On Solutions of the one-dimensional Holstein Model Feng Pan and J. P. Draayer Liaoning Normal Univ. Dalian China 23rd International Conference on DGM in Theoretical Physics, Aug ,05 Tianjin Louisiana State Univ. Baton Rouge USA

I. Introduction II. Brief Review of What we have done III. Algebraic solutions the one-dimensional Holstein Model IV. Summary Contents

Introduction: Research Trends 1) Large Scale Computation (NP problems) Specialized computers (hardware & software), quantum computer? 2)Search for New Symmetries Relationship to critical phenomena, a longtime signature of significant physical phenomena. 3) Quest for Exact Solutions To reveal non-perturbative and non-linear phenomena in understanding QPT as well as entanglement in finite (mesoscopic) quantum many-body systems.

Exact diagonalization Group Methods Bethe ansatz Quantum Many-body systems Methods used Quantum Phase transitions Critical phenomena

Goals: 1) Excitation energies; wave-functions; spectra; correlation functions; fractional occupation probabilities; etc. 2) Quantum phase transitions, critical behaviors in mesoscopic systems, such as nuclei. 3) (a) Spin chains; (b) Hubbard models, (c) Cavity QED systems, (d) Bose-Einstein Condensates, (e) t-J models for high Tc superconductors; (f) Holstein models.

All these model calculations are non- perturbative and highly non-linear. In such cases, Approximation approaches fail to provide useful information. Thus, exact treatment is in demand.

(1) Exact solutions of the generalized pairing (1998)Exact solutions of the generalized pairing (1998) (3) Exact solutions of the SO(5) T=1 pairing (2002)Exact solutions of the SO(5) T=1 pairing (2002) (2) Exact solutions of the U(5)-O(6) transition (1998) (4) Exact solutions of the extended pairing (2004)Exact solutions of the extended pairing (2004) (5) Quantum critical behavior of two coupled BEC (2005) Quantum critical behavior of two coupled BEC (2005) (6) QPT in interacting boson systems (2005)QPT in interacting boson systems (2005) II. Brief Review of What we have done (7) An extended Dicke model (2005)An extended Dicke model (2005)

Origin of the Pairing interaction Seniority scheme for atoms (Racah) (Phys. Rev. 62 (1942) 438) BCS theory for superconductors (Phys. Rev. 108 (1957) 1175) Applied BCS theory to nuclei (Balyaev) (Mat. Fys. Medd. 31(1959) 11 Constant pairing / exact solution (Richardson) (Phys. Lett. 3 (1963) 277; ibid 5 (1963) 82; Nucl. Phys. 52 (1964) 221)

General Pairing Problem

Some Special Cases constant pairing separable strength pairing c ij =A  ij + Ae -B(  i -  i-1 ) 2  ij+1 + A e -B(  i -  i+1 ) 2  ij-1 nearest level pairing

Exact solution for Constant Pairing Interaction [1] Richardson R W 1963 Phys. Lett [2] Feng Pan and Draayer J P 1999 Ann. Phys. (NY)

Next Breakthrough? Solvable mean-field plus extended pairing model

Different pair-hopping structures in the constant pairing and the extended pairing models

Bethe Ansatz Wavefunction: Exact solution Mk w

Eigen-energy: Bethe Ansatz Equation:

Energies as functions of G for k=5 with p=10 levels  1 =1.179  2 =2.650  3 =3.162  4 =4.588  5 =5.006  6 =6.969  7 =7.262  8 =8.687  9 =9.899  10 =10.20

Higher Order Terms

Ratios: R  = /

P(A) =E(A)+E(A-2)- 2E(A-1) for Yb Theory Experiment “Figure 3” Even A Odd A Even-Odd Mass Differences

6 6

Nearest Level Pairing Interaction for deformed nuclei In the nearest level pairing interaction model: c ij =G ij =A  ij + Ae -B(  i -  i-1 ) 2  ij+1 + A e -B(  i -  i+1 ) 2  ij-1 [9] Feng Pan and J. P. Draayer, J. Phys. A33 (2000) 9095 [10] Y. Y. Chen, Feng Pan, G. S. Stoitcheva, and J. P. Draayer, Int. J. Mod. Phys. B16 (2002) 2071 Nilsson s.p.

Nearest Level Pairing Hamiltonian can be written as which is equivalent to the hard-core Bose-Hubbard model in condensed matter physics

Eigenstates for k-pair excitation can be expressed as The excitation energy is 2 n dimensional n

Binding Energies in MeV Th U Pu

Th U Pu First and second 0 + excited energy levels in MeV

Th Pu U odd-even mass differences in MeV

Th U Pu Moment of Inertia Calculated in the NLPM

Models of interacting electrons with phonons have been attracting much attention as they are helpful in understanding superconductivity in many aspects, such as in fullerenes, bismuth oxides, and the high-T c superconductors. Many theoretical treatments assume the adiabatic limit and treat the phonons in a mean-field approximation. However, it has been argued that in many CDW materials the quantum lattice fluctuations are important. [1]A. S. Alexandrov and N. Mott, Polarons and Bipolarons (World Scientific, Singapore, 1995). [2] R. H. McKenzie, C.J. Hamer and D.W. Murray, PRB 53, 9676 (96). [3] R. H. McKenzie and J. W. Wilkins, PRL 69, 1085 (92). III. Algebraic solutions the one-dimensional Holstein Model

Here we present a study of the one- dimensional Holstein model of spinless fermions with an algebraic approach. The Hamiltonian is The model (1)

Analogue

(3) (4) (5)

Let us introduce the differential realization for the boson operators with (7) For i=1,2,…,p. Then, the Hamiltonian (1) is mapped into (8) Solutions

According to the diagonalization procedure used to solve the eigenvalue problem (2), the one-fermion excitation states can be assumed to be the following ansatz form: (9) Where |0> is the fermion vacuum and

By using the expressions (8) and (9), the energy eigen- equation becomes (11)

which results in the following set of the extended Bethe ansatz equations: for ¹ = 1, 2, …, p, which is a set of coupled rank-1 Partial Differential Equations (PDE’s), which completely determine the eigenenergies E and the coefficients. Though we still don’t know whether the above PDE’s are exactly solvable or not, we can show there are a large set of quasi-exactly solutions in polynomial forms. The results will be reported elsewhere.

Once the above PDEs are solved for one-fermion excitation, according to the procedure used for solving the hard-core Fermi- Hubbard model, the k-fermion excitation wavefunction can be orgainzed into the following from: (13) with (14)

The corresponding k-fermion excitation energy is given by (15)

In summary (1) General solutions of the 1-dim Holstein model is derived based on an algebraic approach similar to that used in solving 1-dim hard-core Fermi-Hubbard model. (2) A set of the extended Bethe ansatz equations are coupled rank-1 Partial Differential Equations (PDE’s), which completely determine the eigenenergies and the corresponding wavefunctions of the model. (3) Though we still don’t know whether the PDE’s are exactly solvable or not, at least, these PDE’s should be quasi-exactly solvable.

Thank You !

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