appendix.
To show the validity of perturbation theory, we accurately calculate the scaled ground-state energy E g /η and the scaled mean photon number N ph = ⟨a † a⟩/η of the Hamiltonian (1) by exact numerical diagonalization.In NP (λ/λ c < 1), the excitation of photons N ph tends to zero because of zero excitation, while it increases in the superradiant phase (λ/λ c > 1).For the transition strength ratio γ = 0.1 < γ TCP in figures 2(a) and (d), both dE g (λ)/dλ and N ph are discontinuous at the critical coupling strength λ = λ c , revealing the first-order nature of QPT.When γ = 1 > γ TCP in figures 2(c) and (f), N ph becomes continuous, while d 2 E g (λ)/d 2 λ is discontinuous, indicating a second-order phase transition.The first-and second-order SR phase transitions are consistent with the analysis using Landau theory.At QCTP, γ TCP = 1/ √ 2, figures 2(b) and (e) show smooth behavior of both E g and N ph .Moreover, the critical point approaches the critical value λ c in equation ( 6) when η grows from 20 to 500, demonstrating the finite-frequency effect.

Universal scaling and critical exponents
To gain universal features of different phase transitions, it is of great interest to explore the critical exponents and universality classes of the QTCP and the second-order QCP in the tricritical Rabi model.It is well-known that different systems can exhibit similar quantum criticality, giving rise to universality.Finite-size scaling is a topic of major interest in QPT systems and has been firmly established since the development of a general theory [1,40,41].
The finite-size scaling ansatz for a physical quantity Q in the critical region takes the following scaling law form where ν is a universal critical exponent but independent of the physical quantity, F Q (x) is the scaling function of Q, and β Q is the critical exponent for Q.The scaling form dependent on η is similar to finite-size scaling in the thermodynamic phase transitions, which is known as finite-frequency scaling.
At the critical point λ c , one obtains the log-log relation as where lnF Q (0) is a constant.The critical exponent β Q /ν is obtained as the slope of the linear dependence.It yields the finite-frequency scaling relation as Q(η, λ c ) ∝ η −βQ/ν .We consider the observable Q as the variance (∆P) 2 = ⟨P 2 ⟩ − ⟨P⟩ 2 of the momentum quadrature P = i(a † − a), which accounts for the quantum fluctuations.Figures 3(a) and (b) shows the dependence of ∆P on the coupling strength λ.As η increases, the quantum fluctuations diverge around the critical point.The finite-frequency scaling function for ∆P is calculated dependent on η in a log-log relation, as illustrated in figure 4(b).From the slope of the lines, the critical exponent for the tricritical point is equal to β Q /ν = 1/2, but equals to 1/3 for the 2nd-order critical point.Thus, various finite-frequency scaling laws are obtained as (∆P) 2 (η, λ c ) ∝ η −1/3 for the 2nd-order critical point and (∆P) 2 (η, λ TCP ) ∝ η −1/2 for the QTCP, respectively.The scaling function in equation ( 8) should be universal for large η in the critical regime, which is independent of η.According to the critical exponent β Q /ν = 1/3 for the second-order phase transition, figure 4(e) shows the universal scalings of (∆P) 2 η 1/3 as a function of (λ − λ c )η 1/ν for different η with γ = 1.Remarkably, excellent collapse in the critical regime is observed according to the scaling function of the curve for η = 1000, 2000, 5000.It demonstrates that the universal critical exponent is ν = 3/2 for the second-order phase transition, which is the same as that in the quantum Rabi model and Dicke model [26,42].Meanwhile, for the QTCP, the universal scaling function (∆P) 2 η 1/2 as a function of (λ − λ TCP )η 1/ν is shown in figure 4(f).It is observed that the curves with the universal critical exponent ν = 1 collapse together.Thus, the universal scaling function of ∆P at the QTCP and the 2nd-order critical point are obtained explicitly as Table 1.Various critical exponents βQ, βQ/ν and ν obtained using the finite-frequency scaling function for the variance of momentum (∆P) 2 and the average photon number N ph for the QTCP and 2nd-order critical point, respectively.

