J Mol Model (2014) 20:2237 DOI 10.1007/s00894-014-2237-1

ORIGINAL PAPER

Complexity measure and quantum shape-phase transitions in the two-dimensional limit of the vibron model Elvira Romera & Manuel Calixto & Ágnes Nagy

Received: 5 February 2014 / Accepted: 7 April 2014 # Springer-Verlag Berlin Heidelberg 2014

Abstract We obtain a characterization of quantum shapephase transitions in the terms of complexity measures in the two-dimensional limit of the vibron model based on the spectrum generating algebra U(3). Complexity measures (in terms of the Rényi entropies) have been calculated for different values of the control parameter for the ground state of this model giving sharp signatures of the quantum shape-phase transition from linear to bent molecules. Keywords Complexity measures . Molecular vibron model . Quantum phase transitions

Introduction Quantum phase transitions (QPT) are a fundamental part of quantum many-body theory and can be considered as an extension of classical phase transitions to zero absolute temperature. QPTs result from the variation of quantum This paper belongs to Topical Collection QUITEL 2013 E. Romera Instituto Carlos I de Fisica Teorica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain E. Romera (*) Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain e-mail: [email protected] M. Calixto Departamento de Matematica Aplicada, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain Á. Nagy Department of Theoretical Physics, University of Debrecen, 4010 Debrecen, Hungary

fluctuations. Generally speaking, one finds different quantum phases connected to specific geometric configurations of the ground state and related to distinct dynamic symmetries of the Hamiltonian. The QPT occurs as a function of a control parameter ξ that appears in the Hamiltonian H. For us, it will appear in the form of a convex combination H(ξ)=(1−ξ)H1 + ξH2. At ξ=0 the system is in phase I, characterized by the dynamical symmetry G1 of H1, and at ξ=1 the system is in phase II, characterized by the dynamical symmetry G2 of H2. At some critical point ξc ∈(0,1), there is an abrupt change in the symmetry and structure of the ground state wavefunction. This is the case of the so-called “vibron models” (see e.g., [13–16, 22, 23]), interacting boson models, which exhibit a second-order shape-phase transition from linear to bent. These models have been used to study the rovibrational properties in diatomic and polyatomic molecules. The vibron model was introduced by Iachello in the 1980s for the description, in an algebraic manner, of molecular structure (see standard textbooks [11, 12] on the subject). It is based on the spectrum generating algebra U(4) and it includes a full description of rovibrational degrees of freedom for a diatomic molecule. The description of polyatomic molecular species is achieved through the coupling of several U(4) algebras. Two limits of this model were later introduced, to avoid the mathematical complexities of the full rovibrational description, the onedimensional limit (based on the U(2) dynamical algebra) and the two-dimensional limit (based on the U(3) dynamical algebra), which is the simplest model retaining angular momentum quantum numbers. We chose the latter model to perform calculations of the statistical complexity across the shapephase transition from linear to bent, two configurations that appear in triatomic molecules. The importance of shape QTPs in molecular modeling relies on the fact that molecular species exist with a nonrigid structure, intermediate between rigid linear and bent, and a unified description of all configurations within the framework of a single approach (as the vibron

2237, Page 2 of 5

model does) is very challenging, as most common approaches so far have concentrated on linear or quasi-linear geometries. On the other hand, information entropies and statistical complexities have a role of growing importance in describing quantum phenomena. Different simple atomic systems [7, 8, 10, 29, 30, 32, 34] were studied with Fisher [9] and Shannon [31] information [1, 25, 33]. These systems display remarkable properties when the so-called LMC complexity [6, 18] is computed on them [21, 30]. Recently, it has been shown that Shannon and Rényi entropies [4, 24, 27], Wehrl entropies [3, 26], entanglement entropies [5], and Rényi-Fisher products [20] detect QPTs in several atomic and chemical quantum models. In this article, we show that the generalized complexity measures [19] are also excellent descriptors of QPT in the vibron model for molecules. Complexity is a general indicator of structure and correlation. Therefore, it is useful in chemistry or in molecular modeling. We mention by passing that the special case of LMC complexity (that is, α=1 and β= 2) has proved to be useful in studying several chemical and physical properties. In [2], it was found that molecules with a low number of electrons possess low complexities, while a larger number of electrons possess larger complexities. Studying hardness and ionization potentials, they concluded that molecules that are more stable chemically possess low complexity values. This article is organized as follows: in the following section we remind the reader of the two-dimensional limit of the vibron model; in the last but one section we study the description of the QPT in this model by means of a complexity measure. Finally some conclusions are given.

