EEmergent Universality in a Quantum Tricritical Dicke Model
Youjiang Xu and Han Pu Department of Physics and Astronomy, and Rice Center for Quantum Materials,Rice University, Houston, Texas 77251-1892, USA
We propose a generalized Dicke model which supports a quantum tricritical point. We map outthe phase diagram and investigate the critical behaviors of the model through exact low-energyeffective Hamiltonian in the thermodynamic limit. As predicted by the Landau theory of phasetransition, the order parameter shows non-universality at the tricritical point. Nevertheless, as aresult of the separation of the classical and the quantum degrees of freedom, we find a universalrelation between the excitation gap and the entanglement entropy for the entire critical line includingthe tricritical point. Here the universality is carried by the emergent quantum modes, whereas theorder parameter is determined classically.
Introduction —
Tricritical point was first proposed byGriffiths within the Landau theory of phase transition [1].A tricritical point is where ordinary critical manifolds in-tersect [2]. In the physically accessible phase diagram, itcan appear as a point where a first-order phase transitionboundary and a second-order one meet [1, 2]. As for thecritical behaviors, the tricritical point normally belongsto a universality class different from that of other pointson the critical line [3, 4].Quantum phase transition[5] has been under intensivestudy over many years, and is a central subject in thestudy of numerous important solid state materials such ashigh temperature superconductors and heavy fermions.Systems that support quantum tricritical point (QTP)are, however, very rare. Recently it has been found thatQTP exists in certain magnetic materials [6, 7]. In thepresent work, we construct a generalized Dicke modelwhich not only supports a QTP, but that the QTP ex-hibits a special feature: Despite the non-universal criticalexponent that distinguishes the QTP from other criticalpoints, there exists a universal relation between the exci-tation gap and the entanglement entropy of the system,which applies to all the critical points of the model. Thisuniversal relation characterizes the quantum fluctuationsand the emergent collective modes of the model.The Dicke model [8, 9] describes an ensemble of two-level systems interacting with a quantized bosonic mode.Though originated as a model of atom-light interaction,the Dicke model can be realized in various experimen-tal settings, including quantum gases [10–13], super-conducting circuit [14–16], and solid state systems [17].The Dicke model features the famous superradiant phasetransition [18], where the bosonic mode becomes macro-scopically occupied if the atom-light interaction strengthexceeds a threshold value and the system enters the su-perradiant phase. While the ground-state phase diagramcan be determined classically through a mean-field ap-proach, the superradiant phase transition is associatedwith a divergent entanglement entropy [19, 20] whichsuggests non-trivial effects induced by quantum fluctu-ations. In the generalized Dicke Hamiltonian we studyin this work, defined in Hamiltonian (1) below, an ad- ditional dimension is present, such that the generalizedmodel extends the critical point in the Dicke model intoa line and the second-order superradiant phase transi-tion can be tuned into a first-order one across a QTP.As a consequence, we shall call the model under studythe quantum tricritical Dicke model . We will explore thephase diagram and the critical behavior of this model atzero temperature in the thermodynamic limit.