Critical exponent
It shows that the universal exponent at QTCP ν = 1 is different from ν = 3/2 at the second-order critical point.
To show the universal critical exponent ν independently of the observables, we investigate the universal scaling of the average photon number N ph .Figure 4(a) shows N ph as a function of η on a log-log scale.The slope of the line at QTCP gives the critical exponent β Q /ν = 1/2, which is different from β Q /ν = 2/3 at the 2nd-order critical point.Around 2nd-order critical point, curves of the scaling function for different scales of η collapse into a single curve with γ = 1 in figure 4(c), which gives the universal critical exponent ν = 3/2.Around the QTCP, we calculate the universal scaling function N ph η 1/2 dependent on (λ − λ TCP )η 1/ν in figure 4(d).A collapse with ν = 1 is achieved for different η.Meanwhile, it is interesting to calculate the finite-frequency scaling of N ph dependent on γ − γ TCP at the tricritical point.Figure 5 shows an excellent collapse of the scaling function N ph η 1/2 around the QTCP for different η, giving the universal exponent ν = 1.Thus, for the average photon number N ph , the scaling functions around the 2nd-order critical point and the QTCP are obtained, respectively.
Based on universal scaling analysis, we have successfully captured various critical exponents that govern two types of phase transitions.Table 1 presents the critical exponents obtained using the finite-frequency scaling function.The critical exponent β Q varies for different observables N ph and ∆P.In contrast, the critical exponent ν is a universal constant that is independent of the physical quantity.Both ∆P and N ph predict the same value of ν = 3/2 for the 2nd-order critical point.In comparison, the universal critical exponent at the QTCP is equal to ν = 1.It indicates that the QTCP belongs to a nontrivial universality class with different critical exponents, which goes beyond the second-order superradiant phase transition in the conventional quantum Rabi and Dicke models.Recently, a chiral tricritical point has also exhibited a distinct universality class of phase transitions [11,36].It demonstrates that it is nontrivial to explore quantum critical phenomena at the tricritical points.

Conclusions
In summary, we have investigated the first-and second-order superradiant phase transitions in the tricritical quantum Rabi model.According to Landau theory, the ground-state energy is obtained up to the sixth-order perturbation, which displays local minima for the first-and second-order phase transitions.The tricritical point arises at the intersection of the boundaries for the first-and second-order phase transitions.We perform finite-frequency scaling analysis to calculate the universal scaling of observables.We find that the superradiant phase transition at the tricritical point belongs to a different universality class with a different universal critical exponent.The generalization of the well-known quantum Rabi model can serve as a valuable platform for exploring critical phenomena and more intricate critical behaviors in few-body systems.
Clearly, the zero-th energy correction is E (0) = ε 1 .Moreover, due to the symmetry of the Hamiltonian, the first-order correction is ⟨ε 1 |D|ε 1 ⟩ = 0.The second-order correction can be calculated as follows: Similarly, the fourth-order correction of the ground-state energy is obtained as Since α is a small value, the second term of the above energy is approximated as α 2 γ 2 (−1 + α 2 γ 2 ).It leads to the approximated ground-state energy Furthermore, the ground-state energy can be given up to the sixth-order correction With the sixth-order correction, the ground-state energy of the effective Hamiltonian (A1) can be approximately given up to the order of α 6 where the coefficients are c 1 = 1/(4λ 2 ) − γ 2 , c 2 = γ 2 (γ 2 − 1 2 ), and c 3 = −γ 2 (1 − 7γ 2 + 8γ 4 )/4 .The energy is given in equation (4).
Figure 6 depicts the ground-state energy E SR /η as a function of α for various values of λ and γ.By adjusting the transition ratio γ < γ TCP , E SR locates at three minimal value of α, predicting a first-order phase transition.

Figure 1 .
Figure 1.(a)An setup of the tricritical Rabi system.A three-level atom is coupled to a single-mode cavity.Three levels of the atom are coupled by cavity-assisted Raman transitions, for which the atomic transition ratio γ is tuned from the side by two driving lasers in red and yellow lines.(b) Average value of the atom energy ⟨h⟩ in the γ − λ plane for the phase transitions from the NP (λ < λc) to the second-order SR and first-order SR phases (λ > λc), respectively.The white solid line is a second-order critical line while the blue dashed line is a first-order critical line, respectively.The QTCP is marked by a red dot.In all our calculations, we set ω = 1 as the units for frequency.

Figure 2 .
Figure 2. (a)-(c) Scaled ground-state energy Eg/η and the mean photon number N ph as a function of the dimensionless coupling strength λ/λc for γ = 0.1 (a) and (d), γ = γTCP = 1/ √ 2 (b) and (e), and γ = 1.0 (c) and (f) for different η = 20, 100, 500.The first derivatives of the energy dEg/dλ is list in the inset (a).The second derivatives of the energy d 2 ESR/d 2 λ are listed in the insets (b) and (c).In all our calculations, the truncated number of photons is Ntr = 800.

Figure 3 .
Figure 3.The variance of momentum (∆P) 2 as a function of λ/λc for the second phase transition with γ = 1 (a) and for the QTCP with γ = γTCP (b).