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arXiv:quant-ph/0507033v1 4 Jul 2005
Quantum breathers in a nonlinear Klein Gordon lattice
Laurent Proville∗
Service de Recherches de M´ etallurgie Physique,
CEA-Saclay/DEN/DMN 91191-Gif-sur-Yvette Cedex, France
(Dated: February 1, 2008)
Abstract
The quantum modes of a nonlinear Klein Gordon lattice have been computed numerically [L.
Proville, Phys. Rev. B 71, 104306 (2005)]. The on-site nonlinearity has been found to lead to
a phonon pairing and consequently some phonon bound states. In the present paper, the time
dependent Wannier transform of these states is shown to exhibit a breather-like behavior, i.e., it is
spatially localized and time-periodic. The typical time the lattice may sustain such breather states
is studied as a function of the trapped energy and the intersite lattice coupling.
PACS numbers: 63.20.Ry, 03.65.Ge, 11.10.Lm, 63.20.Dj
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I.NONLINEAR LATTICE MODES
The discrete breather solutions are currently a matter of intensive research (see
Refs.1,2,3,4,5,6,7,8,9,10,11,12,13,14). The distinctive property of those lattice modes is to gather
the spatial localization and the time periodicity so they lead to a energy trapping and thus
a delay in the equipartition5. As a general consequence of anharmonicity, the emergence
of breathers may be recognized as a paradigm of physics since it occurs at different scales
in various contexts, e.g., in macroscopic networks as a chain of coupled pendulums, in mi-
croscopic Josephson arrays15as well as in molecules16, polymers17and crystals as the PtCl
ethylene diamine chlorate18.
The nonlinear excitations in materials have been studied for several decades. In the late
fifties19, the possible existence of a two-phonon bound state was pointed out in the infrared
(IR) spectroscopy of H2solid. The vibrational and rotational nonlinear excitations in the H2
crystal have been thoroughly investigated both experimentally20and theoretically21. About
the same period22, the spectrum anomalies of the crystalline acetanilide (ACN) was revealed
and later interpreted with different theories (see Refs.23,24,25and for a historical survey see
Ref.26). Early in the sixties27, in the HCl solid, the anharmonicity of the first overtone of
hydrogen vibration has been measured by IR adsorption. It has been interpreted as a two
phonon bound state, namely a biphonon28, in regard of the earlier theoretical work of V.M.
Agranovich29,30. The triphonon has also been identified in the spectrum of HCl28. In sev-
enties, similar phonon bound states have been recognized in several molecular crystals such
as CO2, N2O and OCS31, as well as in water ice32by measuring the anharmonic IR absorp-
tions. F. Bogani achieved some convincing simulations of these anharmonic spectra33by
using the technics of renormalized perturbation theory. For the last decade, the nonlinearity
has emerged in several other materials:
- The inelastic neutron scattering (INS) has revealed the phonon bound states in the
metal hydrides as PdH34or TiH and ZrH35.
- The INS has also permitted to infer proton dynamics in the molecular crystals as
polyglycine36and 4-methylpyridine37. In the latter, the bound states of the methyl
group rotational modes proved to last several days (see Refs.37,38and Ref.26for a
survey of the theory).
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- The stretch overtone of carbon monoxide adsorbed on Ru(100) has been found
to exhibit a strong anharmonicity at low surface coverage40,41. Several theoretical
approaches40,41,42have been attempt to analyze the IR spectroscopy on Ru(100):CO.
The previous list is probably not complete but it is sufficient to emphasize that the nonlinear
excitations have been worked out in many different materials, whether it is a molecular
crystal33, a hydrogen-bonded crystal28or a metal hydride34. Furthermore the nonlinearity
may occur in one37, two40and three-dimensional33systems. In most of the above cited
examples, the phonon dispersion may be evaluated as smaller than 10% of the fundamental
optical excitation and the anharmonicity proves to reach less than 5%. The latter estimation
holds for the first overtone whereas for higher orders the strength of the anharmonicity may
increase as it is the case in HCl solid28or stabilize as in PdH34. The phonon bound states
are the siblings of breathers as they all stem from anharmonicity (see Refs.14,26and Refs.
therein). The quantum breather may be viewed as a Wannier transform, applied to the
phonon bound states that participate to a same energy band. Presently, the purpose of our
work is to study this idea within the nonlinear KG model.
