Entropies and IPR as Markers for a Phase Transition in a Two-Level Model for Atom–Diatomic Molecule Coexistence

  Citation: Baena, I.; Pérez-Fernández, P.; Rodríguez-Gallardo, M.; Arias, J.M. Entropies and IPR as Markers for a Phase Transition in a Two-Level Model for Atom–Diatomic Molecule Coexistence. Entropy 2022,24, 113. https://doi.org/10.3390/e24010113 Academic Editors: Miguel Angel Bastarrachea-Magnani and Adolfo del Campo Received: 15 November 2021 Accepted: 6 January 2022 Published: 12 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). entropy Article Entropies and IPR as Markers for a Phase Transition in a Two-Level Model for Atom–Diatomic Molecule Coexistence Ignacio Baena 1, Pedro Pérez-Fernández 2,3,* , Manuela Rodríguez-Gallardo 1,3 and José Miguel Arias 1,3,* 1Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain; [email protected] (I.B.); [email protected] (M.R.-C.) 2Departamento de Física Aplicada III, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, 41092 Sevilla, Spain 3Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain *Correspondence: [email protected] (P.P.-F.); [email protected] (J.M.A.) Abstract: A quantum phase transition (QPT) in a simple model that describes the coexistence of atoms and diatomic molecules is studied. The model, which is briefly discussed, presents a second-order ground state phase transition in the thermodynamic (or large particle number) limit, changing from a molecular condensate in one phase to an equilibrium of diatomic molecules–atoms in coexistence in the other one. The usual markers for this phase transition are the ground state energy and the expected value of the number of atoms (alternatively, the number of molecules) in the ground state. In this work, other markers for the QPT, such as the inverse participation ratio (IPR), and particularly, the Rényi entropy, are analyzed and proposed as QPT markers. Both magnitudes present abrupt changes at the critical point of the QPT. Keywords: quantum phase transitions; exactly solvable models; inverse participation ratio; Rényi entropy 1. Introduction The study of phase transitions in quantum systems is a topic of present interest, usually referred to as quantum phase transitions (QPT) [ 1 – 3 ]. Since the seminal Gilmore and collaborators works [ 4 – 6 ], there have been numerous papers characterizing QPTs in two-level quantum systems of different dimensionalities used to model nuclear and molecular systems, such as the interacting boson model (IBM) and the vibron model (see [7–9] and references therein). In connection with this, particular Hamiltonians based on algebraic structures that are exactly solvable have been proposed, such as the Lipkin [ 10 ], the Bose-Hubbard [ 11 ], the Jaynes–Cummings, the Tavis–Cummings and the Dicke models [ 12 – 14 ], just to cite a few of them. These models present specific dynamical symmetries that correspond to different equilibrium configurations of the system in the ground state. The algebraic structure of these models allows for simple solutions in some cases, which provide important references for more complex systems In this work, a solvable two-level model that represents the coexistence of atoms and homo-nuclear diatomic molecules is used to study QPTs [ 15 – 18 ]. The model is briefly presented in Sect. II, where the matrix elements relevant in the model Hamiltonian are given explicitly on a basis with two labels: the number of molecules and the number of atoms. The eigenvalues and eigenvectors of the Hamiltonian are easily obtained by diagonalizing the corresponding matrix. The thermodynamic or large particle number limit of the model is also presented so as to classify the QPT and analyze the critical transition point. The model has one control parameter that drives the system from a molecular condensate, in one phase, to a new phase in which atoms and molecules coexist. Usual markers for the critical point are the ground state energy and the behavior of an order parameter that is zero in one phase and different from zero in the other one. Usually this Entropy 2022,24, 113. https://doi.org/10.3390/e24010113 https://www.mdpi.com/journal/entropy Entropy 2022,24, 113 2 of 16 order parameter is the expected value in the ground state of the number of atoms (or the number of molecules). In this work, we propose the use of the inverse participation ratio (IPR) and the Rényi entropy as other good markers for the critical point. They are presented in Section 3. Section 4is for conclusions. 