Model —
The quantum tricritical Dicke model is ob-tained by partially breaking the exchange symmetry be-tween the two-level atoms in the Dicke Hamiltonian H Dicke through an additional term H SB H = H Dicke + H SB , (1) H Dicke = ωb † b + N (cid:88) i =1 (cid:34) δ σ ( z ) i + g (cid:0) b + b † (cid:1) √ N σ ( x ) i (cid:35) , (2) H SB = ε N (cid:88) i =1 ( − i σ ( x ) i . (3)Here the operator b represents the annihilation operatorfor the bosonic light mode, σ i ’s are Pauli matrices de-scribing the i th atom. ω, δ and g represent the light fre-quency, the atom excitation energy, and the atom-lightinteraction strength, respectively. Without loss of gener-ality, all these parameters are taken to be non-negative.In H Dicke , all atoms are identical. This symmetry is,however, broken by H SB which separates the atoms intotwo groups: one group experiences an effective Zeemanfield along the x -axis, while the other group sees the Zee-man field in the opposite direction. We choose the to-tal number of atoms N to be even. As we will see, thesecond-order quantum phase transition in the conven-tional Dicke model can be tuned into a first-order one byincreasing the strength ε of the symmetry breaking term.In Fig. 1, we present a potential experimental realizationof our model, which involves Raman transition [10] intwo cavities linked by optical fiber [21, 22]. If N = 1, ourmodel reduces to the asymmetric Rabi model [23], whichhas received much attention recently, partially due to itsrelevance in circuit QED [24].To proceed, we carry out a series expansion of the a r X i v : . [ qu a n t - ph ] A p r FiberCavity 1 Cavity 2AtomsRaman Driving LightAtomsMicrowave Coupling |0 ⟩ (a) (b) |1 ⟩ FIG. 1. (a) A schematic representation of potential realizationof the tricritical Dicke model. We have two identical cavitieslinked by an optical fiber and atoms are trapped within eachcavity. We assume that the fiber coupled cavity system con-tains one normal mode that is near resonant with the atomictransition and all other modes can be neglected. (b) An ex-ternal light, together with the dominant cavity mode, drivesa Raman transition between two low-energy states labelledas | (cid:105) and | (cid:105) , which realizes the Dicke coupling as proposedin [10]. In addition, a microwave field directly couples thetwo spin states. The two microwaves for each cavity have aphase difference of π , and serve as the effective Zeeman fieldin Eq. (3). Hamiltonian in terms of 1 /N , so that a solvable low-energy effective Hamiltonian can be obtained. To thisend, we introduce the shifted bosonic operator b ≡ b − ψ .Here ψ is a c -number, which can be regarded as arbitraryfor now. After rotating the Pauli matrices, we can recastthe Hamiltonian into the following form H = ω b † b + ω ψ (cid:16) b + b † (cid:17) + ω ψ + (cid:88) i, even ω σ ( z ) i + g (cid:16) b + b † (cid:17) √ N (cid:16) sin θ σ ( z ) i + cos θ σ ( x ) i (cid:17) + (cid:88) i, odd ω σ ( z ) i + g (cid:16) b + b † (cid:17) √ N (cid:16) sin θ σ ( z ) i + cos θ σ ( x ) i (cid:17) , where ω ≡ ω ,ω , ≡ (cid:114) δ + (cid:16) gψ/ √ N ± ε (cid:17) ,θ , ≡ tan − [(2 gψ/ √ N ± ε ) /δ ] . We then define two collective atomic angular momentumoperators for the two groups of atoms: J ( x,y,z )2 ≡ (cid:88) i, even σ ( x,y,z ) i , J ( x,y,z )3 ≡ (cid:88) i, odd σ ( x,y,z ) i . Without loss of generality, we restrict the Hilbert space tothe subspace with maximum J and J . These operators can be represented by two new bosonic operators b , b by means of the Holstein-Primakoff mapping [25]: J ( z ) i = b † i b i − N/ , J (+) i = b † i (cid:113) N/ − b † i b i , i = 2 , . By expanding J ( ± ) i in powers of 1 /N , the following effec-tive Hamiltonian of H can be constructed: H eff = ω (cid:16) b † b + ψ (cid:17) − N ( ω + ω ) / (cid:104) ω ψ − g √ N (sin θ + sin θ ) / (cid:105) (cid:16) b + b † (cid:17) + (cid:88) i =2 , (cid:20) ω i b † i b i + g cos θ i √ (cid:16) b i + b † i (cid:17) (cid:16) b + b † (cid:17)(cid:21) . (4)We label the set of states satisfying (cid:104) b † i b i (cid:105) = o ( N ) , i = 2 , V , and H − H eff = o ( H eff ) holds only in V when N → ∞ . H eff is quadratic and solvable for arbitrary ψ .However, if we want V to contain the low-energy statesof H and H eff , the second line in Eq. (4) is necessarilysmall. This can be achieved by choosing ψ to coincidewith the expectation value (cid:104) b (cid:105) , which can be identified asthe order parameter in the mean-field theory, as we showbelow.The mean-field order parameter minimizes the dimen-sionless mean-field energy-per-atom functional [26]: f ( z ) = z /y − √ xz + z − √ − xz + z , (5)where x ≡ ε/ω and y ≡ g / ( ωω ) are two dimensionlesssystem parameters with ω ≡ √ ε + δ , and z = 2 g (cid:104) b (cid:105) / ( ω √ N ) , (6)is the normalized order parameter. As a result, the coef-ficient of the term linear in b and b † in Eq. (4) vanishessince ω ψ − g √ N (sin θ + sin θ ) / √ N gf (cid:48) ( z ) / . (7)Consequently, the eigenstates of H eff satisfies (cid:104) b (cid:105) = 0,which self-consistently yields ψ = (cid:104) b (cid:105) . Low-energy effective Hamiltonian and phase diagram—
With ψ given by the mean-field theory, H eff becomes H eff = H C + H Q , (8) H C = N ω f ( z ) , (9) H Q = (cid:88) i =1 , , ω i b † i b i + (cid:88) i =2 , g cos θ i √ (cid:16) b + b † (cid:17) (cid:16) b i + b † i (cid:17) . (10)If we regard z as a classical degree of freedom when wesearch for the ground state of H eff in Eq. (8), then bytaking the thermodynamic limit, the classical degree offreedom becomes fully separated from the quantum ones, Order Parameter z
Normal PhaseSuperradiant Phase y = g / ( ω √ ( ε + δ ) ) x = ε / √(ε + δ ) FIG. 2. The phase diagram of the tricritical Dicke model. Theorder parameter z vanishes in the normal phase and is finite inthe superradiant phase. The quantum tricritical point (QTP)is marked by a red dot, which is located at the intersectionof the second-order phase transition boundary (red solid line)and the first-order phase transition boundary (green dashedline). in the sense that H Q = o ( H C ) when N → ∞ . As aresult, z is fully determined by the classical part H C ,independent from the quantum part H Q . The separationof the two kinds of degrees of freedom contributes to theemergence of a new universality as we will show when wediscuss the critical behavior of the model.By minimizing H C , we obtain the order parameter z ,from which we can map out the phase diagram [27] in the xy -parameter space as shown in Fig. 2. The normal andthe superradiant phases are characterized by z = 0 and z >
0, respectively. The entire phase boundary is splitinto a solid line and a dashed line, which mark the 2nd-and the 1st-order phase transition, respectively. Thesetwo lines join together at the QTP marked as a red dotin the figure. The position of the QTP is given by( x tc , y tc ) = (cid:16) / √ , / (cid:17) . (11)The presence of the QTP is one of the main results ofour work.While H C determines the order parameter, H Q inEq. (10) gives the quantum fluctuation above the groundstate, from which we can find the excitation gap and theground state atom-light entanglement entropy. It is con-venient to define the generalized position and momentumoperators as X i = b i + b † i √ ω i , P i = (cid:114) ω i b i − b † i i , i = 1 , , , in terms of which, H Q takes the form of a Hamiltonianthat describes a 3-dimension harmonic oscillator: H Q = 12 (cid:88) ij P i + 12 (cid:0) Ω (cid:1) ij X i X j − ω i , (12) Ω ≡ ω λ λ λ ω λ ω , λ ij ≡ (cid:114) ω i ω j g cos θ j . Here X and P represent the original photonic degrees