Pair formation in the Hubbard-Davydov model

L. Cruzeiro-Hansson, J.C. Eilbeck, J.L. Marın and F.M. Russell

Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, U.K.

(December 22, 1999)

Abstract

We study a model 1-D system which describes two quasi-particles interacting

with a nonlinear lattice. The quasi-particles are treated quantum mechani-

cally, whereas the lattice is treated classically. For a range of model param-

eters, we find pairing between the two quasi-particles. We investigate the

dynamical stability of such pairs, and comment on the possible applications

to models of HTSC.

Typeset using REVTEX

1

I. INTRODUCTION.

In spite of the many studies [1,2] made since it was first discovered [3], the mechanism

underlying high temperature superconductivity (HTSC) remains a challenge. The nature

of the carriers and the mechanism behind pair formation are still unclear. While some

argue that, similarly to conventional superconductivity, the electron-phonon interaction is

the main operating force, others think that charge and/or spin interactions are paramount.

The model we study, in which a Hubbard Hamiltonian is coupled to the dispersive phonons,

includes both effects. While our ultimate aim is to understand HTSC, here we propose a

specific mechanism for pair formation that involves the interaction of a polaron with a bare

quasiparticle. We study the stability of such a pair with respect to uncoupled polarons and

as a function of the quasiparticle-quasiparticle interaction.

Th paper is laid out as follows: in §II we present the Hamiltonian used, and the com-

bined classical-quantum approach. In §III we discuss our numerical results for a variety of

parameters and calculational strategies. In §IV the results and their implications for coupled

Discrete Breather/quasi-particle states are surveyed.

II. THE HUBBARD-DAVYDOV HAMILTONIAN.

The Hamiltonian H we use has three parts:

H = Hqp + Hqp-ph +Hph (1)

where Hqp is the Hamiltonian for a quasiparticle with spin 1/2, Hqp-ph describes the

interaction of the quasiparticle with the lattice and Hph is the lattice Hamiltonian.

The Hamiltonian for the quasiparticle is the 1D Hubbard Hamiltonian:

Hqp = ε∑n,σ

(a†nσanσ

)+ γ

∑n

a†n↑an↑a†n↓an↓ (2)

−t∑n,σ

(a†nσan−1σ + a†nσan+1σ

)

2

where the sums are over the sites n, going from 1 to N and σ refers to the spin and can be

up or down. The symbol a†nσ stands for the creation operator for a quasiparticle of spin σ at

site n, ε is the self-energy of the quasiparticle, and t the transfer term for the quasiparticle

to move between neighbouring sites. We depart from the usual notation in that the on-site

quasiparticle-quasiparticle coupling is here designated as γ (and not U) to avoid confusion

with the variables {un} used for lattice displacement (see below). Both negative and positive

values of γ will be considered, corresponding to the attractive and repulsive Hubbard models,

respectively.

As in the Davydov model for energy transfer in proteins [4], Hqp-ph, the Hamiltonian

for the interaction of the quasiparticle with the lattice includes the coupling to acoustic (or

Debye) phonons:

Hqp-ph = χ∑n,σ

(un+1 − un−1) a†nσanσ (3)

where χ is a parameter which describes the strength of the quasiparticle-lattice interaction.

The phonon Hamiltonian is given by

Hph = Hcoph +Hos

ph (4)

Hcoph =

κa2

72

N∑n=1

( a

a+ un − un−1

)12

− 2

(a

a+ un − un−1

)6

Hosph = κ′

N∑n=1

(1

2u2n +

1

4u4n

)+

1

2M

N∑n=1

p2n

where un is the displacement from equilibrium position of site n, pn is the momentum of site

n, a is the equilibrium distance between sites, κ is the elasticity of the nonlinear lattice and

κ′ is a similar constant for the on-site potential. Here, the coupling interactions between

sites are described by a Lennard-Jones potential Hcoph, a potential commonly used to describe

interactions between atoms. The on-site potential Hosph is as used in many breather studies

[5]. It can be considered to represent the effect, in a mean field approach, of the rest of the

crystal on the one dimensional chain whose states are studied explicitly.

Our Hamiltonian, (2-4) includes two sources of nonlinear effects. One comes from the

intrinsic nonlinearity of the Lennard-Jones potential, Hcoph, and of the on-site potential,

3

Hosph. A second source of nonlinearity, that is extrinsic, comes from the interaction of the

quasiparticle with the lattice (cf. Eq. 3). The former is the source of nonlinearity in studies

of discrete breathers [5] and the latter is the source of localization in polaron theory.