The accurate computation of nonlinear modes, whether it is in a classical lattice or in
a quantum one requires the use of numerics. Recently43, we proposed a numerical method
that permits to compute the nonlinear quantum modes in a Klein Gordon lattice (KG) for
different type of nonlinearity. In the present paper, we use those developments to study the
Wannier transform14of the lattice eigen-modes that exhibits a quasi-particle spectrum, i.e.,
a narrow energy band. The time dependant Wannier transform of these states is found to
exhibit a breather-like behavior, i.e., it is spatially localized and time-periodic. The life time
of such breather states is studied as a function of their energy. We found that the higher
the energy spike is, the longer it remains localized. That study has been carried out for
different model parameters, including the case where the phonon dispersion is larger than
the anharmonicity.
After a brief introduction of the nonlinear KG lattice model, our computing method is
spelled out in Sec. II. In Sec. III and Sec. IV, we present and discuss our results on phonon
bound states and breathers, respectively. Some perspectives are given in Sec. V.
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FIG. 1:The plot of energy spectrum of a 1D chain, composed of N = 13 atoms for A4= 0.2,
versus the dimensionless coupling C. The eigenvalues are plotted as empty circles excepted the
phonon bound states energies, plotted as full black circles. The tags indicate the order of phonon
bound states.
II.LATTICE MODEL AND NUMERICAL METHOD
The energy of a lattice made of identical particles is expressed as a Hamiltonian operator:
H =
?
l
[p2
2m+ V (xl) +
l
?
j=<l>
W(xl− xj)].(1)
where xland plare displacement and momentum of the particle at site l, in a one-dimensional
lattice. Such a lattice may prove relevant to model the quasi-one-dimensional networks of
quantum particles in ZrH or in PtCl. The quantum particle of mass m evolves in a on-site
potential V , being coupled to its nearest neighbors, j =< l > by the interaction W. The
on-site potential V is developed to the fourth order whereas W is modelled by a quadratic
term:
V (xl) = a2x2
l+ a3x3
l+ a4x4
l
W(xl− xj) = −c(xl− xj)2.(2)
Higher order terms could have been added with no difficulty for our theory. It is possi-
ble to fixe the coefficients of V within a first principle calculation as done for PdH44and
confirmed by the analyze of the INS spectrum34. For simplicity, we choose to fixe a3= 0.
Introducing the dimensionless operators Pl= pl/√m?Ω, Xl= xl
?mΩ/? and the frequency
Ω =?2(a2− 2.c)/m, the Hamiltonian is rewritten as follows:
H = ?Ω
?
l
P2
2
l
+X2
l
2
+ A4X4
l+C
2Xl
?
j=<l>
Xj
(3)
where the dimensionless coefficients are A4= a4
?
m2Ω3 and C =
4c
mΩ2. The first step of our
method is concerned with the exact diagonalization of the Hamiltonian where no interaction
couples displacements. The procedure has been detailed in Ref.43. Arranging the on-site
eigenvalues in increasing order, the αth eigenstate is denoted φα,iand its eigenvalue is γ(α).
In case of a negligible inter-site coupling, the H eigenstates can be written as some Bloch
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FIG. 2:Energy spectrum of a 1D chain which model parameters are A4= 0.2 and C = 0.05.
The chain is composed of N = 13 sites. Four energy regions have been reported: (a) phonons,
(b) biphonons, (c) triphonons and (d) quadriphonons. The eigen-energies are plotted as empty
symbols and the phonon bound states energies have been signalized by full symbols.
(c)(d)
waves as follows:
B[Πiαi](q) =
1
?A[Πiαi]
?
j
e−iq.jΠiφαi,i+j
(4)
where A[Πiαi]ensures the normalization. The label [Πiαi] identifies a single on-site state
product Πiφαi,iamong the different products that may be derived from the present one by
translation. The set of states {B[Πiαi](q)}q,Ncut, including the uniform state Πiφ0,iat q = 0,
form a truncated basis where Ncut fixes the upper boundary on the on-site excitations:
?