offreedom, while X , and P , represent the atomic degreesof freedom.From Hamiltonian (12), it follows that the lowest ex-citation energy, i.e., the excitation gap, ∆, is given bythe smallest eigenvalue of Ω , and the ground state wavefunction Ψ G is a Gaussian of the formΨ G ( X ) = (cid:18) det Ω π (cid:19) / exp (cid:18) − Ω ij X i X j (cid:19) , (13)from which we can calculate the reduced density matrixof the light field by integrating out the atomic degrees offreedom: ρ ( X , X (cid:48) ) = C exp (cid:18) − A + (cid:0) X + X (cid:48) (cid:1) + A − X X (cid:48) (cid:19) , (14)where A ± ≡ (cid:16) Ω ± det Ω Ω Ω − Ω (cid:17) and C is a normaliza-tion factor. The von Neumann entropy, which measuresthe entanglement between the light and atoms, can becalculated as [19] S ≡ − Tr ( ρ ln ρ ) = γe γ − − ln (cid:0) − e − γ (cid:1) , (15)where γ ≡ cosh − ( A + /A − ). In the limit γ (cid:28)
1, we have S ≈ − ln γ . We calculate ∆ and S numerically anddisplay the results in Fig. 3. These two quantities, unlikethe order parameter or H C which only depends on x and y , also depend on λ ≡ ω/ω like H Q . Therefore the fulldiagram should be 3-dimensional. In Fig. 3, we plot ∆and S on the ( x, y )-plane for λ = 0 . , ,
10. Althoughit is difficult to distinguish the two phases (normal andsuperradiant) through ∆ and S , the phase boundary isquite clear in the plots. On the 2nd-order phase transi-tion boundary, the gap closes and the critical entangle-ment entropy diverges logarithmically. By contrast, onthe 1st-order phase transition boundary, both ∆ and S have finite jumps across the phase boundary. Critical behavior —
Let us now turn to the criticalbehavior of the tricritical Dicke model. One is often con-cerned with how the order parameter behaves near thecritical line (i.e., the 2nd-order phase boundary). Con-sider a point ( x, y ) in the superradiance region and closeto the critical line, if we draw a line perpendicular to thecritical line through this point and intercepts the critical
Entanglement Entropy S y = g / ( ω √ ( ε + δ ) ) y = g / ( ω √ ( ε + δ ) ) y = g / ( ω √ ( ε + δ ) ) x = ε / √(ε + δ ) λ x = ε / √(ε + δ ) λ = 10λ = 1λ = 0.1 FIG. 3. The atom-light entanglement entropy S (left panel)and the lowest excitation energy ∆ (right panel) as functionsof x and y for λ ≡ ω/ω = 0 .
1, 1, 10 (from top to bottom).The QTP is marked by the red dot as in Fig. 2. line at ( x c , y c ), then the order parameter at ( x, y ) can beobtained by expanding f ( z ) in powers of zz = 2 (cid:113) y − c + 4 x c y c − x c n + o ( n ) , (16)where n is the distance between ( x, y ) and the criticalline. Hence, the critical exponent α defined by z ∝ n α is 1/2. However, if the line through ( x, y ) intercepts thecritical line at the QTP ( x tc , y tc ), we have a differentscaling: z = 5 √ n + o ( n ) , (17)which yields an exponent α = 1 / H C . Now let us examine the behavior of the excitation gap ∆ and the entangle-ment S , both of which are governed by H Q . To thisend, we need to find the matrix elements of Ω . It canbe shown that, on the critical line, Ω has eigenvalues 0, ω and √ λ ω . The smallest eigenvalue is 0 whichindicates that the gap ∆ vanishes, as expected. Further-more, the entropy S diverges logarithmically accordingto Eq. (15). Near the critical line, to the leading order indet (Ω /ω ), we have∆ /ω ∼ (cid:0) λ (cid:1) − / det (Ω /ω ) , (18) S ∼ −
12 ln (cid:34) (cid:0) λ + 1 (cid:1) det (Ω /ω ) λ (cid:35) , (19)which establishes a universal relation between S and ∆in the critical region as S ∼ −