We adopt a mixed quantum-classical approach in which the lattice is treated classically,

while the quasiparticle is treated quantum mechanically. Accordingly, the displacements

un and momenta pn are real variables. The quasiparticle variables are operators, a distinc-

tion which is marked by the hats above the operators. In a classical system, equilibrium

thermodynamic quantities that do not involve velocities do not depend on the mass of its

components. A consequence in a classical lattice is that isotopic effects on the transition

temperatures do not arise. In some cases, this is a limitation of the mixed quantum-classical

approach. The importance of quantum effects of the lattice can be assessed by considering

the full quantum model, but such an investigation would make the calculations presented

here much more difficult to tackle.

Ultimately, the need for a full quantum treatment comes from comparison with experi-

mental results. Evidence for isotopic effects is as controversial as a mechanism for HTSC,

with results both for [6] and against [7]. The consensus seems to be that isotopic effects are

weak, which constitutes an extra a priori justification for a classical treatment of the lattice.

An isotopic dependence of magnetic quantities [8], on the other hand, is not necessarily

evidence for quantum lattice effects, as magnetic variables are dependent on velocities and

thus on the mass of sites even in a classical system. Thus, as a first approximation, we

restrict ourselves to the mixed quantum-classical regime and study the behaviour of a pair

of quasiparticles, coupled to a nonlinear lattice.

With these assumptions, the exact two quasiparticle wavefunction for Hamiltonian (1-4)

is:

|ψ(t) >=∑

n,m=1,...,N

φnm({un}, {pn}, t) a†n↑ a†m↓|0 > (5)

where φnm is the probability amplitude for a quasiparticle with spin up to be in site n and a

quasiparticle with spin down to be in site m. The probability amplitude is dependent on the

4

lattice displacements and momenta in a way that is not specified a priori and is determined

by the equations of motion. Similarly to other systems [9], the equations of motion for

probability amplitudes φnm are derived by inserting the wavefunction (5) in the Schrodinger

equation for the Hamiltonian (2-4), and the equations for the displacements and momenta

are derived from the Hamilton equations for the classical functional E2 =< ψ|H|ψ >. They

are:

ıhdφjldt

= −t (φj−1l + φj+1l + φjl−1 + φjl+1) + γφjlδjl + (6)

χ (uj+1 − uj−1 + ul+1 − ul−1)φjl

dpjdt

= −∂Hph∂uj

− χ(|ϕ↑j−1|2 − |ϕ↑j+1|2 + |ϕ↓j−1|2 − |ϕ↓j+1|2

)(7)

where |ϕ↑j |2 is the probability for the quasiparticle with spin up to be in site j and |ϕ↓j |2 is

the probability for the quasiparticle with spin down to be in the same site. These are given

by:

|ϕ↑j |2 = < ψ|a†j↑aj↑|ψ >=N∑l=1

|φjl|2 (8)

|ϕ↓j |2 = < ψ|a†j↓aj↓|ψ >=N∑l=1

|φlj|2 (9)

III. RESULTS.

We consider the case in which the quasiparticle density is low and the starting point

is that of an isolated quasiparticle interacting with the lattice. We want to find whether

the addition of a second quasiparticle with opposite spin to that state can lead to a pairing

of the two quasiparticles and how the relative stability of the paired state depends on the

quasiparticle-quasiparticle interaction γ.

The initial state is thus that of a single quasiparticle. The wavefunction is

|ψ1σ >=

∑n

φ1na†nσ|0 > (10)

5

Minimum energy states for the one quasiparticle can be found by numerical minimization of

the energy functional E1 =< ψ1|H|ψ1 > with respect to the probability amplitude for a single

quasiparticle in site n, φ1n and to the displacements un [10]. Two kinds of minimum energy

states are found. For sufficiently large quasiparticle-lattice interaction χ, the quasiparticle

states are localized and there is an associated lattice distortion. Below a threshold value for

χ, the states are delocalized, as in the usual Bloch states, and the lattice is undistorted. The

energy difference between two one quasiparticle states and the paired state in the latter case

is zero, for γ = 0. For delocalized states, the binding energy is thus zero. In the next section

we consider the binding energy to one quasiparticle state with finite lattice distortion. We

have considered a value of χ and other parameters such that the initial one quasiparticle

polaron state is neither too weak nor too stable when compared with delocalized, Bloch

states for the same values.