U(3) vibron model and shape QPT The two-dimensional vibron model, based on the U(3) symmetry spectrum generating algebra, describes a planar molecular system containing a dipole degree of freedom [13]. It is an appropriate model for the study of critical properties because it is the simplest model that still retains the basics of more complicated threedimensional models and embodies the necessary physical ingredients to reproduce the spectroscopic signatures of rigidly linear, rigidly bent, and floppy molecular configurations [15, 16]. The basic physical realization of such a system is the bending vibrational mode of a molecule [14, 23]. In the U(3) vibron model, one finds different shape phases connected to specific geometric configurations of the ground state and related to distinct dynamic symmetries of the Hamiltonian [22]. Let us give a brief explanation of the basic ingredients to construct this Hamiltonian. In this model, the elementary

J Mol Model (2014) 20:2237

excitations are described by creation and annihilation twodimensional vector τ -bosons {τ†x ,τ†y ,τx,τy} and a scalar σ -boson {σ†,σ}. It is convenient to introduce circular bosons:
 pffiffiffi τ  ¼ ∓ τ x ∓iτ y = 2 . The essential Hamiltonian of this system is written in terms of bilinear products of these creation and annihilation operators, which lead to up to nine generators of the corresponding U(3) algebra. Concretely, the essential Hamiltonian [22] is only written in terms of the following five generators: b n ¼ τ †þ τ þ þ τ †− τ − ; b ns ¼ σ† σ; bl ¼ τ †þ τ þ −τ †− τ − ;  pffiffiffi pffiffiffi
 b þ ¼ 2 τ †þ σ−σ† τ − ; D b − ¼ 2 −τ † σ þ σ† τ þ D −

ð1Þ

where b n and b ns are number operator of vector and scalar bosons, respectively, bl is the two-dimensional angular b  is the dipole operator. We have to momentum, and D take into account that the total number of bosons b ¼b N nþb nσ and the two-dimensional angular momentum bl are conserved. With these ingredients, the essential Hamiltonian is given by b N ðN þ 1Þ−W b ¼ ð1−ξÞb H nþξ N −1

2

ð2Þ

where N is the “size” of the system [it labels the totally symmetric (N+1)(N+2)/2 -dimensional representation [N] of   b2 ¼ D b− þ D b þ =2 þ bl 2 is the b þD b −D U(3)], the operator W squared angular momentum of the SO(3) subalgebra, and ξ∈[0,1] is a control parameter. In the thermodynamic (mean field or large size) limit, where number of bound states becomes large, N→∞, there is a shape QPT at a critical value ξc =0.2 of the control parameter ξ, which marks the boundary between the two geometrical phases: the linear (for ξξc) phases (see later on this section for a more precise explanation). We will consider the following state space basis in order to solve the eigenvalue problem
† N −n  †  nþl2
†  n−l2 σ τþ τ− jN ; n; l >¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ffi j0 > nþl n−l ðN −nÞ! ! ! 2 2

ð3Þ

where the bending quantum number n=N,N−1,N−2,…, 0 and the angular momentum l=±n,±(n−2),…,±1 or 0 (n= odd or even) are the eigenvalues of b n and bl ,