2. The Model for the Atom–Diatomic Molecule Coexistence A simple two-level model designed to describe a system of two coexisting components, individual atoms and diatomic homo-nuclear molecules, is worked out. In Figure 1, the model is represented schematically. ω0− ω a b Figure 1. Schematic representation of the model used in this work for the atom–diatomic coexistence. This is a two-level model. Diatomic molecules ( b ) are in the lower level, while single atoms ( a ) are in the upper level. The quantity ω0−ω represents the energy needed for separating the molecule into its two single atoms. This figure was taken from [16]. Each component in the model is represented in terms of bosons. There are two boson types: a and b . Boson type a represents individual atoms of energy ¯hω0/ 2, whereas b− bosons represent diatomic molecules of energy ¯hω . Atoms and molecules interact, and the proposed Hamiltonian is (¯h=1 is used in this work) [16] H=ω0 2a†a+ωb†b+λ √2M(b†aa +ba†a†), (1) where M=2nb+na(2) is the total number of atoms and is a conserved quantity. This magnitude gives the size of the system. Moreover, ˆ na=a†a is the particle number operator of bosons of type a (atoms) and ˆ nb=b†b is the particle number operator of b− bosons (diatomic molecules). The expected values of these operators are the numbers of particles of the two boson types ( na and nb ). To make everything simpler, only even M-values are considered in this work. Furthermore, λ is a control parameter that drives the system from one phase to the other. Given that ω0>ω , for λ= 0 the ground state is just a molecular condensate without any single atom. However, as λ increases, the interaction produces a more balanced atom– molecule distribution. Thus, depending on the control parameter λ , the system has two phases: one with just molecules and another with molecules and atoms. 2.1. Exact Solution of the Eigenvalue Problem An obvious basis for studying the Hamiltonian (1) is obtained by giving the number of molecules nb and the number of individual atoms na : |na , nbi . Since M= 2 nb+na is conserved, one can use alternatively the notation |M , nbi with M being fixed and defining the system. The matrix elements of (1) in the mentioned basis are trivial and produce a tridiagonal matrix that can be easily diagonalized for each selected M -value. The relevant matrix elements are: Entropy 2022,24, 113 3 of 16 hM,nb|a†a|M,n0 bi=δnb,n0 b(M−2nb), (3) hM,nb|b†b|M,n0 bi=δnb,n0 bnb, (4) hM,nb|b†aa|M,n0 bi=δnb,n0 b+1pM−2nb ×pM−2nb−1pnb+1, (5) hM,nb|ba†a†|M,n0 bi=δnb,n0 b−1pM−2nb+1 ×pM−2nb+2√nb. (6) For a given M , the matrix to be diagonalized is of dimensions (M/ 2 + 1 ) , since one can have from zero molecules (only M atoms) to M/ 2 molecules (no atoms). A simple diagonalization of the corresponding tridiagonal matrix will provide one with all Hamiltonian eigenvalues and eigenfunctions. In particular, given an M -number, this diagonalization allows one to obtain the ground state energy and the corresponding wavefunction as a function of the control parameter λ . This can be used to study the ground state phase transition of the system as a function of λ. In particular, once the ground state wavefunction is obtained, |gs(λ)i , one can use it to calculate the expected value of the number of atoms hgs(λ)|ˆ na|gs(λ)i . We show in the next subsection that this magnitude behaves as an order parameter. It is zero in one phase and different from zero in the other one. In a later section we will show other observables that can be used as markers for the critical point of the phase transition. In order to study a reference for the phase transition, a mean field study of the model is presented next. 2.2. Mean Field for the Model Hamiltonian In order to develop a mean field study for this model, it is useful to introduce the operators: K+=1 2a†a†, (7) K−=1 2(aa), (8) K0=1 2a†a+1 2, (9) which close under the su( 1,1 ) commutation relations. Using the Holstein–Primakoff expansion [19], a new c−boson can be introduced as K+=c†1 2+c†c1/2 , (10) K−=1 2+c†c1/2 c, (11) K0=c†c+1 4. (12) In terms of bosons band c, the Hamiltonian (1) can be written as H=ω0c†c+ωb†b(13) +λ √2M"1 2+c†c1/2 cb†+bc†1 2+c†c1/2#. Entropy 2022,24, 113 4 of 16 To perform a semiclassical analysis of the system, the usual relation with atom coordinates and momenta (x , p) and diatomic molecule coordinates and momenta (y , q) from the harmonic oscillator are introduced: c √M=1 √2(x+ip);c† √M=1 √2(x−ip); (14) b √M=1 √2(y+iq);b† √M=1 √2(y−iq). (15) These are canonical transformations, and