A. Relative stability of a quasiparticle pair.

Figure 1 shows the eigenstates of the paired quasiparticles in the presence of the lattice

distortion produced by one quasiparticle. That is, we first find the minimum energy one

quasiparticle states by minimizing the energy functional E1, for values of the parameters

that lead to a localized state, with an associated lattice distortion. Then we take the

displacements un induced by a single quasiparticle as defined by the latter state and find the

lower eigenstate of the two particle wavefunction (5) for those values of the displacements.

We then plot the dependence of this eigenstate on the quasiparticle-quasiparticle interaction

γ. These lower energy eigenstates, determined for a fixed lattice distortion as described, we

designate as the minimum energy unrelaxed states of two quasiparticles.

At low values of the quasiparticle-quasiparticle interaction γ, the states of the two quasi-

particles are localized (only the probability distribution for one quasiparticle is given because

the other quasiparticle has the same distribution). As the repulsion between the two quasi-

particles increases, the distribution is broadened and for sufficiently large values of γ, i.e.

6

for a large repulsion, the distribution starts to split.

The minimum energy states for two quasiparticle are determined by minimizing the

functional E2 =< ψ|H|ψ >, with respect to both the probability amplitudes φnm and the

displacements un. The results are shown in Fig. 2. The starting states for this minimization

are the unrelaxed states displayed in Fig. 1. We designate the states in Fig. 2 by minimum

energy relaxed states of the two quasiparticles. They are not necessarily global energy

minima of E because the space of stationary states of the two quasiparticles has many

nearly degenerate local minima. Numerical minimization of E for a given set of system

parameters leads to different results depending on the starting state. The states in Fig. 2

are only the minima which were nearest to the particular starting states considered.

At low values of γ, these minimum energy relaxed states for the quasiparticle pair are

more localized than those in Fig. 1. We also find windows of γ in which different regimes

are found. For small values of γ, as γ increases, the distribution becomes broadened. Then,

for γ/t ≈ 2 for the parameters specified in the figure, there is a transition to a different

distribution, in which the probability for the quasiparticle to be in adjacent sites is equal.

When γ/t approaches 3, a second transition is observed in which the distribution is split,

with two peaks, each with its own local lattice distortion. The distance between the two

peaks is as large as is allowed by the size of the lattice, so it is as if, for those values of the

repulsion, the two quasiparticles try to avoid each other as much as possible.

Figure 3 shows the energy saving of the two paired quasiparticles versus two uncorrelated

ones. A negative value means that the pair has a lower energy than two uncorrelated single

quasiparticles. In Figure 3 (a) the total energy saved by the states with unrelaxed lattice

(see Fig. 1) are displayed in the solid lines, while the total energy difference between the

latter and the states with relaxed lattice (see Fig. 2) are displayed by the dotted line. The

three components (2-4) of the total energy are given separately for a better understanding of

the source of the stability. Both the repulsion γ and the energies are given in the same units.

Fig. 3 (a) shows that coupling the two quasiparticles leads to an appreciable energy saving,

even for repulsive interactions. The energy saving decreases as the repulsion increases and

7

for γ/t ≈ 3 it can be approximately half that value. Fig. 3 also shows that the greater

part of the energy saving comes from the correlation energy, which actually increases (in

absolute value) as the repulsion increases. This is because, as the repulsion increase, the

states broaden (see Figs. 1,2), something that is favoured by the quasiparticle Hamiltonian

(2).

Another appreciable contribution comes from the phonon Hamiltonian, with the saving

in lattice potential energy when the two quasiparticles share the same local distortion. The

interaction energy for the paired quasiparticles state with the unrelaxed lattice is actually

higher than that of two uncorrelated quasiparticles. When the lattice is allowed to relax, an

overall extra saving of approximately 1/3 is obtained, which is due solely to the interaction of

the paired quasiparticles with the lattice. This occurs because the relaxed correlation energy

and the lattice potential are greather than those for the unrelaxed paired quasiparticle state.

B. Dynamical states.

In this section, we concentrate on dynamical perturbations of the localized one quasi-

particle state, induced by the presence of a second quasiparticle with opposite spin. The

initial condition for the simulations shown here is the unrelaxed paired quasiparticle state

associated with the lattice distortion produced by the minimum energy one quasiparticle

state, that is, the states shown in Fig. (1). Because the number of variables φnm increases

with the square of the lattice size, in order to be able to integrate the equations of motion

for a sufficiently long time, the size of the lattice was kept relatively short, i.e. the number

of sites N = 20. The aim is to investigate the influence of the strength and sign of the

quasiparticle-quasiparticle interaction γ on the dynamics of the paired quasiparticle states.