J Mol Model (2014) 20:2237

Page 3 of 5, 2237

b 2 can be easily respectively. The matrix elements of W derived (see e.g., [22]): 0 b 2 jN ; n; l >¼ < N ; n ; ljW i h ðN −nÞðn þ 2Þ þ ðN −n þ 1Þn þ l 2 δn0 ;n ð4Þ 1 −½ðN −n þ 2ÞðN −n þ 1Þðn þ l Þðn−l ފ 2 δn0 ;n−2

1
N jN ; r > ≡ pffiffiffiffiffiffi b†c j0 >; N!


 1 b†c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi σ† þ rτ †x ð5Þ 1 þ r2

1

−½ðN −nÞðN −n−1Þðn þ l þ 2Þðn−l þ 2ފ 2 δn0 ;nþ2 The ground state energy in the thermodynamic limit N→∞ is recovered by using ground state ansätze in terms of (semi-classical) coherent states

with r being a free variational real parameter and b†c being the boson condensate. Other rotationally equivalent possibilities can also be considered [17]. The variational parameter r is fixed by minimizing the ground state energy functional “per particle” (see e.g., [22]):

2 D E b ; r N ð N þ 1 Þ− N ; r W N b hN ; rjnjN ; ri εξ ¼ þ ¼ ð1−ξÞ N N ðN −1Þ N  2 r2 1−r2 ¼ ð1−ξÞ þξ : 1 þ r2 1 þ r2 D E b N ; r H N ; r

ð6Þ

From ∂ Eξ(r)/∂r=0 one gets the “equilibrium radius” re and the ground state energy Eξ as a function of the control parameter ξ: 8 0; ffi ξ ≤ ξc ¼ 1=5; > < sffiffiffiffiffiffiffiffiffiffiffiffiffi 5ξ−1 r e ðξ Þ ¼ > : 3ξ þ 1 ; ξ > ξc ¼ 1=5; ð7Þ 8 ξ; ξ ≤ ξc ¼ 1=5; i < h E ξ re ðξÞ ¼ −9ξ2 þ 10ξ−1 ; ξ > ξc ¼ 1=5: : 16ξ

order to calculate complexity measures for the ground state density function f=|ψ|2, we must introduce position (and momentum) representations. Denoting by (a1,a2,a3)≡(σ,τ+,τ−) the three oscillator operators and position and momentum operators way as  are defined in the  standard  bxi ¼ p1ffiffi2 a†i þ ai and b pi ¼ piffiffi2 a†i −ai , respectively, for i=1,2,3. We should say that the “position” r=(x1,x2,x3) is not directly related with the true space coordinates of the molecule, but it provides a useful representation to write the wave packet. The basis functions (3) are then written in position representation

Then one finds that d2 Eξ[re(ξ)]/dξ2 is discontinuous at ξc = 1/5 and the shape phase transition, from linear (ξ≤1/5) to bent (ξ>1/5) is said to be of second order. For triatomic molecules, the (dimensionless) algebraic variational coordinate r has been related to a physical angular displacement [14, 15, 22] (or “bending angle”) θ∼r/a, with a the equilibrium bond length, which reflects the degree of distortion of the molecular framework from linearity (r=0). For a finite number N of bound states, a numerical diagonalization of the Hamiltonian (2) must be done for each value of the control parameter ξ. The ground state wave function is given

ϕNn;l ðrÞ

2−N =2 π−3=4 e−r =2 ¼ 〈 r jN ; n; l 〉¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ffi ; ð9Þ nþl n−l ðN −nÞ! ! ! 2 2 2

ðN Þ

jψξ

〉¼

N X n X n¼0

NÞ cðn;m ðξÞjN ; n; l ¼ n−2m〉

ð8Þ

m¼0

where the coefficients c(N) n,m(ξ) are obtained by diagonalization of the Hamiltonian (2) in terms of the basis vectors (3). In

H N −n ðx1 ÞH nþl ðx2 ÞH n−l2 ðx3 Þ: 2

in terms of Hermite polynomials Hn(x). In this model, wave functions in position and momentum representation are the same. This is easy to check taking into account the properties 2 of the Fourier transform of the e−x H n ðxÞ and the fact that the system is parity invariant. As position and momentum wave functions are the same in this particular model, in the following we will only consider position space expressions. We will (N) write ψ(N) ξ (r)= the corresponding wave function in 2 position representation and fξ(N)(r)=|ψ(N) ξ (r)| (or simply f(r) when there is no confusion) the ground state position density distribution.