Hartree approximation of the true eigenstates. The perturbation theory might be applied
iαi≤ Ncut. In case of a non-zero coupling, the states Eq.(4) may be thought as some
to the intersite coupling so as to estimate the eigenspectrum. However, we have chosen
to carry out a computation as accurate as possible. Thus the Bloch wave basis is used to
expand the Hamiltonian in. As the waves with different q, are not hybridized by H, the
Hamiltonian can be expanded separately for each q. It can be achieved analytically whereas
the diagonalization of the resulting matrix has been realized numerically with a standard
method, from a numerical library45. The accuracy of our calculations has been tested both
in a anharmonic43and harmonic46chain. For these two comparisons, a very good agreement
has been found in the two-phonon energy region and lower.
We denote by ψλ(q) and Eλ(q) the H eigenstates and the corresponding eigen-energies,
respectively. The subscript λ fixes the correspondence between a eigenstate and its eigen-
energy. Our numerical technics allows us to compute the scalar product Vλ,[Πiαi](q) between
ψλ(q) and B[Πiαi](q). Among the Bloch waves B[Πiαi](q), we note those bearing a single on-
site excitation of order αj> 0, all the other lattice sites l, being such as αl= 0. For those
states the label [Πiαi] reduces to α. In case of Vλ,α(q) > 0.5, we choose to distinguish the
eigenstate ψλby setting λ = α. It simply means that the Bloch wave Bα(q) has a dominant
contribution into ψλ(q). At C = 0, one notes that Vα,α(q) = 1. As it may be expected, the
scalar product Vα,α(q) decreases as C increases but its variation is smooth as found in Fig. 1
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FIG. 3: Energy spectrum of a 1D chain which model parameters are A4= 0.2 and C = 0.3. The
chain is composed of either N = 13 sites. Two energy regions have been reported: (a) phonons
and (b) quadriphonons. The eigen-energies are plotted as empty symbols and the phonon bound
states energies have been signalized by full symbols.
(a) (b)
(the results shown in this figure are examined thoroughly in the following). The eigenstates
ψα(q) correspond to the α phonon bound states. That terminology may be rightly thought
as ambiguous since a binding energy usually refers to a groundstate rather than to some
excited states. However it is convenient as the excitation order α appears in the name. This
order corresponds, indeed, to the energy level of the anharmonic on-site potential.
III.PHONON BOUND STATES
In lattices, treated in Ref.43, the sites number was N = 33 for a basis cutoff Ncut= 4,
which proves sufficient for the study of the two-phonon energy region. Here, we would like
to extend our study to the case of a four phonon bound state (quadriphonon). We thus
increased Ncutbut the number of Bloch waves, involved in our basis for N = 33, would
overload our computer’s memory, so we had to work with smaller lattices. For N = 13 and
Ncut= 6, the rank of our basis reaches 6564 which can be managed within a reasonable
time. We worked also with a even smaller lattice, N = 7 which allows us to increase again
Ncutas large as Ncut= 9. That case serves us as a reference in order to test the precision of
our computations on the larger lattice.
Varying C from the anti-continuous limit4, i.e., C = 0 we plotted in Fig. 1 the eigenspec-
trum as a function of C. Every circle symbol represents a single eigenvalue in the half first
Brillouin zone. The eigenvalues that correspond to the eigenstates ψα(q) (described in Sec.II)
have been plotted as full circles in Fig. 1, instead of empty ones for the other eigenstates.