The parameters of the simulations in Figs. 4-11 are the same, except for the quasiparticle-

quasiparticle interaction γ. In Fig. 4 we set γ/t = −10 in an attractive Hubbard model.

Addition of a second quasiparticle leads to a localized state for the pair, with a very slight

peak oscillation, hardly visible in the figure. (The probability for the second quasiparticle is

8

the same as that shown and is not displayed). The lattice, however, sets into a breather-like

oscillation [5], i.e., a localized excitation with an internal oscillation. Indeed, at the site

of the initial lattice distortion, oscillations are clearly visible in the lattice displacements

and momenta. A striking observation is that the amount of radiation generated is very

small, and most of the energy of the lattice is associated with the breather. Fig. 5, which

displays another 6 ps period of the dynamics at a later time, demonstrates the stability of

this solution.

A Hubbard Hamiltonian with a much weaker attraction, corresponding to a ratio of

γ/t = −0.5, is considered in Fig. 6, where the last 6 picoseconds of a 42 picosecond simulation

are displayed. A modulation of the peak of the probability distribution is now clearly

seen, which has the same frequency as the main modulation of the lattice breather. The

modulation of the quasiparticle probability is associated with a periodic change of shape

in which a lower peak with a slight tail appears. Even at this comparatively much weaker

interaction, the amount of radiation is very small and most of the lattice energy is in the

breather. The frequency of the main modulation of the breather is as for γ/t = −10.

In Fig. 7 a repulsive Hubbard Hamiltonian is considered, with γ/t = +0.5. The modu-

lations and the associated tails of the probability distribution for the quasiparticle are now

more pronounced, but their main frequency is unchanged. Although there is a slight increase

in the radiation in the lattice, the stability of the breather and of the quasiparticle solution

is apparent.

In Fig. 8 the repulsive interaction is increased to γ/t = +1. The modulations in the

probability distribution for the quasiparticles lead to greater periodic changes of shape, still

with the same frequency as for the other values of γ. The radiation in the lattice is now

more visible, but the breather remains stable.

In Figs. 9 and 10, a large repulsive value, corresponding to γ/t = 5 is taken. This leads

to a change in the probability distribution for the quasiparticles, from a single site peak

into a two site peak, with periodic oscillations which make one probability at one site larger

than the other. The lattice variables show that, concurrently with the appearance of the

9

breather, a considerable amount of radiation is generated. Also noticeable is the fact that

the frequency of the modulations has changed. Fig. 10 shows that the new quasiparticle

probability distribution is stable, as well as the lattice breather, even if the noise which

results from successive passes of the radiation through the periodic boundaries, constitutes

a significant part of the lattice energy.

In Figs. 11-12 a repulsive interaction corresponding to γ/t = 10 is used. Fig. 11 shows

that a drastic transformation takes place in which the initial distribution changes into a

two peak distribution. One of the peaks is located where the initial lattice distortion was

and the second peak is as far away from it as it can be in this lattice. Also, while the

peak that is located at the original lattice distortion site remains unmodulated in time, as

well as its associated lattice distortion, the second peak oscillates with approximately the

same frequency as that in Figs. 9 and 10. The momenta in Fig. 11 show clearly that the

second peak has an associated lattice breather, while the first peak is associated with a

distortion that is essentially static. After some time, because of the repeated reflection of

the radiation from the boundaries, this picture is not so clear. Both peaks show oscillations

in the displacements and the momenta of the lattice are rather noisy. However, Fig. 11 does

illustrate the stability of the two peak solution, even in the presence of such relatively large

amount of noise.

IV. DISCUSSION

Our aim was to investigate the relative stability of a correlated pair of quantum quasi-

particles with opposite spins with respect to their uncorrelated states. The starting point

was thus the state of a single quasiparticle and we studied the static and dynamic states

which arise when a second quasiparticle is added to the first state. The Hamiltonian used

includes several physical ingredients. On the one hand, it contains two sources of nonlin-

earity, one intrinsic to the lattice and another which arises from the quasiparticle lattice

interaction. Such nonlinear lattices have been shown to possess generic solutions known as