2237, Page 4 of 5

J Mol Model (2014) 20:2237

Fig. 1 (Color online) ðα;β Þ Complexity measure C f ξ for

1.34

the exact (numerical) ground state of the vibron model as a function of ξ for (α,β) (left) (1,3/2) and (right) (3/4,3/2) for N=8, 16, and 20

1.33

1.66

N=8 N=16 N=20

N=8 N=16 N=20

1.64 1.62

1.32

C(3/4,3/2)

C(1,3/2)

1.6 1.31 1.3

1.58 1.56

1.29

1.54

1.28

1.52

1.27

1.5 0.2

0.2 ξ

Complexity and description of the QPT in the vibron model Consider a D-dimensional nonnegative function f(r). A complexity measure of f can be defined as [19]: ðα;β Þ

Cf

ðαÞ

≡eR f

ðβ Þ

−R f

;

0 < α; β < ∞;

ð10Þ

where R(α) f is the Rényi entropy of the function f given by ðαÞ

Rf

¼

1 ln∫f α ðrÞdr with 0 < α < ∞ and α≠1 1−α

ð11Þ

and R(1) f =Sf =−∫f(r)lnf(r)dr with Sf the so-called Shannon entropy of f [31]. That is, the Rényi entropy reduces to the Shannon entropy if α→1. Eq. (10) is the generalization of the so-called LMC complexity measure [6]: C LMC ¼ H f D f

ξ

translations, rescaling transformations and replication, and near continuity (for an extensive description see [19]). The generalized complexity is an increasing function of β for a fixed α. That means that increasing the value of β we can “enlarge” the changes in the complexity. This is the advantage of examining complexity instead of the Rényi entropy. In studying the QPT, it can be useful to magnify the vicinity of the transition point. It can especially be helpful if we want to apply the generalized complexity as a control parameter. Figure 1 shows the complexity measure for the ground state density as a function of ξ for (α,β)=(1,3/2) (left) and (3/4,3/2) (right), for N=8, 16, and 20. The density has been computed numerically. The complexity measure is quasi-constant in the linear phase and is a decreasing function for ξ≥0.2. That is, C(α,β) is quasi-constant for linear and quasi-linear molecules f and a decreasing function of ξ for bent molecules. So the complexity measure is a detector of the quantum phase transition and could be considered as an order parameter in the system to distinguish different types of molecules. We would like to point out that we have considered other values of the parameters α and β and the results are analogous.

ð12Þ Conclusions

H f ¼ eS f is the Shannon entropy power, while ðβ¼2Þ

is the disequilibrium. A oneD f ¼ ∫f 2 ðrÞdr ¼ e−R f parameter extension of the LMC complexity measure [28] is given by the selection β=2 in Eq. (10). The most general form of this kind of family of complexities can be defined by Eq. (10). This definition leads to several appealing features. The complexity measure (10) verifies the mathematical properties of inversion symmetry, monotonicity, invariance under

In this work, a complexity measure has been proposed to describe a QPT in the U(3) vibron model for molecules. We have shown that this information measure has different behaviors for each phase in this model: (1) it is quasi-constant in the linear phase and (2) it is a decreasing function of the control parameter in the model in the bent phase. This analysis can be extended to other systems where QPTs occur to prove the general validity of the results.