As far as we increased the coupling C (see comment in Ref.47), in Fig.1, for a given order α
and a fixed wave vector q, we found a unique eigenstate that verifies Vλ,α(q) > 0.5. This is
the numerical proof that the nonlinear excitations may be continued from C = 0 to larger
coupling. This involves that the solutions ψα(q) conserve some features similar to the Bloch
waves Bα(q). Such a behavior could have been expected43while the energy gaps of the zero
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coupling spectrum remain. The hybridization between bound and unbound phonon states
is, indeed, thought to be weak in that case. The point that was very unexpected is that
even though C is large enough for gaps to close (between the triphonon and the surround-
ing unbound phonon bands, for instance) we found a dominant contribution of Bα(q) into
ψα(q). Moreover, for parameters in Fig.1 this property does not depend on the order of the
excitation α. It holds for phonons as for higher order phonon bound states. What differs,
however, for the latter is their band width which increases with C much smoother. In Figs.2
(a-d), at a fixed coupling, the eigenspectrum is plotted for different energy regions versus
the wave vector. The same symbols as in Fig.1 are used. We note that the larger the energy
is, the narrower the band of the phonon bound states is. Indeed, the phonon band width is
about 0.06, whereas for the biphonon it is less than 0.01, for triphonon it is around 10−3and
quadriphonon the band width falls to 10−5, in our energy unit. Although we approach only
the very first energy excitations, up to the fourth order, we may reasonably extrapolate our
results to higher energies. We then expect that the band width of the phonon bound states
becomes exponentially narrower as the eigen-energy increases. In Figs.3 (a-b), the coupling
parameter is such as the energy gaps close at high energy. We note that even though the
energy spectrum exhibits no gap, we find some eigenstates ψλ(q) that verify Vλ,α> 0.5 for
α = 4. In that case, the binding energy of the so called α phonon bound states vanishes.
However a strong component of Bα(q) takes part in ψα(q). For that reason, we propose to
dub the ψα(q) eigenstates as nonlinear α phonons to emphasize that these states differ from
the linear superposition of phonons, as well as to stress their quantized feature. In Fig.3
(b), the band width of the nonlinear four phonons is around 0.03 instead of 10−5in Fig.2
(d). Although the width of that band increases substantially with the coupling, it is yet one
order of magnitude below the phonon band width which is roughly 0.45. The exponential
decrease noted at low coupling seems to be no longer valid at larger coupling. This point
deserves a thorough study that we propose to report in a future work. In Figs.2 (d) and 3
(b), the band of the quadriphonon does not exhibit the anomaly which appears when Ncut
is diminished and that consists in a breaking of the band continuity. As far as C < 0.3, our
numerical approach seems to be reliable to treat the first phonon bound states. When the
nonlinear parameter A4is small, i.e., of the order of 10−2in our dimensionless model, the
scalar product Vλ,αfalls below 1/2 for a sufficiently large C which depends on α. The larger
α is, the larger the transition coupling Cαis. Moreover the Cαis found to depend on the
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wave vector q. At the edge of the lattice Brillouin zone, Cαis larger than in the center. In
the limit where A4equals zero, the strictly harmonic eigenstates verify Vλ,α= 0 as soon as
C is switched on. In that particular case, Cα= 0 for all α but for non zero A4, the Cαare
larger than zero, even for α = 1 which corresponds to the single phonon.
In Ref.46, the author attempted to compare his theoretical computations, similar to Fig.1,
to some experimental measures in H220solid and Ru(100):CO40. Although such a exercise
was based on qualitative considerations, it is worthy to complete these comparison by not-
ing that some spectral bands are due to the linear superposition of a biphonon and a single
phonon (see Fig.1 in the present paper and Fig. 3 in Ref.46). The signature of these states
has been measured in the IR spectrum of HCl solid28. Following a theoretical approach
proposed earlier33, C. Gellini et al. carried out the computation28of renormalized Green
functions to interpret the HCl spectrum. Although such a theory would be inadequate for
strong intersite coupling, for molecular crystals whose the molecule’s bond anharmonicity
dominates the inter-molecular coupling, as crystalline CO2or HCl for instance, the renormal-
ized Green functions seems relevant to capture the main physical properties. A convincing
demonstration has been given by Bogani in Ref.48where a precise fit of the IR adsorp-
tion spectrum has been achieved in several molecular crystals. The earlier work of V.M.
Agranovich introduced initially the concept of phonon bound states and more specifically of
biphonon for interpreting some experiments where anharmonic modes had been measured
(see Ref.29for infrared and Ref.49for neutron spectra). The lattice model of Agranovich
involves several energy terms that can be described briefly as follows. The elementary ex-
citation is proportional to the on-site product of Bose-Einstein operators a+
iai, while the
tunneling between neighboring sites i and j is modelled by a hopping term Ca+
jai. The
on-site Hubbard interaction between the boson pairs simulates the lattice anharmonicity by
adding locally the energy operator U(a+
i)2a2
i. Some other terms can be incorporated in the
model to modify, for instance, the biphonon tunneling30or the triphonon energy35. These
energy contributions are parameterized by independent coefficients, e.g., C, U. For instance,
the Hubbard model for boson has been used to interpret the INS in metal hydrides35. Here
the model parameters have been adjusted to exhibit the same energy resonances as the INS
spectrum. It is possible to achieve a similar work within the KG model, as shown in Ref.50.