10

discrete breathers (DBs) [5]. The study of systems in which nonlinear lattices are coupled to

one quantum quasiparticle, on the other hand, is just beginning [11,12]. To our knowledge,

this is the first time that the coupling of two quantum quasiparticles to a nonlinear lattice

has been considered. Indeed, a second ingredient is the inclusion of quasiparticle-lattice in-

teractions, as in the polaron model, as well as quasiparticle-quasiparticle interactions. These

can represent Coulomb interactions, and/or spin-spin interactions and be either attractive

or repulsive. One interesting point is that even for fairly repulsive interactions, correspond-

ing to γ/t ≈ +3, it is energetically favourable for the two quasiparticles to share the same

lattice distortion (see Fig. 3). Keeping all parameters fixed and just varying quasiparticle-

quasiparticle interaction leads to windows of γ in which the states are similar. This is seen

in Fig. 1 and particularly in Fig. 2 where three different types of states are found. The fig-

ure does not show results for negative values of γ, i.e., attractive quasiparticle interactions,

because they are similar to what is found for the small repulsive interactions displayed.

The dynamical simulations indicate that for these extended systems also, DBs are generic

solutions. These lattice breathers are shown to stabilise localized, paired, quasiparticle

states, for a large range of γ values. Also in the dynamical simulations, windows of γ were

found for which similar solutions are obtained. Thus, for a ratio of γ/t between −10,+1

(Figs. 4-8), DBs are found in the lattice and in the quasiparticle, with the same main

modulation frequencies. For larger values of γ/t, two different solutions were found (see

Figs. 9-12). In one solution the quasiparticles distribution is split into equal values in two

neighbouring sites and in the second a two peak distribution, with the peaks as far apart

as possible in the lattice used, is observed. This is similar to what is observed in the static

simulations when the lattice is allowed to relax in order to find a local minimum energy state

from an initial distortion due to one quasiparticle only. The dynamical simulations conserve

the energy and the states in Figs. 4-12 have higher energies than those seen in Fig. 2, but

the ‘steady state’ quasiparticle distributions are similar for the corresponding values of γ.

Both the static and the dynamical simulations indicate that there is a threshold in γ above

which the quasiparticles split up.

11

The stability of the quasiparticle pair has different sources according to whether the lat-

tice is allowed to relax or not. When the lattice is kept at the deformation induced by a single

quasiparticle, the main contribution to the energy saving on formation of a correlated pair

comes from the quasiparticle Hamiltonian (the Hubbard Hamiltonian in this study). This is

so even when the quasiparticle-lattice interaction is relatively large, as in this study, in which

v = tκ/χ2 ≈ 0.2 in Figs. 1-3 and 0.1 in the remaining figures. The next big energy saving

is in the lattice potential, which contributes more than half the saving of the quasiparticle

energy (see Fig. 3). In the unrelaxed lattice scenario, the quasiparticle-lattice interaction,

per se, favours uncorrelated, unpaired quasiparticles. On the other hand, when the lattice is

allowed to relax, the energy saving comes mostly from the quasiparticle-lattice interaction

and, depending on the value of γ, also from the quasiparticle-quasiparticle interaction. (The

energy saving of the relaxed state with respect to a delocalized two quasiparticle state can

be obtained by summing the energies displayed in the solid lines with that of the dotted

lines in Fig. 3).

This Hamiltonian includes the two main physical causes for quasiparticle pairing that

have been considered in HTSC and allows for interpolation between them, by varying the

strength of the relevant parameters. According to our results, quasiparticle-lattice interac-

tions in pair formation should be more important in systems for which the dynamics of the

lattice dynamics is fast enough compared to the quasiparticle dynamics, so that the lattice

relaxes when the two quasiparticles meet. Correspondingly, quasiparticle-quasiparticle in-

teractions should be more important in systems in which the lattice dynamics is much slower

than the quasiparticle dynamics.

An implicit assumption in this study is that the nonlinear character of the lattice plays

an important role in HTSC. In conventional superconductors, the lattice distortions are

weak and thus the lattice dynamics can be approximately described as a linear system.

We suggest that in HTSC these distortions are such that the lattice goes into a nonlinear

regime. This may be the reason why the sound velocity decreases by a few parts per million

in conventional superconductors, whereas in the high Tc material there is an increase which

12

is two or three orders of magnitude larger than the former case.

The breather-like solutions found in the dynamical simulations are a signature of the

nonlinear dynamics of the lattice. The possibility that breathers are associated with HTSC

has been suggested elsewhere [13,14]. Our study indicates that DBs are generic excitations

in systems governed by the Hamiltonian used here. Morevover, within a certain range of

the parameters, the states in which two quasiparticles are paired and coupled to a DB are

energetically more favourable than those of uncorrelated quasiparticles. Hence, this study

substantiates the possibility that DBs are important in HTSC.

