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arXiv:1409.5596 · Journal of Physics A Mathematical and Theoretical

Parity-symmetry-adapted coherent states and entanglement in quantum phase transitions of vibron models

Manuel Calixto, E Romera, R del Real

Published Aug 20, 2012• 25 citations
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Abstract

We propose coherent (‘Schrödinger cat-like’) states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite ( N )-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value ξ c of a control parameter. These trial cat states reveal a sudden increase in vibration–rotation entanglement linear ( L ) and von Neumann ( S ) entropies from zero to (to be compared with L ( N ) max (ξ) = 1 − 1/( N + 1)) and , respectively, above the critical point, ξ > ξ c , in agreement with exact numerical calculations. We also compute inverse participation ratios, for which these cat states capture a sudden delocalization of the ground-state wave packet across the critical point. Analytic expressions for entanglement entropies and inverse participation ratios of variational states, as functions of N and ξ, are given in terms of hypergeometric functions.

DOI
10.1088/1751-8113/45/36/365301

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We propose coherent (‘Schrödinger cat-like’) states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite ( N )-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value ξ c of a control parameter. These trial cat states reveal a sudden increase in vibration–rotation entanglement linear ( L ) and von Neumann ( S ) entropies from zero to (to be compared with L ( N ) max (ξ) = 1 − 1/( N + 1)) and , respectively, above the critical point, ξ > ξ c , in agreement with exact numerical calculations. We also compute inverse participation ratios, for which these cat states capture a sudden delocalization of the ground-state wave packet across the critical point. Analytic expressions for entanglement entropies and inverse participation ratios of variational states, as functions of N and ξ, are given in terms of hypergeometric functions.
We study the entanglement properties of the shape phase transitions between different geometric limits of two coupled equivalent molecular benders modeled with the two-dimensional limit of the vibron model. This system has four possible geometric configurations: linear, cis-bent, trans-bent, and nonplanar. We show how the entanglement, accessed through the calculation of the linear entropy, between benders and between rotational and vibrational degrees of freedom changes sensitively in the critical regions of this two-fluid bosonic model. The numeric calculation is complemented with a variational approach to the ground-state wave function in terms of symmetry-adapted coherent states.
The Husimi distribution is proposed for a phase-space analysis of quantum phase transitions in the two-dimensional U(3) vibron model for NNN-size molecules. We show that the inverse participation ratio and Wehrl's entropy of the Husimi distribution give sharp signatures of the quantum (shape) phase transition from linear to bent. Numerical results are complemented with a variational approach using parity-symmetry-adapted U(3) coherent states, which reach the minimum Wehrl entropy N(3+2N)(N+1)(N+2)\frac{N(3+2N)}{(N+1)(N+2)}(N+1)(N+2)N(3+2N)​, in the rigidly linear phase, according to a generalized Wehrl-Lieb conjecture. We also propose a characterization of the vibron-model quantum phase transition by means of the zeros of the Husimi distribution.
We study the phase transition properties of the ground state of the Dicke model by means of a variational procedure together with the catastrophe formalism.The proposed variational state is the tensorial product of the SU(2) and Weyl coherent states, and the expectation value of the Dicke Hamiltonian with respect this state is calculated.The stability properties of the energy surface exhibit a phase transition from the normal to the super-radiant behavior of the two-level atoms, when ωa ωF = ±4 γ 2 .At the same time, the analytic form of the ground state in terms of the minimum critical points of the system is obtained.The Dicke Hamiltonian is invariant under transformations of the point group C 2 = {I, exp (iπ Λ)}, with Λ the excitation number operator.To restore this parity symmetry, we separate the variational proposed states into orthogonal even and odd components, which give rise to analytic expressions for the ground and first excited states.The fidelity of these projected variational states is verified by studying their overlap with the exact quantum results.
In most cases, excited-state quantum phase transitions can be associated with the existence of critical points (local extrema or saddle points) in a system's classical limit energy functional. However, an excited-state quantum phase transition might also stem from the lowering of the asymptotic energy of the corresponding energy functional. One such example occurs in the two-dimensional (2D) limit of the vibron model, once an anharmonic term in the form of a quadratic bosonic number operator is added to the Hamiltonian. This case has been studied in the broken-symmetry phase [P\'erez-Bernal and \'Alvarez-Bajo, Phys. Rev. A 81, 050101 (2010)]. In the present work we delve further into the nature of this excited-state quantum phase transition and we characterize it in the symmetric phase of the model, making use of quantities such as the effective frequency, the expected value of the quantum number operator, the participation ratio, the density of states, and the quantum fidelity susceptibility. In addition to this, we extend the usage of the quasilinearity parameter, introduced in molecular physics, to characterize the phases in the spectrum of the anharmonic 2D limit of the vibron model and a practical analysis is included with the characterization of the critical energies for the linear isomers HCN and HNC.
We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N→∞ limit. The D=3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D=2 atomic coherent states, to define the coherent state representation Q_{ψ} (Husimi function) of a symmetric N-qudit state |ψ〉 in the phase space CP^{D-1} (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N→∞ limit, where the discrete parity symmetry Z_{2}^{D-1} is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2^{D-1} different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.
We study in detail the relation between the two-dimensional Euclidean dynamical E(2) symmetry and the quantum phase transition in the two-dimensional limit of the vibron model, called the U(3) vibron model. Both geometric and algebraic descriptions of the U(3) vibron model show that structures of low-lying states at the critical point of the model with a quartic potential as its classical limit can be approximately described by the E(2) symmetry. We also fit the finite-size scaling exponent of the energy levels and E1 transition rates in the F(2) model, which is exactly the E(2) model but with truncation in its Hilbert subspace, as well as those at the critical point in the U(3) vibron model. The N-scaling power law around the critical point shows that the E(2) symmetry is well preserved even for cases with finite number of bosons. In addition, two kinds of experimentally accessible effective order parameters, such as the energy ratios E21/E11, E31/E 11 and E1 transition ratios B(E1;21→1 1)/B(E1;11→11), B(E1;02→ 11)/B(E1;11→01), are proposed to identify the second-order phase transition in such systems. Possible empirical examples exhibiting approximate E(2) symmetry are also presented. © 2010 IOP Publishing Ltd.