J Mol Model (2014) 20:2237 Acknowledgments This work was supported by the projects FIS201124149, FIS2011-29813-C02-01 (Spanish MICINN), FQM.1861 (Junta de Andalucía) and CEI-BIOTIC project P-V-8 and PP2012-PI04 (Universidad de Granada). The work was also supported by the Hungarian Social Renewal Operational Program (TAMOP 4.2.2.A-11/1/KONV2012-0036). The project was co-financed by the European Union and the European Social Fund. Grant from the Hungarian Scientific Research Fund (OTKA No. K100590) is also gratefully acknowledged.

References 1. Angulo JC, Antolí J, Sen KD (2008) Phys Lett A 372:670 2. Angulo JC, Antolin J, Esquivel RO (2011) in Statistical Complexity ed. K. D. Sen. Springer, Berlin, p 167 3. Calixto M, del Real R, Romera E (2012) Phys Rev A 86:032508 4. Calixto M, Nagy Á, Paraleda I, Romera E (2012) Phys Rev A 85: 053813 5. Calixto M, Romera E, del Real R (2012) J Phys A 45:365301 6. Catalan RG, Garay J, López-Ruiz R (2002) Phys Rev E 66:011102 7. Chatzisavvas KC, Moustakidis CC, Panos CP (2005) J Chem Phys 123:174111 8. Coffey MW (2003) J Phys A, Math Gen 36:7441 9. R. A. Fisher, Proc. Cambridge Philos. Soc.22, 700 (1925). 10. Gadre SR, Sears SB, Chakravorty SJ, Bendale RD (1985) Phys Rev A 32:2602 11. Iachello F, Arima A (1987) The Interacting Boson Model. Cambridge University Press, Cambridge

Page 5 of 5, 2237 12. Iachello F, Levine RD (1995) Algebraic Theory of Molecules. Oxford University Press, Oxford 13. Iachello F, Oss S (1996) J Chem Phys 104:6956 14. Iachello F, Pérez-Bernal F, Vaccaro PH (2003) Chem Phys Lett 375: 309 15. Larese D, Iachello F (2011) J Mol Struct 1006:611–628 16. Larese D, Pérez-Bernal F, Iachello F (2013) J Mol Struct 1051:310– 327 17. Leviatan A, Kirson MW (1988) Ann Phys NY 188:142 18. López-Ruiz R, Mancini HL, Calbet X (1995) Phys Lett A 209:321 19. López-Ruiz R, Nagy Á, Romera E, Sañudo J (2009) J Math Phys 50: 123528 20. Nagy Á, Romera E (2012) Physica A 391:3650 21. Nagy Á, Sen KD, Montgomery HE (2009) Phys Lett A 373:2552 22. Pérez-Bernal F, Iachello F (2008) Phys Rev A 77:032115 23. Pérez-Bernal F, Santos LF, Vaccaro PH, Iachello F (2005) Chem Phys Lett 414:398 24. Romera E, Calixto M, Nagy Á (2012) Europhys Lett 97:20011 25. Romera E, Dehesa JS (2004) J Chem Phys 120:8906 26. Romera E, del Real R, Calixto M (2012) Phys Rev A 85:053831 27. Romera E, del Real R, Calixto M, Nagy S, Nagy Á, Math J (2013) Chem 51:620–636 28. Romera E, López-Ruiz R, Sanudo J, Nagy Á (2009) Int Rev Phys 3:207 29. Romera E, Sánchez-Morena P, Dehesa JS (2005) Chem Phys Lett 414:468 30. Sañudo J, López-Ruiz R (2008) Phys Lett A 372:5283 31. Shannon CE (1948) Bell Syst Tech J 27:379 32. Szabó JB, Sen KD, Nagy Á (2008) Phys Lett A 372:2428 33. Vignat C, Bercher JF (2003) Phys Lett A 312:27 34. Yánez RJ, Van Assche W, Dehesa JS (1994) Phys Rev A 32:3065