The Hubbard model for boson involves to neglect the energy terms that do not conserve
the total boson number, despite the fact that these terms stem from the potential energy
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FIG. 4: Energy spectrum of a single anharmonic oscillator versus the eigenvalue rank for different
parameters: (a) A4= 0.2 and (b) A4= 0.4. The semi-classical calculation (empty square symbols,
dashed line) is compared to the Hamiltonian diagonalization (full circle symbols, solid line) onto
the truncated Einstein basis (see Ref.43) The Y axis unit is ?Ω.
of atoms and molecules. The consequence of such an approximation is exemplified in com-
puting the phonon dispersion law. For a one-dimensional lattice, the boson Hubbard lattice
would exhibit a phonon branch of the form (1+Ccos(q)), whereas the form
?1 + 2Ccos(q)
would be expected from the harmonic approximation with similar parameters. The latter
case corresponds to the exact diagonalization of the linear KG Hamiltonian which includes
only the quadratic potential energy of atoms. Since the two formula above diverge as C
increases, the Hubbard model for boson proves inappropriate to treat the normal modes at
strong coupling. It might however be relevant to work out the high order nonlinear modes as
proposed in Ref.51(see also the contribution of G.P. Tsironis in the present volume). Then
the skipping of the non-conservative boson terms might find some substantiation in the fact
that we found nonlinear narrow bands in continuous spectra as in Fig.1. A comparison be-
tween the boson Hubbard and our KG model would be very interesting to tentatively infer
the properties of the high order phonon bound states.
IV. QUANTUM BREATHERS
One introduces the time dependent Wannier state Wα(t,n), which is constructed from
a combination of the α phonon bound states ψα(q). We recognize these eigenstates among
others ψλ(q) by the fact that they verify Vλ,α(q) > 0.5, for a fixed α. This definition permits
us to build a Wannier state even though the energy spectrum has no gap. Then the Wannier
transform is written as follows:
|Wα(t,k) >=
1
√N
?
q
e−i(q×k+Eα(q)Ωt)|ψα(q) > . (5)
The subscript k indicates the lattice site where is centered the Wannier transform. In Fig.1,
we found that the band of the α phonon bound states contains a single state per wave vector
so that the sum over q in the Wannier transform is complete. In case of a small intersite
coupling, the Bloch wave Bα(q) is a good approximate of the α phonon bound state with
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the same wave vector. This may be thought as a Hartree approximation. To a first order in
C, we found43that the Eα(q) dependence on q is negligible provided that α > 1 and V is a
single well potential. Then the Wannier state |Wα(t,k) > can be rewritten as:
|Wα(t,k) >= e−i(EαΩt)φα,kΠl?=kφ0,i. (6)
Such a state is localized and time periodic so it may be considered as the quantum counter-
part of the breather solutions for the classical nonlinear discrete KG lattice. These classical
breather solutions have two important features that are first their spatial localization and
second their time periodicity with a frequency and its overtones that are out of the linear
classical phonon branch2. Our proposition could be verified by comparing the energies of
a localized time periodic Wannier state and the semi-classical quantization of the classi-
cal breather orbits, in same lattice. In the simple case of zero inter-site coupling, such a
comparison has been carried out in Ref.43and Fig.4, for different on-site potentials. The
remarkable agreement allows to expect that our proposition on breather quantum counter-
part holds for larger values of C. To enforce our arguments, we dwell upon Fig.1. To a fixed
α > 1, the energy of the corresponding Wannier state, given by the bracket of H, equals the
mean energy˜Eα, defined as the sum of Eα(q)/N over the first Brillouin zone. According
to Fig.1, that mean energy does not vary much with C. Indeed, provided that α > 1 and
C < 0.3 (the upper boundary on C to obtain a satisfactory precision), the bisecting line of
the phonon bound states band is roughly parallel to the X axis. Consequently, the energy
of the Wannier state of order α is comparable to the same quantity computed at C = 0.