in the thermodynamic limit, i.e., M→∞ , the operator’s position and momentum commute. In addition, in this limit the factor 1 / 2 can be negligible in comparison with a term multiplied by M . Then, by introducing these relations in the Hamiltonian, it is written as H=H0+H1+H2(16) with H0=ω0M 2(x2+p2) + ωM 2(y2+q2), (17) H1=λM 2q(x2+p2)(xy +pq +ipy −iqx), (18) H2=λM 2(xy +pq −ipy +iqx)q(x2+p2). (19) To analyze the properties of the ground state of the system in the thermodynamic limit, it is useful to rewrite the Hamiltonian (16) using polar coordinates: x=rcos α;p=rsin α;y=scos β;q=ssin β. Then it can be shown that the Hamiltonian (16) can be written as H=M[1 2(ω0r2+ωs2) + λr2scos(α−β)]. (20) Itseems clearfromthisHamiltonianthattheminimumenergy corresponds to cos(α−β) = −1 (it is the value that makes the second term, and therefore, the H minimum, since the other terms are positive, r2 and s2 ). This corresponds to α−β=π . Any choices of α and β such that they differ by π give the minimum energy. A possible combination is α= 0 and β=−π , which correspond to p= 0 and q= 0, but any other selection of α and β (and, correspondingly, of p and q ) that satisfies α−β=π will give the same minimum energy surface per particle: H/M=1/2(ω0r2+ωs2)−λr2s(21) which is Equation (22) in terms of x and y , taking α= 0 and β=−π (or equivalently p=q=0), V(x,y,λ) = Mhω0 2x2−ω 2y2−λx2yi. (22) This equation can be obtained in a more straightforward way from Equation (2) using coherent boson states. However, it is interesting to illustrate some tools, such as those presented above, that potentially can be used to extract finite size effects in the system (expanding the potential energy surface in 1/M powers), thereby going beyond the mean field description. Anyway, Equation (22) gives the classical energy surface associated with this model. On the other hand, the Mconservation leads to the condition x2+p2+y2+q2=1, (23) Entropy 2022,24, 113 5 of 16 which for p , q→ 0 gives x2+y2= 1. This allows us to reduce the original two dimensional problem to another one with only one effective degree of freedom, x∈[− 1,1 ] . Taking into account that the sign selection y=−√1−x2 produces lower energy than the positive sign, the energy surface per particle can be written as v(x,λ) = V(x,λ) M=ω 2+∆ω 2x2−λx2p1−x2, (24) where ∆ω=ω0−ω. The condition for the minimum is dv(x,λ) dx xmin =0, (25) and provides two solutions, xmin =(x1=0 which is always a solution ; x2=12λ2−∆ω2−∆ω√∆ω2+12λ2 18λ21/2 .(26) This last solution provides energy lower than x1=0 when λis larger than a critical value that we call λc . It is also a solution of the problem of the expression of x2 with a positive sign in front of the square root, but the written expression, with the minus sign, always gives lower energy. A value for λccan be obtained to make x2=0: 12λ2 c−∆ω2±∆ωq∆ω2+12λ2 c=0, (27) which gives the critical point for the transition, λc=ω0−ω 2=∆ω 2. (28) For given values of ω and ω0 (this fixes λc ), the minimum energy per particle as a function of λis obtained: e0(λ) = E0(λ) M=(ω 2λ≤λc, ω 2+∆ω 2x2 2−λx2 2q1−x2 2λ>λc.(29) Equation (29), with x2 from Equation (26), gives an analytic expression for the minimum of the energy surface per particle as a function of the control parameter λ . In Figure 2, the large-M limit of the ground state energy per particle (panel a), its first derivative (panel b) and its second derivative (panel c) are represented, respectively, for the cases ω0= 2 and ω= 1. In the three plots it is clear that at λ= 0.5 there is a structural change in the system. This value is the λc given in Equation (28) . Furthermore, the discontinuity of the second derivative indicates that this is a second-order (or continuous) phase transition. From Figure 2it is clear that the system undergoes a second-order QPT at λc. Since we can solve the problem exactly for a finite M , in Figure 3the mean field result for the ground state energy per particle is represented, together with the exact numerical calculations with M= 50 and M= 700 for the cases ω0= 2 and ω= 1 that produce λc= 0.5. The mean field calculation is depicted by a black line; the exact numerical results for M= 50 are shown as a green line; and for M= 700, the exact numerical results are shown as a dashed red line. For a system of size M= 50, the exact numerical result fits quite well with the analytical mean-field, except in a small region close to the critical point (finite-size effects). Nevertheless, the bigger the system is, the better the agreement with the mean field