Acknowledgements. JLM acknowledges a Marie Curie TMR fellowship from the EU

(no. ERBFMBICT972761). JCE is also grateful for EU support from RTN proposal no.

RTN1-1999-00370.

13

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[4] A. Scott. Davydov’s soliton. Physics Reports, 217:1–67, 1992.

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1991.

[7] G.V.M. Williams, J.L. Tallon, J.W. Quilty, H.J Trodahl, and N.E. Flower. Absence

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percarriers in the copper oxide superconductors La2−xSrxCuO4. Nature, 385:236–239,

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[9] Leonor Cruzeiro-Hansson and Shozo Takeno. Davydov model: The quantum, mixed

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14

with intrinsic and extrinsic nonlinearities. submitted to Physica D, 1999.

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15

FIGURES

05

1015

20

n

1020

3040

50γ

0.1

0.2

0.3

0.4

0.5

0.6

ϕ1

FIG. 1. Lower eigenstate for the two quasiparticle state with the distortion of the minimum

energy one quasiparticle state for the following parameter values: t = 19 10−22J, , χ = 100pN,

κ = 1N/m, κ′ = 2κ and a = 4.5A.

16

1116

n

1020

3040

50γ

0.2

0.4

0.6

0.8

1

ϕ1

FIG. 2. Local minimum for the two quasiparticle state. Parameters are as in Fig. 1.

16

-40

-30

-20

-10

0

0 20 40 60

Bin

ding

Ene

rgy

γ

-60

-40

-20

0

20

0 20 40 60

Hex

γ

-80

-40

0

40

0 20 40 60

Hin

t

γ

-20

0

20

40

0 20 40 60

Hph

γ

FIG. 3. Energy difference of the two quasiparticle state in Fig. 1 with respect to 2

non-interacting one polaron states (solid line) and the minimum energy state with respect to

the former (dotted line). The three components of the Hamiltonian are showed separately.

17

5101520

n

02

46

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

02

46

t-0.4-0.2

00.20.4

un

5101520

n

02

46

t

-1-0.5

00.5

1

pn

FIG. 4. Time dependence for (a) the probability for one quasiparticle to be in site n,

(n = 1 · · ·N , N = 20), (b) the lattice displacement and (c) the momentum of site n. Time is

in picoseconds. The parameters are as in Fig. 1, except that t = 10 10−22J and γ = −100 10−22J.

18

5101520

n

3638

4042

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

3638

4042

t-0.4-0.2

00.20.4

un

5101520

n

3638

4042

t

-1-0.5

00.5

1

pn

FIG. 5. Same as Fig. 4, at a later time.

19

5101520

n

3638

4042

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

3638

4042

t-0.4-0.2

00.20.4

un

5101520

n

3638

4042

t

-1-0.5

00.5

1

pn

FIG. 6. Same as Fig. 4, but with γ = −5 10−22J.

20

5101520

n

3638

4042

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

3638

4042

t-0.4-0.2

00.20.4

un

5101520

n

3638

4042

t

-1-0.5

00.5

1

pn

FIG. 7. Same as Fig. 4, but with γ = +5 10−22J.

21

5101520

n

3638

4042

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

3638

4042

t-0.4-0.2

00.20.4

un

5101520

n

3638

4042

t

-1-0.5

00.5

1

pn

FIG. 8. Same as Fig. 4, but with γ = +10 10−22J.

22

5101520

n

02

46

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

02

46

t-0.4-0.2

00.20.4

un

5101520

n

02

46

t

-1-0.5

00.5

1

pn

FIG. 9. Same as Fig. 4, but with γ = +50 10−22J.

23

5101520

n

3638

4042

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

3638

4042

t-0.4-0.2

00.20.4

un

5101520

n

3638

4042

t

-1-0.5

00.5

1

pn

FIG. 10. Same as Fig. 9, but at a later time.

24

5101520

n

02

46

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

02

46

t-0.4-0.2

00.20.4

un

5101520

n

02

46

t

-1-0.5

00.5

1

pn

FIG. 11. Same as Fig. 4, but with γ = +100 10−22J.

25

5101520

n

3638

4042

t

0.20.40.60.8

1

|ϕ1n|2

5101520

n

3638

4042

t-0.4-0.2

00.20.4

un

5101520

n

3638

4042

t

-1-0.5

00.5

1

pn

FIG. 12. Same as Fig. 9, but at a later time.

26