12 ln (cid:34) (cid:0) λ + 1 (cid:1) / ∆ λω (cid:35) . (20)Equation (20) represents another key result of this work.Two important remarks are in order here. First, Eq. (20)does not explicitly contain z , which is due to the sepa-ration of the classical and the quantum degrees of free-dom aforementioned. The harmonic oscillator modes, de-picted by H Q , are collective modes involving both lightand atoms, emerging above the mean-field ground stateof H C in the thermodynamic limit, and Eq. (20) is solelydetermined by these modes, therefore we can call Eq. (20)an emergent quantum universality. Second, Eq. (20) isvalid near all the critical points despite of the fact thatpoints around the QTP exhibit different scaling behaviorfor the order parameter. It is even valid in the normalphase region below the critical line where the order pa-rameter vanishes.Given a point ( x, y ) sufficiently close to, and a distance n away from, the critical line, the key factor det (Ω /ω )in Eq. (19) can be expressed by n asdet (cid:0) Ω /ω (cid:1) /λ = β (cid:113) y − c + 4 x c y c n + o ( n ) , (21)where the coefficient β takes different values in differentcritical regions. If ( x, y ) is located in the superradiantphase, then β = 2 unless ( x, y ) approaches the QTP,in which case β = 4. If ( x, y ) is located in the normalphase where z = 0, then β = 1. The scaling exponentbetween det (Ω /ω ) and n, is always the same while thescaling amplitude varies. Consequently, we have ∆ ∝ n / and the entropy diverges logarithmically in termsof n . Another point to remark is that, as a functionof λ , the critical entanglement entropy takes the form S ( λ ) ≈ − ln (cid:0) λ + λ − (cid:1) + const, which indicates thatthe entanglement between light and atom is maximizedunder the resonance condition λ = 1.In our model, as in the conventional Dicke model, thestrengths of the rotating and the counter-rotating termsare equal. Previous studies have considered a Dicke-typemodel where these two strengths can have different val-ues and found that there exists a multicritical point inthe ground state phase diagram [28]. However, in thepresence of dissipation, the multicritical point disappears[29]. This is related to the disappearance of the super-radiance phase in the presence of dissipation when thecounter-rotating terms are absent. Due to the presenceof the counter-rotating terms, we expect that the QTPin our model should be robust against dissipation. Nev-ertheless, how the dissipation affect the universal scalingrequires further study. Conclusion —
In conclusion, we have constructed ageneralized Dicke model that supports a QTP. The phaseboundary and the position of the QTP in the parameterspace, as well as the scaling behavior of the order param-eter, can be determined from the mean-field theory andare found analytically. From this, we explicitly show thatthe QTP belongs to a different universality class thanother points on the critical line. We further investigatedthe quantum fluctuations above the mean-field groundstate, and calculated the excitation gap and the entangle-ment entropy and their critical behavior near the criticalline. We established a new universal relation between theexcitation gap and the entanglement entropy in the entirecritical regime that includes the QTP. The universalityis the result of the separation of the quantum and theclassical degrees of freedom in the thermodynamic limit,being the property of the emergent collective quantummodes. Our model could be realized using atoms andcavities, or maybe other platforms, with current technol-ogy. Our work opens up new opportunities to investigatequantum tricriticality.We acknowledge the support from the NSF and theWelch Foundation (Grant No. C-1669). [1] R. B. Griffiths, Phys. Rev. Lett. , 715 (1970).[2] T. S. Chang, A. Hankey, and H. E. Stanley, Phys. Rev.B , 346 (1973).[3] E. K. Riedel, Phys. Rev. Lett. , 675 (1972).[4] M. Henkel, Conformal Invariance and Critical Phenom-ena (Springer-Verlag Berlin Heidelberg, 2013).[5] S. Sachdev,
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