calculation. This is shown in Figure 3for M= 700, which is basically indistinguishable from the mean-field result. In order to show better the convergence, an inset is included in Figure 3representing a function εdefined as: Entropy 2022,24, 113 6 of 16 ε=E(M) 0−EMean−Field 0 EMean−Field 0 , (30) as a function of λfor different Msizes. Figure 2. Large–M limit (mean–field) results of the system as a function of the control parameter λ for the cases of ω0= 2 and ω= 1. In panel ( a ) the ground state energy per particle is represented. In panel ( b ), its first derivative with respect to λ is plotted. Finally, in panel ( c ), the second derivative of the ground state energy per particle is given as a function of λ . The system undergoes a QPT for λc=0.5. Figure 3. Ground state energy per particle in the large–M limit of the system as a function of the control parameter λ for the cases of ω0= 2 and ω= 1. The system undergoes a QPT for λc= 0.5. The mean field calculation is depicted in black full line; meanwhile, the exact numerical results for M= 50 (full green line) and M= 700 (dashed red line) are also presented. In order to show the convergence to the mean field with M , the inset represents the difference between the exact M calculation and the mean field result. Different Msizes are shown. Entropy 2022,24, 113 7 of 16 In addition to the energy, one can calculate analytically at the mean field level (large M limit) the expected value for the number of atoms of type a . From Equations (9) and (12), one gets the relation na= 2 nc . Using the definitions of c† and c as a function of x and p , and by taking the classical limit ( p→ 0 and [x , p] = 0), one obtains easily that the number of individual atoms per particle, na/M , is x2 . The expected value of this observable in the system ground state is then hgs|ˆ na/M|gsi=hgs|x2|gsi=0λ≤λc, x2 2λ>λc,(31) with x2 being given in Equation (26). This expression can be compared with real finite-M calculations to check how fast is the convergence to the large-M limit, and consequently how large are the finite-M effects. In Figure 4, the large M limit of the expected value of ˆ na/M in the ground state is plotted as a function of λ . It is clear that this observable is an order parameter for the phase transition, since it is zero in one phase and different from zero in the other one. When λ→∞ , this order parameter tends to 2/3, as given by Equations (31) and (26). The latter means means that there would be a coexistence phase of atoms and molecular particles in which it is equally likely for an atom to either be chemically bonded or to remain unbound. In Figure 5, the exact expected value for ˆ na/M is presented for M= 700, along with three different selections for ∆ω : 1, 2 and 3, that lead to λc : 0.5, 1.0 and 1.5, respectively. For this large M value, the plots match the mean field result given by Equations (31) and (26). It is clearly seen in Figures 4and 5that this order parameter marks the critical point (represented in Figure 5with filled dots for each ωselection). Figure 4. The large-M value for na/M , number of atoms of type a per particle, as a function of the control parameter λ for the cases of ω0= 2 and ω= 1. The system undergoes a QPT for λc= 0.5. This observable behaves as an order parameter. It is zero for λ<λc and different to zero for larger values of λ. Entropy 2022,24, 113 8 of 16 Figure 5. Numerical exact calculation for the expected value of ˆ na/M . Number of atoms of type a per particle, as a function of the control parameter λ for three different cases: ∆ω= 1 (full black line), ∆ω= 2 (dashed red line) and ∆ω= 3 (dotted and dashed blue line). In all these cases, the critical value for λ has been marked: λc= 0.5, λc= 1, and λc= 1.5, respectively. All calculations were done for M=700. In all cases, we have checked that the numerical results tend to the mean field approximation expressions as M is increased, and that the critical point corresponds to Equation (28). In our model, as in the Tavis–Cummings and Jaynes–Cummings models [ 20 ], quantum fluctuations are zero, and consequently, these fluctuations cannot be responsible for the corresponding vacuum instability. In this respect, some researchers consider that this is not a quantum phase transition. However, this model possesses non-analyticity in the ground state, in agreement with a continuous quantum phase transition. As such, it is a matter of taste whether the transition should be termed quantum or not. 3. Other Markers for the QPT In this section, we