In turn, the latter approaches very well the semi-classical quantization (see Fig.4) so the
energy of the Wannier state and the one of semi-classical breather orbits do not differ in a
significant manner provided that C remains weak. It seems reasonable in the following to
call α breather what is indeed the Wannier state of order α. It would be worth carrying
out the semi-classical quantization at non zero coupling in order to evaluate to what extend
our expectations might be confirm. We think it should not contradict our arguments unless
the classical breather becomes unstable, i.e., its frequency or one of its overtones fall in the
spectrum of the classical normal modes.
We now study the dynamics of a α breather as a function of the order α and for a non
zero coupling parameter. To that purpose, we integrate the time evolution of the on-site
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FIG. 5: The time evolution of kinetic energy of 17 atoms in one-dimensional KG chain, for a
Wannier state made of phonon. The model parameters are A4= 0.2, A3= 0 and C = 0.05. The
lattice sites are reported on the Y axis while the X axis bears the time scale. The time unit is the
inverse of Ω.
time
FIG. 6: Profile of the 3D-plot described in Fig.5 for different breathers made of: (a) phonon, (b)
biphonon, (c) triphonon and (d) quadriphonon. The parameters are same as in Fig.5. The time is
reported on the X axis. The Y axis unit is ?Ω).
(c)(d)
kinetic energy P2
j/2. The expectation of this operator is given by the bracket:
< Wα(t,k)|P2
j
2|Wα(t,k) >=1
N
?
q,q′
< ψα(q′)|P2
j
2|ψα(q) > ×e−i((q−q′)×k+(Eα(q)−Eα(q′))Ωt). (7)
For α = 1, the Wannier state Eq.(5) is constructed from phonons. In the 3D plot (see
Fig.5) of the kinetic energy time evolution, one notes that the energy is initially localized
and quickly spreads over the lattice. In Fig.6 (a), the profile of the 3D plot shows that
after 40 time unit, the energy is no longer localized. The same profile plot for α = 2, in
Fig.6 (b) shows that the quantum breather made of biphonons may last 10 times longer
than for phonons. With the Wannier transform of triphonons, the life time of the localized
excitation is again raised by one order of magnitude (see Fig.6 (c)). The life time of the
fourth quantum breather (i.e., the Wannier transform of the quadriphonons) overpasses
the first case in Fig.6 (a), by three orders. Conclusively, we found that the nonlinear KG
lattice may sustain a high energy spike for longer than 103times the typical relaxation of
low energy excitations, imposed by phonons. This behavior is related to the dispersion of
the phonon bound states since the thinner the band is, the longer the Wannier transform
remains coherent. According to our results, the breather life time increases exponentially
with respect to α. As noted previously (see Sec.III), the band width decreases as the order of
phonon bound states increases, even though at high energy the spectrum becomes continuous
(see Fig.1). According to our results, this continuity does not involve a particular decay in
the breather life time. It is noteworthy that the decay of a quantum breather is athermal
as it stems from the decoherence of the phonon bound states.
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FIG. 7: The same as in Fig.6 but for C = 0.3 and different sizes: (a-b) N = 13 and (c-d-e) N = 7.
The Wannier states are made of either (a-c) phonon, (b-d) quadriphonon or (e) pentaphonon. The
time is reported on the X axis and the Y axis unit is ?Ω. In the insets, some distinct time intervals
have been magnified.
(e)
As C increases, the α breather life time decreases, in agreement to the band width
enlargement shown in Fig.1. This can be worked out from the comparison of Fig. 6 (a) and
Fig. 6 (d) for C = 0.05 to the left hand side insets in Fig. 7 (a) and Fig. 7 (b) for C = 0.3.
Each couple of figures concern the cases α = 1 and α = 4. For α = 1, one sees that the
life time of the localized excitation decreases from 40 to 4 time units while for α = 4, the
breather life time decreases from 8×104to 8×101. The drop is sharper for the higher order.
However for a fixed value of C, whatever this value is, the band width of phonon bound
states decreases as the energy increases so that the life time of the corresponding breather
increases too. In Fig. 7 (e), the life time of the fifth breather is still two order of magnitude
larger than for α = 1. We thus expect that for a sufficiently high energy spike, the breathing
mode survives noticeably even though the intersite coupling is large.