propose other markers for the critical point in the QPT. 3.1. Inverse Participation Ratio The inverse participation ratio (IPR) is defined as IPR =P(k)=1 ∑i|c(k) i|4. (32) This magnitude measures the degree of delocalization of a quantum state within a specific basis. The coefficients c(k) i are the coefficients of the state k in the used basis. On one hand, in case of full localization, the k state is one of the basis states, so only one ci= 1 and the IPR will be close to 1. On the other hand, if the state k is equally distributed among all basis states, then the normalization condition is ∑ i|c(k) i|2=D|c(k) 1|2=1, (33) with Dbeing the dimension of the matrix diagonalized. Then |c(k) i|=1 √D. (34) Entropy 2022,24, 113 9 of 16 In this case, the maximum IPR is obtained IPR max =M/ 2 + 1. Consequently, any values of IPR between 1 and M/2 +1 are expected in general. For the model discussed here, an IPR = 1 is expected for λ= 0, since in this case our Hamiltonian eigenstates are those of the harmonic oscillator. For other λ− values, the Hamiltonian eigenstates will be a mixture of harmonic oscillator states and the IPR will increase. However, not every state of the harmonic oscillator will "participate" in the eigenstate of the coexistence phase. Only a linear combination of states in which the expected number of atoms is 2/3 of M will contribute. Thus, IPR will reach a constant but smaller size than the maximum possible value. The numerical results from the exact diagonalization of the system Hamiltonian have already been presented, and these were compared to the mean-field results in the preceding section. For a given M , the exact diagonalization produces the ground state, and consequently provides the coefficients cgs M,nb . With these, one can calculate the IPR (32). In Figure 6, the ground state IPR values for M= 700 and for different ∆ω choices as a function of the control parameter λare presented. Figure 6. IPR for the ground state as a function of the control parameter λ for M= 700 and for three different ω selections: ∆ω= 1 (full black line), ∆ω= 2 (dashed red line) and ∆ω= 3 (dotted and dashed blue line). The IPR marks clearly the transition of the system at the corresponding λc . The ground state is well localized (small IPR) in the harmonic oscillator basis for λ below to the QPT critical point, whereas it tends to be delocalized for values of λ above the critical value. Indeed, an abrupt change of IPR occurs at λc=∆ω/2 in the QPT. A natural question in relation to Figure 6is: what are the asymptotic values for λ→ 0 and λ→∞ ? In order to show the λ→∞ , we plot in Figure 7the IPR for the cases ∆ω= 1 and M=700. It is seen that the IPR for large λis around 30. Whilst an IPR = 1, or approximately 1, is expected for λ≤λc , for which the ground state is close to a molecular condensate (the state is basically |M , na= 0, nb=M/ 2 i ). For larger λ− values the Hamiltonian eigenstates will be mixtures of harmonic oscillator states, in which there will be more than one relevant state, and therefore, the IPR will increase. The limit of M/2+1 is obtained when all the basis states are contributing with equal weight. However, this is not reasonable, and states in which the number of atoms is na= 2 M/ 3, and consequently nb=M/ 6 (we notice that na+ 2 nb=M ), are expected to have larger weights. In fact, if one assumes for the wavefunction coefficients a binomial distribution with D = M /2 components, |M , na , nbi , whose probability of nb is p= 1 / 3, the corresponding IPR would be around 31. Although the distribution in our ground state is not exactly binomial, something similar is expected. In that case, the IPR will not reach the maximum possible value, and an IPR around 31 is expected for M= 700. Entropy 2022,24, 113 16 of 16 Funding: This work is part of the I+D+i projects PID2019-104002GB-C22 and PID2020-114687GB-I00 funded by MCIN/AEI/10.13039/501100011033. This work is also part of grant Group FQM-160, EU FEDER funds US-1380840 and the project PAIDI 2020 with reference P20_01247, funded by the Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía (Spain) and “ERDF—A Way of Making Europe”. Acknowledgments: We thank J. Gómez-Camacho for discussions. Conflicts of Interest: The authors declare no conflict of interest. References 1. Sachdev, S. Quantum Phase Transitions; Cambridge University Press: Cambridge, UK, 1999. 2. Carr, L. Understanding Quantum Phase Transitions; CRC Press: Boca Raton, FL, USA, 2011. 3. 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