For a fixed order α, the breather life time does not depend on the lattice size, as found in
comparing Fig. 7 (a) to Fig. 7 (c) and Fig. 7 (b) to Fig. 7 (d). The Figs. 7 (a-b) have been
obtained for a 13 sites lattice and the Figs. 7 (c-d) for a 7 sites lattice. Another interesting
feature revealed by these results is the time recurrence of breather. Indeed, one notes that
a certain time after the energy spike has spread, the energy backs to its initial trapped
state, similar to the original one (see right hand side insets in Figs. 7 (a-e)). The breather
is then bear by few sites although it has not exactly the same amplitude as initially. The
retrapping process occurs for a time twice larger than the breather life time because of the
time inversion symmetry. According to our computations, there is no exact frequency for the
breather recurrence as no regular behavior may be depicted in Figs. 7 (a-e). Though, we note
that the recurrence occurs sooner in a smaller lattice, which is demonstrated by comparison
either of Fig. 7 (a) to (c) or Fig. 7 (b) to (d). In a macroscopic crystal, the recurrence
is thus expected never to take place. In contrast, the breather recurrence might occur in a
single molecule as benzen. To that respect, the breather recurrence might be worth studying
thoroughly. Eventually, the shortest time interval upon which the recurrence occurs after
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starting the breather dynamics seems to increase with α as shown by comparing Fig. 7 (c)
to Fig. 7 (d) or Fig. 7 (d) to Fig. 7 (e).
V.CONCLUSION AND POSSIBLE DEVELOPMENTS
As a summary, we attempted to work out the breather modes in the quantum KG lattice.
We provided a numerical method to estimate their life time and spatial expansion. At the
quantum scale, it proves that the breathers are closely related to what has been called earlier,
the phonon bound states that are anharmonic eigenmodes. It is, indeed, well-known in
condensed matter physics that a narrow band excitations may be viewed as a quasi-particle
through a Wannier transform. We applied that theory to the phonon bound states and
showed that the lattice may sustain the corresponding breather for a time which increases
as the magnitude of the energy spike. At low intersite coupling, we found that the breather
life time increases exponentially with the trapped energy. This variation softens at larger
coupling, mainly because of the hybridization between the phonon bound states and the
linear superpositions of lower energy modes. For seak of simplicity, we only treated a quartic
nonlinearity. We found nonlinear excitations for all couplings we tested, i.e., up to C = 0.3
which corresponds to a dispersion that is larger than the anharmonicity. In the classical
counterpart of our KG lattice, a similar result is obtained since the discrete breathers occur
at all coupling too because their frequency is higher than the normal modes band. However,
the cubic nonlinearity is known to modify significantly this feature as the breather frequency
should be smaller than the normal modes. Consequently, for a given breather solution, that
is for a fixed frequency, there is a coupling threshold above which the breather is no longer
stable. This transition occurs when the breather frequency or one of its overtones fall into
the classical normal band2. A similar behavior is expected in the quantum case, which will
be studied in a future work.
The results we obtained in a KG lattice are rather encouraging for a possible future study
of the quantum acoustic lattices, as the FPU52chain. Even though we did not address
precisely that case, we found that in energy spectra where no gap occurs, the nonlinear
excitations may yet be distinguished and still exhibit a particle-like energy branch. Such
excitations are expected to emerge in the quantum FPU chain too, under the condition that
they corresponds to a sufficiently large energy. Our numerical theory should be tractable
13
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on the one-dimensional FPU lattice with only few sites, even though the rank of our basis
might increase dramatically. Then, one could yet achieve the Hamiltonian diagonalization
with a iterative procedure as the Lanczos method.
Alongside the present work, we carried out the calculation of the dynamical structure
factor of the nonlinear KG lattice50. A simulation of the inelastic scattering has been
achieved so as to compare our theory to practical cases.
Acknowledgments
I gratefully acknowledge S. Aubry who introduced me to the theory of breathers, at coffee
breaks in Laboratoire L´ eon Brillouin (CEA-Saclay) .
∗Electronic address: lproville@cea.fr
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